Applied Energy 93 (2012) 742–749

Contents lists available at SciVerse ScienceDirect

Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

Fractal dimension of wind speed time series

Tian-Pau Chang a,⇑, Hong-Hsi Ko a, Feng-Jiao Liu b, Pai-Hsun Chen a, Ying-Pin Chang b, Ying-Hsin Liang a,Horng-Yuan Jang a, Tsung-Chi Lin a, Yi-Hwa Chen a

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

a r t i c l e i n f o

Article history:Received 14 February 2011Received in revised form 9 August 2011Accepted 10 August 2011Available online 16 September 2011

Keywords:Fractal dimensionWind speedWind fluctuationProbability density functionWeibull function

0306-2619/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.apenergy.2011.08.014

⇑ Corresponding author. Fax: +886 49 2561408.E-mail address: t118@nkut.edu.tw (T.-P. Chang).

a b s t r a c t

The fluctuation of wind speed within a specific time period affects a lot the energy conversion rate ofwind turbine. In this paper, the concept of fractal dimension in chaos theory is applied to investigate windspeed characterizations; numerical algorithms for the calculation of the fractal dimension are presentedgraphically. Wind data selected is observed at three wind farms experiencing different climatic condi-tions from 2006 to 2008 in Taiwan, where wind speed distribution can be properly classified to high windseason from October to March and low wind season from April to September. The variations of fractaldimensions among different wind farms are analyzed from the viewpoint of climatic conditions. Theresults show that the wind speeds studied are characterized by medium to high values of fractal dimen-sion; the annual dimension values lie between 1.61 and 1.66. Because of monsoon factor, the fluctuationof wind speed during high wind months is not as significant as that during low wind months; the value offractal dimension reveals negative correlation with that of mean wind speed, irrespective of wind farmconsidered. For a location where the wind distribution is well described by Weibull function, its fractaldimension is not necessarily lower. These findings are useful to wind analysis.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Wind power is proportional to the cube of wind speed; a littlechange of the wind speed might cause extreme instability in elec-tricity generation through wind turbine. Studying about windspeed characterizations for a particular location is pretty importantwhile utilizing wind potential energy.

Two-parameter Weibull probability density function (pdf) hascommonly been applied to model wind speed distribution in liter-ature considering generally long-term measurements, e.g. monthly,seasonal or annual data [1–11]. Weibull pdf even became a refer-ence distribution in commercial wind energy software such asWind Atlas Analysis and Application Program [12]. However thesimilarity or irregularity of wind speeds for different time periodscannot be analyzed through the Weibull pdf. Recently the conceptof chaos theory is gradually adopted in many applications includingwind field based on the analysis of fractal dimension [13–19]. Someresearchers use it to quantify wind speed’s fluctuation within aspecific time period, e.g. a day; the fractal dimension calculated isabout 1.5–1.7 [20–23]. However the fractal research about windenergy is still few in literature; the relevant analyses consideringdifferent climatic factors and wind speed distributions are not

ll rights reserved.

really detailed; besides effective method for estimating fractaldimension has seldom been proposed previously.

In this paper, detailed numerical procedures for the calculationof fractal dimension for arbitrary signal are presented graphically,that would help the engineers to easily establish the computer pro-gram. The relationship between the fractal dimension and winddistribution will be investigated from the viewpoint of climaticconditions. Wind data selected are observed each 10 min from2006 to 2008 at three wind farms having different climatic condi-tions, which are handled by the Ministry of Economic Affairs. Thereare 3–8 wind turbines for each wind farm; anemometer is installedon the wind turbine with different heights. Raw speed dataobtained has a precision of 0.1 m/s. For the consideration of studyquality, some wind data are excluded when analyzing if theturbine is in maintenance or shut down during typhoon. The10-min wind speed measurements are transferred to hourly dataand are averaged over the 3 years before doing subsequent analy-ses. This data set has been used in many researches recently tostudy wind characterizations such as in [2,24]. The first wind farmDayuan (with longitude 121.1�E, latitude 25.1�N) is located at thenorthwestern plain of Taiwan; the northeastern monsoon is espe-cially active in winter months; the southwestern monsoon prevailsin summer but wind velocity becomes lower; wind turbine is man-ufactured by General Electric (GE, USA), the height of the anemom-eter is 64.7 m above ground level. The second wind farm Hengchun(120.7�E, 21.9�N) is located at the southern peninsula experiencing

Nomenclature

D fractal dimensionf time series signalg Weibull probability density functionG Weibull cumulative distribution functionH Hurst exponentL box lengthM number of data points in log–log plotN number of boxes to cover an objectn number of nonzero data points in calculating Weibull

parametersp sample size in calculating rectangle area

ti time stepv wind speed (m/s)Y theoretical value of regressed line

Greek letters/ rectangle area to cover a signalDt width of rectanglek residual value in doing regressione quadratic errora Weibull shape parameter, dimensionlessb Weibull scale parameter (m/s)

tΔ

t

)(tf

t

)(tf

Fig. 1. Arbitrary signal covered by rectangles with different time widths inestimating fractal dimension.

T.-P. Chang et al. / Applied Energy 93 (2012) 742–749 743

more stable weather conditions throughout the year, with thesame turbine specifications as Dayuan. The third one Penghu(119.6�E, 23.6�N) is at a small island in the Taiwan Strait experi-encing the highest wind in winter and spring among the threelocations studied; wind turbine is manufactured by Enercon (Ger-many), the height of anemometer is 46 m above ground level. Allthe wind speeds were transformed by using one-seventh powerlaw to the same height [10,11], 50 m above the ground level, onthe subsequent calculations.

2. Fractal dimension

The fractal dimension (D) is a numerical measure of the self-similarity of an object, it can be used to analyze the irregularityof a set of time series data; the larger the fractal dimension themore random the data. The commonly used definition about thefractal dimension in literature is the Hausdorff–Besicovitch dimen-sion given as below [25–27]:

D ¼ limL!0

log NðLÞlogð1LÞ

ð1Þ

where N(L) is the smallest number of boxes of side L to cover thedata (i.e. called box-counting method). To calculate the fractaldimension of wind speed time series data, a modified box-countingmethod is used in the present study; the fractal dimension isexpressed as [28–30]:

D ¼ 2� Hð/Þ ð2Þ

where H(/) is the Hurst exponent representing the degree of self-similarity of data. As mentioned in Harrouni and Guessoum [28]and in Kavasseri and Nagarajan [31], values of H(/) in the range(0, 0.5) characterize anti-persistence, whereas those in the range(0.5, 1) characterize persistent correlations, and represents uncorre-lated noise when H(/) = 0.5, that implies the fractal dimension isusually a non-integer value for real time series data measured innature. For arbitrary signal, as shown in Fig. 1, it can be coveredby rectangles with various widths of time interval Dt, relevantHurst exponent is given by:

Hð/Þ ¼ limDt!0

log /ðDtÞlogðDtÞ ð3Þ

where /(Dt) is the total rectangle area corresponding the coveredsignal and is calculated by:

/ðDtÞ ¼Xp�1

i¼0

f ðti þ DtÞ � f ðtiÞj jDt ð4Þ

where f(ti) is the value of the signal at time step ti, p is the number ofdata points (signal length), then |f(ti + Dt) � f(ti)| reflects the

fluctuation of the signal for a specific interval Dt. Substituting H(/)into previous equation, the fractal dimension becomes:

D ¼ limDt!0

2� log /ðDtÞlogðDtÞ

� �¼ lim

Dt!0

log /ðDtÞ=Dt2� �

logð1=DtÞ

( )ð5Þ

For a signal with limited data points, its fractal dimension canbe practically determined by the following equation with the man-ner of least-squares linear regression shown as:

log½/ðDtÞ=Dt2� ffi D logð1=DtÞ þ k; as Dt ! 0 ð6Þ

where k is the residual value while doing regression. Note that theslope of the regression line among the data points (logð1=DtiÞ,

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Wind speed (m/s)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Prob

abilit

y de

nsity

func

tion

0

0.2

0.4

0.6

0.8

1

Cum

ulat

ive

dist

ribut

ion

func

tion

observed wind speedobserved cdfWeibull pdfWeibull cdf

Dayuan

Fig. 2. Wind speed frequency and cumulative distribution function for stationDayuan.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Wind speed (m/s)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Prob

abilit

y de

nsity

func

tion

0

0.2

0.4

0.6

0.8

1

Cum

ulat

ive

dist

ribut

ion

func

tion

observed wind speedobserved cdfWeibull pdfWeibull cdf

Hengchun

Fig. 3. Wind speed frequency and cumulative distribution function for stationHengchun.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Wind speed (m/s)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Prob

abilit

y de

nsity

func

tion

0

0.2

0.4

0.6

0.8

1

Cum

ulat

ive

dist

ribut

ion

func

tion

observed wind speedobserved cdfWeibull pdfWeibull cdf

Penghu

Fig. 4. Wind speed frequency and cumulative distribution function for stationPenghu.

Table 1Wind speed statistics for station Dayuan.

Period Mean(m/s)

Standarddeviation(m/s)

Speedrange(m/s)

Skewness Kurtosis

January 10.25 3.15 16.52 �1.04 3.34February 7.41 4.13 17.91 0.33 1.87March 6.87 4.01 15.20 0.05 1.60April 6.67 3.40 16.02 0.36 2.52May 4.67 2.88 15.12 0.95 3.32June 4.39 2.58 14.60 0.76 2.80July 5.94 3.01 16.16 0.53 2.48August 5.11 2.93 15.81 0.89 2.76September 8.32 3.91 21.33 0.20 2.24October 11.74 2.40 16.22 �0.34 3.22November 13.32 2.37 16.84 �0.45 3.16December 12.28 1.92 12.32 0.04 2.84October–March

(high windperiod)

10.35 3.91 18.46 �0.72 2.72

April–September(low windperiod)

5.84 3.42 18.44 0.73 2.88

Yearly 8.09 4.31 21.64 0.07 1.81

744 T.-P. Chang et al. / Applied Energy 93 (2012) 742–749

log½/ðDtiÞ=Dt2i �) is just the value of fractal dimension. Basically the

maximum value of time interval Dti used in determining the fractaldimension cannot exceed the half of data length, in the presentstudy the maximum time interval is set to be the value that makesthe total quadratic error e between each data point and theregressed line minimum, which is defined as below:

e ¼XM

i¼1

log½/ðDtiÞ=Dt2i � � Y

� �2 ð7Þ

where Y is the value calculated from the regressed line with thesame abscissa of logð1=DtiÞ; M is the number of data points usedin the line regression in log–log plot.

The magnitude of fractal dimension depends not only on prob-lem itself but also on the number of data points considered. Inwind application it is worth to investigate the variation of windenergy available within a day; in this context, hourly mean windspeeds, 24 data points a day, are generally used in the literatures

[31–33]. In this paper, the fractal dimension presented is calcu-lated using hourly data that represents the hourly fluctuation ofwind speed within a day. Several maximum values concerningtime interval had been examined using the whole time series dataand the optimal one is found to be 10 for all the three wind farmsstudied.

3. Weibull function

Weibull function has frequently been used to describe the windspeed distribution. Its probability density function (pdf) is given as[2–5]:

gðvÞ ¼ ab

vb

� a�1

exp � vb

� a� �ð8Þ

The corresponding Weibull cumulative distribution function(cdf) is expressed by:

Table 2Wind speed statistics for station Hengchun.

Period Mean(m/s)

Standarddeviation(m/s)

Speedrange(m/s)

Skewness Kurtosis

January 9.21 3.89 20.50 0.10 2.39February 9.37 4.21 22.21 0.19 2.03March 7.92 3.62 21.32 0.51 2.58April 7.27 2.93 17.64 0.31 2.89May 7.14 2.83 15.37 0.01 2.68June 6.19 2.74 16.44 0.12 2.45July 6.62 3.23 16.13 0.18 2.19August 5.28 2.80 14.27 0.36 2.19September 6.87 3.00 16.12 0.33 2.48October 6.87 2.88 18.40 0.77 3.07November 8.03 4.77 22.90 0.53 2.29December 9.99 4.09 21.63 0.09 2.27October–March

(high windperiod)

8.56 4.08 21.06 0.40 2.38

April–September(low windperiod)

6.56 3.00 15.90 0.22 2.50

Yearly 7.55 3.72 22.90 0.55 2.89

Table 3Wind speed statistics for station Penghu.

Period Mean(m/s)

Standarddeviation(m/s)

Speedrange(m/s)

Skewness Kurtosis

January 13.47 3.11 15.80 �0.47 3.02February 10.51 5.34 22.94 0.21 2.00March 10.54 6.04 24.32 0.60 2.22April 7.67 4.50 21.01 0.45 2.73May 7.14 3.01 19.63 0.85 4.69June 7.67 3.06 16.18 �0.03 2.50July 5.65 1.98 10.52 �0.33 2.42August 5.04 3.29 23.32 1.71 6.90September 7.39 3.70 18.16 0.55 2.87October 12.30 4.16 21.32 �0.65 3.35November 14.79 3.76 23.82 �0.47 3.72December 13.93 5.14 22.68 �0.27 2.14October–March

(high windperiod)

12.61 4.96 24.32 �0.22 2.41

April–September(low windperiod)

6.75 3.49 20.54 0.76 3.77

Yearly 9.63 5.17 24.84 0.43 2.36

0 30 60 90 120 150 180 210 240 270 300 330 360

Day of year

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Frac

tal d

imen

sion

Dayuan

Fig. 5. Variation of daily fractal dimension for station Dayuan.

0 30 60 90 120 150 180 210 240 270 300 330 360

Day of year

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3Fr

acta

l dim

ensi

onHengchun

Fig. 6. Variation of daily fractal dimension for station Hengchun.

T.-P. Chang et al. / Applied Energy 93 (2012) 742–749 745

GðvÞ ¼ 1� exp � vb

� a� �ð9Þ

where v is the wind speed, a is the shape parameter (dimension-less), b is the scale parameter having the same unit as wind speed.Weibull shape parameter reflects the width of data distribution, thelarger the shape parameter the narrower the distribution and thehigher its peak value. Scale parameter influences the abscissa scaleof a plot of data distribution. Two Weibull parameters can beobtained by using the maximum likelihood method given as [2,9]:

a ¼Pn

i¼1vai lnðv iÞPn

i¼1vai

�Pn

i¼1 lnðv iÞn

" #�1

ð10Þ

b ¼ 1n

Xn

i¼1

vai

!1=a

ð11Þ

where vi is the wind speed at time step i and n is the number of non-zero data points. All the procedures are implemented using com-puter program written in MATLAB languages.

4. Results and discussion

Figs. 2–4 show the yearly observed wind speed histograms forthe three stations studied. The Weibull probability density function(pdf) as well as cumulative distribution function (cdf, referred tothe right ordinate) calculated using relevant Weibull parametersare also plotted for comparison in the figures. The annual Weibullshape parameters calculated are 1.98, 2.16 and 1.97 for the Dayuan,Hengchun and Penghu, respectively; corresponding scale parame-ters are 9.13, 8.53 and 10.87 m/s respectively.

It is found that the histogram in Hengchun (Fig. 3) is best fittedby the theoretical curve of Weibull pdf due to its more stable cli-matic conditions throughout the year, thus the Weibull cdf hasthe smallest difference with the observed cdf. Tables 1–3 summa-rize the descriptive statistics for different time periods. While con-sidering the annual period, the mean wind speed is 8.09, 7.55 and9.63 m/s for the three stations respectively; the station Hengchunreveals the highest kurtosis coefficient, 2.89, among the three

0 30 60 90 120 150 180 210 240 270 300 330 360

Day of year

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Frac

tal d

imen

sion

Penghu

Fig. 7. Variation of daily fractal dimension for station Penghu.

0 2 4 6 8 10 12 14 16 18 20 22 24Hour of day

0

1

2

3

4

5

6

7

8

9

10

Win

d sp

eed

(m/s

)

Dayuan

Fig. 8. Example of wind speed observed at Dayuan with larger fractal dimension.

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Log (1/Δt)

-2

-1

0

1

2

3

4

5

Log

( Φ( Δ

t)/Δt

2 )

Dayuan

slope = 1.99R2=0.9436

Fig. 9. Estimation of fractal dimension by least-squares regression (related toFig. 8).

0 2 4 6 8 10 12 14 16 18 20 22 24Hour of day

10

11

12

13

14

15

16

17

18

19

20W

ind

spee

d (m

/s)

Dayuan

Fig. 10. Example of wind speed observed at Dayuan with smaller fractal dimension.

746 T.-P. Chang et al. / Applied Energy 93 (2012) 742–749

stations (Table 2). Additionally the wind speeds for the threestations could be roughly distinguished as the high wind periodfrom October to March and low wind period from April to Septem-ber resulting from the climatic monsoon factors, although this phe-nomenon is not very obvious in Hengchun.

Figs. 5–7 show the variations of averaged daily fractal dimen-sion for the entire year. For both Dayuan and Penghu stations,the fractal dimensions reveal significant oscillation and have largervalues during the low wind period; whereas the oscillation inHengchun is gentler among the stations.

To clearly demonstrate how the fractal dimension is estimatedand how the dimension value can be related to the fluctuation ofwind speed within a day, Fig. 8 being an example shows the windspeed data observed at Dayuan on 28 July 2007 (in low wind per-iod), that corresponds to a larger fractal dimension, 1.99, i.e. theslope of straight line shown in Fig. 9. Similarly Figs. 10 and 11 showthe relevant data observed at Dayuan on 27 November 2007 (highwind period) in which the fractal dimension is smaller, 1.11.

According to such greater determination coefficients (R-squared)while regressing the straight lines as provided in the figures, wecan conclude that the method presented is suitable to estimatethe fractal dimension; the wind data considered reveal fractalbehaviors; the more the fluctuation of wind speed the larger thefractal dimension.

Table 4 lists the averaged fractal dimension and its range forvarious time periods. It is shown that the fractal dimensionsmostly lie between 1.55 and 1.70; the yearly value for Dayuan,Hengchun and Penghu is 1.632, 1.661 and 1.607 respectively. Notethat the same characterization can be found at all the three sta-tions studied: the fractal dimension for high wind period is smallerthan that for low wind period; i.e. the value of fractal dimensionpresents reverse correlation with that of mean wind speed. Thisis because the change of wind speed within a day is less significantwhile the stronger northeast monsoon is prevailing in winter sea-son, especially in wide-open ocean Penghu where the fractal

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Log (1/Δt)

-2

-1

0

1

2

3

4

5Lo

g (Φ

(Δt)/

Δt2 )

Dayuan

slope = 1.11R2=0.9975

Fig. 11. Estimation of fractal dimension by least-squares regression (related toFig. 10).

Table 4Fractal dimensions computed for different time periods and stations.

Periods Dayuan Hengchun Penghu

January 1.480 (1.280–1.840)a

1.614 (1.338–1.989)

1.488 (1.351–1.613)

February 1.654 (1.331–1.999)

1.674 (1.303–1.849)

1.580 (1.377–1.811)

March 1.638 (1.235–1.997)

1.603 (1.380–1.965)

1.609 (1.352–1.913)

April 1.680 (1.337–1.977)

1.670 (1.295–1.928)

1.592 (1.404–1.897)

May 1.748 (1.448–1.968)

1.666 (1.287–1.880)

1.688 (1.519–1.938)

June 1.737 (1.315–1997)

1.667 (1.434–1.908)

1.681 (1.484–1.986)

July 1.705 (1.496–1.946)

1.735 (1.360–1.905)

1.714 (1.529–1.976)

August 1.718 (1.494–1.963)

1.649 (1.404–1.956)

1.722 (1.492–1.966)

September 1.690 (1.338–1.931)

1.663 (1.366–1.993)

1.649 (1.411–1.983)

October 1.538 (1.315–1.744)

1.655 (1.409–1.967)

1.507 (1.351–1.892)

November 1.544 (1.227–1.876)

1.676 (1.440–1.936)

1.480 (1.334–1.744)

December 1.453 (1.303–1.695)

1.654 (1.257–1.881)

1.549 (1.358–1.829)

October–March (highwind period)

1.550 (1.227–1.999)

1.646 (1.257–1.989)

1.535 (1.334–1.913)

April–September (lowwind period)

1.713 (1.315–1.997)

1.675 (1.287–1.993)

1.674 (1.404–1.986)

Yearly 1.632 (1.227–1.999)

1.661 (1.257–1.993)

1.607 (1.334–1.986)

a Range of fractal dimension (minimum–maximum).

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6Fractal dimension

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Rel

ativ

e fre

quen

cy

Dayuan

Fig. 12. Yearly relative frequency of fractal dimension for station Dayuan.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6Fractal dimension

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4R

elat

ive

frequ

ency

Hengchun

Fig. 13. Yearly relative frequency of fractal dimension for station Hengchun.

T.-P. Chang et al. / Applied Energy 93 (2012) 742–749 747

dimension is only 1.535. On the other hand, a contrary result couldbe predicted if the wind speed becomes lower in summer whenairflow is unstable, e.g. in Dayuan the fractal dimension reaches1.713. Though the wind speed distribution in Hengchun is most fit-ted by the Weibull pdf, but its mean fractal dimension is the largestone.

Figs. 12–14 show the yearly relative frequency of fractal dimen-sion for the three stations, these distributions indicate that thewind speeds in Taiwan are characterized by medium to high valuesof fractal dimension, implying that the hourly wind speeds studied

exhibit relatively high fluctuation. We also found that the value offractal dimension increases by about 8% if the data length used toestimate the fractal dimension becomes double.

Fig. 15 illustrates the hourly wind speeds averaged over theyear for three stations. The peak values lie in the afternoon. Thefluctuation of wind speed throughout the day in both Dayuanand Hengchun is more significant; the slightest fluctuation inPenghu objectively confirms its smallest fractal dimension amongthe stations (1.607 annually). The wind energy available for agiven site is proportional to the cube of wind speed; a little var-iation of wind speed within a particular time period might causeobvious instability in power generation by wind turbine. Themost ideal site for wind energy production is the area where con-tinuous or steady state wind condition is dominant over a signif-icant percentage of a given period of time. From the analysesaforementioned, we conclude that studying about the fractaldimension of wind speed would help engineers to assess thewind energy potential.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6Fractal dimension

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Rel

ativ

e fre

quen

cy

Penghu

Fig. 14. Yearly relative frequency of fractal dimension for station Penghu.

0 2 4 6 8 10 12 14 16 18 20 22 24Hour of day

5

6

7

8

9

10

11

12

13

14

15

Mea

n w

ind

spee

d (m

/s)

DayuanHengchunPenghu

Fig. 15. Hourly mean wind speeds averaged for the whole year.

748 T.-P. Chang et al. / Applied Energy 93 (2012) 742–749

5. Conclusions

Knowing about wind speed distribution is an essential stepbefore utilizing wind resources. In the present study, the fluctua-tion of wind speed within a day had been investigated throughthe analysis of fractal dimension by considering climate factors.The graphs illustrating how the fractal dimension relates to thewind fluctuations had been shown as well. The conclusions canbe summarized as follows:

(a) The annually averaged fractal dimension values lie between1.61 and 1.66 for the three wind farms studied that impliesthe wind speeds reveal relatively high fluctuation.

(b) The value of fractal dimension presents reverse correlationwith that of mean wind speed; the change of wind speedwithin a day is less significant while the stronger northeastmonsoon is prevailing in winter season, independent of loca-tions considered.

(c) Though the wind distribution may be well described by theconventional Weibull function for someplace as in Hengc-hun, its mean fractal dimension calculated is not necessarilylower than other locations.

(d) The value of fractal dimension increases by about 8% if thedata length considered to estimate the fractal dimensionbecomes double.

(e) Studying about fractal dimension enables us to understandwind’s fluctuation, some findings of the present work basedon the analysis of climate factors are useful to windapplications.

Acknowledgments

The authors would deeply appreciate the Central WeatherBureau and Ministry of Economic Affairs for providing observationdata and deeply thank Dr. Wu CF and Dr. Huang MW, researchersof the Institute of Earth Sciences, Academia Sinica, Taiwan, for theirprecious comments. This study was partly supported by theNational Science Council under contract NSC99-2221-E-252-011.

References

[1] Fyrippis I, Axaopoulos PJ, Panayiotou G. Wind energy potential assessment inNaxos Island, Greece. Appl Energy 2010;87:577–86.

[2] Chang TP. Performance comparison of six numerical methods in estimatingWeibull parameters for wind energy application. Appl Energy2011;88:272–82.

[3] Akpinar EK, Akpinar S. Determination of the wind energy potential for Maden-Elazig, Turkey. Energy Convers Manage 2004;45:2901–14.

[4] Kwon SD. Uncertainty analysis of wind energy potential assessment. ApplEnergy 2010;87:856–65.

[5] Yang HX, Lu L, Burnett J. Weather data and probability analysis of hybridphotovoltaic-wind power generation systems in Hong Kong. Renew Energy2003;28:1813–24.

[6] Akdag SA, Bagiorgas HS, Mihalakakou G. Use of two-component Weibullmixtures in the analysis of wind speed in the Eastern Mediterranean. ApplEnergy 2010;87:2566–73.

[7] Celik AN. A statistical analysis of wind power density based on the Weibull andRayleigh models at the southern region of Turkey. Renew Energy2003;29:593–604.

[8] Jowder FAL. Wind power analysis and site matching of wind turbinegenerators in Kingdom of Bahrain. Appl Energy 2009;86:538–45.

[9] Chang TJ, Wu YT, Hsu HY, Chu CR, Liao CM. Assessment of wind characteristicsand wind turbine characteristics in Taiwan. Renew Energy 2003;28:851–71.

[10] Ucar A, Balo F. Evaluation of wind energy potential and electricity generationat six locations in Turkey. Appl Energy 2009;86:1864–72.

[11] Zhou W, Yang HX, Fang ZH. Wind power potential and characteristic analysisof the Pearl River Delta region, China. Renew Energy 2006;31:739–53.

[12] Carta JA, Ramirez P, Velazquez S. A review of wind speed probabilitydistributions used in wind energy analysis Case studies in the CanaryIslands. Renew Sust Energy Rev 2009;13:933–55.

[13] Mandelbrot B. The fractal geometry of nature. New York: W.H. Freemann andCompany; 1983.

[14] Mandelbrot B. Self-affine fractals and fractal dimension. Phys Scr1985;32:257–60.

[15] Jimenez-Hornero FJ, Pavon-Dominguez P, Gutierrez de Rave E, Ariza-VillaverdeAB. Joint multifractal description of the relationship between wind patternsand land surface air temperature. Atmos Res 2011;99:366–76.

[16] Karakasidis TE, Charakopoulos A. Detection of low-dimensional chaos in windtime series. Chaos Soliton Fract 2009;41:1723–32.

[17] Liaw SS, Chiu FY. Fractal dimensions of time sequences. Physica A2009;388:3100–6.

[18] Shih TY, Hwang JT, Tsai TJ. The fractal properties of sea surface topographyderived from TOPEX/POSEIDON (1992–1996). Comput Geosci1999;25:1051–8.

[19] Dubuc B, Quiniou JF, Roques-Carmes C, Tricot C, Zucker SW. Evaluating thefractal dimension of profiles. Phys Rev A 1989;39:1500–12.

[20] Zhong L, Li J. Fractal characteristic analysis of wind speed time history. JHuazhong Univ Sci Technol (Urban Sci Ed) 2008;25:273–5.

[21] Feng T, Fu ZT, Deng X, Mao JY. A brief description to different multi-fractalbehaviors of daily wind speed records over China. Phys Lett A2009;373:4134–41.

[22] Syu CY, Kirchhoff RH. The fractal dimension of the wind. J Sol Energy Eng1993;115:151–4.

[23] Tijera M, Cano JL, Cano D, Bolster D, Redondo JM. Filtered deterministic wavesand analysis of the fractal dimension of the components of the wind velocity. IlNuovo Cimento 2008;31:653–67. doi:10.1393/ncc/i2008-10324-3.

T.-P. Chang et al. / Applied Energy 93 (2012) 742–749 749

[24] Chang TP. Estimation of wind energy potential using different probabilitydensity functions. Appl Energy 2011;88:1848–56.

[25] Soille P, Rivest JF. On the validity of fractal dimension measurements in imageanalysis. J Vis Commun Image Represent 1996;7:217–29.

[26] Turcotte DL. Fractals and chaos in geology and geophysics. 2nded. Cambridge: Cambridge University Press; 1997.

[27] Viscek T. Fractal growth phenomena. Singapore: World Scientific; 1989.[28] Harrouni S, Guessoum A. Using fractal dimension to quantify long-range

persistence in global solar radiation. Chaos Soliton Fract 2009;41:1520–30.[29] Maafi A, Harrouni S. Preliminary results of the fractal classification of daily

solar irradiances. Sol Energy 2003;75:53–61.

[30] Rehman S, Siddiqi AH. Wavelet based hurst exponent and fractal dimensionalanalysis of Saudi climatic dynamics. Chaos Soliton Fract 2009;40:1081–90.

[31] Kavasseri RG, Nagarajan R. A multifractal description of wind speed records.Chaos Soliton Fract 2005;24:165–73.

[32] Kocak K. Examination of persistence properties of wind speed records usingdetrended fluctuation analysis. Energy 2009;34:1980–5.

[33] Harrouni S. Fractal analysis to quantify wind speed fluctuations. In:Proceedings of the 3rd conference on nonlinear science and complexity,Ankara, Turkey, 2010 [Article ID: 14].