# 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

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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