Taylor Law in Wind Energy Data

Abstract

The Taylor power law (or temporal fluctuation scaling), is a scaling relationship of the form σ ~ (P)λ where !! is the standard deviation and hPi the mean value of a sample of a time series has been observed for power output data sampled at 5 min and 1 s and from five wind farms and a single wind turbine, located at different places. Furthermore, an analogy with the turbulence field is performed, consequently allowing the establishment of a scaling relationship between the turbulent production IP and the mean value (P).

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Resources 2015,4, 787-795; doi:10.3390/resources4040787 OPEN ACCESS
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Article
Taylor Law in Wind Energy Data
Rudy Calif 1,* and François G. Schmitt 2
1EA 4935, LARGE Laboratoire en Géosciences et Énergies, Université des Antilles, 97159
Pointe-à-Pitre, Guadeloupe, France
2CNRS, UMR 8187 LOG Laboratoire d’Océanologie et de Géosciences, Université de Lille 1,
28 avenue Foch, Wimeureux 62930, France; E-Mail: francois.schmitt@univ-lille1.fr
*Author to whom correspondence should be addressed; E-Mail: rcalif@univ-ag.fr;
Tel.: +33-590483112; Fax: +33-590483105.
Academic Editor: Witold-Roger Poganietz
Received: 29 May 2015 / Accepted: 19 October 2015 / Published: 27 October 2015
Abstract: The Taylor power law (or temporal fluctuation scaling), is a scaling relationship of
the form σ∼ hPiλwhere σis the standard deviation and hPithe mean value of a sample of
a time series has been observed for power output data sampled at 5 min and 1 s and from five
wind farms and a single wind turbine, located at different places. Furthermore, an analogy
with the turbulence field is performed, consequently allowing the establishment of a scaling
relationship between the turbulent production IPand the mean value hPi.
Keywords: Taylor power law; temporal fluctuation scaling; wind energy; wind farms;
wind turbine
1. Introduction
Wind energy is a complex process in constant growing. Its complexity results from interactions
between weather dynamics, particularly atmospheric turbulence and wind turbines located at different
positions in wind farms. This energy resource exhibits high fluctuations at all temporal and spatial
scales. Such complex processes are ubiquitous in many research fields. Despite their complexity, scaling
properties can be highlighted. In this study, we investigate a scaling relationship between the standard
deviation σand the mean:
σ=constant ×meanλ(1)

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This scaling relationship called Taylor law was established by L.R. Taylor in 1961 in the field of
ecology [1] and observed for the first time by H. F. Smith (1938) [2]. The Taylor law has been highlighted
in various fields of research such as ecology [1–5], networks [6–11], economy [12–14], climatology [15,16]
and life sciences [17–19]. A review is given in [12]. In physics, De Menezes and Barabási (2004)
qualified this by “fluctuation scaling” [6,7]. According to them, the exponent value λcan fluctuate
between two universal classes λ= 1/2and λ= 1 [6,12].
This scaling relationship has been highlighted for the power output delivered by a wind farm [20].
Here, as an extension of this early study, fluctuation scaling is investigated for the power output delivered
by five wind farms and a single turbine.
2. Wind Power Output Data
In this study, we consider time series of power output measurements delivered by wind farms and
a single turbine. The wind farms are located in the Guadeloupean Archipelago (French West Indies)
situated at 16◦150N latitude and 60◦300W longitude, in the eastern Caribean sea. The power output
measurements delivered by the wind farms labelled n◦1, 2, 3 and 4, are collected continuously with
a sampling rate of Ts= 5 min over more than one year period: this corresponds to 125,942 data
points. These power output data are collected and provided by the French operator of electricity grid
Electricité de France (EDF). The power output measurements delivered by the wind farm labelled n◦5,
are collected continuously with a sampling rate of Ts= 1 s during approximately four months: this
corresponds to 6,529,000 data points. The power output measurements from the single turbine, are
collected continuously with a sampling rate of Ts= 1 s during more than six months: this corresponds
to 12,257,600 data points. This wind turbine is located at Risø Campus, Roskilde, Denmark and is
a three-bladed stall regulated Nordtank, NTK 500/41 wind turbine. Figure 1gives an example of
power output delivered by this wind turbine during two days. Table 1gives a description of following
characteristics, sampling frequency, number of continuously data points, implementation site, installed
capacity, for each dataset.
0 8 16 24 32 40 48
0
200
400
600
τ (hour)
P (kW)
Figure 1. An example of power output sequence p(t)delivered by the single wind turbine
during 48 h.

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Table 1. Description of characteristics (sampling frequency, number of continuously data
points, implementation site, installed capacity) for each dataset.
Dataset Sampling Frequency Number of Implementation Site Installed Capacity
(Hz) Data Points Pinst
Wind farm◦13.3×10−3125,942 plateau 2.6 MW
Wind farm◦23.3×10−3125,942 plain 2.9 MW
Wind farm◦33.3×10−3125,942 plateau 1.9 MW
Wind farm◦43.3×10−3125,942 plain 3 MW
Wind farm◦51 6,529,000 cliff 10 MW
Single wind turbine 1 12,257,600 plain 500 kW
3. Taylor Law, a Scaling Relationship between the Mean Value and the Standard Deviation
3.1. Definition of the Taylor Power Law
The study of complex systems in many fields such as ecology, physics, life sciences and engineering
sciences [12], has highlighted the universality of the Taylor power law established by L.R. Taylor in
1961 [1]. This relationship has been observed by De Menezes and Barabási with data internet traffic [6]
and named later “temporal fluctuation scaling” [10]. Taylor power law is characterized by a relationship
between the standard deviation σ=q1
N−1PN−1
n=1 (x(t)− hxi)2of a signal x(t)and its mean value
hxi=1
NPN
n=1 x(t)estimated over a sequence of length Nof the considered signal x(t):
στ=C0hxiλτ(2)
with h.idefining the statistical average, N=L/τ + 1,Lis the total period of the signal x(t)and τis the
time window corresponding to the time scales explored, C0is a constant and λTthe Taylor exponent.
In this study, the Taylor power law is investigated for the power output data from wind farms and a
single wind turbine.
3.2. Taylor Power Law in Wind Energy Data
To verify the existence of Taylor power law for our datasets, the mean value hPiτand the standard
deviation στare computed for time scales (or window sizes) τranging between approximately 4 h and
7 days for data sampled at 5 min, and τranging between approximately 16 min and 1 day for data
sampled at 1 s. The choice of the lower limit of these time intervals is justified by the value of a Pearson
correlation r > 0.8between the map (Pr,στ/C0) and a non parametric kernel smoothing regression fit.
Meanwhile, the choice of the upper limit is defined by a time window having a significant sample number
for the estimation of the Taylor exponent λτ. Hence, each dataset is splitted over a time window of length τ.
In Figure 2a, to compare the datasets considered here with the straight line of 1/2slope, στ/C0versus
Pr(Pr=hPi/hPimax with hPimax the maximum of the mean value for the dataset considered) is
illustrated for τ= 5 h, in log-log representation. One can observe that the map (Pr, στ/C0)can be
modeled by a relationship of the form:

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log στ
C0=λτlog (Pr) + c(3)
This leads to a power law of the form στ
C0
= (Pr)λτ(4)
where C0= 10c.
10−4 10−3 10−2 10−1 100
10−3
10−2
10−1
100
101
Pr
σ/C0
wind farm n°1
wind farm n°2
wind farm n°3
wind farm n°4
wind farm n°5
single wind turbine
1/2 slope
(a)
103104
0.4
0.45
0.5
0.55
0.6
0.65
τ (× 5 min)
λτ
102104
0.4
0.45
0.5
0.55
τ (seconds)
λτ
(b)
Figure 2. (a) Evolution of στ/C0versus the mean Pr. Evolution of the standard deviation
στ/C0versus the adimensioned mean value Prfor the power output from find wind farms
and a single wind turbine. στand hPiτare computed with a time window τ= 5 h. The map
(Pr,στ/C0)is fitted by a non parametric kernel regression (straight line); (b) Evolution of
the Taylor exponent λτversus the time scales τ, for the power output data sampled at five
minutes (in the inset for the power output data sampled at 1 s).
The Taylor exponent whose values correspond to each data set, are drawn up in Table 2. In all the
cases, the Taylor power law given in Equation (4) is verified and a scaling behavior is visible over more
than four orders of magnitude. Furthermore, the values of λτare close to 1/2. To investigate a possible
dependence of the exponent λτwith the time scales τ, we plot in Figure 2b, the evolution of the Taylor
exponent λversus the time scale τfor the data sampled at 5 min and in the inset for data sampled at 1 s.
One can observe no dependence of the exponent λτwith the time scales τ: it stays between 0.4and 0.57
with a range of variation which does not depend on τ.
Globally, the Taylor exponent λτvaries between 0.45 and 0.55 for times scales τbetween 1 h and
a one week and C0can be considered as a parameter characterizing the wind farm or the single turbine
considered. Indeed, Figure 3illustrates the evolution of the parameter C0versus the installed capacity
of the single wind turbine and the wind farms considered. The parameter C0increases as the installed
capacity Pinst, excepted for the wind farms n◦2 and 3 having installed capacities very close.

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Table 2. Taylor exponent λτand C0estimated for each dataset with τ= 5 h: the values
obtained are close to 1/2.C0can be considered as a parameter characterizing the wind farm
or the single turbine considered.
Data λτC0
Wind farm◦10.48 ±0.07 260.05
Wind farm◦20.49 ±0.07 213.05
Wind farm◦30.50 ±0.08 335.45
Wind farm◦40.55 ±0.07 439.84
Wind farm◦50.48 ±0.05 901.57
Single wind turbine 0.50 ±0.05 116.41
0 1 2 3 4 5 6 7 8 9 10 11
200
400
600
800
1000
Pinst (MW)
C0
Figure 3. Evolution of parameter C0versus the installed capacity Pinst of the wind farm and
the single wind turbine considered.
3.3. Turbulent Production Intensity IP
Here an analogy is made with the field of the turbulence. Classically, in the turbulence field, a
turbulent intensity parameter Iexpressing the ratio standard deviation to the mean value of the wind
speed, is a metric to characterize a turbulence level for flows with high variability, such as wind tunnel or
atmospheric wind [21–23]. Here this parameter can be used for measuring the degree of variability of the
wind power output and for classifying the variability level. In the turbulence field, one can distinguish
three classes: (i) 0< I < 5% corresponds to a weak level of variability, (ii) 5% < I < 10% corresponds
to a medium level of variability and I > 10% corresponds to a high level of variability. In this study, we
translate this coefficient to wind power and hence we introduce a new way to measure the variability of
wind power data. From the Taylor relationship defined in Equation (4), a scaling relationship between
the turbulent production intensity Ip=σ/hPiand the mean value hPi, is established by replacing hPi
by Pr:
Ip=σ
Pr
=C0Pλ
r
Pr(5)

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This leads
Ip=C0(Pr)α(6)
where the exponent α=λ−1is here negative.
Figure 4illustrates the evolution of the turbulent production intensity IPdivided by C0versus the
adimensioned mean value Pr, in log-log scale. As expected, the turbulent production intensity IP
decreases following a −1/2power law with the mean value hPi. Hence, taking into account the value
obtained for the Taylor exponent λτ≈1/2,α≈ −1/2for time scales 1h < τ < 7days. Table 3draws
up the values of the exponent αestimated for each dataset with τ= 5 h. It can be seen that the values
are generally corresponding to IP>10% which is a very high level of turbulence and that larger values
of Prcorrespond to smaller values of Ip.
10−4 10−3 10−2 10−1 100
10−5
10−4
10−3
10−2
10−1
Pr
IP/C0
wind farm n°1
wind farm n°2
wind farm n°3
wind farm n°4
wind farm n°5
single wind turbine
−1/2 slope
Figure 4. Evolution of the adimensioned turbulent production intensity IP/C0versus the
value Prcompared to the −1/2slope, in log-log scale.
Table 3. Exponent αestimated for each dataset with τ= 5 h.
Data α
Wind farm◦1−0.48 ±0.07
Wind farm◦2−0.49 ±0.07
Wind farm◦3−0.50 ±0.08
Wind farm◦4−0.55 ±0.05
Wind farm◦5−0.48 ±0.05
Single wind turbine −0.50 ±0.05
4. Conclusions and Discussions
In this study, we have investigated the existence of a Taylor power law or temporal fluctuation scaling
of power output delivered by five wind farms and a single wind turbine, with different installed capacity.

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The analyzed data are sampled at 1 s and five minutes, and recorded over different periods. A Taylor
power law has been highlighted for all the datasets. Furthermore, a universal scaling exponent λτclose
to 1/2, is observed for time scales 1h < τ < 7days for the data sampled at 5 min, and 5min < τ < 6h
for the data sampled at 1 s.
The existence of such Taylor law has been shown in multiple disciplinary fields. This universality
has conducted many authors to suggest the existence of a universal mechanism for its emergence.
Various approaches and theoretical investigations have been dedicated to possible explanations of
the origin of Taylor law. In the framework of a complex systems whose dynamics is the result of
interactions of many components belonging to a network [6,7], the value of λgives an information
on the mechanism governing the fluctuations involved in the process: λ≈1/2describes processes or
systems where internal factors drive dynamics and λ≈1describes processes where external factors
drive the dynamics [6–8]. This was investigated for internet traffic data and complex networks [6–8].
This result was experimentally based and cannot be used for understanding wind energy dynamics.
Recently, Fronczak and Fronczak (2010) [24] attempted to provide an interpretation of Taylor’s relation
based on the second law of thermodynamics (the maximum entropy principle) and the number of states.
Kendal and Jørgensen (2011) [25] proposed the Tweedie Convergence Theorem to give a possible
explanation of the origin of Taylor law. They show that Tweedie convergence theorem, a generalization
of Central Limit Theorem, provides an explanation for the genesis of Taylor laws. They also showed that
Taylor law is a scaling relationship, compatible with the presence 1/f scaling and multifractal properties,
characteristic of a self-similar process. On the other hand, several authors have shown the presence of
1/f scaling [26] and recently multifractal properties for wind energy data [20,27–29]. A way to highlight
multifractal properties is the use of a multi-scaling analysis including qth- order central moments versus
the mean value, a natural generalization of Taylor law where q= 2, or multifractal analysis [8,27,29].
Although the Tweedie Convergence Theorem seems offer a promising explanation, there is currently no
generally accepted theory to explain the emergence of Taylor’s relation.
The existence of Taylor’s law with 1/2exponent should help to provide an estimation of the mean
fluctuations for a mean value fixed of wind power data, but also to propose a statistical model of wind
power data using Tweedie model PDF. Our findings may be useful for developers and operators of
wind parks.
Acknowledgments
We thank the network operator E.D.F. (Electricité de France) for providing the aggregate power
output delivered by the wind farms. The power output data from the wind turbine, has been downloaded
at www.winddata.com.

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Author Contributions
Rudy Calif contributed to perform the analyzes and the writing of the manuscript. François G. Schmitt
contributed to the general design of the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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