Power spectrum and multifractal detrended fluctuation analysis of high-frequency wind measurements in mountainous regions

Applied Energy 162 (2016) 1052–1061

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

Power spectrum and multifractal detrended fluctuation analysis of high-frequency wind measurements in mountainous regions Luciano Telesca a,⇑, Michele Lovallo b, Mikhail Kanevski c a

Consiglio Nazionale delle Ricerche, Istituto di Metodologie per l’Analisi Ambientale, C.da S.Loja, 85050 Tito, PZ, Italy ARPAB, 85100 Potenza, Italy c IDYST, University of Lausanne, Switzerland b

h i g h l i g h t s
High-frequency records of wind speed measured in Switzerland are investigated.
Daily and half-daily periods are linked with temperature and pressure variation.
Higher harmonics of 8 h and 6 h are not negligible.
Two timescale ranges, with crossover at about 7 days characterize all the series.
Persistence and multifractality feature the wind speed at large timescales.

a r t i c l e

i n f o

Article history: Received 27 August 2015 Received in revised form 24 October 2015 Accepted 30 October 2015 Available online 18 November 2015 Keywords: Wind Fisher information measure Shannon entropy

a b s t r a c t The main objective of the study was to investigate the temporal features of the wind speed in complex mountainous terrains that are important for the assessment and development of wind energy. Six high-frequency records of 10-min averages of wind speed measured in Switzerland are investigated in order to better characterize their inner time dynamics. All the wind speed time series are modulated by components periods of 1 day and 12 h, linked with the temperature and pressure daily variation due to the sunset and sunrise. Furthermore the time dynamics of the wind speed is characterized by the presence of two different timescale ranges, separated by a crossover at about 7 days: persistent and multifractal at larger timescales and antipersistent and monofractal (or weakly multifractal) at smaller ones. The found features do not seem to depend on the altitude, because all the wind speed series share the same dynamical characteristics. The results of this comprehensive study can be utilized to better understand the mechanisms governing the time dynamics of wind speed and to perform a better wind energy assessment and management. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Demand of clean and sustainable energy is continuously growing, stimulating research studies for developing clean energy technologies [1], and more and more innovating sustainable future energy systems [2]. Among renewable energy sources, wind energy has been becoming the most promising energy source for its very competitive cost of production, and, more importantly, for its capability to overcome efficiently the well-know environmental problems linked with the use of more traditional energy sources [3]. Wind speed strongly influences wind energy; in fact, wind power is a function of the cube of wind speed. Thus, investigating the time ⇑ Corresponding author. E-mail address: [email protected] (L. Telesca). http://dx.doi.org/10.1016/j.apenergy.2015.10.187 0306-2619/Ó 2015 Elsevier Ltd. All rights reserved.

structure of wind speed time series is important not only to design more properly and efficiently wind power plants, but also to better understand the dynamical mechanisms governing its variability. For instance, it should be noted that wind speed studies besides their importance to the renewable energy topics have a fundamental relationships to the study of the boundary layer, especially to the understanding of atmospheric turbulence [4]. A deep statistical analysis of wind speed in terms of averages and distributions would be generally required to evaluate wind power potential [5], but its intermittent nature in a wide range of time and space scales represents the most crucial issue to address in order to reduce as much as possible the effects of wind variability on power systems. This implies that the analysis of the inner structure of wind speed time series could furnish information about the underlying dynamical mechanisms governing the

L. Telesca et al. / Applied Energy 162 (2016) 1052–1061

variability of wind speed, and, thus, contribute to a better designing of wind power plant by means of assessment of more reliable predictive models [6]. The time series of wind speed has been generally investigated in terms of estimation of parameters of a Weibull distribution function that is commonly used in wind energy applications. However, Chang [7] showed that wind data actually observed do not necessarily follow a Weibull distribution, and suggested several methods to evaluate Weibull parameters, concluding that the maximum likelihood method provides more accurate estimation in case wind speed does not fit well a Weibull distribution. On the other side, the veracity of the Weibull distribution model to describe offshore wind data measured by three meteorological stations located in Hong Kong was investigated by Shu et al. [8], who concluded that the Weibull model furnishes an adequate representation of the frequencies of actual wind speed. Astolfi et al. [9] analyzed wind speed series by proposing data mining models to evaluate the performance of onshore wind farms and using indexes that capture and describe mainly the mechanical characteristics of wind speed. In the context of forecasting methods, recently a great attention was paid to the application to wind speed series of nonlinear data driven models based on machine learning algorithms [10–12]. A different approach to investigate time series of wind speed at several spatial and temporal scales is based on the concept of fractal. The fractal dimension has been recently proposed to investigate the inner time structure of wind speed. Chang et al. [13] analyzed the wind speed time series observed at three wind farms in Taiwan with different climatic conditions by means of the fractal dimension D calculated by using the box-counting method. Their findings were the existence of an inverse correlation between the mean wind speed and the fractal dimension. The box-counting method, however, describes the fractal properties of a time series in a geometrical sense. To quantify, instead, the long-term correlation properties of wind speed records, de Oliveira Santos et al. [14] applied the detrended fluctuation analysis (DFA) to average and maximum hourly wind speed time series measured at four weather stations in Brazil, and found that both these two observables are characterized by almost identical power-law behavior, with two different scaling regimes and two different DFA scaling exponents. From the value of the fractal dimension D, Fortuna et al. [15] derived the value of the Hurst exponent H (D = 2  H). H is used to quantify the long memory in a time series; in particular, if H varies between 0 and 0.5, the time series is antipersistent (an increase/decrease of the series in one period is likely followed by a decrease/increase in the next period); if it varies between 0.5 and 1, the time series is persistent (an increase/decrease of the series in one period is likely followed by a increase/decrease in the next period); if H = 0.5, the time series is uncorrelated and no dependence can be found among its values. Fortuna et al. [15] found that the wind speed time series recorded in several weather stations in Italy and USA share the feature of persistence. They, furthermore, found that the daily and hourly wind speed time series are characterized by different shape of the power spectrum that, even if in both cases is a power-law, is featured by different spectral exponent, being 0.46 for the daily series and 1.37 for hourly series. In all the cases mentioned above, the wind speed time series are modeled as mono-fractal series, thus indicating that only one scaling exponent (the fractal dimension D, the Hurst exponent, the spectral exponent, the DFA scaling exponent, or any other quantity that could be defined to describe the scaling characteristics of the series) is enough to get all the information concerning the dynamics of fluctuations of wind speed. Monofractals are suited to model homogeneous series, meaning that the scaling properties do not change within the series and are characterized by a single singularity exponent [16]. More scaling

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exponents are, instead, necessary to describe the dynamics of series that are more complex and more heterogeneous than monofractals; and this happens when different scaling exponents have to be calculated for many interwoven fractal subsets into which the original series can be decomposed [17]. These time series are called multifractal, and are generally characterized by a spiky dynamics, with sudden and intense bursts of high frequency fluctuations [18]. Multifractality in wind speed has been becoming an important topic only recently; and, thus, not many studies have been performed so far. Up to our knowledge the first study that showed the presence of multifractal dynamics in wind speed was performed by Kavasseri and Nagarajan [19]; they investigated four time series of hourly means of wind speed in USA, finding that the binomial cascade multiplicative model could represent a close fit to the data. Telesca and Lovallo [20] analyzed the hourly wind speed time series at several heights from the ground, in the range between 50 m and 213 m, and in all the examined cases they found that the most of multifractality of the wind speed series is due to the different long-range correlations for small and large speed fluctuations. Fortuna et al. [15] applied the multifractal detrended fluctuation analysis (MFDFA) to several hourly wind speed series in Italy and USA and all of them were characterized by similar values of the multifractal width (width of the multifractal spectrum, measuring the degree of the multifractality) ranging between 0.39 and 0.59. de Figueirêdo et al. [6] applied the MFDFA to the mean and the maximum of four wind speed time series in Brazil and found that both the types of series are persistently correlated with a larger multifractality for the maximum than for the mean. Meteorological interpretation of the annual variation of some key multifractal parameters of wind speed was furnished by Piacquadio and de la Barra [21] suggesting their use as local indicator of climate change. The most of the analyses that have been carried out on the wind speed series (especially those concerning the investigation of spectral and multifractal properties) were focused on, at most, hourly averages of wind speed, with obvious limitations in the range of investigated temporal scales. Using higher frequency sampled data would allow to explore the time dynamics of wind speed at timescales lower than those that have been generally investigated so far. In the present study, in fact, we intend to analyze the spectral and multifractal fluctuations of 10-min averages of six wind speed time series recorded during 2013 in Switzerland by weather stations located in sites with different altitudes and geomorphologic conditions. The high-frequency sampling of the wind data and the different site locations of the wind sensors would enable to get a more complete picture of the complex time dynamics of wind speed in different ranges of timescales down to very small ones, at which the mechanisms governing its variability might be different from those at large timescales. In particular, the power spectrum and the multifractal analysis will be jointly used to study the temporal fluctuations of wind speed, pointing out its persistence properties the first, and its heterogeneity features the second. These properties, which were not deeply analyzed for high frequency wind speed time series so far, are of crucial importance because they are typical of atmospheric turbulence that has important impact on wind power generation performance, turbine loads, fatigue and wake effects [22]. Therefore, the study presented in this paper would contribute to enrich the background for a better understanding of atmospheric processes in a complex mountainous regions and for renewable energy assessments.

2. Preliminary data analysis We analyzed the time series of 10 min averages of wind speed measured during 2013 at six stations in Switzerland

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(Lugano – LUG, Evolène – EVO, Geneva – GVE, Sion – SIO, Payern – PAY, Luzern – LUZ). The stations are located at different altitudes. The data were collected from IDAWEB server of Meteosuisse (http://www.meteosuisse.admin.ch). Fig. 1 shows a map of the altitude in Switzerland along with the locations of automatic stations measuring wind speed used in this study. All the analyzed series do not have data missings. Fig. 2 shows the time variation of the wind speed. The Weibull distribution can be used to feature wind speed regimes in terms of its probability density function and it is usually utilized to estimate and to assess wind energy potential [23]. The probability density function of a Weibull distribution is given by
k1
ðv Þk  the following formula f WB ðv Þ ¼ kc vc e c , where the factors k and c are the shape and the scale parameters, respectively, which are determined for each measurement site. The function fWB(v) is the probability of observing a particular value v of the wind speed. Fig. 3 shows the histograms of the wind speed of the six stations along with the fitting Weibull functions. Table 1 reports the altitude, geographical coordinates of the six stations and the values of scale and shape parameters of the Weibull functions fitting the wind speed series. 3. Spectral data analysis Each time series is characterized by the presence of cyclic oscillations that can be identified by using the power spectrum, which represents the frequency distribution of a signal power. Fig. 4 shows, as an example, the power spectra of two series, EVO and LUG, computed by using the periodogram method. On the base of the power spectrum analysis, all the wind speed series are modulated by the daily and half-daily cycles. However, few stations, like EVO, are also characterized by the presence of higher harmonics with lower amplitude at 8 h, 6 h and even less. Diurnal and semi-diurnal cycles are clearly linked with the atmospheric temperature cycles driven by the sun. It is, in fact, well known that atmospheric heating by sun, along with other local forcings, produces internal atmospheric gravity waves, that cause the atmospheric pressure, temperature and wind fields to oscillate [24], at periods (24 h and 12 h) that are integer fractions of a solar day [25]. The contribution to the power of the wind speed series of the higher harmonics, like those at 8 h and 6 h, is lower than that given by the diurnal and semi-diurnal cycles; however, such contribution cannot be neglected: in fact, considering the spectrum of EVO series, for instance, the ratio between the power of the 8-h period to that of 24-h period, P8-h/P24-h  10%, and analogously

P6-h/P24-h  4%, while P8-h/P12-h  20%, and P6-h/P12-h  10%. These ratios indicate that the contribution of the higher harmonics is rather significant, contrarily to what asserted in Dai and Deser [25] and Cook [26]. Another feature can be identified by the analysis of the power spectrum. For purely random processes, like white noise, the power spectral density is approximately constant for any frequency, and this indicates that the series is totally uncorrelated, any value is totally independent of the others, and, thus, no memory can exist. A power-law (also defined ‘‘scaling”) power spectrum, instead, reveals that long-range correlation exists in the series. This behavior is very common in geophysical series and from the estimation of the spectral exponent information about type and strength of the temporal fluctuations in that series can be deduced [27]. Looking at the power spectrum of EVO series (Fig. 4b), we could see that two frequency regimes are present, one involving the low frequency range and the other the high frequency range, with a crossover at about 7 days. From a visual inspection, at low frequencies the spectrum is almost flat (typical of an uncorrelated process), while at higher frequencies it is linearly (in log–log scales) decreasing. Such linear decreasing in log–log scales in the high frequency range indicates a power-law (scaling) behavior at these frequencies, and, thus, that long-range correlation exists at these frequencies. The value of the exponent a of the power-law of the power spectrum, P(f)  fa, furnishes relevant information about the temporal fluctuations of the series [28–31]: – a < 1, the series is stationary;
in particular, if 1 < a < 0, the series is antipersistent (an increase/decrease of the series in one period is very likely followed by a decrease/increase of the series in the next period)
if a = 0, the series is uncorrelated;
if 0 < a < 1 the series is persistent (an increase/decrease of the series in one period is very likely followed by an increase/decrease of the series in the next period) – a > 1, the series is nonstationary; in particular if 1 < a < 3, the series is nonstationary with stationary increments. In case of stationary series, a relationship between the spectral exponent and the Hurst exponent H (a well-know parameter used to describe the persistence of a series) can be established, a = 2H  1 [32]. Thus, the persistence/antipersistence of the series could be deduced from H that is larger or smaller than 0.5, where 0.5 represents the value of H for pure randomness.

Fig. 1. (Left) Map of Central Europe. Switzerland is colored in red. (Right) Location of the measuring stations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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LUG

EVO

14

10

12

8 10

6

m/s

m/s

8 6

4 4

2

2 0

0

-2 0

1x10

4

2x10

4

3x10

4

4x10

4

5x10

4

6x10

4

0

1x10

4

2x10

t (10 min)

4

3x10

4

4x10

4

5x10

4

6x10

4

6x10

4

6x10

4

t (10 min)

GVE

SIO

18

14

16 14

10

12

8

10

m/s

m/s

12

6

8 6

4

4 2

2

0

0

-2

-2 0

1x10

4

2x10

4

3x10

4

4x10

4

5x10

4

6x10

4

0

1x10

4

2x10

t (10 min)

4

3x10

4

4x10

4

5x10

4

t (10 min)

PAY

12

LUZ

16 14

10

12 10 8

6

m/s

m/s

8

6

4

4 2

2 0

0

-2 0

1x10

4

2x10

4

3x10

4

4x10

4

5x10

4

6x10

4

0

1x10

4

t (10 min)

2x10

4

3x10

4

4x10

4

5x10

4

t (10 min) Fig. 2. Wind speed time series recorded in 2013.

From all above, it is clear that the value of the spectral exponent (given by the slope of the line fitting the power spectrum plotted in log–log scales), which conveys information on type and strength of the temporal fluctuations in a series, has to be estimated very accurately. The power spectrum, computed by using the periodogram method (and shown in Fig. 4 for EVO and LUG series), is very irregular especially at high frequencies; thus, the linear regression performed in this frequency range could lead to a poorly accurate estimation of the spectral exponent. In the following section, we will describe the multifractal detrended fluctuation analysis (MFDFA) that will allow not

only to reveal more complex structure in the wind speed series, but also to provide a good estimation of the spectral exponent.

4. Multifractal data analysis The MFDFA [33] is a simple and effective method to detect multifractal behavior in time series. Given the time series x(i), for i = 1, 2, . . . , N (N being the length of the series), firstly the average xave is subtracted from the series, then the ‘‘profile” is constructed by integration

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Fig. 3. Histograms and Weibull fitting functions for each wind speed series.

Table 1 Location and Weibull parameters of the wind speed series. Series

Altitude

Latitude

Longitude

k

c

LUG EVO GVE SIO PAY LUZ

273 m 1825 m 412 m 482 m 490 m 454 m

46°000 46°070 46°150 46°130 46°490 47°020

8°580 7°310 6°080 7°200 6°570 8°180

1.8388 1.6704 2.6189 2.3302 2.0632 1.6084

1.1628 1.535 1.268 1.0673 1.3082 1.4248

profile y(i) at the end may remain. Therefore, the same procedure is repeated starting from the opposite end. Thereby, 2Nm segments are obtained altogether. Then the polynomial local trend for each of the 2Nm segments is calculated by a least square fit of the series, and the variance is computed as follows

F 2 ðm; mÞ ¼

m 1X 2 fy½ðm  1Þm þ i  ym ðiÞg m i¼1

ð2Þ

for each segment m, m = 1, . . . , Nm and

yðiÞ ¼

i X

½xðkÞ  xav e :

F 2 ðm; mÞ ¼ ð1Þ

k¼1

The profile is divided into Nm = int(N/m) nonoverlapping windows of equal length m. Since the length N of the time series could not be a multiple of the timescale m, a short part of the

m 1X 2 fy½N  ðm  Nm Þm þ i  ym ðiÞg m i¼1

ð3Þ

for m = Nm+1, . . . , 2Nm. Here, ym(i) is the fitting polynomial of a certain degree p in segment m. The fitting polynomial of degree p removes all the non-stationarities in the profile of order up to p, and, thus, up to p  1 in the original signal [33].

L. Telesca et al. / Applied Energy 162 (2016) 1052–1061

P (f)

10 10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

-9

10

-10

10

-11

LUG

7d

0

1d

12 h

(a) 10

-5

10

-4

10

-3

10

-2

10

-1

10

0

10

0

P (f)

f (1/10min) 10

2

10

1

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

-9

10

EVO 1 d 12 h

8h 6h 5h

(b)

-10

10

-5

10

-4

10

-3

10

-2

10

-1

f(1/10min) Fig. 4. Power spectrum of two wind speed time series: (a) LUG and (b) EVO.

Then, we average over all segments to obtain the q-th order fluctuation function

( F q ðmÞ ¼

2N m h i2q 1 X F 2 ðm; mÞ 2Nm m¼1

)1q ð4Þ

where in general, the index variable q can take any real value except zero. The parameter q enhances the small fluctuations if is negative, otherwise, if it is positive it enhances the large fluctuations of the time series. Repeating the procedure described above, for several time scales m, Fq(m) will increase with increasing m. Then analyzing log–log plots Fq(m) versus m for each value of q, we determine the scaling behavior of the fluctuation functions. If the series xi is longrange power-law correlated, Fq(m) increases for large values of m as a power-law

F q ðmÞ  mhq :

ð5Þ

The value h0 corresponds to the limit hq for q ? 0, and is obtained through the logarithmic averaging procedure:

(

2Nm h i 1 X F 0 ðmÞ  exp ln F 2 ðm; mÞ 4Nm m¼1

)  mh0 :

ð6Þ

In general the exponent hq will depend on q. For stationary series, h2 is the Hurst exponent H. Therefore, hq is called the generalized Hurst exponent. hq independent of q characterizes

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monofractal series. If the scaling behavior of small fluctuations is different from that of the large ones, hq depends on q and indicates that the series is multifractal. The exponent h2 is related with the spectral exponent a by a = 2 h2  1; therefore, in particular, the power-law behavior of the fluctuation function F2(m) allows to estimate the spectral exponent a. For stationary series h2 = H, thus, from the numerical value of h2 the persistence properties of the series can be deduced [33]. For non-nonstationary (a > 1) but with stationary increments, according to Movahed et al. [34,38], the relation between h2 and the Hurst exponent H is H = h2  1. The MFDFA is sensitive to the presence of periodic components in the series [35]; thus, we firstly filtered out the main cycles and standardized each residual time series subtracting its mean and dividing by its standard deviation. Fig. 5 shows the residual speed time series. In order to illustrate how the estimation of the spectral exponent could be better performed by analyzing the power-law behavior of the fluctuation function F2(m) instead of the periodogram-based power spectrum, Fig. 6 shows the F2(m)  m for LUG series. The series shows two different scaling regimes, with a crossover at about m = 1000 (about 7 days) (black arrow). This twofold scaling is also visible in the power-law shape of the power spectrum (Fig. 4a); however, the estimation of the spectral exponent from F2(m) is more accurate due to the lower scatter along the fitting line. For LUG series, we estimated h2  1.185 and 0.60 for timescales respectively smaller and larger than the crossover. From these values, we can derive a  1.37 and 0.2 for frequencies respectively larger and smaller than the crossover. In relationship with the persistence, the series is characterized by significantly different persistent features in the two timescale ranges: in the higher timescale range the series is stationary and the value of h2  0.60 (which is identical to the Hurst exponent H) indicates that it is characterized by weak persistence. Different observations have to be done for the lower timescale range: at these timescales, the series is nonstationary (a > 1), and H is H = h2  1  0.18, which indicates that the series is antipersistent at lower timescales (that correspond to the higher frequencies in the power spectrum). Fig. 7 shows the results of the MFDFA performed on the residual LUG wind speed series, as a pedagogical example. We applied the MFDFA in the timescale range between m = 10 samples (100 min) and m = 104 samples (about 70 days), considering the q-range between 5 and 5 with 0.5 step and the degree of the detrending polynomial p = 1, 2, 3. Fig. 7a shows the fluctuation functions for the degree of the detrending polynomial p = 1. It is clearly visible that the series is characterized by two different scaling regimes, with a crossover at about m = 1000 (about 7 days) (vertical dotted line): for timescales lower than the crossover, the slopes of the fluctuation curves do not seem to differ significantly, while for timescales higher than the crossover, the slope of the fluctuation curves visibly changes with q. This indicates that the series at timescales higher than the crossover is more multifractal than at lower timescales. Such twofold dynamical behavior is also present for higher degrees of the detrending polynomial p (Fig. 7b and c). Fig. 7d shows the root mean square error (rmse) of the estimation of the slope of the fluctuation functions plotted in Fig. 7a: the rmse does not exceed 0.035 in the lower timescale range, while it varies between about 0.03 and 0.07 in the higher range. Such larger estimation error in the higher timescale range, because narrower that the lower timescale range, is still acceptable and the slope estimation in the higher timescale range can be considered reliable. Therefore, this finding confirms what observed in Fig. 7a, that is the time series is more multifractal at lower frequencies (higher timescales) than at higher frequencies (lower timescales). On the basis of this observation, we analyzed separately the two timescale

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LUG

8

8

normalized variable

6

normalized variable

EVO

10

4

2

0

6 4 2 0 -2

-2 -4 0

1x10 4

2x10 4

3x10 4

4x10 4

5x10 4

6x10 4

0

1x10 4

t (10 min)

2x10 4

3x10 4

4x10 4

5x10 4

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t (10 min)

GVE

SIO

8

6

normalized variable

normalized variable

6 4

2

0

4

2

0

-2 -2

-4 0

1x104

2x104

3x104

4x104

5x104

6x104

0

1x104

t (10 min)

2x104

3x104

4x104

5x104

6x104

t (10 min)

LUZ

PAY 12

6

normalized variable

normalized variable

10 4

2

0

8 6 4 2 0

-2

-2 0

1x10 4

2x10 4

3x10 4

4x10 4

5x10 4

6x10 4

t (10 min)

0

1x10 4

2x10 4

3x10 4

4x10 4

5x10 4

6x10 4

t (10 min)

Fig. 5. Residual of the wind speed time series plotted in Fig. 2, after removing the main periodicities.

ranges: (i) from m = 10 to m = 500 (lower range) and (ii) from m = 1500 to m = 10,000 (higher range). The upper timescale of the first range and the lower timescale of the second range are sufficiently smaller and larger, respectively, than the crossover of about 7 days, in order to guarantee that the estimation of the generalized Hurst exponents is performed on a linear (in log–log scales) timescale range of the fluctuation functions that are slightly bending around the crossover. The fluctuation functions for p = 2 and p = 3 have almost the same slope and, even, rather overlapped for q < 0 (Fig. 7b and c), and this suggests that p = 2 is sufficient to get insight the multifractality of the series; therefore hereafter we

will analyze the multifractality in wind speed time series using p = 2. Fig. 8 shows the comparison of the spectrum of generalized Hurst exponents hq in the two timescale ranges: it is very clear the large range of variation of the Hurst exponents shown by the time series in the higher timescale range, indicating that the series can be clearly characterized as a multifractal, while substantially the series appears monofractal in the lower timescale range with a mean generalized Hurst exponent hhqi  1.15. Fig. 9 shows the hq spectrum for the other five stations. All the stations share the same characteristics, and are multifractal at higher timescales while substantially weakly multifractal or monofractal at lower

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LUG p=2

2.5

h2=0.60 +0.02

1.2

2.0 1.0

1.0 0.5

hq

log10(F2(m))

1.5

h2=1.185 + 0.002

0.0

0.8

0.6

-0.5

LUG higher range lower range

0.4

-1.0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-6

log10 (m)

2

4

5. Conclusions High-frequency records of 10-min averages of wind speed at six different weather monitoring stations in Switzerland are examined. The main findings of our study are the following:

3.0

LUG p=1

2.5

LUG q=5 p=1 p=2 p=3

2.0

log10(Fq(m))

1.5 1.0 0.5

1.5 1.0 0.5

0.0 0.0

-0.5 -1.0

-0.5

(a)

(c)

-1.0

-1.5 1.0

1.5

2.0

2.5

3.0

3.5

1.0

4.0

1.5

2.0

LUG q=-5

3.5

4.0

Low range High range

0.06 0.05

1.0

rmse

log10(Fq(m))

1.5

3.0

0.07

p=1 p=2 p=3

2.0

2.5

log10(m)

log10(m) 2.5

0.5

0.04

0.0 0.03

-0.5 -1.0

0.02

(b)

-1.5 1.0

1.5

2.0

6

and are consistent with a multifractal model; in particular LUG, GVE and SIO presents the higher multifractality among all the series in the higher range. All the stations show a more or less intense antipersistence in the lower timescale range and persistence in the higher timescale range.

2.0

log10(Fq(m))

0

Fig. 8. Comparison between the generalized Hurst exponents in the lower range and higher range for station LUG.

timescales. Following López and Contreras [36], we calculated for each timescale range and for each time series the mean (hhqi) and standard deviation (rq) for hq and the largest relative difference between the values of the Hurst exponents and the corresponding mean, Dmax = |max{hq}  hhqi|/hhqi; in particular rq and Dmax are used to quantify the multifractality of a time series. Table 2 shows the values of the three quantities along with the values of h2 and Hurst exponent H. The behavior of all the stations in the lower range seems to be consistent with a monofractal signal or weakly multifractal. In the higher range, the size rq and Dmax is much larger than the corresponding values in the lower range,

2.5

-2

q

Fig. 6. F2(m)  m for LUG series.

3.0

-4

2.5

log10(m)

3.0

3.5

4.0

(d)

0.01 -10

-5

0

q

Fig. 7. Fluctuation functions for the station LUG for p = 1 (a); for q = 5 (b) and for q = 5 (c).

5

10

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L. Telesca et al. / Applied Energy 162 (2016) 1052–1061 1.2 1.3 1.2

1.2 1.1

1.0 1.0

hq

hq

hq

1.0

0.8

0.8

0.9 0.8 0.7

0.6

0.6

EVO

0.6 SIO

GVE

higher range lower range

higher range lower range

0.5

higher range lower range

0.4 -6

-4

-2

0

2

4

6

-6

-4

-2

q

0

2

4

6

-6

-4

-2

0

2

4

6

q

q 1.1

1.2 1.0 1.1

hq

hq

0.9 1.0

0.9

0.8

0.7 LUZ

PAY higher range lower range

higher range lower range

0.8

0.6 -6

-4

-2

0

2

4

6

-6

-4

-2

q

0

2

4

6

q

Fig. 9. Comparison between the generalized Hurst exponents in the lower range and higher range for stations EVO, GVE, SIO, PAY and LUZ.

Table 2 Multifractal parameters of the wind speed series. Series

hhqi

rq

Dmax

h2

H

LUG (lower range) LUG (higher range) EVO (lower range) EVO (higher range) GVE (lower range) GVE (higher range) SIO (lower range) SIO (higher range) PAY (lower range) PAY (higher range) LUZ (lower range) LUZ (higher range)

1.15 0.83 1.07 0.73 1.23 0.77 1.24 0.71 1.21 0.92 1.04 0.77

0.02 0.28 0.02 0.07 0.02 0.17 0.07 0.16 0.02 0.07 0.02 0.07

0.03 0.43 0.03 0.12 0.02 0.34 0.06 0.35 0.02 0.13 0.03 0.11

1.2 0.61 1.06 0.68 1.24 0.65 1.21 0.58 1.22 0.87 1.06 0.72

0.2 0.61 0.06 0.68 0.24 0.65 0.21 0.58 0.22 0.87 0.06 0.72

(1) All the wind speed time series are modulated by cyclic components with periods of 1 day and 12 h. These periods are integer fractions of the solar day, indicating that the wind speed is influenced by the day/night atmospheric temperature and pressure variations induced by the sun. (2) Some wind speed series are also modulated by cyclic components at 8 h, 6 h and lower periods. These periodic components should not be discarded in a deep analysis of the time dynamics of the wind speed, because their power is not negligible with respect to that of the daily and halfdaily cycles. Furthermore, these periods should be strictly linked with the pressure and temperature variations. He et al. [37] showed a comparative analysis of wind speed, temperature and pressure series and found cyclic components of 8 h and 6 h in the power spectrum of temperature and pressure, possibly modulating the wind speed. (3) Two timescale regimes are clearly identified in the wind speed series, with a crossover at about 7 days. This twofold scaling behavior in wind speed is in agreement with Kavasseri and Nagarajan [39], who analyzed the hourly wind speed data recorded at three potential wind generation sites

in North Dakota (USA) by using the DFA. They found two scaling regions in the fluctuation function F2(m) separated at about 100 h (4 days) with scaling exponent 1.4 at short timescales (consistent with a Brownian-like model) and 0.7 at larger timescales (consistent with a persistent series). In our case, the scaling exponent at larger timescales ranges between 0.58 and 0.87, while that at smaller timescales varies between 1.06 and 1.24; these values indicate that the wind speed is weakly persistent or persistent at larger timescales and weakly antipersistent or antipersistent at smaller timescales. (4) Two different multifractal behaviors characterize the wind speed at short and large timescales; in particular, at short timescales, wind speed is monofractal or weakly multifractal, while at larger timescales is multifractal. This indicates that the wind speed is characterized by a more heterogeneous structure at large timescales than at short ones. This result was never obtained so far and represents the basis on which to continue the investigation in the future in order to understand the physical forcings that are responsible of this different time dynamics at short and large timescales.

Acknowledgments The authors acknowledge Meteosuisse for the access to the wind speed data. L.T. acknowledges the financial support of Herbette foundation of the University of Lausanne.

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