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A parametric model for wind turbine power curves incorporating environmental conditions
Yves-Marie Saint-Drenan, Romain Besseau, Malte Jansen, Iain Staffell, Alberto Troccoli, Laurent Dubus, Johannes Schmidt, Katharina Gruber, Sofia G. Simões, Siegfried Heier
TL;DR
Reliable, site-specific turbine power curves are difficult to obtain because outdoor assessment is challenging and available databases often lack key environmental information. The paper develops a parameterised model using turbine characteristics and environmental conditions, then validates it against database curves. The model identifies nominal power, rotor area, and maximal Cp as the most influential parameters, while producing realistic curves for most turbine models.
Problem
Power-curve assessment is difficult in real outdoor conditions, and many available curves lack environmental reference information needed for reliable turbine calculations.
Method
The paper develops a physically based model with 12 parameters that adapts power curves using turbine characteristics, turbulence intensity, and air density.
Results
Nominal power, rotor area, and maximal Cp are the most influencing parameters, and validation produces realistic curves for most turbine models.
Takeaways & Limitations
The model provides a way to estimate power curves for existing or hypothetical turbines under specified conditions.
Takeaways & Limitations
Real outdoor conditions make robust power-curve assessment difficult because of spatial and temporal variations.
Abstract
from arXiv · showhide
A wind turbine's power curve relates its power production to the wind speed it experiences. The typical shape of a power curve is well known and has been studied extensively; however, the power curves of individual turbine models can vary widely from one another. This is due to both the technical features of the turbine (power density, cut-in and cut-out speeds, limits on rotational speed and aerodynamic efficiency), and environmental factors (turbulence intensity, air density, wind shear and wind veer). Data on individual power curves are often proprietary and only available through commercial databases. We therefore develop an open-source model which can generate the power curve of any turbine, adapted to the specific conditions of any site. This can employ one of six parametric models advanced in the literature, and accounts for the eleven variables mentioned above. The model is described, the impact of each technical and environmental feature is examined, and it is then validated against the manufacturer power curves of 91 turbine models. Versions of the model are made available in MATLAB, R and Python code for the community.
1. Introduction
The paper addresses limited access to reliable, site-specific turbine power curves by proposing a parameterised model that incorporates turbine characteristics and environmental conditions. It motivates this approach because outdoor assessment is difficult and missing curve information creates uncertainty in power calculations.
- Power curves are important because wind-farm construction and operation risks depend directly on their accuracy.
- Outdoor power-curve assessment is difficult because modern turbines cannot be tested in wind tunnels and measurements vary spatially and temporally.
- Commercial power-curve databases are not freely available, while many available curves omit reference turbulence intensity or air density.
- Missing information introduces non-negligible uncertainty, or can make turbine power calculations impossible, especially for prospective energy-mix analyses.
- The paper proposes a physically based, parameterised model using rated power, rotor dimensions, operating characteristics, aerodynamic-efficiency functions, and explicit environmental factors.
- Statistical models require historical training data and capture net power affected by wakes, orography, turbine availability, and wind-speed errors rather than gross turbine production.
- The model is systematically studied through sensitivity and statistical analyses, validated against database power curves, and implemented in Python, R, and MATLAB.
2. Methodology
The methodology represents turbine power curves across four operating regions and models region II using parametric aerodynamic relationships. It incorporates rotor and environmental effects, including turbulence intensity, air density, wind shear, and wind veer.
- Operating regions: Power curves are divided into four operating regions bounded by cut-in, rated, and cut-off wind speeds.Region I is below cut-in, Region II spans cut-in to rated speed, Region III maintains rated power, and Region IV exceeds cut-off speed.
- Operating regions: Above cut-in, power increases with the cube of wind speed until the turbine reaches its rated power.Rated power is designed not to be exceeded; pitch control can maintain it by adjusting blade pitch in Region III.
- Parametric power curve: The wind-power equation combines air density, rotor area, wind speed, and the power coefficient Cp(λ, β).The power coefficient represents the recoverable fraction of wind-flow power and depends on tip-speed ratio λ and blade pitch angle β.
- Parametric power curve: Six literature parameter sets are used to represent the power coefficient, while the approach can be extended to other parametric models or numerical data.Rotor area and turbine product-sheet parameters are typically available, although missing parameters can be estimated.
- Parametric power curve: Region II is modeled by setting blade pitch to zero and targeting the tip-speed ratio λopt that maximizes Cp, subject to rotor-speed limits.The zero-pitch assumption excludes Region II regulation strategies intended to limit noise or mechanical effects.
3. Analysis of the sensitivity of the power curve to the model parameters
The sensitivity analysis varies each model parameter independently across a typical range, showing that rotor area, nominal power, and Cp,max most strongly affect predicted power, while rotational and cut-in/cut-off effects are more localized.
- Analysis limitations: The analysis varies one parameter at a time and assesses sensitivity qualitatively rather than quantifying interactions with Morris or Sobol methods.Sensitivity is represented visually using differently coloured curves in Figure 8.
- Overall sensitivity: Rotor area and nominal power produce the largest sensitivity in output power and should receive the greatest attention, although both are usually known design parameters.The authors note that manufacturers commonly include these parameters directly in turbine names.
- Rotor area and rotational speed: Small rotors can depart from cubic power growth and even decline at high wind speeds when the maximum rotational-speed limit reduces the power coefficient.The effect follows increasing rotational speed toward the optimal TSR until the maximum rotational speed is reached.
- Cut-in and cut-off speeds: Cut-off wind speed has the largest direct effect among the two threshold speeds, while cut-in sensitivity is moderate; both remain important for annual energy estimation.Their importance is lower than that of nominal power and rotor area, partly because wind-speed frequency differs near the two thresholds.
- Rotor area and rotational speed: Minimum rotational speed mainly affects 3–9 m/s winds, whereas maximum rotational speed acts near nominal wind speed and is generally less visible.The minimum rotational speed deserves careful selection because winds in the 3–9 m/s interval occur frequently.
- Power coefficient: The Cp parameterisation has little effect on the curve shape, but Cp,max strongly affects the curve, making its accurate value decisive for power estimation.The model output is insensitive to the selected scaled Cp form but responds significantly to Cp,max variation.
4. Statistical analysis of the most sensitive model input parameters
The statistical analysis uses turbine and power-curve data to recommend default values for sensitive parameters that may be unavailable. It identifies typical distributions for Cp,max and cut-in/cut-off speeds, while rotational speeds depend strongly on rotor diameter.
- Maximum power coefficient: Cp,max is most frequently 0.44, with 80 % of values between 0.4 and 0.5; the authors recommend 0.44 when unavailable.No clear dependence of Cp,max on further turbine characteristics was identified in the available data.
- Threshold wind speeds: Cut-in wind speeds range from 1 to 5 m/s, with 90 % between 2 and 4 m/s and most values around 3 m/s.The distribution comes from the European thewindpower.net dataset.
- Threshold wind speeds: Cut-off wind speeds range from 15 to 30 m/s, with 20 and 25 m/s most frequent at 12 % and 70 %, respectively.When cut-in or cut-off data are missing, the recommended values are 3 and 25 m/s.
- Rotational speeds: Minimum and maximum rotational speeds depend strongly on rotor diameter, so the authors fit exponential functions to estimate them when characteristics are missing.The fitted expressions provide estimates for both rotational-speed limits.
5. Validation of the parametric power curve model
The model is validated qualitatively against manufacturer and database power curves because turbine-specific turbulence intensity is unavailable for quantitative calibration. Across the comparison, it reproduces the behaviour of most curves, while mismatches expose data-quality and modelling uncertainties.
- Validation design: The validation compares model outputs with power curves from 91 turbines, using visual rather than quantitative assessment because turbulence intensity is unknown.Model outputs are generated over turbulence-intensity values from 0 to 10 %, preventing a direct quantitative validation without calibration.
- Validation results: Three example turbines show close correspondence between model outputs and database curves, suggesting realistic synthesis across a variety of turbines.The authors note that this similarity does not completely exclude systematic modelling error.
- Environmental inputs: The model can control turbulence intensity and air density, enabling exploration of power-curve changes across altitude, temperature, and season.This control is presented as an advantage because the validation database lacks turbulence-intensity information.
- Sources of mismatch: Some mismatches arise from disagreement between the modelled and actual maximum power coefficients, including a turbine curve that exceeds the Betz limit.The authors associate that case possibly with measurement error and stress careful screening of database power curves.
- Sources of mismatch: The validation includes curves with unusual shapes whose causes remain unclear, so further work is needed to assess model performance and uncertainty.Possible explanations include modelling error, turbulence correction, and shaded wind measurements.
- Overall validation: After rotor-area normalisation, many database curves overlap between 3 and 10 m/s, and the model reproduces most power-curve behaviour across the wind industry.The authors state that this remains true even for cases with the largest errors.
6. Conclusion
The proposed model estimates wind-turbine power curves from turbine characteristics while adapting turbulence intensity and air density to site conditions. Validation found realistic curves for most database turbines, but some discrepancies and scope limitations remain.
- Model and sensitivity: The model estimates a wind turbine’s power curve from its main characteristics and offers twelve parameters for adapting turbulence intensity and air density to site conditions.Nominal power, rotor area, and maximal Cp are identified as the most influential parameters.
- Validation: Qualitative validation against a wind-turbine dataset found realistic modelled power curves for most turbines.The validation compared model outputs with database power curves.
- Validation: Large and suspicious differences occurred for a limited number of turbines, and their causes could reflect modelling issues or database data-quality issues.The authors therefore advise caution when using power curves found online.
- Validation: Database power curves corresponded to different turbulence-intensity levels, whereas the proposed model can avoid uncertainty from missing turbulence information.The database generally does not report the relevant turbulence intensity.
- Applications and scope: The model is intended as additional information for cross-checking results or estimating existing and hypothetical turbine performance, not as a replacement for measurement campaigns.Its value is especially noted for estimating regional wind-power production when only limited turbine characteristics are available.
- Limitations and future work: The approach assumes turbines operate for maximum possible output, so future improvements could incorporate sub-optimal control strategies and require further validation.Examples include noise-emission limits and smooth disconnection at cut-off.
- Implementation and applications: Implementations that generate power curves are provided in MATLAB, R, and Python as supplementary material.The model is presented for wind-power-production simulation and energy-mix analysis.
Appendix A. Parametric power coefficient models Cp(λ, β)
The appendix describes power-coefficient parameterisations based on tip-speed ratio and blade pitch angle. It frames these expressions as empirical alternatives to experimental or numerical evaluation and illustrates comparisons among models and turbine curves.
- Power coefficient: The power coefficient Cp represents the fraction of wind power extracted by a turbine and is generally modelled as a function of tip-speed ratio and blade pitch angle.The appendix introduces Cp parameterisations for the turbine model.
- Power coefficient: Cp can be evaluated experimentally, calculated numerically with BEM, CFD, or GDW models, or approximated using empirical relations from the literature.The appendix presents numerical approximations as a convenient alternative.
- Literature parameterisations: Table A.2 lists coefficients for the different literature parameterisations of Cp.
- Validation comparisons: Figures 12 and 13 compare model outputs with power curves for three high-quality and two lower-quality turbine models, respectively.The cited turbine sets are 258, 263, and 270, and 429 and 408.
- Validation comparisons: Figure 14 compares the model output with the power curve of turbine 404, while Figure 15 compares database curves with model outputs for selected power-density ratios.The ratios shown are 0.25, 0.375, and 0.5 kW/m2.
- Literature parameterisations: Figure A.16 compares the different literature Cp models for blade pitch angles of 0, 1, 3, and 5°.