# Analytical Modeling of Wind Farms: A New Approach for Power Prediction

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## Abstract

**:**

## 1. Introduction

## 2. Description of the New Analytical Wind Farm Model

#### 2.1. Analytical Model for the Velocity Deficit

#### 2.2. Turbulence Intensity Model

#### 2.3. Power Prediction

## 3. Case Description

^{2}. It is located in the North Sea, approximately 15 km off the westernmost point of Denmark. Each turbine has a rotor diameter of $d=80\text{m}$ and a hub height of ${H}_{hub}=70\text{m}$ (above sea level). Figure 3 shows a schematic of the Horns Rev wind farm layout. The wind farm has a rhomboid shape with wind turbines arranged in 8 columns (aligned with the East-West direction) and 10 rows (turned approximately 7° counterclockwise from the North-South direction). The turbines are regularly spaced, with a minimum spacing between two consecutive turbines of 7 rotor diameters.

## 4. Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Wake growth rate for the V-80 turbine in boundary layer flow with different streamwise turbulence intensities at hub height.

**Figure 2.**Power curve of the V-80 wind turbine. Red circles correspond to the manufacturer’s data and the blue line represents a polynomial fit.

**Figure 3.**Layout of the Horns Rev wind farm. Distances are normalized by the rotor diameter $d=80$ .

**Figure 4.**Measured and simulated power curve and thrust coefficient curve of the Vestas V-80 2 MW wind turbine, for a range of wind speeds (Source: Wu and Porté-Agel, 2014).

**Figure 5.**Distribution of the normalized Horns Rev wind farm power output obtained with the new analytical model and LES for different wind directions.

**Figure 6.**Comparison of the wind-farm power output for ${\theta}_{wind}=270\xb0$ obtained using LES as well as the new analytical model with both energy and velocity deficit superpositions.

**Figure 7.**Comparison of the power output for ${\theta}_{wind}$ = 270° obtained with the new analytical model using a constant wake growth rate and a variable wake growth rate based on the local streamwise turbulence intensity.

**Figure 8.**Comparison of the simulated and observed power output centered on three mean wind directions ${\theta}_{wind}$ = 270° (

**a**); 222° (

**b**) and 312° (

**c**). Symbols, lines, and dashed lines denote the observed, new analytical model, and WAsP data, respectively. Blue, red, and black colors represent ±5°, ±10°, and ±15° wind sectors, respectively.

**Figure 9.**Comparison of the time-averaged streamwise velocity at a horizontal plane at hub height: (

**a**) LES; (

**b**) new analytical model and (

**c**) top-hat model.

**Figure 10.**Comparison of the streamwise turbulence intensity at hub height immediately upstream of each turbine row, obtained with LES and three simple models.

© 2016 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/).

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**MDPI and ACS Style**

Niayifar, A.; Porté-Agel, F. Analytical Modeling of Wind Farms: A New Approach for Power Prediction. *Energies* **2016**, *9*, 741.
https://doi.org/10.3390/en9090741

**AMA Style**

Niayifar A, Porté-Agel F. Analytical Modeling of Wind Farms: A New Approach for Power Prediction. *Energies*. 2016; 9(9):741.
https://doi.org/10.3390/en9090741

**Chicago/Turabian Style**

Niayifar, Amin, and Fernando Porté-Agel. 2016. "Analytical Modeling of Wind Farms: A New Approach for Power Prediction" *Energies* 9, no. 9: 741.
https://doi.org/10.3390/en9090741