Multifractal analysis of validated wind speed time series Multifractal analysis of validated wind...

$Page 1: Multifractal analysis of validated wind speed time series Multifractal analysis of validated wind speed time series$
Multifractal analysis of validated wind speed time seriesA P Garciacutea-Mariacuten J Esteacutevez F J Jimeacutenez-Hornero and J L Ayuso-Muntildeoz Citation Chaos 23 013133 (2013) doi 10106314793781 View online httpdxdoiorg10106314793781 View Table of Contents httpchaosaiporgresource1CHAOEHv23i1 Published by the American Institute of Physics Related ArticlesLong-time dynamics for a class of Kirchhoff models with memory J Math Phys 54 021505 (2013) Investigating the effect of suspensions nanostructure on the thermophysical properties of nanofluids J Appl Phys 112 114315 (2012) Characterization of complexities in combustion instability in a lean premixed gas-turbine model combustor Chaos 22 043128 (2012) Optimal and suboptimal networks for efficient navigation measured by mean-first passage time of random walks Chaos 22 043129 (2012) Shape analysis using fractal dimension A curvature based approach Chaos 22 043103 (2012) Additional information on ChaosJournal Homepage httpchaosaiporg Journal Information httpchaosaiporgaboutabout_the_journal Top downloads httpchaosaiporgfeaturesmost_downloaded Information for Authors httpchaosaiporgauthors

Multifractal analysis of validated wind speed time series

A P Garcıa-Marın1a) J Estevez1 F J Jimenez-Hornero2 and J L Ayuso-Mu~noz1

1Department of Rural Engineering University of Cordoba PO Box 3048 14080 Cordoba Spain2Department of Graphic Engineering and Geomatics Gregor Mendel Building (3rd floor)Campus Rabanales University of Cordoba 14071 Cordoba Spain

(Received 5 September 2012 accepted 11 February 2013 published online 1 March 2013)

Multifractal properties of 30 min wind data series recorded at six locations in Cadiz (Southern

Spain) have been studied in this work with the aim of obtaining detailed information for a range of

time scales Wind speed records have been validated applying various quality control tests as a

pre-requisite before their use improving the reliability of the results due to the identification of

incorrect values which have been discarded in the analysis The scaling of the wind speed moments

has been analysed and empirical moments scaling exponent functions K(q) have been obtained

Although the same critical moment (qcrit) has been obtained for all the places some differences

appear in other multifractal parameters like cmax and the value of K(0) These differences have

been related to the presence of extreme events and zero data values in the data series analysed

respectively VC 2013 American Institute of Physics [httpdxdoiorg10106314793781]

This work studies the multifractal properties of wind speed

time series at different places finding a common value for

the critical moment also showing a self-organized critical-

ity (SOC) nature for the wind speed time series independ-

ently of their origin The multifractal theory studies the

mathematical scale invariance of objects processes or sys-

tems So when a system exhibits a multifractal behavior

all of its events behave the same at any temporal scale The

SOC systems reach a statistically steady state which is sup-

ported by temporal fluctuations characterized by the

energy that they release in an indefinitely repeated cycle

The steady state can be described in terms of power laws

(eg Bak 1997) with exponents explaining the statistical

properties of complex systems These exponents are con-

nected to one of the multifractal parameters (qcrit) obtained

when the multifractal nature of a system is analyzed using

the scaling exponent function K(q)

I INTRODUCTION

Wind data are crucial to a variety of scientific and tech-

nical applications ranging from hydrological studies (eg

soil erosion modelling fight against desertification and ref-

erence evapotranspiration estimations) and civil engineering

design (eg bridge and building constructions) to renewable

energy uses (Chellali et al 2012 DeGaetano 1997 Flores

et al 2005 WMO 2008) Wind singularities affect melt

production soil moisture groundwater recharge stream dis-

charge and vegetation patterns (Flerchinger and Cooley

2000 Marks et al 2002 Pohl et al 2006)

Wind speed varies in time and space making it necessary

its detailed characterization and modelling Furthermore

many wind models used for energy production are very sensi-

tive to wind speed fluctuations (Calif et al 2011) For this

reason several studies have been performed to describe wind

speed frequency distributions using a variety of probability

density function (Chang 2011 Tar 2007)

In addition increasingly high quality climate input data-

sets for accurate and reliable estimates and assessments are

required for improving these applications and models

(Graybeal 2006) Validation of meteorological data ensures

that needed information has been properly generated identify-

ing incorrect values and detecting problems that require im-

mediate maintenance attention (Estevez et al 2011)

Accuracy of wind speed measurements is difficult to assess

unless duplicate instruments are used (Allen 1996) which is

quite rare in weather station networks Nevertheless different

methods can be applied to ensure quality of wind data peri-

odic maintenance of stations and field sensor checks sensor

calibration and validation of data (Estevez et al 2011)

Consistent and low wind speeds can indicate dirty anemome-

ter bearings that will increase anemometer wind speed thresh-

old and might eventually seize and stop the anemometer

altogether Wind speeds from failed anemometers will usu-

ally appear as small or constant values (ASCE-EWRI 2005)

OrsquoBrien and Keefer (1985) proposed a set of three computer-

based rules which were applied by Meek and Hatfield (1994)

to validate meteorological data These rules include computa-

tion of fixed or dynamic highlow bounds for each variable

use of fixed or dynamic rate of change limits for each variable

and a continual no-observed-change in time limit A set of

validation procedures summarized by Estevez et al (2011)

was applied to wind speed data including range test step

test internal consistency test and persistence test

Meteorological variables as indicators of fluctuations in

general atmospheric circulation and climate system show frac-

tal nature by taking the self-similar structure over a wide range

of time scales (Feng et al 2009) Even though this similarity

exists there are many more small events than large ones for

these meteorological variables These larger events are those

that more influence the changes of complex systems (Bak

1997) being the basis for the self-organized criticality theory

a)Author to whom correspondence should be addressed Electronic mail

amandagarciaucoes

1054-1500201323(1)0131339$3000 VC 2013 American Institute of Physics23 013133-1

CHAOS 23 013133 (2013)

Usually a single fractal dimension might not always be enough

to describe complex and heterogeneous behavior of some me-

teorological variables In this case it is necessary to consider a

set of fractal dimensions (Mandelbrot 1974) Multifractal

description of these variables is an efficient tool to study the

time structure of these variables (Jimenez-Hornero et al2011) SOC systems and multifractals can be related taking the

combination of scaling with hyperbolic probabilities as the

defining feature of self-organized criticality (Bak et al 1987)

and some parallels can be made between both behaviors

(Schmitt et al 1994) Many multifractal approach can be used

with this purpose such us the turbulence formalism (Schertzer

and Lovejoy 1987) the multifractal detrended fluctuation

(Kantelhardt et al 2002) the strange attractor formalism (eg

Hentschel and Procaccia 1983) and the joint multifractal

approach (Meneveau et al 1990) among others Wind speed

data records have been recently started to study from a scale

invariance point of view (eg Kavasseri and Nagarajan 2005

Feng et al 2009 Jimenez-Hornero et al 2011 Telesca and

Lovallo 2011) The objective of the present work is improving

the knowledge on wind speed time series by applying the mul-

tifractal analysis to data recorded at six different locations in

the province of Cadiz (Southwestern Spain) This area has

been selected because wind speed is one of its most character-

istic meteorological variables affecting tourism agriculture

wind farms and daily life of its inhabitants

II MATERIALS AND METHODS

A Data

Semihourly wind speed data (30-min scale) used in this

work were obtained from the Agroclimatic Information

Network of Andalusia (RIAA in Spanish) This network was

installed by the Spanish Ministry of Agriculture Fisheries and

Food (MAPYA) and now it is managed by IFAPA

(Agricultural Research Institute of Regional Government of

Andalusia) Each automated weather station is controlled by a

CR10X datalogger (Campbell Scientific) and is equipped with

a wind monitor RM Young 050103 to measure wind speed

and direction Data provided by six stations located through-

out Cadiz province (Fig 1) were used in this study The wind

speed time series corresponding to four years (2002ndash2005) are

depicted in Fig 2 for each of the selected locations Site eleva-

tions ranged from 20 to 171 m above mean sea level longi-

tude from 5384 to 6131W and latitude from 36286 to

36845N Table I shows latitude longitude elevation above

mean sea level average wind speed maximum wind speed

standard deviation wind speed skewness coefficient kurtosis

coefficient percentage of calms and percentage of wind

speed data higher than six times the mean value Wind calms

correspond to wind speed below 05 m s1 (eg Builtjes

1983 Jimenez-Hornero et al 2011) As can be checked in

Table I average wind speed for semihourly values varies

from 122 to 295 m s1 maximum wind speed values range

from 931 to 1797 m s1 and wind calms percent varies from

800 at Vejer station to 3582 at Conil station Skewness

and kurtosis coefficients are positive therefore wind speed

values are grouped below the mean and they are ldquopeakedrdquo

distributed Finally standard deviation values vary from

1235 m s1 at Jerez station to 2427 m s1 at Vejer station

B Validation procedures

For identifying erroneous data from sensor measure-

ments some validation methods were applied to wind speed

FIG 1 Spatial distribution of stations located in Cadiz province (Andalusia Southern Spain)

013133-2 Garcıa-Marın et al Chaos 23 013133 (2013)

records Initially they are necessary to verify that all possi-

ble data have been collected and that record structure is cor-

rect complete and without any gaps in the data files

Meteorological data not fulfilling the above requirements are

flagged as erroneous Data were screened according to the

procedures reported in Table II Thus the range test is based

on a combination of performance specifications of wind

monitor and physical extremes for wind speed The step test

is a quality control procedure based on time consistency and

it compares the difference between successive measure-

ments If the difference exceeds an allowed value different

for each parameter both observations are flagged as suspect

(Shafer et al 2000) Therefore this test checks the excessive

rate of change of two consecutive values (Meek and

Hatfield 1994) The internal consistency test is based on the

verification of physics or climatologic consistency of each

observed parameter or on the relation between two measured

variables (Greurouter et al 2001) To validate wind speed data

using direction data the following conditions will be true

wind speedfrac14 0 and wind directionfrac14 0 wind speed 6frac14 0 and

wind direction 6frac14 0 (DeGaetano 1997 Zahumensky 2004)

The persistence test checks the variability of the

FIG 2 Semihourly wind speed time se-

ries registered at Conil Jerez Jimena

Puerto Vejer and Villamartin stations

for four years (2002ndash2005)


measurements In this sense a continual no-observed-change

in time limit was used for validating wind speed data

No gaps were detected for all datasets analyzed No data

were flagged applying the range and step tests Nevertheless

internal consistency routines together with persistence test

were able to detect anomalous data (Table III) These anom-

alous data flagged by the tests were not considered in the

multifractal analysis ensuring the quality of wind speed data

series and therefore the reliability of the results obtained in

this study

C Multifractality of wind

Multifractal analysis is applicable to variables that can

be regarded as multifractal measures ie variables self-

similarly distributed on a geometric support that is repre-

sented by a line (ie time series) plane volume or fractal

set (Feder 1988) Different methodologies exist to identify

multifractality being the turbulence formalism developed by

Shertzer and Lovejoy (1987) one of the most widely used in

hydrology (eg Shertzer and Lovejoy 1988 De Lima and

Grassman 1999 De Lima and De Lima 2009 Garcıa-Marınet al 2013) The theoretical basis for applying this multi-

fractal approach is the assumption that the variability of the

process could be directly modeled as a stochastic (or ran-

dom) turbulent cascade process (Schertzer and Lovejoy

1987 Gupta and Waymaire 1993 Over and Gupta 1994

Lovejoy and Schertzer 1995)

Following this methodology the multifractal temporal

structure of a process can be investigated by studying the

(multiple) scaling of its statistical moments The scaling of

the moments can be described by the exponent function

K(q) which satisfies (Schertzer and Lovejoy 1987)

heqki kKethqTHORN (1)

where heqki is the average qth moment of the intensity of the

process at a scale k (the ratio between the length of the data

set and any time interval) and K(q) is the so-called empirical

moments scaling exponent function

K(q) can be regarded as a characteristic function of scal-

ing behaviour (eg Svensson et al 1996) For simple scaled

or (mono) fractal processes (eg Mandelbrot 1972) the plot

of K(q) versus q is a straight line but does not pass through

the origin showing only an intensity level for the process If

K(q) is linear through the origin the measure is self-similar

However if the moment scaling function is nonlinear the

measure is multifractal (Veneziano et al 2006) The inter-

cept of the corresponding linear section in the K(q) function

is an estimate of c(cmax) (de Lima and de Lima 2009) being

c(c) the codimension function of the process c the order of

singularity and cmax its maximum value The function c(c)

describes the scaling of the probability distribution of the

TABLE I Summary of weather station site characteristics used in this study (Cadiz province)

Stations Lat ()a Long ()b Elev (m)c Avg (m s1)d Std (m s1)e Csf Ck

g Max (m s1)h Calmsi Datagt 6Avgj

Jerez de la Frontera 36643 6012 32 1571 1235 1490 6260 10460 1739 0017

Villamartın 36845 5621 171 1898 1601 1434 5495 12120 1695 0012

Conil de la Frontera 36337 6130 26 1231 1304 1533 5046 9310 3582 0055

Vejer de la Frontera 36286 5838 24 2956 2427 1308 4997 17970 800 0004

Jimena de la Frontera 36413 5384 53 2215 2364 1351 4254 15400 2884 0037

Puerto de Sta Marıa 36617 6151 20 2581 2007 1403 5841 15930 1077 0004

aLat LatitudebLong LongitudecElev Elevation above mean sea leveldAvg Average wind speedeStd Standard deviation wind speedfCs Skewness coefficientgCk Kurtosis coefficienthMax Maximum wind speediCalms Percentage of wind speed data lower than 05 m s1jDatagt 6Avg Percentage of wind speed data higher than six times average wind speed value

TABLE II Validation procedures applied to wind speed (Estevez et al 2011)

Range test Step test Internal consistency test Persistence test

0ltUshlt 603 m s1 jUsh-Ush-2jlt 10 m s1 Speedfrac14 0 and directionfrac14 0 Speed 6frac14 0 and direction 6frac14 0 Ush 6frac14Ush-2 6frac14Ush-4 6frac14Ush-6

TABLE III Percentage of anomalous data detected () at each automated

weather station

Stations

Range

test

Step

test

Internal

consistency test

Persistence

test

Jerez de la Frontera 0000 0000 0004 0000

Villamartın 0000 0000 0021 0000

Conil de la Frontera 0000 0000 0174 0218

Vejer de la Frontera 0000 0000 0011 0007

Jimena de la Frontera 0000 0000 0004 0000

Puerto de Sta Marıa 0000 0000 0018 0001


process intensity and indicates how the histograms change

with resolution (de Lima 1998) Both multifractal functions

are related by a Legendre transform (Frisch and Parisi 1985)

and establish a correspondence between orders of singularity

c and statistical moments q

KethqTHORN frac14 maxcfqc cethcTHORNgcethcTHORN frac14 maxqfcq KethqTHORNg

(2)

For the high wind speeds the empirical function K(q) shows

a linear section when moments q exceed a critical value qcrit

FIG 3 Log-log plot of the qth moments of wind speed intensity ek on the time scales from 05 h (kfrac14 65536) up to 32 768 h (almost four years kfrac14 1) versus

the scale ratio k at all the locations (a) For moments higher than 1 (b) for moments lower than 1


q gt qcrit Such a discontinuity in the first or second deriva-

tive of K(q) arises either because of divergence of moments

at qcrit (called qD in the first order case) or due to the inad-

equate sample size (at qs in the second order case) cmax is

the corresponding largest singularity order present in the

sample data (eg Schertzer and Lovejoy 1987 Tessier

et al 1993 Lovejoy and Schertzer 1995 de Lima and de

Lima 2009) and can be determined from

cmax frac14 maxethK0ethqTHORNTHORN (3)

In terms of K(q) the low wind speeds are characterized by

Keth0THORN frac14 Cs Cs is the codimension of the support being

the corresponding fractal dimension of the set over which

the measure is carried out D frac14 1 Cs

III RESULTS AND DISCUSSION

To analyse the multifractal behaviour of wind speed

time series the scaling of the moments q has been deter-

mined For all the locations Fig 3 shows the log-log plot of

the average qth moments of the wind speed intensity ek on

the time scales from 05 h (kfrac14 65 536) up to 32 768 h (almost

four years kfrac14 1) against the scale ratio k For each place

the top plot shows moments larger than 1 and the bottom one

shows those smaller than one In both cases only some of

the q moments calculated are plotted in order to clarify the

figure and the results As it can be seen in Fig 3 the scaled

behaviour was detected from 24 h to three months for all the

places This scaling time interval from daily to seasonal

could be related to wind dynamics and its dependence on

pressure gradients turbulence topography and temperature

variations during night and day and between seasons (eg

Kavasseri and Nagarajan 2005 Telesca 2011) It also cor-

roborates the long term analysis of wind speed data needed

for deciding the best site for a wind farm (eg Elamouri and

Bel Amar 2008 Ucar and Balo 2009) The absence of a

typical time scale for less than 24 h highlights the importance

of knowing the wind speed behaviour for short time dura-

tions in order to guaranty the stability of electrical supply

from these wind farms (eg Calif et al 2011) In summary

the detected scaling behaviour seems to be a wind speed fea-

ture and should be considered for modelling the correspond-

ing time series

Despite the similar scaling behaviour found some dif-

ferences appear for moments lower than one As it can be

seen in Fig 3 for all the q values represented the average

qth moments for all the time scales show different proximity

depending on the place These results demonstrate that small

moments amplify the effects of the lowest values in the time

series highlighting singularities of lower order (eg de

Lima and Grassman 1999) Thereby the average moments

plot appears to be more disperse in the case of Conil com-

pared to Jimena that is closely followed by Jerez The nar-

rowest average qth moments plot is corresponding to Vejer

pursued by Puerto and Villamartin The same trend can be

found when analysing the percentage of calms in the wind

data series (Table I) being this percentage related to the low-

est wind speed values

The clear different behavior of qth moments for Conil

could also be related to its wind pattern characterized by

very frequent values of low wind speed (Table I) This cir-

cumstance has to be considered when discussing the proper

probability distribution function for this location The

Weibull distribution has been widely used to describe the

probability density of wind speed in different regions over

the world (Burton et al 2001 Carta et al 2009) Recently

a new family of distributions based on the principle of the

maximum entropy has been proposed as a new alternative to

FIG 4 Empirical moments scaling exponent functions k(q) for the range of scales from 24 h to three months and considering the qth moments of the wind

speed intensity


fit wind speed probability density providing better results

(Li and Li 2004) Locations where low wind speed values

are very significant (including calm conditions) cannot be

properly adjusted by Weibull distribution because of its own

mathematical definition (Ben Amar and Elamouri 2011)

However distributions based on maximum entropy have

provided better fits at many sites (Chang 2011 Ramırez and

Carta 2006 Zhou et al 2010)

Figure 4 shows the K(q) functions obtained for scales

from 24 h to three months As it can be seen on the left plot

of the figure all the functions are convex meaning that the

wind speed time series are multifractal The empirical

moments scaling functions for the lowest values of q are

plotted on the right side of Fig 4 As it can be checked the

behavior of the K(q) functions around the mean (qfrac14 1) is

similar at all the locations being Keth1THORN 0 showing the con-

servation condition heki frac14 1 (de Lima 1998) For Keth0THORNsome differences have been found between Conil and the

rest of wind speed time series analyzed (Table IV) These

values are related to process ldquozerosrdquo (de Lima 1998) and are

in agreement with the percentage of wind speed data of

00 m s1 (zeros in Table IV) that for Conil (285) is sig-

nificantly greater compared to the rest of sites (from 004 to

022) The same values also agree with the fractal dimension

D being close to 1 for ldquosaturatedrdquo data series (eg de Lima

and de Lima 2009) or what is the same with very few zero

data values

A detailed plot of the K(q) functions is shown in Fig 5

where the same value of the critical moment qcrit was yielded

for all the locations This critical moment has been tradition-

ally related to the algebraic decay of the probability distribu-

tion of the extreme events size (eg Schertzer et al 1993

Schertzer and Lovejoy 1994 Tessier et al 1996 Garcıa-

Marin et al 2008) and suggests that values of qcrit higher

than 2 are present in multifractal data sets According to

Pandey et al (1998) multifractals naturally contain singular-

ities of extreme orders associated with multifractal phase

transitions (eg Schertzer et al 1993 Schertzer and

Lovejoy 1994) and generally exhibit algebraic decays

of extreme events Bak et al (1987 1988) consider both fea-

tures (associated with a finite value of qcrit) as the main char-

acteristics of SOC systems

FIG 5 Detailed empirical moments scaling exponents functions K(q) for all the locations indicating the values for qcrit and cmax

TABLE IV Characterization of empirical function K(q) at qfrac14 0 and values

of the fractal dimension (D) and percentage of zero wind speed values (

Zero) for all the locations

Locations K(0) D Zero

Conil de la Frontera 00252 09748 285

Jerez de la Frontera 00002 09998 004

Jimena de la Frontera 00003 09997 010

Puerto de Sta Marıa 00000 10000 018

Vejer de la Frontera 00000 10000 004

Villamartin 00000 10000 022


The values of cmax range from 0239 (Puerto and

Villamartin) to 0324 (Conil) Intermediate values were

found for Vejer (0254) Jerez (0314) and Jimena (0316)

These values could provide some information about extreme

and rare events in the time series (eg Tessier et al 1993

Boutet 2000 Veneziano and Furcolo 2002 Garcıa-Marınet al 2012) According to Table I the top percentage of

wind speed records higher than six times the average wind

speed value corresponds to Conil station (0055) and the

lowest belongs to Puerto (0004) Vejer (0004) and

Villamartin (0012) Intermediate values are newly found

for Jerez (0017) and Jimena (0037) This consistency of

results shows the correspondence between cmax and the vio-

lent (ie rare and extreme) behavior of wind speed at the

locations analyzed (eg de Lima and de Lima 2009)

IV SUMMARY AND CONCLUSIONS

The analysis of validated wind speed time series at six

locations of Cadiz (Andalusia) has shown their multifractal

nature The scaling behavior of the moments was found from

24 h to around three months for all the places being this

range relevant for models or processes dealing with wind

speed data The influence of calms in the time series has been

also detected with the study of the moments smaller than one

Thereby the location with higher percentage of calms in its

wind speed records showed a wider log-log plot of the aver-

age qth moments against the scale ratio k This behavior high-

lights how the small moments amplify the effects of the

lowest values The appearance of the moments log-log plots

can also be related to low wind speed pattern and helping to

know the proper probability distribution function for a place

The convex shape of the empirical moments scaling

exponent function yielded for all the locations indicates the

presence of multiple scaling in the wind speed times series

All the exponent functions are linear after the same critical

value higher than 2 and behave the same around the mean

Nevertheless some differences appear for the functions at

the value of zero The lowest values are obtained for the pla-

ces with less zero data in their wind speed series and the

fractal dimension is then close to one indicating saturation

A qualitative characterization of extreme and rare wind

speed events could be done according to the values of the

highest singularities present in the data Thereby as singular-

ity order increases a larger amount of high wind speed

records is detected in the time series

ACKNOWLEDGMENTS

We applied the ldquosequence-determines-creditrdquo (SDC)

approach for the sequence of authors F J Jimenez-Hornero

acknowledges the support from the Spanish Ministry of

Economy Competitiveness ERDF Projects No AGL2009-

12936-C03-02 J Estevez also thanks the IFAPA (Junta de

Andalucıa) for providing meteorological data to carry out

this work

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piration estimationrdquo J Irrig Drain Eng 122 97ndash106 (1996)

ASCE-EWRI The ASCE Standardized Reference EvapotranspirationEquation Environmental and Water Resources Institute of the ASCEStandardization of Reference Evapotranspiration EvapotranspirationTask Committee (American Society of Civil Engineers Reston Virginia

2005) p 216

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planation of 1f noiserdquo Phys Rev Lett 59(4) 381ndash384 (1987)

Bak P Tang C and Wiesenfeld K ldquoSelf-organized criticalityrdquo Phys

Rev A 38 364ndash374 (1988)

Ben Amar F and Elamouri M ldquoA new theoretical model for modeling the

wind speed frequency distributionrdquo Int J Renewable Energy Res 1(4)

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Boutet S ldquoMultifractality and multifractal phase transitions in turbulencerdquo

Term Paper for Physics 464 1ndash16 (2000)

Builtjes P J H ldquoTurbulent diffusivities and dispersion coefficients

Applicacion to calm wind conditionsrdquo Sci Total Environ 23 107ndash118

1982

Burton T Sharpe D Jenkins N and Bossanyi E Wind Energy HandBook (John Wiley and Sons 2001)

Calif R Emilion R and Soubdhan T ldquoClassification of wind speed dis-

tributions using mixture of Dirichlet distributionsrdquo Renewable Energy

36 3091ndash3097 (2011)

Carta J A Ramırez P and Velazquez S ldquoA review of wind speed proba-

bility distributions used in wind energy analysis Case studies in the

Canary Islandsrdquo Renewable Sustainable Energy Rev 13(5) 933ndash955

(2009)

Chang T P ldquoEstimation of wind energy potential using different probabil-

ity density functionsrdquo Appl Energy 88 1848ndash1856 (2011)

Chellali F Khellafb A Belouchranic A and Khannichea R ldquoA com-

parison between wind speed distributions derived from the maximum en-

tropy principle and Weibull distribution Case of study six regions of

Algeriardquo Renewable Sustainable Energy Rev 16 379ndash385 (2012)

De Lima M I P and de Lima J L M P ldquoInvestigating the multifractality

of point precipitation in the Madeira archipielagordquo Nonlinear Processes

Geophys 16 299ndash311 (2009)

De Lima M I P and Grasman J ldquoMultifractal analysis of 15-min and daily

rainfall from a semi-arid region in Portugalrdquo J Hydrol 220 1ndash11 (1999)

De Lima M I P ldquoMultifractals and the temporal structure of rainfallrdquo

PhD dissertation (Wageningen Agricultural University The

Netherlands 1998)

DeGaetano A T ldquoA quality control procedure for hourly wind datardquo

J Atmos Ocean Technol 14 308ndash317 (1997)

Elamouri M and Ben Amar F ldquoWind energy potential in Tunisiardquo

Renewable Energy 33 758ndash768 (2008)

Estevez J Gavilan P and Giraldez J V ldquoGuidelines on validation

procedures for meteorological data from automatic weather stationsrdquo

J Hydrol 402 144ndash154 (2011)

Feder J Fractals (Plenum New York 1988)

Feng T Fu Z Deng X and Mao J ldquoA brief description to different

multi-fractal behaviours of daily wind speed records over Chinardquo Phys

Lett A 373 4134ndash4141 (2009)

Flerchinger G N and Cooley K R ldquoA ten-year water balance of a moun-

tainous semi-arid watershedrdquo J Hydrol 237 86ndash99 (2000)

Flores P Tapia A and Tapia GldquoApplication of a control algorithm for

wind speed prediction and active power generationrdquo Renewable Energy

30 523ndash536 (2005)

Frisch U and Parisi G ldquoOn the singularity structure of fully developed

turbulencerdquo in Turbulence and Predictability in Geophysical FluidDynamics and Climate Dynamics edited by Ghil M Benzi R and

Parisi G (North-Holland New York 1985) pp 84ndash88

Garcıa-Marın A P Ayuso-Mu~noz J L Jimenez-Hornero F J and

Estevez J ldquoSelecting the best IDF model by using the multifractal

approachrdquo Hydrolog Process 27 433ndash443 (2013)

Garcıa-Marın A P Jimenez-Hornero F J and Ayuso J L ldquoApplying

multifractality and the self-organized criticality theory to describe the

temporal rainfall regimes in Andalusia (southern Spain)rdquo Hydrolog

Process 22 295ndash308 (2008)

Graybeal D Y ldquoRelationships among daily mean and maximum wind

speeds with application to data quality assurancerdquo Int J Climatol 26

29ndash43 (2006)

Greurouter E Heuroaberli C Keuroung U Mumenthaler P Mettler J Bassi M

Konzelmann T and Deuroosseger R ldquoThe next generation of quality


control tools for meteorological data at MeteoSwissrdquo in Proceedings of

the Deutsch-Oesterreichisch-Schweizerischen Meteorologentagung

(DACH Vienna Austria 2001)

Gupta V K and Waymire E C ldquoA statistical analysis of mesoscale rain-

fall as a random cascaderdquo J Appl Meteorol 32 251ndash267 (1993)

Hentschel H G E and Procaccia I ldquoThe infinite number of generalized

dimensions of fractals and strange attractorsrdquo Physica D 8 435ndash444

(1983)

Jimenez-Hornero F J Pavon-Domınguez P Gutierrez de Rave E and

Ariza-Villaverde A B ldquoJoint multifractal description of the relationship

between Wind patterns and land surface temperaturerdquo Atmos Res 99

366ndash376 (2011)

Kantelhardt J W Zschiegner S A Koscielny-Bunde E Havlin S

Bunde A and Stanley H E ldquoMultifractal detrended fluctuation analy-

sis of nonstationary time seriesrdquo Physica A 316 87ndash114 (2002)

Kavasseri R G and Nagarajan R ldquoA multifractal description of wind

speed recordsrdquo Chaos Solitons Fractals 24 165ndash173 (2005)

Li M and Li XldquoOn the probabilistic distribution of wind speeds

Theoretical development and comparison with datardquo Int J Energy 1(2)

237ndash255 (2004)

Lovejoy S and Schertzer D ldquoMultifractals and rainrdquo in New UncertaintyConcepts in Hydrology and Water Resources edited by Kundzewicz Z

W (Cambridge University PressndashUNESCO International Hydrology

Series New York 1995) pp 61ndash103

Mandelbrot B B ldquoPossible refinement of the lognormal hypothesis con-

cerning the distribution of energy dissipation in intermittent turbulencerdquo

in Statistical Models and Turbulence Lectures Notes in Physics Vol 12

(Springer-Verlag New York 1972) pp 333ndash351

Mandelbrot B B ldquoIntermittent turbulence in self-similar cascades

Divergence of high moments and dimension of the carrierrdquo J Fluid

Mech 62 331ndash358 (1974)

Marks D Winstral A and Seyfried M ldquoSimulation of terrain and forest

shelter effects on patterns of snow deposition snowmelt and runoff over

a semi-arid mountain catchmentrdquo Hydrolog Process 16 3605ndash3626

(2002)

Meek D W and Hatfield J L ldquoData quality checking for single station

meteorological databasesrdquo Agric Forest Meteorol 69(1ndash2) 85ndash109

(1994)

Meneveau C Sreenivasan K R Kailasnath P and Fan M S ldquoJoint

multifractal measuresmdashTheory and applications to turbulencerdquo Phys

Rev A 41 894ndash913 (1990)

OrsquoBrien K J and Keefer T N ldquoReal-time data verification Computer

applications in water resourcesrdquo in Proceedings of the ASCESpecification Conference Buffalo NY ASCE New York NY (1985) pp

764ndash770

Over T M and Gupta V K ldquoStatistical analysis of mesoscale rainfall de-

pendence of a random cascade generator on large scaling forcingrdquo

J Appl Meteorol 33 1526ndash1543 (1994)

Pandey G Lovejoy S and Schertzer D ldquoMultifractal analysis of daily

river flows including extremes for basins of five to two million square kil-

ometres one day to 75 yearsrdquo J Hydrol 208 62ndash81 (1998)

Pohl S Marsh P and Liston G E ldquoSpatial-temporal variability in turbu-

lent fluxes during spring snowmeltrdquo Arctic Antarctic Alp Res 38(1)

136ndash146 (2006)

Ramırez P and Carta J A ldquoThe use of wind probability distributions

derived from the maximum entropy principle in the analysis of wind

energy A case studyrdquo Energy Convers Manage 47 2564ndash2577 (2006)

Schertzer D and Lovejoy S ldquoPhysical modelling and analysis of rain and

clouds by anisotropic scaling multiplicative processesrdquo J Geophys Res

[Atmos] 92 9693ndash9714 doi101029JD092iD08p09693 (1987)

Schertzer D and Lovejoy S ldquoMultifractal simulations and analysis of rain

and clouds by anisotropic scaling multiplicative processesrdquo Atmos Res

21 337ndash361 (1988)

Schertzer D and Lovejoy S ldquoMultifractal generation of self-organized

criticalityrdquo in Fractals in the Natural and Applied Sciences edited by

Novak M M (Elsevier North-Holland 1994) pp 325ndash339

Schertzer D Lovejoy S and Lavallee D ldquoGeneric multifractal phase

transitions and self-organized criticalityrdquo in Cellular AutomataProspects in Astrophysical Applications edited by Perdang J M and

Lejeune A (World Scientific 1993) pp 216ndash227

Shafer M A Fiebrich C A Arndt D S Fredrickson S E and Hughes

T W ldquoQuality assurance procedures in the Oklahoma Mesonetrdquo


Schmitt F Schertzer D Lovejoy S and Brunet Y ldquoEmpirical study of

multifractal phase transitions in atmospheric turbulencerdquo Nonlinear

Processes Geophys 1 95ndash104 (1994)

Svensson C Olsson J and Berndtsson R ldquoMultifractal properties of

daily rainfall in two different climatesrdquo Water Resour Res 32 2463ndash

2472 doi10102996WR01099 (1996)

Tar K ldquoSome statistical characteristics of monthly average wind speed at

various heightsrdquo Renewable Sustainable Energy Rev 12(6) 1712ndash1724

(2008)

Telesca L and Lovallo M ldquoAnalysis of time dynamics in wind records by

means of multifractal detrended fluctuation analysis and Fisher-Shannon

information planerdquo J Stat Mech Theory Exp 2011 P07001

Tessier Y Lovejoy S Hubert P Schertzer D and Pecknold S

ldquoMultifractal analysis and modelling of rainfall and river flows and scal-

ing causal transfer functionsrdquo J Geophys Res [Atmos] 101 26427ndash

26440 doi10102996JD01799 (1996)

Tessier Y Lovejoy S and Schertzer D ldquoUniversal multifractals in rain

and clouds Theory and observationsrdquo J Appl Meteorol 32 223ndash250

(1993)

Ucar A and Balo F ldquoEvaluation of wind energy potential and electricity

generation at six locations in Turkeyrdquo Appl Energy 86 1864ndash1872

(2009)

Veneziano D and Furcolo P ldquoMultifractality of rainfall and scaling of

intensity-duration-frequency curvesrdquo Water Resour Res 38(12) 42

(2002)

Veneziano D Langousis A and Furcolo P ldquoMultifractality and rainfall

extremes A reviewrdquo Water Resour Res 42(6) W06D15 doi101029

2005WR004716 (2006)

World Meteorological Organization Guide to Meteorological Instrumentsand Methods of Observations WMO-No8 Geneva Switzerland 2008

Zahumensky I Guidelines on Quality Control Procedures for Data fromAutomatic Weather Stations WMO-No 955 Geneva Switzerland 2004

Zhou J Erdem E Li G and Shi J ldquoComprehensive evaluation of wind

speed distribution models A case study for North Dakota sitesrdquo Energy

Convers Manage 51 1449ndash1458 (2010)


Multifractal analysis of validated wind speed time series

A P Garcıa-Marın1a) J Estevez1 F J Jimenez-Hornero2 and J L Ayuso-Mu~noz1

1Department of Rural Engineering University of Cordoba PO Box 3048 14080 Cordoba Spain2Department of Graphic Engineering and Geomatics Gregor Mendel Building (3rd floor)Campus Rabanales University of Cordoba 14071 Cordoba Spain

(Received 5 September 2012 accepted 11 February 2013 published online 1 March 2013)

Multifractal properties of 30 min wind data series recorded at six locations in Cadiz (Southern

Spain) have been studied in this work with the aim of obtaining detailed information for a range of

time scales Wind speed records have been validated applying various quality control tests as a

pre-requisite before their use improving the reliability of the results due to the identification of

incorrect values which have been discarded in the analysis The scaling of the wind speed moments

has been analysed and empirical moments scaling exponent functions K(q) have been obtained

Although the same critical moment (qcrit) has been obtained for all the places some differences

appear in other multifractal parameters like cmax and the value of K(0) These differences have

been related to the presence of extreme events and zero data values in the data series analysed

respectively VC 2013 American Institute of Physics [httpdxdoiorg10106314793781]

This work studies the multifractal properties of wind speed

time series at different places finding a common value for

the critical moment also showing a self-organized critical-

ity (SOC) nature for the wind speed time series independ-

ently of their origin The multifractal theory studies the

mathematical scale invariance of objects processes or sys-

tems So when a system exhibits a multifractal behavior

all of its events behave the same at any temporal scale The

SOC systems reach a statistically steady state which is sup-

ported by temporal fluctuations characterized by the

energy that they release in an indefinitely repeated cycle

The steady state can be described in terms of power laws

(eg Bak 1997) with exponents explaining the statistical

properties of complex systems These exponents are con-

nected to one of the multifractal parameters (qcrit) obtained

when the multifractal nature of a system is analyzed using

the scaling exponent function K(q)

I INTRODUCTION

Wind data are crucial to a variety of scientific and tech-

nical applications ranging from hydrological studies (eg

soil erosion modelling fight against desertification and ref-

erence evapotranspiration estimations) and civil engineering

design (eg bridge and building constructions) to renewable

energy uses (Chellali et al 2012 DeGaetano 1997 Flores

et al 2005 WMO 2008) Wind singularities affect melt

production soil moisture groundwater recharge stream dis-

charge and vegetation patterns (Flerchinger and Cooley

2000 Marks et al 2002 Pohl et al 2006)

Wind speed varies in time and space making it necessary

its detailed characterization and modelling Furthermore

many wind models used for energy production are very sensi-

tive to wind speed fluctuations (Calif et al 2011) For this

reason several studies have been performed to describe wind

speed frequency distributions using a variety of probability

density function (Chang 2011 Tar 2007)

In addition increasingly high quality climate input data-

sets for accurate and reliable estimates and assessments are

required for improving these applications and models

(Graybeal 2006) Validation of meteorological data ensures

that needed information has been properly generated identify-

ing incorrect values and detecting problems that require im-

mediate maintenance attention (Estevez et al 2011)

Accuracy of wind speed measurements is difficult to assess

unless duplicate instruments are used (Allen 1996) which is

quite rare in weather station networks Nevertheless different

methods can be applied to ensure quality of wind data peri-

odic maintenance of stations and field sensor checks sensor

calibration and validation of data (Estevez et al 2011)

Consistent and low wind speeds can indicate dirty anemome-

ter bearings that will increase anemometer wind speed thresh-

old and might eventually seize and stop the anemometer

altogether Wind speeds from failed anemometers will usu-

ally appear as small or constant values (ASCE-EWRI 2005)

OrsquoBrien and Keefer (1985) proposed a set of three computer-

based rules which were applied by Meek and Hatfield (1994)

to validate meteorological data These rules include computa-

tion of fixed or dynamic highlow bounds for each variable

use of fixed or dynamic rate of change limits for each variable

and a continual no-observed-change in time limit A set of

validation procedures summarized by Estevez et al (2011)

was applied to wind speed data including range test step

test internal consistency test and persistence test

Meteorological variables as indicators of fluctuations in

general atmospheric circulation and climate system show frac-

tal nature by taking the self-similar structure over a wide range

of time scales (Feng et al 2009) Even though this similarity

exists there are many more small events than large ones for

these meteorological variables These larger events are those

that more influence the changes of complex systems (Bak

1997) being the basis for the self-organized criticality theory

a)Author to whom correspondence should be addressed Electronic mail

amandagarciaucoes

1054-1500201323(1)0131339$3000 VC 2013 American Institute of Physics23 013133-1

CHAOS 23 013133 (2013)



























A Data








































































this study











































Villamartın 36845 5621 171 1898 1601 1434 5495 12120 1695 0012










weather station

Stations

Range

test

Step

test

Internal

consistency test

Persistence

test


Villamartın 0000 0000 0021 0000












(2)













































ing time series



























speed intensity



























data values







































































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)

































A Data








































































this study











































Villamartın 36845 5621 171 1898 1601 1434 5495 12120 1695 0012










weather station

Stations

Range

test

Step

test

Internal

consistency test

Persistence

test


Villamartın 0000 0000 0021 0000












(2)













































ing time series



























speed intensity



























data values







































































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)











































this study











































Villamartın 36845 5621 171 1898 1601 1434 5495 12120 1695 0012










weather station

Stations

Range

test

Step

test

Internal

consistency test

Persistence

test


Villamartın 0000 0000 0021 0000












(2)













































ing time series



























speed intensity



























data values







































































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)
















this study











































Villamartın 36845 5621 171 1898 1601 1434 5495 12120 1695 0012










weather station

Stations

Range

test

Step

test

Internal

consistency test

Persistence

test


Villamartın 0000 0000 0021 0000












(2)













































ing time series



























speed intensity



























data values







































































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)













(2)













































ing time series



























speed intensity



























data values







































































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)














































ing time series



























speed intensity



























data values







































































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)
































data values







































































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)


















































ACKNOWLEDGMENTS







this work




2005) p 216





Rev A 38 364ndash374 (1988)



306ndash313 (2011)





1982




36 3091ndash3097 (2011)




(2009)














Netherlands 1998)







J Hydrol 402 144ndash154 (2011)




Lett A 373 4134ndash4141 (2009)





30 523ndash536 (2005)













29ndash43 (2006)











(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)














(1983)




366ndash376 (2011)








237ndash255 (2004)










Mech 62 331ndash358 (1974)




(2002)



(1994)



Rev A 41 894ndash913 (1990)



764ndash770









136ndash146 (2006)









21 337ndash361 (1988)















2472 doi10102996WR01099 (1996)



(2008)







26440 doi10102996JD01799 (1996)



(1993)



(2009)



(2002)



2005WR004716 (2006)







Date post:	04-Dec-2023
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