# Taylor Law in Wind Energy Data

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## Abstract

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## 1. Introduction

## 2. Wind Power Output Data

**Figure 1.**An example of power output sequence $p\left(t\right)$ delivered by the single wind turbine during 48 h.

**Table 1.**Description of characteristics (sampling frequency, number of continuously data points, implementation site, installed capacity) for each dataset.

Dataset | Sampling Frequency (Hz) | Number of Data Points | Implementation Site | Installed Capacity ${P}_{\mathrm{inst}}$ |
---|---|---|---|---|

Wind farm${}^{\circ}$1 | $3.3\times {10}^{-3}$ | $125,942$ | plateau | 2.6 MW |

Wind farm${}^{\circ}$2 | $3.3\times {10}^{-3}$ | $125,942$ | plain | 2.9 MW |

Wind farm${}^{\circ}$3 | $3.3\times {10}^{-3}$ | $125,942$ | plateau | 1.9 MW |

Wind farm${}^{\circ}$4 | $3.3\times {10}^{-3}$ | $125,942$ | plain | 3 MW |

Wind farm${}^{\circ}$5 | 1 | $6,529,000$ | 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

#### 3.2. Taylor Power Law in Wind Energy Data

**Figure 2.**(

**a**) Evolution of ${\sigma}_{\tau}/{\mathcal{C}}_{0}$ versus the mean ${P}_{r}$. Evolution of the standard deviation ${\sigma}_{\tau}/{\mathcal{C}}_{0}$ versus the adimensioned mean value ${P}_{r}$ for the power output from find wind farms and a single wind turbine. ${\sigma}_{\tau}$ and ${\langle P\rangle}_{\tau}$ are computed with a time window $\tau =5$ h. The map (${P}_{r}$,${\sigma}_{\tau}/{\mathcal{C}}_{0})$ is fitted by a non parametric kernel regression (straight line); (

**b**) Evolution of the Taylor exponent ${\lambda}_{\tau}$ versus 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).

**Table 2.**Taylor exponent ${\lambda}_{\tau}$ and ${\mathcal{C}}_{0}$ estimated for each dataset with $\tau =5$ h: the values obtained are close to $1/2$. ${\mathcal{C}}_{0}$ can be considered as a parameter characterizing the wind farm or the single turbine considered.

Data | ${\lambda}_{\tau}$ | ${\mathcal{C}}_{0}$ |
---|---|---|

Wind farm${}^{\circ}$1 | $0.48\pm 0.07$ | $260.05$ |

Wind farm${}^{\circ}$2 | $0.49\pm 0.07$ | $213.05$ |

Wind farm${}^{\circ}$3 | $0.50\pm 0.08$ | $335.45$ |

Wind farm${}^{\circ}$4 | $0.55\pm 0.07$ | $439.84$ |

Wind farm${}^{\circ}$5 | $0.48\pm 0.05$ | $901.57$ |

Single wind turbine | $0.50\pm 0.05$ | $116.41$ |

**Figure 3.**Evolution of parameter ${\mathcal{C}}_{0}$ versus the installed capacity ${P}_{inst}$ of the wind farm and the single wind turbine considered.

#### 3.3. Turbulent Production Intensity ${I}_{P}$

**Figure 4.**Evolution of the adimensioned turbulent production intensity ${I}_{P}/{\mathcal{C}}_{0}$ versus the value ${P}_{r}$ compared to the $-1/2$ slope, in log-log scale.

Data | α |
---|---|

Wind farm${}^{\circ}$1 | $-0.48\pm 0.07$ |

Wind farm${}^{\circ}$2 | $-0.49\pm 0.07$ |

Wind farm${}^{\circ}$3 | $-0.50\pm 0.08$ |

Wind farm${}^{\circ}$4 | $-0.55\pm 0.05$ |

Wind farm${}^{\circ}$5 | $-0.48\pm 0.05$ |

Single wind turbine | $-0.50\pm 0.05$ |

## 4. Conclusions and Discussions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Taylor, L.R. Aggregation, variance and the mean. Nature
**1961**, 189, 732–735. [Google Scholar] [CrossRef] - Smith, H.F. An empirical law describing heterogeneity in the yields of agricultural crops. J. Agric. Sci.
**1938**, 28, 1–23. [Google Scholar] [CrossRef] - Keitt, T.H.; Stanley, H.E. Dynamics of North American breeding bird populations. Nature
**1998**, 393, 257–260. [Google Scholar] [CrossRef] - Kerkhoff, A.J.; Ballantyne, F. The scaling of reproductive variability in trees. Ecol. Lett.
**2003**, 6, 850–856. [Google Scholar] [CrossRef] - Xu, M.; Schuster, W.S.; Cohen, J.E. Robustness of Taylor’s law under spatial hierarchical groupings of forest tree samples. Popul. Ecol.
**2015**, 103, 1–11. [Google Scholar] [CrossRef] - De Menezes, M.; Barabási, A.L. Fluctuations in Networks Dynamics. Phys. Rev. Lett.
**2004**, 92, 028701. [Google Scholar] [CrossRef] [PubMed] - De Menezes, M.; Barabási, A.L. Separating Internal and External Dynamics of Complex Systems. Phys. Rev. Lett.
**2004**, 93, 068701. [Google Scholar] [CrossRef] - Eisler, Z.; Kertész, J.; Yook, S.G.M.; Barabási, A.L. Multiscaling and non-universality in fluctuations of driven complex systems. Europhys. Lett.
**2005**, 69, 664–670. [Google Scholar] [CrossRef] - Eisler, Z.; Kerész, J. Random walks on complex networks with inhomogeneous impact. Phys. Rev. E
**2005**, 71, 057104. [Google Scholar] [CrossRef] - Eisler, Z.; Kertész, J. Scaling theory of temporal correlations and size-dependent fluctuations in the traded value of stocks. Phys. Rev. E
**2006**, 73, 040109. [Google Scholar] [CrossRef] - Šuvakov, M.; Tadić, B. Transport processes on homogeneous planar graphs with scale-free loops. Phys. A
**2006**, 372, 354–361. [Google Scholar] [CrossRef] - Eisler, Z.; Bartos, I.; Kertész, J. Fluctuation scaling in complex systems: Taylor’s law and beyond. Adv. Phys.
**2008**, 57, 89–142. [Google Scholar] [CrossRef] - Lee, Y.; Amaral, L.A.N.; Canning, D.; Meyer, M.; Stanley, H.E. Universal features in the growth dynamics of complex organizations. Phys. Rev. Lett.
**1998**, 81, 3275–3278. [Google Scholar] [CrossRef] - Jiang, Z.Q.; Guo, L.; Zhou, W.X. Endogenous and exogenous dynamics in the fluctuations of capital fluxes: An empirical analysis of the Chinese stock market. Eur. Phys. J. B
**2007**, 57, 347–355. [Google Scholar] [CrossRef] - Jánosi, I.M.; Gallas, J.A. Growth of companies and water-level fluctuations of the river Danube. Phys. A
**1999**, 271, 448–457. [Google Scholar] [CrossRef] - Dahlstedt, K.; Jensen, H.J. Fluctuation spectrum and size scaling of river flow and level. Phys. A
**2005**, 348, 596–610. [Google Scholar] [CrossRef] - Azevedo, R.B.; Leroi, A.M. A power law for cells. Proc. Natl. Acad. Sci.
**2001**, 98, 5699–5704. [Google Scholar] [CrossRef] [PubMed] - Nacher, J.C.; Ochiai, T.; Akutsu, T. On the relation between fluctuation and scaling-law in gene expression time series from yeast to human. Mod. Phys. Lett. B
**2005**, 19, 1169–1177. [Google Scholar] [CrossRef] - Kendal, W.S. An exponential dispersion model for the distribution of human single nucleotide polymorphisms. Mol. Biol. Evol.
**2003**, 20, 579–590. [Google Scholar] [CrossRef] [PubMed] - Calif, R.; Schmitt, F.G. Analyse de séries temporelles de production éolienne: Loi de Taylor et propriétés multifractales. In Compte Rendu de la 15e Rencontre du Non-Linéaire; Falcon, E., Josserand, C., Lefranc, M., Letellier, C., Eds.; Non-Linéaire Publications: Paris, France, 2012; pp. 61–66. [Google Scholar]
- Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000; p. 792. [Google Scholar]
- Petersen, E.K.; Mortensen, N.G.; Landberg, L.; Højstrup, J.; Frank, H.P. Wind power meteorology. Wind Energy
**1999**, 1, 25–45. [Google Scholar] [CrossRef] - Calif, R.; Emilion, R.; Soubdhan, T. Classification of wind speed distributions using a mixture of Dirichlet distributions. Renew. Energ.
**2011**, 36, 3091–3097. [Google Scholar] [CrossRef] - Fronczak, A.; Fronczak, P. Origins of Taylor’s power law for fluctuation scaling in complex systems. Phys. Rev. E
**2010**, 81, 066112. [Google Scholar] [CrossRef] - Kendal, W.S.; Jørgensen, B. Tweedie convergence: A mathematical basis for Taylor’s power law, 1/f noise, and multifractality. Phys. Rev. E
**2011**, 84, 066120. [Google Scholar] [CrossRef] - Apt, J. The spectrum of power from wind turbines. J. Power Sources
**2007**, 169, 369–374. [Google Scholar] [CrossRef] - Calif, R.; Schmitt, F.G.; Huang, Y. The multifractal description of wind power fluctuations using arbitrary order Hilbert spectral analysis. Phys. A
**2013**, 392, 4106–4120. [Google Scholar] [CrossRef] - Milan, P.; Wächter, M.; Peinke, J. Turbulent character of wind energy. Phys. Rev. Lett.
**2013**, 110, 138701. [Google Scholar] [CrossRef] [PubMed] - Calif, R.; Schmitt, F.G. Multiscaling and joint multiscaling description of the atmospheric wind speed and the aggregate power output from a wind farm. Nonlinear Proc. Geophys.
**2014**, 21, 379–392. [Google Scholar] [CrossRef] [Green Version]

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Calif, R.; Schmitt, F.G.
Taylor Law in Wind Energy Data. *Resources* **2015**, *4*, 787-795.
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Calif R, Schmitt FG.
Taylor Law in Wind Energy Data. *Resources*. 2015; 4(4):787-795.
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**Chicago/Turabian Style**

Calif, Rudy, and François G. Schmitt.
2015. "Taylor Law in Wind Energy Data" *Resources* 4, no. 4: 787-795.
https://doi.org/10.3390/resources4040787