# Methods to Apply a 3-Parameter Logistic Model to Wind Turbine Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data Filtering

- To discard wind speed values that are lower than the cut-in wind speed.
- To discard wind speed values that are 1.5 m/s above the cut-in wind speed, when they provide an output power lower than a 5% of the rated power.
- To discard wind speed values that are 1 m/s above the rated wind speed, when they provide an output power lower than a 75% of the rated power.

## 3. Methods Proposed

#### 3.1. Spline

- Divide the data into intervals, one every 0.5 m/s. The identification of each interval is an integer or the mean value of two consecutive integers. Notice that the number of data on each interval may be different.
- Obtain the mean power on each interval and assign that value to the identification of the interval on each one, too. The result is a number of pairs of values (wind speed, output power).
- Finally, the spline according to Equation (1) is obtained. Using MATLAB (R2019a, Mathworks Inc.), all the points for the spline are provided.

#### 3.2. Clustering

#### 3.3. Cloud Data

#### 3.4. Cluster Simulation

- Divide the data in intervals, one every 0.5 m/s. The identification of each interval is an integer or and the mean value of two consecutive integers. This method is just an alternative when the number of data on each interval is different, as usual.
- Obtain the mean power and the standard deviation for each interval.
- Afterwards, a Normal distribution of the data on each interval is assumed and, with the parameters (µ, σ) obtained, values of power for each interval are simulated. The number of values generated for each interval has to be the same and has to be representative of all the possibilities (i.e. 200) The simulation has to be performed using MATLAB.
- Using the simulated values as data, the process depicted in the method Cloud data is applied.

#### 3.5. Maximum Error Cluster

## 4. Method Assessment

- Cloud Data MAPE: it measures the difference, in absolute value, between the value provided by the model and the output power value (from the filtered data). It is obtained according to Equation (4).$$CloudDataMAPE=\frac{{\displaystyle \sum}abs(Pmode{l}_{j}-Pdat{a}_{j})}{Prated}$$
- Mean Values MAPE: it measures the difference, in absolute value, between the value provided by the model and the corresponding mean value of the power. In order to obtain the mean values, intervals of 0.5 m/s were taken. It is obtained according to Equation (5).$$MeanValuesMAPE=\frac{{\displaystyle \sum}abs(Pmode{l}_{j}-Pmea{n}_{j})}{Prated}$$

## 5. Case Study

- In the reference case the MV MAPE is always 0 because it is the definition of the reference. However, when obtaining the CD MAPE, the value obtained is close to 2%. The reason is that all the pairs of points do not correspond exactly with the spline models, there is some variability.
- In all cases, the MV MAPE is lower than the CD MAPE because the latter measures the errors of all the pairs of points with respect to the model, so the variability is higher. There is an exception to the previous rule and is in the case of the cloud data because, in that case, all the pairs of points participate with the same weight in the model.
- The errors provided by the Clustering and the Cluster simulation methods are very low in both cases. In fact, in both cases the difference between the CD MAPE and the MV MAPE is lower than the corresponding one of the spline.
- Comparing Clustering and Cluster simulation methods, they provide very similar results in this case.
- In the case of the Max error cluster method, the values of errors are a bit higher in all cases.

- The CD MAPE for the Spline is a bit lower than in the case of the first wind turbine. The reason may be that the variability of the output power of that wind turbine is lower than in the first one which can be due to performance reasons.
- The MV MAPE Cloud data is higher than in the first wind turbine while the CD Cloud data is lower. The reason may be the same as in the first comment, the pairs of points are not very dispersed, therefore, they provide a very good model, when assessing all data.

- It is the wind turbine with the lower errors from a general point of view. The MV MAPE Clustering and the MV MAPE Cluster simulation errors are close to 1%.
- Its behavior and levels of errors are very similar to the ones of the first one.

- The CD MAPE for the spline is over 2% (very high compared with the others).
- The CD MAPE Clustering is also very high (more than 3%).
- The MV MAPE for Clustering and Cluster simulation are very similar to the rest of wind turbines.
- The rest of errors are out of the ranges of the other three wind turbines.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Manufacturer | Model | Rated Power (kW) | Hub Height (m) | Rotor Diameter (m) | Rated Wind Speed (m/s) | Cut-in Wind Speed (m/s) | Cut-out Wind Speed (m/s) |
---|---|---|---|---|---|---|---|

Senvion | MM82 | 2050 | 80 | 82 | 14.5 | 3.5 | 25 |

2013 | 2014 | 2015 | 2016 | 2017 | |
---|---|---|---|---|---|

CD MAPE spline (%) | 1.71 | 1.64 | 2.14 | 1.94 | 2.08 |

MV MAPE spline (%) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

CD MAPE clustering (%) | 2.62 | 2.60 | 3.27 | 3.17 | 3.23 |

MV MAPE clustering (%) | 1.16 | 1.40 | 1.29 | 1.38 | 1.81 |

CD MAPE cloud data (%) | 2.26 | 2.11 | 2.77 | 2.57 | 2.66 |

MV MAPE cloud data (%) | 6.86 | 6.34 | 5.87 | 7.07 | 6.94 |

CD MAPE cluster simulation (%) | 2.57 | 2.53 | 3.15 | 3.05 | 3.23 |

MV MAPE cluster simulation (%) | 1.21 | 1.39 | 1.34 | 1.44 | 2.02 |

CD MAPE max error cluster (%) | 2.70 | 2.68 | 3.20 | 3.13 | 4.71 |

MV MAPE max error cluster (%) | 1.76 | 1.62 | 2.24 | 2.52 | 4.63 |

2013 | 2014 | 2015 | 2016 | 2017 | |
---|---|---|---|---|---|

CD MAPE spline (%) | 1.50 | 1.30 | 1.79 | 1.68 | 1.76 |

MV MAPE spline (%) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

CD MAPE clustering (%) | 2.61 | 2.49 | 2.99 | 2.95 | 2.81 |

MV MAPE clustering (%) | 1.25 | 1.36 | 1.52 | 1.35 | 1.46 |

CD MAPE cloud data (%) | 2.01 | 1.73 | 2.34 | 2.24 | 2.33 |

MV MAPE cloud data (%) | 8.85 | 8.60 | 6.91 | 7.77 | 6.81 |

CD MAPE cluster simulation (%) | 2.51 | 2.41 | 2.91 | 2.79 | 2.69 |

MV MAPE cluster simulation (%) | 1.30 | 1.42 | 1.54 | 1.43 | 1.48 |

CD MAPE max error cluster (%) | 2.66 | 2.52 | 3.29 | 3.11 | 2.65 |

MV MAPE max error cluster (%) | 1.60 | 1.59 | 1.53 | 1.55 | 2.05 |

2013 | 2014 | 2015 | 2016 | 2017 | |
---|---|---|---|---|---|

CD MAPE spline (%) | 1.50 | 1.42 | 1.76 | 1.64 | 1.75 |

MV MAPE spline (%) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

CD MAPE clustering (%) | 2.42 | 2.42 | 2.80 | 2.82 | 2.78 |

MV MAPE clustering (%) | 1.11 | 1.26 | 1.27 | 1.18 | 1.18 |

CD MAPE cloud data (%) | 2.03 | 1.89 | 2.35 | 2.22 | 2.35 |

MV MAPE cloud data (%) | 6.57 | 5.99 | 5.09 | 6.33 | 6.11 |

CD MAPE cluster simulation (%) | 2.31 | 2.30 | 2.69 | 2.64 | 2.70 |

MV MAPE cluster simulation (%) | 1.14 | 1.32 | 1.29 | 1.24 | 1.19 |

CD MAPE max error cluster (%) | 2.54 | 2.68 | 3.51 | 2.89 | 3.04 |

MV MAPE max error cluster (%) | 1.41 | 1.28 | 1.46 | 1.64 | 2.19 |

2013 | 2014 | 2015 | 2016 | 2017 | |
---|---|---|---|---|---|

CD MAPE spline (%) | 2.04 | 1.62 | 2.33 | 2.16 | 2.27 |

MV MAPE spline (%) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

CD MAPE clustering (%) | 4.69 | 2.85 | 3.62 | 3.49 | 3.50 |

MV MAPE clustering (%) | 1.60 | 1.51 | 1.52 | 1.40 | 1.27 |

CD MAPE cloud data (%) | 3.48 | 2.14 | 3.00 | 2.83 | 2.96 |

MV MAPE cloud data (%) | 11.45 | 6.88 | 6.36 | 7.08 | 7.37 |

CD MAPE cluster simulation (%) | 4.93 | 2.78 | 3.42 | 3.33 | 3.39 |

MV MAPE cluster simulation (%) | 1.50 | 1.53 | 1.58 | 1.47 | 1.29 |

CD MAPE max error cluster (%) | 3.84 | 2.83 | 4.06 | 4.15 | 3.71 |

MV MAPE max error cluster (%) | 3.97 | 2.04 | 1.86 | 3.30 | 2.96 |

Optimization Method | Cloud Data MAPE (%) | Mean Values MAPE (%) |
---|---|---|

Spline | 1.80 | 0.00 |

Clustering | 3.01 | 1.36 |

Cloud data | 2.41 | 7.06 |

Cluster simulation | 2.92 | 1.41 |

Maximum error of cluster | 3.19 | 2.16 |

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**MDPI and ACS Style**

Villanueva, D.; Sixto, A.; Feijóo, A.; Fernández, A.; Miguez, E.
Methods to Apply a 3-Parameter Logistic Model to Wind Turbine Data. *Appl. Sci.* **2020**, *10*, 3317.
https://doi.org/10.3390/app10093317

**AMA Style**

Villanueva D, Sixto A, Feijóo A, Fernández A, Miguez E.
Methods to Apply a 3-Parameter Logistic Model to Wind Turbine Data. *Applied Sciences*. 2020; 10(9):3317.
https://doi.org/10.3390/app10093317

**Chicago/Turabian Style**

Villanueva, Daniel, Adrián Sixto, Andrés Feijóo, Antonio Fernández, and Edelmiro Miguez.
2020. "Methods to Apply a 3-Parameter Logistic Model to Wind Turbine Data" *Applied Sciences* 10, no. 9: 3317.
https://doi.org/10.3390/app10093317