# Estimating Air Density Using Observations and Re-Analysis Outputs for Wind Energy Purposes

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theory

#### 2.2. Data

#### 2.3. Interpolation of Power Curves

#### 2.4. Extrapolation of Power Curves

## 3. Results

- Equation (3) but using T instead of ${T}_{v}$, ${P}_{1}=101325$ Pa and $L=-0.0065$ K m${}^{-1}$ (WAsP 11).

- Equation (3) using ERA5 outputs with L estimated from the nearest grid point using the closest pressure level and a pressure level 50 hPa above (WAsP 12 ERA5 L).

#### 3.1. The Effect of Humidity

#### 3.2. The Effect of a Varying Lapse Rate

#### 3.3. Using Re-Analysis Outputs

#### 3.4. Example

## 4. Conclusions and Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AEP | annual energy production |

CFSR | climate forecast system re-analysis |

DTU | Technical University of Denmark |

ECMWF | European Centre for Medium-Range Weather Forecasts |

ERA5 | fifth major re-analysis of the European Centre for Medium-Range Weather Forecasts |

IEC | International Electrotechnical Commission |

WAsP | Wind Atlas Analysis and Application Program |

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**Figure 2.**Scatter plot of three different methods (see text). Note that the points from the Zugspitze station were omitted because it had a very low $\rho $, which reduced the readability of the plot.

**Figure 4.**Scatter plot of the different modelling approaches. Note that the point from the Zugspitze station is omitted because it had a very low $\rho $ which reduced the readability of the plot.

**Figure 5.**Scatter plot of the different modelling approaches. Note that the points from the Zugspitze station were omitted because it had a very low $\rho $ and reduced the readability of the plot. (

**a**) Mean absolute errors in estimated $\rho $ compared to the observations as a function of horizontal distance for the models WAsP 12, WAsP 12 CFSR and WAsP 12 ERA (see Section 3); (

**b**) As panel (

**a**), but instead as a function of vertical distance and using the WAsP 12, WAsP 12 ERA5 and the WAsP ERA5 L model.

**Table 1.**Summary error metrics from the methods presented in this paper. The modelled and observed air density at site i of the total number of sites $N=77$ are denoted as ${x}_{i}$ and ${y}_{i}$, respectively. Then, the mean absolute error is defined as $\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}|100({y}_{i}-{x}_{i})/{x}_{i}|$, the correlation coefficient $R=\frac{1}{{\sigma}_{x}{\sigma}_{y}}\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}({x}_{i}-\overline{x})({y}_{i}-\overline{y})$, where the overbar and $\sigma $ denote the mean and standard deviation, respectively.

Method | Mean Abs. Error (%) | R |
---|---|---|

WAsP 11 | 0.203 | 0.9975 |

IEC 61400–12–1 | 0.185 | 0.9977 |

WAsP 12 | 0.186 | 0.9977 |

WAsP 12 CFSR | 0.241 | 0.9987 |

WAsP 12 ERA5 | 0.127 | 0.9993 |

WAsP 12 ERA5 L | 0.114 | 0.9994 |

**Table 2.**Annual energy production [GWh] calculated by Wind Atlas Analysis and Application Program (WAsP) 12 CFSR for the turbines with the lowest and highest air density and for the entire Parque Fictio test wind farm using power curves adapted to individual turbine sites or selected reference air densities.

Turbine | $\mathit{z}+{\mathit{z}}_{\mathbf{hub}}$ [m] | ${\mathit{\rho}}_{\mathbf{site}}$ [kg/m^{3}] | ${\overline{\mathit{U}}}_{\mathbf{site}}$ [m/s] | Interpol. | Reference Air Density ${\mathit{\rho}}_{\mathbf{ref}}$ | ||
---|---|---|---|---|---|---|---|

1.12 kg/m${}^{3}$ | 1.15 kg/m${}^{3}$ | 1.225 kg/m${}^{3}$ | |||||

T4 | 670 | 1.134 | 8.81 | 6716 | 6664 (−0.77%) | 6773 (+0.85%) | 7043 (+4.87%) |

T8 | 544 | 1.148 | 8.31 | 6416 | 6316 (−1.56%) | 6423 (+0.11%) | 6692 (+4.30%) |

All | 51,348 | 50,782 (−1.10%) | 51,638 (+0.56%) | 53,782 (+4.74%) |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Floors, R.; Nielsen, M.
Estimating Air Density Using Observations and Re-Analysis Outputs for Wind Energy Purposes. *Energies* **2019**, *12*, 2038.
https://doi.org/10.3390/en12112038

**AMA Style**

Floors R, Nielsen M.
Estimating Air Density Using Observations and Re-Analysis Outputs for Wind Energy Purposes. *Energies*. 2019; 12(11):2038.
https://doi.org/10.3390/en12112038

**Chicago/Turabian Style**

Floors, Rogier, and Morten Nielsen.
2019. "Estimating Air Density Using Observations and Re-Analysis Outputs for Wind Energy Purposes" *Energies* 12, no. 11: 2038.
https://doi.org/10.3390/en12112038