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Intermittency study of high frequency global solar radiation
sequences under a tropical climate
Rudy Calif
a,⇑
, Francßois G. Schmitt
b
, Yongxiang Huang
c
, Ted Soubdhan
a
a
EA 4539, LARGE – laboratoire en Ge
´osciences et E
´nergies, Universite
´des Antilles et de la Guyane, 97170 P-a
´-P, France
b
CNRS, UMR 8187, LOG – Laboratoire d’Oce
´anologie et de Ge
´osciences, Universite
´de Lille 1, 28 avenue Foch, 62930 Wimereux, France
c
Shanghai Key Laboratory of Mechanics in Energy and Environment Engineering, Shanghai Institute of Applied Mathematics and Mechanics,
Shanghai University, Shanghai 200072, China
Received 20 March 2013; received in revised form 12 September 2013; accepted 13 September 2013
Communicated by: Associate Editor Jan Kleissl
Abstract
A good knowledge of the intermittency of global solar radiation is crucial for selecting the location of a solar power plant and
predicting the generation of electricity. This paper presents a multifractal analysis study of 367 daily global solar radiation sequences
measured with a sampling rate of 1 Hz over one year at Guadeloupean Archipelago (French West Indies) located at 16°150N latitude
and 60°300W longitude. The mean power spectrum computed follows a power law behavior close to the Kolmogorov spectrum. The
intermittent and multifractal properties of global solar radiation data are investigated using several methods. Under this basis, a char-
acterization for each day using three multifractal parameters is proposed.
Ó2013 Elsevier Ltd. All rights reserved.
Keywords: Global solar radiation; Fourier spectrum; Intermittency; Multifractal analysis; Empirical Mode Decomposition (EMD); Hilbert spectral
analysis
1. Introduction
The installed capacity for energy from solar farm, is con-
stantly increasing in response to worldwide interest in low-
emissions power sources and a desire to decrease the depen-
dence on petroleum. The variability and unpredictability of
solar energy over short time scales remains a major problem
as its penetration of this energy into the electric grid is lim-
ited. This can provoke voltage flicker and fluctuations that
can trigger automated line equipment on distribution feed-
ers, leading to larger maintenance costs for utilities (Lave
et al., 2012). In small non-interconnected grids, such as
those that exist on islands, i.e., Guadeloupean Archipelago
(FWI), these variations can cause instabilities. If some
energy storage devices are installed in the non-intercon-
nected electric network, as a real-world demonstration has
done, the power fluctuations of the network due to the pho-
tovoltaic arrays could effectively be suppressed. Thus, the
stability of the network could be greatly enhanced. How-
ever, the fluctuations of aggregated power output from
photovolatics systems vary following global solar radiation
fluctuations. Because of stochastic courses and distribution
of clouds, the solar radiation signal contains huge intraday
fluctuations reaching 700 W/m
2
and occurring within time
scales from a few seconds to a few minutes in tropical cli-
mates. Hence, a good knowledge of global solar radiation
variations is of real practical importance in managing the
electrical network integrating this kind of stochastic energy.
Numerous studies have been devoted to classifying daily
global solar radiation under different regimes of cloud cov-
erage, including fractal dimensions (Maafi and Harrouni,
0038-092X/$ - see front matter Ó2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.solener.2013.09.018
⇑
Corresponding author.
E-mail address: rcalif@univ-ag.fr (R. Calif).
www.elsevier.com/locate/solener
Available online at www.sciencedirect.com
ScienceDirect
Solar Energy 98 (2013) 349–365
2003), mixtures of Dirichlet distributions (Soubdhan et al.,
2009), and mathematical morphological techniques
(Gasto
´n et al., 2011). These methods and classification cri-
teria assume a limited number of classes. In (Czekalski
et al., 2012), a parametrization of daily solar radiation
was projected with the use of classical indexes: a mathemat-
ical THD (Total Harmonic Distorsion) and a statistical one
(based on the concept of standard deviation). This statisti-
cal estimator describes a medium level of fluctuations
(Schertzer et al., 1997; Morales et al., 2011). However, they
are not sufficient to fully describe the fluctuations of a pro-
cess, at all intensities and all scales, taking into account
extreme events.
Multifractal properties have been highlighted in several
disciplinary fields, such as turbulence (Bo
¨ttcher et al., 2007;
Frisch, 1995; Sreenivasan and Antonia, 1997; Warhaft,
2000; Lohse and Xia, 2010), finance (Ghashghaie et al.,
1996; Schmitt et al., 1999; Calvet and Fisher, 2002), phys-
iology (Ivanov et al., 1999), rain-fall (Schertzer and Love-
joy, 1987, 1998, 2006), and geophysical fields
(Kantelhardt et al., 2006; Mauas et al., 2008; Huang
et al., 2009). A multifractal framework may be relevant
for characterizing the high variability of daily global solar
radiation sequences. Global solar radiation recorded at the
ground level is a complex natural phenomenon which is a
combination of many nonlinear dynamics processes involv-
ing turbulence such as cloud fields and atmospheric wind in
the atmospheric boundary layer. Some works have already
considered related quantities in a multifractal framework.
In (Sachs et al., 2002; Watson et al., 2009), the authors
highlighted scaling and multifractal properties of clouds
over a wide range of scales. In (Chigirinskya et al., 1994;
Lovejoy et al., 2001), the authors investigated scaling and
multifractal properties of atmospheric wind for mesoscale
range in the boundary layer. This was the motivation
behind the present multifractal analysis of a daily global
solar radiation sequence, which is the first full analysis of
the multifractal properties of radiation fluctuations.
Here solar radiation data is studied in the framework of
universal multifractals (Schertzer and Lovejoy, 1987)in
order to describe the fluctuations at all scales and at all
intensities (low-level, medium level and high level) using
only three relevant parameters. For this, we considered
the scaling exponent f(q) that characterizes the scaling
behavior or measures the distance between a monofractal
and multifractal process. Indeed if f(q) is linear, the process
considered is monofractal, if f(q) is nonlinear, the process is
multifractal. Furthermore, the concavity gives information
on the intermittency degree: the more concave the curve is,
the more intermittent the process (Frisch, 1995; Schertzer
et al., 1997; Vulpiani and Livi, 2004). Several methods exist
for estimating scaling exponents f(q): structure function
analysis, wavelet-based methods (wavelet leader, wavelet
transform modulus maxima), detrended fluctuation analy-
sis or multifractal detrended fluctuation analysis and a new
method developed by Huang et al. (2008), arbitray-order
Hilbert spectral analysis an extension of Hilbert Huang
Transform (HHT) (Huang et al., 1999). The last method
includes Empirical Mode Decomposition (EMD) which is
a self-adaptive signal processing method for nonstationary
and nonlinear time series. To estimate the scaling expo-
nents here structure function analysis, multifractal detrend-
ed fluctuations analysis, wavelet leaders and arbitrary
order Hilbert spectral analysis to global solar radiation
data were applied.
The structure of this paper is as follows. In Section 2, the
site and the dataset are described. In Section 3, a brief
description of the multifractal framework is given,
Nomenclature
Ainstantaneous amplitude
C
1
fractal co-dimension
Dfractal dimension
E(f) Fourier spectrum
ffrequency (Hz)
f
s
sampling frequency (Hz)
F
q
fluctuations function
g(t) global solar radiation (W/m
2
)
g
norm
(t) normalized global radiation
G
0
(t) extraterrestrial solar radiation (W/m
2
)
hHilbert spectrum
HHurst parameter
I(t) IMF component
jscale index
kposition index
Mmultifractal spectrum
Ntotal length of a sequence
qorder of moment
S
q
structure functions
Ttime scale (s)
T
s
sampling time (s)
T
c
time of passing for clouds (s)
Zqpartition functions
amultifractal Le
´vy parameter
bspectral exponent
gscaling exponents in MFDFA
csingularity exponent
xinstantaneous frequency (Hz)
sRe
´nyi exponent function
nscaling exponent function in the Hilbert space
fscaling exponent function
350 R. Calif et al. / Solar Energy 98 (2013) 349–365
including structure function analysis, wavelet leaders, mul-
tifractal detrended fluctuations analysis and arbitrary-
order Hilbert spectral analysis. In Section 4, the results of
multifractal analysis for a global solar radiation sequence
using several methods, and a full characterization for the
367 days of global solar radiation is provided using three
multifractal parameters, are presented. The objective was
to provide a new way for classifying global solar radiation
sequences.
2. Global solar radiation measurements
In this study, we considered 367 sample days of global
solar radiation g(t) recorded continuously at the University
site of Pointe a
´Pitre in Guadeloupe, an island in the West
Indies, situated at 16°150N latitude and 60°300W longitude.
In such a tropical region, the average solar load for a hor-
izontal surface ranges between 4 kW h/m
2
and 7 kW h/m
2
per day. Constant sunshine combined with the thermal
inertia of the ocean makes the temperature variation quite
weak, between 17 and 33 °C and relative humidity between
70% and 80%. The global solar radiation measurements
g(t) were collected with a sampling frequency f
s
=1Hz
(or a sampling time T
s
= 1 s), over a one year period (from
January 2008 to January 2009); this represents 31,708,800
data points. The global solar radiation data were measured
with a pyranometer from Kipp and Zonen model Sp-Lite.
The response time less than 1 s of this sensor is compatible
with our sampling rate. This pyranometer measures the
solar energy received from the entire hemisphere (i.e.,
180 °C) and is equipped with a leveling device and bubble
gauge so that it can be perfectly aligned. The spectral range
of this sensor extends from 400 to 1100 nm.
In Fig. 1a), a global solar radiation recording over six
consecutive days, is shown. This sequence shows different
regimes of fluctuation depending of passing clouds.
Fig. 1b) illustrates a daily global solar radiation data com-
ing from six consecutive days. Generally, this signal can be
decomposed into a deterministic part, G
0
(t) the extraterres-
trial solar radiation and a stochastic part namely the clear-
ness index k(t)(Iqbal, 1983; Woyte et al., 2007). Fig. 1c)
gives a zoom of daily global solar radiation measurements
displayed in Fig. 1b). This zoom shows a closer view of the
short time scales of this daily global solar radiation
sequence, exhibiting fluctuations stochastically distributed
in time, due to the influence of passing clouds and multiple
diffusion of sunlight in the atmosphere.
3. Multifractal framework
3.1. Defining of scaling exponent function f(q)
Multifractal processes could be seen as an extension of
monofractal processes introduced by Mandelbrot and
characterized by a single exponent such as Hurst parameter
Hor Fractal dimension D(D=2H). The Hurst expo-
nent is not sufficient for describing the dynamics of a mul-
tifractal process, needing a scaling exponent function f(q)
or K(q). The multifractal concepts were introduced with
the multiplicative cascade model, to study the energy dissi-
pation in the context of fully developed turbulence in the
1980’s(Grassberger and Procaccia, 1983; Benzi et al.,
0 1 2 3 4 5 6
0
500
1000
1500
T (s)
g (t) (W/m2)
2.6 2.7 2.8 2.9 33.1 3.2
0
500
1000
1500
T (s)
2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.8 2.81
x 105
0
500
1000
1500
T (s)
(a)
(b)
(c)
g (t) (W/m2)
g (t) (W/m2)
x 105
x 105
Fig. 1. (a) Six consecutive days exhibiting different fluctuating regimes, due to the influence of passing clouds, (b) an example of daily global solar
radiation sequence and (c) a corresponding zoom showing huge fluctuations over short time periods.
R. Calif et al. / Solar Energy 98 (2013) 349–365 351
1984; Parisi and Frisch, 1985). Let xbe a nonstationary sig-
nal with stationary increments. For a scaling process, its
increments Dx=x(t+T)x(t) posses scaling statistics
of the form:
hjDxjqiTfðqÞð1Þ
where Tis the time increment and f(q) is the scale invariant
exponent function, which is non-linear and concave. The
estimation of the (q,f(q)) curve for integer and non-integer
moments, provides a full description of the stochastic pro-
cess x(t) at all intensities and all scales. This function char-
acterizes the nature of its scaling behavior: f(q) is non-
linear and concave for a multifractal process and linear
for a monofractal process. Furthermore, the concavity is
an indication of intermittency. The parameter H=f(1) is
the Hurst parameter. Monofractal processes correspond
to a linear function f(q)=qH, where Brownian motion is
described by H= 1/2, fractional Brownian motion is de-
scribed by 0 < H< 1 and homogeneous non-intermittent
turbulence is described by H= 1/3.
In order to model intermittency, many statistical func-
tions have been proposed to fit f(q) since the introduction
of mulitfractals in the turbulence field. In this study, we
focus on the log-stable model or universal multifractal pro-
posed by Schertzer and Lovejoy (1987) and Kida (1991):
fðqÞ¼qH C1
ða1ÞðqaqÞð2Þ
where H=f(1) the Hurst parameter defines the degree of
smoothness or roughness of the field. The parameter C
1
is the fractal co-dimension of the set giving the dominant
contribution to the mean (q= 1) and bounded between 0
and d(dthe dimension space, here d= 1). It measures
the inhomogeneity mean or the mean intermittency charac-
terizing the sparseness of the field: the larger C
1
, the more
the mean field is inhomogeneous. The multifractal Le
´vy
parameter ais bounded between 0 and 2, where a= 0 cor-
responds to the monofractal case and a= 2 corresponds to
the multifractal lognormal case. The parameter ameasures
the degree of multifractility, i.e, how fast the inhomogene-
ity increases with the order of the moments (Seuront et al.,
1996).
For illustration, in Fig. 2, we have plotted the corre-
sponding exponent function f(q) using the log-stable model
given in (2), with arbitrary fixed parameters H= 0.5,
C
1
= 0.1 and the Le
´vy index afrom 0 to 2 with 0.25 incre-
ment. The concavity of f(q) is a characteristic of the inter-
mittency (Frisch, 1995; Schertzer et al., 1997; Vulpiani and
Livi, 2004).
The computation of multifractal indices a, and C
1
are
obtained from Eq. (2) (Schertzer et al., 1997). For that,
we analyze the following function
RðqÞ¼qf0ð0ÞfðqÞ¼ C1
a1qað3Þ
Thus, the function R(q) versus qwill have a slope aand C
1
can be estimated by the intercept.
In the next section, multifractal analysis techniques to
estimate the empirical curve f(q) from a dataset is given,
in order to extract the multifractal parameters H,C
1
and a.
3.2. Structure function analysis (SF)
Structure function analysis is the traditional multifractal
analysis technique used to estimate the exponent function
f(q), which has been proposed in the turbulence field
(Monin and Yaglom, 1971). This analysis technique has
been widely used in the turbulence research community
(Frisch, 1995; Anselmet et al., 1984) and other research
fields (Schmittbuhl et al., 1995; Schmitt et al., 2011a,bSch-
mitt et al.,2011). We consider a signal x(t
i
) with i=1, 2,
3, ... ,Nand Nthe total length of signal. We define the
time increments of this process written as:
DxTðtiÞ¼xðtiþTÞxðtiÞð4Þ
with Ta time scale. Thus, we study the global solar radia-
tion fluctuations considering their moment of order q
called “structure functions of order q”S
q
(T):
SqðTÞ¼ 1
NT1X
N
i¼1
jDxTðtiÞj
"#
q
¼1
NT1X
N
i¼1
jxðtiþTÞxðtiÞj
"#
q
ð5Þ
where jj means the absolute value. The scaling behavior of
the structures functions S
q
(T) with respect to Tis deter-
mined through the generation of log–log plots of S
q
(T) ver-
sus T, for q= 0–3 and T
s
<T<NT
s
(T
s
is the sampling
time of measurements). A scaling is checked if the existence
of a linear relationship between S
q
(T) and dtis observed in
log–log plot, for the same range for all values of q. Hence,
this highlights a power law of S
q
(T) versus Tas:
0 0.5 1 1.5 2 2.5 3 3.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
q
ζ (q)
α = 2
α = 0
Fig. 2. Illustration of degree multifractility athrough the function f(q),
using the log-stable model. Plots for arbitrary fixed index H= 0.5,
C
1
= 0.1 and afrom 0 to 2 with 0.25 increment.
352 R. Calif et al. / Solar Energy 98 (2013) 349–365
SqðTÞTfðqÞð6Þ
where f(q) is the scaling exponent function. An estimation
of f(q) is obtained through least squares regression fit of
S
q
(T).
3.3. Multifractal detrended fluctuation analysis (MFDFA)
Multifractal detrended fluctuation analysis proposed by
Kantelhardt et al. (2002), is a technique to investigate scal-
ing properties of a time series. It can be described as fol-
lows for a given data series X
j
of length Nwith a
compact support j=1,2,...,N. The integrated series Y(k)
is computed by taking the sum of deviations from the mean
value hXi(Kantelhardt et al., 2002):
YðkÞ¼X
k
j¼1
ðXðjÞhXiÞ ð7Þ
Then, the integrated series is split into S
n
segments of equal
size n. The local trend of each segment vcan be removed by
apth-order polynomial Pp
v. Then, the variances for all seg-
ments vand for all segments of length nare computed by:
F2ðv;nÞ¼1
nX
n
k¼1
Yðv1Þnþk½Pp
vðkÞ
2ð8Þ
The qth-order fluctuation function is defined as:
FqðnÞ¼ 1
2SnX
2Sn
v¼1
½F2ðv;nÞq=2
!
1=q
ð9Þ
The scaling behavior of the fluctuation function is deter-
mined from the log–log plots F
q
(n) versus nfor each value
of q. In case of scale invariance, a power law scaling is
obtained:
FqðnÞngðqÞð10Þ
with g(q) the corresponding scaling exponent function. The
g(q) obtained from MFDFA is related to the Re
´nyi expo-
nent s(q) and the scaling exponents function f(q) by:
qhðqÞ¼sðqÞþ1;qgðqÞ¼qþfðqÞð11Þ
3.4. Wavelet Leaders (WL)
Wavelet leaders is a method based on wavelet analysis
for estimating the scaling exponent or the multifractal spec-
trum. This approach proposed by Jaffard et al. (2006),Jaff-
ard et al. (2008) is used also for extracting the scaling
exponents from a scaling time series. A wavelet transform
is defined as:
wðj;kÞ¼ZR
XðtÞuð2ktjÞdt ð12Þ
where uis the chosen wavelet, w(j,k) is the wavelet coeffi-
cient, kis the position index, jis the scale index and T=2
j
is the corresponding scale. The first way to detect the scale-
invariant properties is to consider the wavelet coefficients:
ZqðkÞ¼hjwðj;kÞjqi2ksðqÞð13Þ
with s(q) the corresponding scaling exponents. Every dis-
crete wavelet coefficient w(j,k) can be associated with the
dyadic interval q(j,k):
qðj;kÞ¼½j2k;ðjþ1Þ2kð14Þ
The wavelet coefficients can be represented as
w(q)=w(j,k). Hence, wavelet leaders are defined as:
lðj;kÞ¼supq03qðj;kÞ;k061jwðq0Þj ð15Þ
where 3q(j,k)=q(j1,k)Sq(j,k)Sq(j+1,k)(Jaffard
et al., 2006). In case of scale of invariance, a power law is
obtained:
ZqðkÞ¼hlðj;kÞqi2ksðqÞð16Þ
where s(q) are the corresponding scaling exponent, called
the Re
´nyi exponent spectrum. The Re
´nyi exponent spec-
trum s(q) can be related to the scaling exponents f(q) by:
fðqÞ¼sðqÞqð17Þ
3.5. Arbitrary-order Hilbert spectral analysis
In order to estimate the scaling exponent f(q), arbitrary-
order Hilbert spectral analysis (Huang et al., 2008,), an
extended version of Hilbert–Huang Transform (Huang
et al., 1999; Huang et al., 1998) has recently been proposed.
The Hilbert–Huang Transform is decomposed into two
steps: (1) Empirical Mode Decomposition and (2) Hilbert
spectral analysis.
3.5.1. Empirical Mode Decomposition (EMD)
Physical processes in nature are mostly nonlinear and
nonstationary, exhibiting simultaneous coexistence of dif-
ferent time scales: this is the case of global solar radiation.
Empirical Mode Decomposition is an efficient tool to ana-
lyze the nonlinear and nonstationary characteristics of time
sequences (Huang et al., 1999; Huang et al., 1998). This
method decomposes an analyzed signal into different oscil-
lations. The high frequency time series is called an Intrinsic
Mode Function (IMF) and the low frequency part is the
residual. This procedure is repeated and applied again to
the residual, providing a new IMF using a spline function
and a new residual. IMF must respect two conditions: (i)
the difference between the number of local extrema where
the number of zero crossings must be zero or one, (ii) the
running mean value of the envelope estimated by the local
maxima is zero (Huang et al., 1999; Huang et al., 1998).
The EMD method extracts IMF modes into signal follow-
ing this algorithm:
1. Identification of local extrema (maxima, minima)
points for a given time sequence g
norm
(t).
2. Construction of the upper envelope e
max
(t) and the
lower envelope e
min
(t) respectively for local maxima
and local minima using a cubic spline algorithm.
R. Calif et al. / Solar Energy 98 (2013) 349–365 353
3. Estimation of the mean M
1
(t) between these two
envelopes:
M1ðtÞ¼emaxðtÞþemin ðtÞ
2ð18Þ
4. Estimation of the first component
h1ðtÞ¼gnormðtÞm1ðtÞð19Þ
5. h
1
(t) can be considered as an IMF, if h
1
(t) respects the
above mentioned conditions to be an IMF. If yes,
h
1
(t) is considered as the first IMF I
1
(t). If no, the
function h
1
(t) is then considered as a new time series
and this sifting process is repeated ktimes, until
h
1k
(t) is an IMF. The first IMF component I
1
(t)is
I1ðtÞ¼h1kðtÞð20Þ
and the first residual r
1
(t)
r1ðtÞ¼gnormðtÞI1ðtÞð21Þ
The EMD procedure is finished when the residuals r
n
(t)
becomes a monotonic function or a local extrema. Thus
the original signal g(t) is decomposed in a sum of n1
IMF modes with the residual r
n
(t)
gnormðtÞ¼X
n1
m¼1
ImðtÞþrnðtÞð22Þ
This shifting process can be stopped by a criterion
(Huang et al., 1999; Huang et al., 1998). More details are
given in (Huang et al., 1999, 1998, 2003,, 2004,, 2005,
2008).
3.5.2. EMD and Hilbert spectral analysis (HSA): Hilbert–
Huang transform
In order to determine the energy-time frequency repre-
sentation from the original signal g
norm
(t), we apply the
Hilbert–Huang transform (HHT) which is a combination
of EMD and HSA. HSA is performed to each obtained
IMF component I
m
(t) extracted by the EMD method
(Huang et al., 1998; Cohen, 1995; Long et al., 1995). The
Hilbert transform is expressed as:
e
ImðtÞ¼1
pUZþ1
1
ImðsÞ
tsds ð23Þ
with Uthe Cauchy principal value (Cohen, 1995; Long
et al., 1995). Then an analytical signal zfor each IMF
mode I
m
(t) is defined as follows:
zm¼Imþje
Im¼A
mðtÞejuiðtÞð24Þ
where AmðtÞ¼jzj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ImðtÞ2þe
ImðtÞ2
qrepresents an ampli-
tude and umðtÞ¼argðzÞ¼arctan eImðtÞ
ImðtÞ
represents the
phase function of IMF modes. Hence, the instantaneous
frequency x
m
(t) is determined from the phase u
m
(t),
xmðtÞ¼ 1
2p
dumðtÞ
dt ð25Þ
The original signal g
norm
(t) can finally be expressed as:
gnormðtÞ¼ReX
N
m¼1
AmðtÞejumðtÞ¼ReX
N
m¼1
AmðtÞejRxmðtÞdt ð26Þ
With Remeaning part real. The Hilbert Huang Transform
(HHT) can be considered as a generalization of the Fourier
transform because of the simultaneously representation of
frequency modulation and amplitude modulation. The en-
ergy in a time–frequency space is estimated from the
Hilbert spectrum, Hðx;tÞ¼A
2ðx;tÞ. The Hilbert spectrum
h(x) being defined as:
hðxÞ¼ZN
0
Hðx;tÞdt ð27Þ
with Nthe total data length. The Hilbert spectrum H(x,t)
gives a measure of the amplitude from each frequency and
time, while the marginal spectrum h(x) gives a measure of
the total amplitude from each frequency. Hence, the mar-
ginal spectrum can be compared to the Fourier spectrum.
3.5.3. Arbitrary-order Hilbert spectral analysis
We give here an extension of HHT, arbitrary-order
Hilbert spectral analysis proposed by Huang et al. in
Huang et al. (2008,) in order to characterize the scale
invariant property of signals. This approach was applied
to turbulence data (Schmitt et al., 2009; Schmitt et al.,
2011; Calif and Schmitt, 2011), and river-flow discharge
data (Huang et al., 2009). For that, the joint PDF
pðA;xÞof the instantaneous frequency xand the ampli-
tude Afor each of these IMF components, is designed
(Huang et al., 2008,). In this frame, the marginal Hilbert
spectrum defined in (27) is rewritten as:
hðxÞ¼ZN
0
pðx;AÞA2dAð28Þ
This expression concerns the second-order statistical mo-
ment. A generalization of this definition is considered to
arbitrary-order statistical moment qP0(Huang et al.,
2008,):
LqðxÞ¼ZN
0
pðx;AÞAqdAð29Þ
Hence, in the Hilbert space, the scale invariance is written
as:
LqðxÞxnðqÞð30Þ
where n(q) is the corresponding scaling exponent in the Hil-
bert space. This scaling exponent function is linked to scal-
ing exponent function f(q) of structure functions, by the
expression (Huang et al., 2008,):
fðqÞ¼nðqÞ1ð31Þ
Here the Hurst exponent is H=n(1) 1.
This approach based on EMD method allows the
extraction of the scaling exponents in the frequency space.
(see Fig. 3).
354 R. Calif et al. / Solar Energy 98 (2013) 349–365
4. Results
Global solar radiation is the sum of a direct part, a dif-
fuse part, and a ground-reflected part. This complex non-
linear process depends strongly on cloud coverage
regimes and can be decomposed into a deterministic part
corresponding to the seasonal effect and a stochastic part
caused notably by cloud coverage regimes. In order to des-
easonalize and detrend the global solar radiation time ser-
ies g(t), we apply to our 367 global solar radiation
sequences, the time series pre-processing given in (Bourb-
onnais and Terraza, 2010; Paoli et al., 2010). In the first
step the time series are divided by daily extraterrestrial
radiation G
0
(t):
yðtÞ¼ gðtÞ
G0ðtÞð32Þ
where y(t) is the clearness index. Although this pretreat-
ment tends to stationarize the time series, a test of Fisher
shows that seasonality was not optimal (Paoli et al.,
2010). To overcome this problem, the obtained ratio is di-
vided by the moving average, more details are given in
(Bourbonnais and Terraza, 2010; Paoli et al., 2010):
gnormðtÞ¼ yðtÞ
1
mPtþm1
2
l¼tm1
2
gðlÞ
ð33Þ
where mis the length of the window (here m= 3 h) and
GðtÞis the mean value over one day. Fig. 4 illustrates a typ-
ical example of global solar radiation sequence g(t) (a) and
the corresponding normalized sequence g
norm
(t) (b) result-
ing to detrend and deseasonality methods. These sequences
exhibit multiple fluctuations regimes caused by the passing
of clouds alternating with clear sky. If we assume that the
passage time of cloud over the sensor gives an indication on
its size: (i) small clouds have a characteristic time of passing
T
c
= 100–300 s (green zone), (ii) and (iii) medium and large
clouds have of passing T
c
= 1000–6000 s (orange zone).
For a better estimation of T
c
, cloud speed (= wind speed
at higher atmospheric layers) should be considered (Lave
and Kleissl, 2013; Bosch and Kleissl, 2013).
4.1. Fourier analysis
Generally, in the solar energy community, the variability
of a stochastic process such as the global solar radiation, is
often characterized by a second order statistic such as the
standard deviation (Perez et al., 2011) or the Fourier spec-
trum E(f)(Curtright and Apt, 2008; Matthew and Kleissl,
2010). The power spectral density separates and measures
the amount of variability occurring in different frequency
bands. Scale invariance can be detected by computing
E(f). For a scale invariant process, the following power
law is obtained over a range of frequencies f:
EðfÞfbð34Þ
where bis the spectral exponent. According to some
authors (Mandelbrot, 1982; Schertzer and Lovejoy, 1987;
Marshak et al., 1994), it contains information about the de-
gree of stationarity of the field:
b< 1, the process is stationary,
b> 1, the process is nonstationary,
1<b< 3, the process is nonstationary with incre-
ments stationary.
It may be also be considered as characterizing the degree
of correlation (Ivanov et al., 2001; Telesca et al., 2003).
We give in Fig. 5 in log–log scale, the Fourier spectrum
E
s
(f) of the normalized fluctuations g
norm
(t) represented in
Fig. 4b. This spectrum shows a power law behavior with
b= 1.68 close to the Kolmogorov spectrum (b= 5/3), for
frequencies 0.7 10
4
6f60.07 Hz, corresponding to
time scales 14 6T614,286 s (approximately 4 h). In the
inset, the mean Fourier spectrum of 367 normalized global
solar radiation sequences. This spectrum displays also a
power law behavior b= 1.70. Fig. 6 illustrates the histo-
Fig. 3. A flowchart describing the considered methodology to estimate the
“universal”multifractal parameters.
R. Calif et al. / Solar Energy 98 (2013) 349–365 355
gram of the spectral exponent bestimated separately for
each of 367 days. The spectral exponent ranges between
1.52 and 2.30. The Fourier spectrum displays a power
law behavior with 1.6 6b61.8 for 65% of sequences,
and bP2 for 3% of sequences. The Fourier spectrum is
a second order statistics providing information on medium
level fluctuations. For a full description of global solar
radiation fluctuations, multifractal analysis techniques are
used.
Multifractal analysis techniques such as wavelet leaders,
multifractal detrended fluctuation, structure function and
arbitrary order Hilbert spectral analyses, are applied in
the following subsection to normalized global solar radia-
tion fluctuations g
norm
(t) sampled at 1 Hz. For illustration,
we have shown the results provided by the four methods,
for a typical daily global solar radiation sequence repre-
sented in Fig. 4 which has a mean value G¼157 kW=m2
and a standard deviation r
g
= 237 kW/m
2
.
4.2. Multifractal analysis techniques: structure functions,
arbitrary order Hilbert spectral, wavelet leaders and
multifractal detrended fluctuations analyzes
To determine the nature of scaling behavior, monoscal-
ing or multiscaling statistics, of global solar radiation
sequences g(t), four multifractal analysis techniques are
compared, the classical structure functions approach, mul-
tifractal detrended fluctuations analysis, wavelet leaders
and arbitrary order Hilbert spectral analyses.
The structure functions S
q
(T), the marginal Hilbert
spectra L
q
(x), the MFDFA fluctuation functions F
q
(T)
and the partition function ZqðTÞgiven respectively in
Eqs. (5), (9), (16) and (29), are estimated for moments q
Fig. 4. An example of typical daily global solar radiation sequence g(t) with different regimes of fluctuations and the corresponding normalized solar
radiation sequence g
norm
(t).
10−3
105
f (Hz)
E (f)
105
EA Averaged Fourier spectrum
slope −1.70
Es Fourier spectrum for a sequence
slope −1.68
Kolmogorov spectrum
100
10-5
10-5 10-4 10-2 10-1 100
100
10-5
10-5 10-4 10-3 10-2 10-1 100
Fig. 5. The power density spectrum E
s
(f)() of a normalized sequence
g
norm
(t) displaying a power law behavior, with a spectral exponent
b= 1.68 (dashed black line). In the inset, the mean Fourier spectrum E
A
(f)
(h) of 367 normalized global solar radiation sequences displaying also a
power law behavior close to the Kolmogorov spectrum (red line). (For
interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
0
10
20
30
40
50
60
70
80
90
spectral exponent β
frequency
Fig. 6. Histogram of the spectral exponent bestimated from Fourier
spectra for the 367 days.
356 R. Calif et al. / Solar Energy 98 (2013) 349–365
−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 00.5
−10
−5
0
5
log10 (ω ( Hz ))
log10 (L
q(ω))
0 0.5 1 1.5 2 2.5 3 3.5
1
1.2
1.4
1.6
1.8
q
ξ (q)
q=0.1
fit q=0.1
q=1
fit q=1
q=2
fit q=2
q=3
fit q=3
(a)
(b)
Fig. 7. Arbitrary order Hilbert spectral approach: (a) The marginal Hilbert spectra L
q
displaying a scaling, in log–log plot, for 0 6q63 with 0.1
increment and for frequencies 0.7 10
4
6f60.07 Hz corresponding to time scales 14 6T614,286 s. (b) The scaling exponents n(q) nonlinear and
concave.
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−4
−2
0
log10 (T)
log10 (Sq(T))
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
q
ζ (q)
q=0.1
fit q=0.1
q=1
fit q=1
q=2
fit q=2
q=3
fit q=3
(b)
(a)
Fig. 8. Structure functions approach: (a) The structure functions S
q
displaying a scaling, in log–log plot, for 0 6q63 with 0.1 increment and for
frequencies time scales 14 6T614,286 s. (b) The scaling exponents f(q) nonlinear and concave.
0 0.5 1 1.5 2 2.5 3 3.5
0
1
2
3
4
q
τ(q)
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−5
0
5
10
log10 (T)
log
10
(Z
q
(T))
q=0.1
fit q=0.1
q=1
fit q=1
q=2
fit q=2
q=3
fit q=3
(b)
(a)
Fig. 9. Wavelet leaders approach: (a) The partition functions Z
q
displaying a scaling, in log–log plot, for 0 6q63 with 0.1 increment and for frequencies
0.7 10
4
6f60.07 Hz corresponding to time scales 14 6T614,286 s. (b) The scaling exponents s(q) nonlinear and concave.
R. Calif et al. / Solar Energy 98 (2013) 349–365 357
from 0 to 3 with increment 0.1, and for time scales
TP14 s. As we can observe in Figs. 7–10, these functions
show a scaling for the sequence g
norm
(t) for q= 0.1, 1, 2 and
3, in log–log scale. The straight lines obtained by a least
square fitting, indicate that the scaling of the relationships
(6), (10), (16) and (36) is well verified. Hence, this allows us
to extract the scaling exponents f(q), h(q), n(q) and s(q)in
physical and frequency spaces, using a least square fitting
algorithm.
For comparison, we estimate the scaling exponents f(q)
for multifractal detrended fluctuations analysis (MFDFA),
wavelet leaders (WL) and arbitrary order Hilbert spectral
analysis (EMD–HSA) using respectively Eqs. (11), (17)
and (31). In this frame, for a monofractal process, the
curve f(q) is linear. Another way to characterize a multi-
fractal process is the singularity spectrum M(c), that is
related to scaling exponents f(q) via Legendre Transform
(Feder, 1988). We recall here that:
c¼dfðqÞ
dq ;MðcÞ¼cqfðqÞþ1ð35Þ
In this frame, for a monofractal process, c=H(Hthe
Hurst exponent) and M(c)=1. Fig. 11a and b illustrate
the obtained scaling exponents f(q) and the corresponding
singularity spectra M(c), for moments order 0 6q63. All
scaling exponents functions observed in Fig. 11a are non-
linear and concave. Furthermore, all singularity spectra
shown represent a set of point. This highlights the multi-
fractal and intermittent properties for this global solar
radiation. We can see that both methods agree on the mul-
tifractal nature of solar radiation time series but if some
methods agree (SF and EMD–HSA), other methods satu-
rate earlier, the worth being WL. The values are given
for f(1) and f(2) in Table 1. We recall here that f(1) gives
an estimation of the Hurst exponent Hand f(2) is linked
to the spectral exponent bby b=1+f(2).
−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
−5
0
5
log10 (T)
log10 (Fq(T))
0 0.5 1 1.5 2 2.5 3 3.5
1
1.5
2
q
η (q)
q=0.1
fit q=0.1
q=1
fit q=1
q=2
fit q=2
q=3
fit q=3
(b)
(a)
Fig. 10. Multifractal detrended fluctuations approach: a) The fluctuation functions F
q
displaying a scaling, in log–log plot, for 0 6q63 with 0.1
increment and for frequencies 0,7 10
4
6f60.07 Hz corresponding to time scales 14 6T614,286 s. (b) The scaling exponents g(q).
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
q
ζ (q)
SF
MFDFA
WL
EMD−HSA
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
γ
M (γ)
SF
MFDFA
WL
EMD−HSA
Fig. 11. The scaling exponents f(q) and the corresponding multifractal spectra M(c) estimated using arbitrary order Hilbert spectral analysis (EMD–
HSA), structure functions analysis (SF), multifractal detrended fluctuations analysis (MFDFA) and wavelet leaders (WL).
358 R. Calif et al. / Solar Energy 98 (2013) 349–365
Huang et al. in Huang et al. (2011) give a comparative
analysis of these multifractal analysis techniques using sim-
ulated time series such as fractional Brownian motion,
multifractal random walk and experimental time series
such as temperature data possessing large structure. This
comparative analysis has shown that arbitrary order Hil-
bert spectral analysis is more suitable for nonlinear and
not stationary time series. Hence, the estimation of multi-
fractal parameters is given in the following by using arbi-
trary order Hilbert spectral method (EMD–HSA).
4.3. Empirical Mode Decomposition and Hilbert spectral
analysis results
Empirical Mode Decomposition is applied to normal-
ized global solar radiation sequence g
norm
(t) sampled at
1 Hz. For illustration purpose, we have presented the
results for the global solar radiation sequence g
norm
(t) dis-
played in Fig. 4b. We can observe an alternation of regime
of fluctuations due to the cloud distributions (small and
large clouds with multiple level dynamics). This sequences
contains fluctuations involving multiple time scales. The
decomposition of signal with EMD algorithm, allows us
to obtain of 17 IMF modes with a residual. For visual con-
venience, Fig. 12 illustrates the odd IMF modes with the
residual, extracted from g
norm
(t). The first modes (I
1
to I
9
)
represent the fast fluctuations while the last modes (I
11
to
I
17
) represent the slower fluctuations. The fluctuations in
these IMF modes may characterize the passage of clouds
which is a nonlinear multiscale process. The decomposition
of g
norm
(t) into multiple time scales using EMD, shows a
decreasing of frequency scales x
m
with the mode index
m, corresponding to an increasing of time scales increases
with the mode index. For that, we estimated the mean fre-
Table 1
The first f(1) and the second order f(2) exponents obtained using the four
methods.
Method f(1) f(2)
EMD–HSA 0.48 0.68
SF 0.48 0.69
MFDFA 0.40 0.62
WL 0.40 0.53
0 1 2 3 4
x 104
−1
0
1
I1
0 1 2 3 4
−0.5
0
0.5
0 1 2 3 4
−0.5
0
0.5
0 1 2 3 4
−0.5
0
0.5
0 1 2 3 4
−0.5
0
0.5
0 1 2 3 4
−1
0
1
0 1 2 3 4
−0.5
0
0.5
0 1 2 3 4
−0.1
0
0.1
0 1 2 3 4
−0.5
0
0.5
t(s)
0 1 2 3 4
0.5
1
1.5
residual
t (s)
x 104
x 104
x 104
x 104x 104
x 104
x 104
x 104
x 104
I3
I5
I7
I9
I11
I13
I15
I17
Fig. 12. Decomposition of the signal g
norm
(t) into 17 IMF modes with a one residual. For visual convenience, we display the odd modes with the residual.
0 2 4 6 8 10 12 14 16 18
−14
−12
−10
−8
−6
−4
−2
m (mode index)
log2 (ωm)
Fig. 13. Illustration of the mean frequency x
m
versus mode index meach.
The fitting slope is 0.61, corresponding to a 1.85-times filter bank.
R. Calif et al. / Solar Energy 98 (2013) 349–365 359
quency x
m
using the following definitions (Huang et al.,
1998):
xm¼R1
0fEmðfÞdf
R1
0EmðfÞdf ð36Þ
where E
m
(f) is the Fourier spectrum of mth IMF mode I
m
.
It is an energy weighted mean frequency in Fourier space
(Huang et al., 1998).
Fig. 13 illustrates the estimated mean frequency x
m
ver-
sus the mode index m, in a log-linear plot. An exponential
decrease is observed and modeled by a relation of the form
x
m
=x
0
g
m
,withx
0
’0.13 Hz and g= 1.85 obtained
using a least square fitting algorithm on the range
16m617. The mean frequency x
m
of IMF I
m
is 1.85
times larger the next one. This characteristic corresponds
to an almost dyadic filter bank property of the EMD algo-
rithm. In previous studies, the dyadic filter bank of EMD
decomposition has been shown for stochastic simulations
(Flandrin and Goncßalve
`s, 2004), for fully developed turbu-
lence data (Huang et al., 2008), for atmospheric wind speed
data (Schmitt et al., 2009; Calif and Schmitt, 2011), and for
aggregate wind power output data (Calif et al., 2013). This
filter bank property shows the adaptiveness of the method.
From the decomposition of signal g
norm
(t) into IMFs, we
have estimated this power spectrum density in Hilbert
space with the equation given in (27).Fig. 14 gives the mar-
ginal Hilbert power spectrum h(x) of the sequence g
norm
(t).
The obtained result is compared with its Fourier power
spectrum E
s
(f). Both spectra show the existence of a scaling
regime fitted by a relation of the form f
b
and x
b
. The
obtained values for the spectral exponent bare very close,
for time scales TP14 s: b= 1.68 in Fourier space and
b= 1.69 in Hilbert space.
4.4. Universal multifractal indices: H, C
1
and a
From the scaling exponents function f(q)=n(q)1in
frequency spaces, we used a least square fitting algorithm,
for obtaining the function R(q) given in Eq. (3).Fig. 15a
and b illustrates the obtained function R(q) for the signal
g
norm
(t), respectively in linear–linear plot and in log–log
representation. Fig. 15a shows that the obtained curve
f(q) is convex. The convexity of R(q) or the concavity of
f(q) and their nonlinearity indicates that the global solar
radiation signal g(t) has intermittent and multiscaling
properties and can be considered as a multifractal process.
The non-analytical behavior of these curves are shown in
Fig. 15b. The proportionalities of R(q)toq
a
confirm the
non-analytical. The values of universal multifractal indices
here are H= 0.48, C
1
= 0.12, a= 1.70 with arbitrary order
Hilbert spectral analysis. Below we perform this for every
day and consider statistics on the values of H,C
1
, and a.
Here, the results for 367 days of normalized global solar
radiation are presented: each day is characterized by the
104
f , ω (Hz)
E (f) , h (ω)
Fourier spectrum
slope −1.68
Hilbert spectrum
slope −1.69
102
100
10-2
10-4
10-6
10-5 10-4 10-3 10-2 10-1 10-0
Fig. 14. The power spectral densities E(f) in Fourier space and h(x)in
Hilbert space for the sequence g
norm
(t). A power law behavior close to the
Kolmogorov spectrum, is observed over three decades for frequencies
f60.07 Hz corresponding to time scales TP14 s.
0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
q
R (q)
0.1 0.6 1.1 1.6 2.1 2.6 3.13.5
101
q
R (q)
qζ’(0)−ζ(q)
fit: 0.17q1.70
100
10-1
10-2
10-3
Fig. 15. Representation of R(q) versus qestimated for qbetween 0 and 3 for time scales tP14 s. Experimental values are plotted in marker (), whereas
continuous lines correspond to power law fits. This indicates the proportionalities of R(q)toq
a
confirming the non-analytical framework.
360 R. Calif et al. / Solar Energy 98 (2013) 349–365
universal multifractal parameters H,C
1
and a, estimated
following the approach given in the flowchart. The meth-
odology presented above for one day of global solar radi-
ation sequence, is performed on 367 days of global solar
radiation.
We first show the effect of varying of multifractal indices
H,C
1
and ain daily global solar radiation sequence.
In Fig. 16a and b, we observe the effect of the Hurst
parameter Hdefining the degree of smoothness or rough-
ness of a measured signal. Fig. 16a illustrates a daily global
solar radiation signal with sharp peaks (indicated by the
arrows) where H= 0.47 while those illustrated in
Fig. 16b are smoother where H= 0.60 (C
1
and aare close
in both cases). In Fig. 16c and d, we can observe the effect
of varying the fractal co-dimension C
1
defining the mean
inohomogeneities or the mean intermittency. We recall that
06C
1
6d(dis the dimension space: here d= 1). C
1
can be
interpreted as quantifying the sparseness of the mean field:
for C
1
= 0 the field is homogeneous whereas the larger C
1
,
the field is more heterogeneous. The signal presented in
Fig. 16d, where C
1
= 0.10 is more sparse than the one pre-
sented in Fig. 16c, where C
1
= 0.06. More precisely, we can
observe an alternation of regime of fluctuations due to the
passing of small (green zone), medium and large clouds
(pink zone) in Fig. 16d, contrary to the signal presented
in Fig. 16c containing one regime of fluctuations caused
by the passing of small clouds alternating with clear sky.
In Fig. 16e and f, we can observe the effect of varying of
the multifractal Le
´vy parameter awhich measures the devi-
ation from the mean of the field values. It indicates that the
signal presented in Fig. 16f is more multifractal than the
signal presented in Fig. 16e.
Fig. 17 illustrates the histrograms for the three parame-
ters, H,C
1
and aestimated for the whole of global solar
radiation sequences.
0 0.5 1 1.5 2 2.5 3 3.5 4
0
1
2
3
4
5
t (s)
gnorm(t) (W/m2)
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
t (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
0.5
1
1.5
2
2.5
t (s)
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
2
2.5
t (s)
0.5 1 1.5 2 2.5 3 3.5 4
0
1
2
3
t (s)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 104
0
1
2
3
t (s)
H ≈ 0.29
C1≈ 0.10
α≈ 1.44
H ≈ 0.60
C1≈ 0.10
α≈ 1.70
H ≈ 0.47
C1≈ 0.10
α≈ 1.66
H ≈ 0.36
C≈ 0.10
α≈ 1.48
(f)
(b)
(d)
(a)
(c)
(e) H ≈ 0.29
C1≈ 0.10
α≈ 1.28
H ≈ 0.33
C1≈ 0.06
α≈ 1.44
gnorm(t) (W/m2)
gnorm(t) (W/m2)
gnorm(t) (W/m2)
gnorm(t) (W/m2)
gnorm(t) (W/m2)
x 104
x 104
x 104
x 104
Fig. 16. Few examples of normalized daily solar radiation sequences illustrating the varying effect of multifractal parameters, H,C
1
and a.
0 1 2 3 4 5 6
0
200
400
α (multifractal parameter)
frequency
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
50
100
C1 (mean intermittency)
frequency
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
50
100
H (Hurst parameter)
frequency
(a)
(b)
(c)
Fig. 17. Histograms of H(a), C
1
(b) and a(c) estimated for the 367 days
of global solar radiation over the considered period and the studied site.
Table 2
The statistics of multifractal parameters H,C
1
and ausing arbitrary order
Hilbert spectral (EMD–HSA) approach: the mean value (mean), the
standard deviation (r), the minimal value (min), the maximum value
(max) and the probable value (P
max
) of considered parameters.
Parameter Mean rmin max P
max
EMD–HSA
H 0.41 0.12 0.20 0.83 0.36
C
1
0.08 0.043 0.01 0.18 0.10
a1.47 0.50 0.23 5.16 1.35
R. Calif et al. / Solar Energy 98 (2013) 349–365 361
Fig. 18. (a) Evolution of the co-dimension parameter C
1
versus the Hurst parameter H. (b) Evolution of the multifractal parameter aversus the Hurst
parameter H.
00.5 11.5 2 2.5 3 3.5 4 4.5
x 104
0
200
400
600
800
1000
1200
1400
t (s)
g(t) (W/m2)
H = 0.51
C1 = 0.16
α = 1.83
Fig. 19. Illustration of sequences with C
1
0.01 (a) and C
1
0.17 (b). Sequences where the multifractal parameter a> 2 indicating regimes close to clear
sky (c) and close to full cloud cover (d).
362 R. Calif et al. / Solar Energy 98 (2013) 349–365
The statistics of a Hurst parameter H, a fractal co-
dimension parameter C
1
and a multifractal Le
´vy parameter
a, obtained for each day, for time scales TP14 s, using
arbitrary order Hilbert spectral approaches, are summa-
rized in Table 2.
For our dataset over this measurement period, the
map (H,C
1
), illustrated in Fig. 18a, does not show
dependence between the estimated multifractal parame-
ters, Hand C
1
. We give in Fig. 19a and b the sequences
having minimal and maximal values of C
1
. As indicates
it Fig. 19b, the presented signal is smoother and more
intermittent than the signal in Fig. 19a, characterizing
an alternation of clear sky with clouds of different size:
(i) characteristic time of passing T
c
100 s (green zone),
(ii) characteristic time of passing T
c
500 s (gray zone)
and (iii) characteristic time of passing T
c
2000 s
(orange zone).
The map (H,a), given in Fig. 18b, shows that sequences
possessing a Hurst parameter H> 0.6, have a multifractal
parameter a> 2. This was found in 5% of cases. We recall
here that ais bounded between 0 and 2, the values of a>2
are not theoretically possible. They are artefacts due to the
automatic numerical method (Lovejoy and Schertzer,
2006). However, this allows us to discriminate “determinis-
tic”meteorological cases for global solar radiation
sequences, i.e., a clear sky day and a completely cloudy
sky day.
In Fig. 19c, global solar radiation sequences are illus-
trated corresponding to normalized global solar radiation
sequences used to estimating the three multifractal
parameters. These sequences characterize a meteorologi-
cal events close to clear sky conditions. Here, the values
of the Hurst exponent are high (0.63 6H60.75) indicat-
ing a weak degree of roughness and a regular shape for
the displayed sequences. In addition, we notice that the
co-dimension index C
1
60.1, is excepted for one
sequence in which C
1
0.16. This indicates an inhomo-
geneity more important than other sequences, due to
the alternation of clear sky with the presence of sharp
peaks characterizing a short passage of small clouds.
These sequences with a high level of sunshine are more
suitable for electricity production obtained with photo-
voltaı
¨cs systems.
In Fig. 19d, we display global solar radiation sequences
corresponding to normalized global solar radiation
sequences characterizing cloudy and overcast sky condi-
tions. Three cases can be distinguished as:
The signal represented by a dotted blue line, is smooth
(H= 0.82) with a weak inhomogeneity (or intermit-
tency) (C
1
= 0.01) highlighting one cloudy regime: a
full cloud cover. Under these conditions, global solar
radiation results from diffuse radiation.
The signal represented by a green line, is less smooth
(H= 0.61) with more inhomogeneity (C
1
= 0.11) high-
lighting two cloudy regimes: an alternation of cloud
cover with big size clouds with a full cloud cover.
The signal represented by a dashed red line, is rough
(H= 0.36) with a weak inhomogeneity (C
1
= 0.01)
highlighting one cloudy regime.
5. Conclusions
A good knowledge of the intermittency of daily global
radiation sequences g(t) is fundamental for management
of solar energy. In this paper, a characterization of var-
iability for g(t) was proposed for 367 daily solar radia-
tion sequences with only three parameters H(Hurst
parameter), C
1
(co-dimension parameter) and a(Le
´vy
parameter) defined as “universal”multifractal parameters
(Schertzer et al., 1997) taking into account transitions
into different regimes of cloud passages. This is moti-
vated by a power law behavior of power spectral densi-
ties for each sequence (E(f)f
b
), with a spectral
exponent branging between 1.5 and 2.3. The mean Fou-
rier spectrum follows a 1.70 slope for time scales
16T610
3
s (approximately 3 h) showing a lack of
characteristic scale. This 1.70 value is close to the non-
intermittent Kolmogorov value b= 5/3. This scale invari-
ance is indirectly linked to scale invariance of cloud field
transported by atmospheric turbulence. In (Sachs et al.,
2002), a power law behavior is also observed for the
spectrum of cloud radiances obtained from ground-based
photography: the spectra exponent b= 2.10 for light
transmitted through clouds, in comparison to b= 2.26
for star light transmitted through interstellar dust,
b= 1.67 for clouds over ocean and b= 1.43 for clouds
over land. We recall here that a power law behavior with
bclose to 5/3, for the spectrum of a cluster of wind tur-
bines has been also observed (Apt, 2007; Calif et al.,
2013; Calif and Schmitt, 2012).
In this paper, scale invariance and multifractal proper-
ties of daily solar radiation sequence data were investi-
gated. Several multifractal analysis techniques (MFDFA,
wavelet leader, structure functions, arbitrary order Hilbert
spectral) have been applied to global solar radiation
sequences, in this study. The exponents which are extracted
characterize the fluctuations at all scales (from one second
to 3 h) and all intensities. The intermittent and multifractal
properties of global solar radiation data, sampled at 1 Hz,
have been highlighted by these methods. A new approach,
arbitrary order Hilbert spectral analysis which is a combi-
nation of Empirical Mode Decomposition and Hilbert
spectral analysis has been applied to 367 daily solar radia-
tion sequences. We confirmed that the Empirical Mode
Decomposition acts as a filter bank (i.e. the modes I
m
of
indices m> 1 are characterized by a set of overlapping
band-pass-filters) due to an exponential decrease obtained
for I
m
versus the mode index m. This showed the adaptive-
ness of this decomposition method on the data.
The log-stable multifractal model was chosen to quan-
tify the curvature of the scaling exponents f(q) through
three parameters H,C
1
and a.
R. Calif et al. / Solar Energy 98 (2013) 349–365 363
In the solar energy community, many studies have
been dedicated to characterization and classification of
daily global solar radiation data. In (Maafi and Harro-
uni, 2003), Maafi and Harrouni classified solar radiation
data in three distinct classes with the fractal dimension
D=2Hand the clearness index K
T
. Here, the pro-
posed method can be viewed as a generalization of
(Maafi and Harrouni, 2003) using the Hurst parameter
Has a point of the scaling exponent curve f(q), which
is used to estimate universal multifractal parameters.
Recently Zeng et al. in Zeng et al. (2013) have high-
lighted the multifractal properties of solar radiation data
using multifractal detrended fluctuations analysis, for
yearly scales. This study do not provide classes contrary
to methods given in (Maafi and Harrouni, 2003; Soubd-
han et al., 2009), but a multifractal parametrization for
each day, considering all cloud coverage regimes with
or without transition. The results of this study show
the relevance of multifractal log-stable model given in
(2) for daily global solar radiation sequences (see
Fig. 15. Moreover, the universal multifractal parameters
obtained for each day may be used for stochastic simu-
lations based on random multiplicative cascade processes
(Schertzer et al., 1997; Schertzer and Lovejoy, 1987).
Such modeling may be useful for estimation and man-
agement of solar energy production. However, to quan-
tify the smoothing effect and to model the variability
of a solar power plant over some spatial extent, this clas-
sification must be applied to other measurement points.
Acknowledgments
This study was financially supported by the Regional
Council of Guadeloupe, European funding (No1/1.4/-
31614). Y. H. was financed in part by the National Natural
Science Foundation of China under Grant No. 11072139.
We thank Professor P. Abry from Laboratoire de Phy-
sique, CNRS and ENS, Lyon (France) for providing his
wavelet leader codes. We thank Peter Magee (<http://
www.englisheditor.webs.com/>) for the English correc-
tions. We thank Dr. Ted Soubdhan for providing global
solar radiation data. We also thank the anonymous refer-
ees and the associate editor for useful comments.
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