# Long-Term Estimation of Wind Power by Probabilistic Forecast Using Genetic Programming

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## Abstract

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## 1. Introduction

- Very short-term forecast: From a few minutes to one hour ahead.
- Short-term forecast: From one hour to several hours ahead.
- Medium-term forecast: From several hours to one week ahead.
- Long-term forecast: From one week to one year or more ahead.

## 2. Materials and Methods

#### 2.1. Wind Speed Probability Distribution and Wind Power

#### 2.2. Data Grouping, Prediction Horizons and Associated Parameters

^{2}calculated from the real data in the chosen time interval. Additionally, we compute the mean wind speed of the Weibull probability distribution $\overline{v}$. The mean wind speed of the fitted Weibull distribution is compared to the average wind speed calculated from the wind speed time series ${v}_{ave}$, as an indicator of whether the fit is good.

#### 2.3. Genetic Programming to Predict Weibull Distributions

_{W}, ${v}_{ave}$, $\overline{v}$ and $\lambda $, all at the time $t-1$. In addition, the generation of random numbers function is included in the terminal set to consider numerical values. Figure 4 shows an example of genotype and encoding used to forecast the Weibull scale parameter. In that specific case, the scale parameter is a function of the solar radiation, the ambient temperature and the atmospheric pressure.

#### 2.4. Estimating Wind Power at Long-Term

#### 2.5. Forecasting Error

## 3. Experiments and Results

#### 3.1. Data Grouping, Prediction Horizons and Associated Parameters

#### 3.1.1. Seasonality

#### 3.1.2. Grouping and Prediction Horizons

#### 30 Days Ahead

#### 25 Days Ahead

#### 20 Days Ahead

#### 15 Days Ahead

#### 10 Days Ahead

#### 3.1.3. Parameters

#### 3.2. Predicting Weibull Distributions

#### 3.3. Estimating Wind Power

#### 3.4. Forecasting Errors

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$v$ | $\mathrm{m}/\mathrm{s}$ | Wind speed |

${f}_{W}\left(v;\lambda ,k\right)$ | − | Weibull probability density function |

${f}_{\mathrm{PDF}}\left(v\right)$ | − | Probability density function |

$\lambda $ | $\mathrm{m}/\mathrm{s}$ | Scale parameter of the Weibull probability density function |

$k$ | − | Shape parameter of the Weibull probability density function |

${T}_{i}$ | − | Percentage of time (of the total time used to construct the Weibull PDF) where an amount of wind power is produced. |

${v}_{l}$ | $\mathrm{m}/\mathrm{s}$ | Lower wind speed in a time interval |

${v}_{u}$ | $\mathrm{m}/\mathrm{s}$ | Upper wind speed in a time interval |

$\overline{v}$ | $\mathrm{m}/\mathrm{s}$ | Mean wind speed from distribution |

${v}_{\mathrm{ave}}$ | $\mathrm{m}/\mathrm{s}$ | Average mean wind speed from data |

$\mathsf{\Gamma}\left(x\right)$ | − | Gamma function |

$I$ | $\mathrm{W}/{\mathrm{m}}^{2}$ | Solar radiation |

$T$ | $\mathrm{C}$ | Ambient temperature |

$P$ | $\mathrm{mbar}$ | Atmospheric pressure |

$H$ | $\%$ | Relative humidity |

${P}_{W}$ | $\mathrm{W}$ | Produced power from wind turbine |

${P}_{\mathrm{average}}$ | $\mathrm{W}$ | Produced power from a wind turbine in an interval of time |

${P}_{\mathrm{turb}}\left(v\right)$ | $\mathrm{W}$ | Power curve of the wind turbine as a function of the wind speed |

${F}_{gp}$ | − | Set of $NF$ functions used by the GP algorithm |

${f}_{gp\_n}$ | − | n function used by the GP algorithm |

${T}_{gp}$ | − | Set of $NT$ terminals used by the GP algorithm |

${a}_{gp\_1}$ | − | n terminal used by the GP algorithm |

${f}_{FIT}$ | − | Fitness function used by the GP algorithm |

$g$ | − | Generation of GP |

$Pop\left(g\right)$ | − | Population at generation $g$ |

$\iota $ | − | Individual of the population |

${R}^{2}$ | − | Coefficient of determination |

${E}_{t}$ | Units of forecasted variable | Forecast or residual error of a variable at a period $t$ |

${Y}_{t}$ | Units of forecasted variable | Measured value of wind speed at a time period |

${F}_{t}$ | Units of forecasted variable | Forecasted value of wind speed at a time period |

$N$ | − | Sample number of measured and forecasted values of variable |

$\mathrm{MAE}$ | Units of forecasted variable | Mean Absolute Error |

$\mathrm{MAPE}$ | % | Mean Absolute Percentage Error |

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**Figure 2.**Weibull probability distribution function for different values of the shape $k$ and scale $\lambda $ parameters. The blue lines correspond to $k=2$ and $\lambda =3,5,7\mathrm{m}/\mathrm{s}$ for the dotted, solid and dash-dotted lines, respectively. The red and yellow solid lines correspond to $k=1,3$, respectively, and $\lambda =5\mathrm{m}/\mathrm{s}$.

**Figure 4.**Example of genotype and its encoding, prefix notation and mathematical model used in the GP.

**Figure 7.**Normalized histogram of wind speeds, in blue, and fitted Weibull distribution, in red, for: (

**a**) 2010; and (

**b**) 2011. The Weibull parameters are $k=2$ and $\lambda =5.99$ and $k=2$ and $\lambda =5.82$, for 2010 and 2011, respectively.

**Figure 8.**Six groups of samples of normalized histograms of wind speed, in blue, and fitted Weibull distribution, in orange, for group periods of 30 days each during WS season. Each graph corresponds to the following period (

**a**) 01/01/2010–01/30/2010; (

**b**) 01/31/2010–03/01/2010; (

**c**) 03/02/2010–03/31/2010; (

**d**) 04/01/2010–04/30/2010; (

**e**) 05/01/2010–05/30/2010; (

**f**) 12/21/2010–01/19/2011.

**Figure 9.**Six groups of samples of normalized histograms of wind speeds, in blue, and fitted Weibull distribution, in orange, for group periods of 25 days each during WS season. Each graph corresponds to the following period (

**a**) 01/01/2010–01/25/2010; (

**b**) 01/26/2010–02/19/2010; (

**c**) 02/20/2010–03/16/2010; (

**d**) 03/17/2010–04/10/2010; (

**e**) 04/11/2010–05/05/2010; (

**f**) 05/06/2010 – 05/30/2010.

**Figure 10.**Six groups of samples of normalized histograms of wind speeds, in blue, and fitted Weibull distribution, in orange, for group periods of 20 days each during WS season. Each graph corresponds to the following period (

**a**) 01/01/2010–01/20/2010; (

**b**) 01/21/2010–02/09/2010; (

**c**) 02/10/2010–03/01/2010; (

**d**) 03/02/2010–03/21/2010; (

**e**) 03/22/2010–04/10/2010; (

**f**) 04/11/2010–04/30/2010.

**Figure 11.**Six groups of samples of normalized histograms of wind speeds, in blue, and fitted Weibull distribution, in orange, for group periods of 15 days each during WS season. Each graph corresponds to the following period (

**a**) 01/01/2010–01/15/2010; (

**b**) 01/16/2010–01/30/2010; (

**c**) 01/31/2010–02/14/2010; (

**d**) 02/15/2010–03/01/2010; (

**e**) 03/02/2010–03/16/2010; (

**f**) 03/17/2010–03/31/2010.

**Figure 12.**Six groups of samples of normalized histograms of wind speeds, in blue, and fitted Weibull distribution, in orange, for group periods of 10 days each during WS season. Each graph corresponds to the following period (

**a**) 01/01/2010–01/10/2010; (

**b**) 01/11/2010–01/20/2010; (

**c**) 01/21/2010–01/30/2010; (

**d**) 01/31/2010–02/09/2010; (

**e**) 02/10/2010–02/19/2010; (

**f**) 02/20/2010–03/01/2010.

**Figure 13.**Predicted scale parameters at different horizons. The blue and the gray dashed lines correspond to the values of the measured and predicted values of the scale parameter in the WS season. The yellow and the gray dashed lines correspond to the values of the measured and predicted values of the scale parameter in the SF season. The horizons of the prediction are as follows: (

**a**) 30 days ahead; (

**b**) 25 days ahead; (

**c**) 20 days ahead; (

**d**) 15 days ahead; and (

**e**) 10 days ahead.

**Figure 14.**Estimated and measured wind power for horizons of: (

**a**) 30 days ahead; (

**b**) 25 days ahead; (

**c**) 20 days ahead; (

**d**) 15 days ahead; and (

**e**) 10 days ahead. The blue and yellow lines correspond to the measured wind power in the WS and SF seasons, respectively, and the grey dashed lines correspond to the estimated wind power.

**Figure 15.**Forecast error $E$ for the estimated wind power for horizons of: (

**a**) 30 days ahead; (

**b**) 25 days ahead; (

**c**) 20 days ahead; (

**d**) 15 days ahead; and (

**e**) 10 days ahead. The blue and yellow lines correspond to the WS and SF seasons, respectively.

GP ( ) |
---|

g ← 0 random initialization of each individual $\mathit{\iota}$∈ Pop(g) ∀$\mathit{\iota}$∈ Pop (g) evaluate f _{FIT}($\mathit{\iota}$)do while $\mathit{g}$ < stopping criterion g ← g−1 Pop(g) ← selection (Pop(g−1)) Pop(g) ← crossover (Pop(g)) Pop(g) ← mutation (Pop(g)) ∀ $\mathit{\iota}$ ∈ Pop(g) evaluate f _{FIT}($\mathit{\iota}$)end do |

**Table 2.**Characteristics of the wind resource per year: average power $P$, average wind speed from data ${v}_{ave}$ and mean wind speed from Weibull distribution $\overline{v}$.

Year | $\mathit{P}$ (W/m^{2})
| ${\mathit{v}}_{\mathit{a}\mathit{v}\mathit{e}}(\mathbf{m}/\mathbf{s})$ | $\overline{\mathit{v}}$ (m/s) |
---|---|---|---|

2010 | 175.56 | 5.15 | 5.314 |

2011 | 160.31 | 5.09 | 5.155 |

**Table 3.**Characteristics of the wind resource per season: scale parameter $\lambda $, average power $P$, average wind speed from data ${v}_{ave}$ and mean wind speed from Weibull distribution $\overline{v}$.

Year | Season | Dates | $\mathit{\lambda}\left(\mathbf{m}/\mathbf{s}\right)$ | $\mathit{P}$ (W/^{m})
| ${\mathit{v}}_{\mathit{a}\mathit{v}\mathit{e}}(\mathbf{m}/\mathbf{s})$ | $\overline{\mathit{v}}$ (m/s) |
---|---|---|---|---|---|---|

2010 | Winter | 01/01–03/20 | 6.97 | 276.03 | 6.11 | 6.18 |

2010 | Spring | 03/21–06/20 | 6.28 | 202.03 | 5.58 | 5.56 |

2010 | Summer | 06/21–09/20 | 4.75 | 87.77 | 4.22 | 4.21 |

2010 | Fall | 09/21–12/20 | 5.23 | 116.71 | 4.60 | 4.63 |

2010 | Winter | 12/21–12/31 | 8.35 | 471.43 | 7.09 | 7.40 |

2011 | Winter | 01/01–03/20 | 6.70 | 244.45 | 5.76 | 5.93 |

2011 | Spring | 03/21–06/20 | 6.04 | 179.68 | 5.61 | 5.35 |

2011 | Summer | 06/21–09/20 | 5.04 | 104.10 | 4.53 | 4.46 |

2011 | Fall | 09/21–12/20 | 5.25 | 117.72 | 4.49 | 4.65 |

2011 | Winter | 12/21–12/31 | 6.34 | 207.33 | 5.40 | 5.61 |

10-day Group | λ (m/s) | k (-) | Average Solar Radiation (W/m ^{2}) | Average Ambient Temerature (C) | Average Relative Humidity (%) | Average Atm Pressure (mbar) | Average Wind Power (m/s) | Average Wind Speed (m/s) | $\overline{\mathit{u}}$ (m/s) |
---|---|---|---|---|---|---|---|---|---|

1 | 4.08 | 2.00 | 180.44 | 8.06 | 59.28 | 774.86 | 55.14 | 3.66 | 3.61 |

2 | 6.27 | 2.00 | 200.99 | 7.02 | 68.77 | 772.57 | 200.70 | 5.35 | 5.56 |

3 | 8.04 | 2.00 | 262.32 | 11.02 | 39.22 | 770.77 | 413.51 | 7.33 | 7.13 |

4 | 6.65 | 2.00 | 184.73 | 7.80 | 66.92 | 769.76 | 239.22 | 6.17 | 5.90 |

5 | 7.00 | 2.00 | 195.05 | 9.51 | 64.07 | 770.81 | 278.25 | 5.71 | 6.20 |

6 | 8.15 | 2.00 | 304.64 | 9.70 | 39.85 | 770.98 | 429.45 | 7.76 | 7.23 |

7 | 7.68 | 2.00 | 246.50 | 12.57 | 38.99 | 771.59 | 363.64 | 6.81 | 6.81 |

8 | 6.55 | 2.00 | 324.38 | 11.24 | 26.15 | 772.40 | 228.36 | 6.10 | 5.80 |

9 | 6.43 | 2.00 | 347.04 | 13.88 | 26.10 | 772.63 | 216.41 | 5.89 | 5.70 |

10 | 6.71 | 2.00 | 304.84 | 16.77 | 30.41 | 772.94 | 245.53 | 6.34 | 5.95 |

**Table 5.**Specifications of the GP process to forecast the scale parameter of wind speed Weibull distributions.

Parameter | Description |
---|---|

Objective: | To find a mathematical function that accurately forecasts the scale parameter of wind speed Weibull distributions. |

Population size: | 100 individuals (mathematical functions). |

Maximum number of tree nodes: | 50 |

Forecast cases (Prediction horizons): | 10, 15, 20, 25 and 30 days ahead groups. |

Fitness function: | MAPE function of the scale parameter (Equation (10)). |

GP evolutionary operations: | Crossover and Mutation. |

Reproduction probability by mutation: | 10% |

Reproduction probability by crossover: | 90% |

Stopping criterion: | 2000 generations |

Prediction Period | Winter-Spring MAE MAPE (m/s) (%) | Summer-Fall MAE MAPE (m/s) (%) | ||
---|---|---|---|---|

30 days ahead | 0.31 | 17.62 | 0.29 | 24.14 |

25 days ahead | 0.46 | 18.92 | 0.28 | 23.47 |

20 days ahead | 0.55 | 20.97 | 0.32 | 25.05 |

15 days ahead | 0.55 | 21.40 | 0.33 | 29.94 |

10 days ahead | 0.52 | 23.74 | 0.33 | 26.93 |

© 2020 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**

Borunda, M.; Rodríguez-Vázquez, K.; Garduno-Ramirez, R.; de la Cruz-Soto, J.; Antunez-Estrada, J.; Jaramillo, O.A. Long-Term Estimation of Wind Power by Probabilistic Forecast Using Genetic Programming. *Energies* **2020**, *13*, 1885.
https://doi.org/10.3390/en13081885

**AMA Style**

Borunda M, Rodríguez-Vázquez K, Garduno-Ramirez R, de la Cruz-Soto J, Antunez-Estrada J, Jaramillo OA. Long-Term Estimation of Wind Power by Probabilistic Forecast Using Genetic Programming. *Energies*. 2020; 13(8):1885.
https://doi.org/10.3390/en13081885

**Chicago/Turabian Style**

Borunda, Mónica, Katya Rodríguez-Vázquez, Raul Garduno-Ramirez, Javier de la Cruz-Soto, Javier Antunez-Estrada, and Oscar A. Jaramillo. 2020. "Long-Term Estimation of Wind Power by Probabilistic Forecast Using Genetic Programming" *Energies* 13, no. 8: 1885.
https://doi.org/10.3390/en13081885