# The Impact of Imperfect Weather Forecasts on Wind Power Forecasting Performance: Evidence from Two Wind Farms in Greece

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## Abstract

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## 1. Introduction

- We measure the impact of imperfect weather forecasts on the wind power forecasting accuracy and the estimation of uncertainty through the evaluation of the point forecasts and the prediction intervals produced by various types of forecasting methods;
- We compare the suitability of statistical versus ML approaches in the task of estimating highly volatile variables when the accuracy of external information varies;
- Assuming varying levels of inaccuracy in the independent variable, we perform a sensitivity analysis with varying distributional assumptions to conclude on the robustness [24] of the forecasting approaches considered and make relevant recommendations.

## 2. Forecasting Methods Considered

#### 2.1. Training and Testing

#### 2.2. Scaling

#### 2.3. Statistical and Machine Learning Methods

**Linear Regression (LR)**: LR is a linear method for modeling the relationships between the target and the predictor variables. The parameters of the forecasting model are estimated directly from the data using closed forms. LR was the first type of regression method to be studied rigorously, and is therefore widely used because of its simplicity, low computational cost, and intuitiveness. Since in this study we consider N weather variables as predictors, LR is implemented in the form of multiple linear regression, as follows:$${\widehat{y}}_{t}=a+{b}_{1}\times {v}_{1}+{b}_{2}\times {v}_{2}...+{b}_{k}\times {v}_{k},$$**Multi-Layer Perceptron (MLP)**: A simple, single hidden layer NN of N input and $2N$ input nodes, constructed so that accurate, yet computationally affordable forecasts are provided [27]. The Scaled Conjugate Gradient method is used for estimating the weights [28] which are initialized randomly. The learning rate is selected between 0.1 and 1 using a tenfold cross-validation procedure (mean squared error minimization), while the maximum iterations are set to 500. The sigmoid activation function is used both for the hidden and the output layers given the lack of trend in the data. We trained 10 models and use the median of their forecasts to mitigate possible variations due to poor weight initializations [29]. The method was implemented using the RSNNS R statistical package [30];**Bayesian Neural Network (BNN)**: This method is similar to the MLP method but optimizes the weights according to the Bayesian concept, as suggested by [31,32]. The Nguyen and Widrow algorithm [33] is used to assign initial weights and the Gauss–Newton algorithm to perform the optimization. Once again, an ensemble of 10 models was considered with a maximum of 500 iterations each. The method was implemented using the brnn R statistical package [34];**Random Forest (RF)**: RTs can be used to perform a treelike recursive partitioning of the input space, thus dividing it into regions, called the terminal leaves [35]. Then, on the basis of the inputs provided, tests are applied to decision nodes in order to define which leave should be used for forecasting. The RF method expands this concept by combining the results of multiple RTs, each one depending on the values of a random vector sampled independently and with the same distribution [36]. In this regard, RF is more robust to outliers and overfitting, even for limited samples of data. In this study, we considered a total of 500 nonpruned trees and sampled the data with replacement. The method was implemented using the randomForest R statistical package [37];**Gradient Boosting Trees (GBT)**: This method is similar to RF, but instead of generating multiple independent trees, it builds one tree at a time, each new tree correcting the errors made by the previously trained one [38]. Thus, although GBT is more specialized than RF in forecasting the target variable, is more sensitive to overfitting [39]. In this study, we constructed a slow learning model with a learning rate of 0.01 and a maximum tree depth of 5. We considered 1000 trees but pruned the constructed model by employing a tenfold cross-validation procedure to mitigate overfitting. The method was implemented using the gbm R statistical package [40];**K-Nearest Neighbor Regression (KNNR)**: KNNR is a similarity-based method, generating forecasts according to the Euclidean distance computed between the points used for training and testing. Given a test sample of N predictor variables, the method picks the closest K observations of the training sample to them and then sets the prediction equal to the average of their corresponding target values. K was selected between 3 and 300 with a step of 3 using a tenfold cross-validation procedure. The method was implemented using the caret R statistical package [41];**Support Vector Regression (SVR)**: SVR generates forecasts by identifying the hyperplane that maximizes the margin between two classes and minimizes the total error under tolerance [42]. We considered $\u03f5$-regression, with $\u03f5$ being equal to 0.01 and a radial basis kernel. The method was implemented using the e1071 R statistical package [43].

## 3. Empirical Evaluation

#### 3.1. Dataset

#### 3.2. Experimental Setup

**Wind speed**. Given that wind speed must be positive, the adjustment was performed by (i) multiplying the original values of the variable with the computed factors and (ii) setting all nonzero forecasts (if any) to zero;**Wind direction**. Given that wind direction must range between 0 and 360 degrees, the adjustment was performed by (i) adding the computed factors, multiplied by 360, to the original values of the variable, and (ii) adding or subtracting 360 to all forecasts that were lower than zero or higher than 360, respectively.

- Given the sample used for training the forecasting methods when producing point forecasts, the random samples are created without replacement (observations used for validation purposes remain unobserved while training);
- For each of the ten random samples, 90% of the observations are used for training the forecasting methods and 10% for estimating the corresponding errors of the point forecasts;
- The empirical distribution of the errors (actual-forecast) is computed using a Kernel density estimator, and the 0.025 and 0.975 quantities of the distribution are determined;
- The forecasting methods are retrained using the complete training sample so that point forecasts are produced for the test sample of interest;
- The 95% prediction intervals are computed by adding the 0.025 and 0.975 quantities to the point forecasts produced in the previous step.

- We randomly split the original dataset of each wind farm into five samples of equal sizes;
- Four out of the five samples are used for constructing a training dataset, which is then scaled;
- The training dataset is randomly split into ten subsamples;
- Nine out of the ten subsamples are used for training the seven forecasting methods considered in this study, with the last one used for producing forecasts and computing the corresponding forecast errors of each method;
- Step 4 is repeated for all the ten possible combinations of subsamples;
- A Kernel density estimator is used to approximate the 0.025 and 0.975 quantities of the error distribution of the methods, as specified through Steps 3, 4, and 5;
- The forecasting models are retrained using the complete training dataset, as specified in Step 2, so that point forecasts are produced for the respective test sample;
- 95% prediction intervals are computed by adding the 0.025 and 0.975 quantities to the point forecasts produced in Step 7;
- Point forecasts and prediction intervals are evaluated using the MAE and MIS measures, respectively;
- Steps 2 to 9 are repeated for all the five possible combinations of samples;
- The results are summarized by averaging the forecasting performance of the forecasting methods for all five samples considered.

#### 3.3. Results

## 4. Discussion

## 5. Conclusions

- The “forecasting horserace” should be expanded to include more methods and accuracy measures;
- Given that the examined dataset includes two predictor variables, i.e., wind speed and direction, bivariate models, inspired by econometric approaches, like VAR models, should be explored;
- “When in doubt combine”: combinations of all (or the top-three methods) for each forecast case should be tested (for both point forecasts and prediction intervals). Alternatively, one could try to go for a clever selection algorithm in between those ML and statistical methods (see for example [59]), or even hybrid approaches [60];
- Temporal aggregation is always a way to self-improve any forecasting method (times series or cross-sectional) and this alternative should be employed in any context, especially when it involves a lot of uncertainty or complexity [61]. This could be achieved either via selecting a single aggregation level [62] or via combining the forecasts produced for multiple aggregation levels [63];
- The provided forecasts should be evaluated “on the money” with real-life (and asymmetric) utility functions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BNN | Bayesian Neural Network |

GBT | Gradient Boosting Trees |

KNNR | K-Nearest Neighbor Regression |

LR | Linear Regression |

ML | Machine Learning |

NN | Neural Network |

MLP | Multi-Layer Perceptron |

RF | Random Forest |

RT | Regression Tree |

SVM | Support Vector Machine |

SVR | Support Vector Regression |

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**Figure 1.**Wind power (measured in MV) compared with wind direction (degrees) and speed (m/s) for the

**Aeolos**farm. The panels represent the empirical power curves of the farm, being subject to wind direction (

**left**) and speed (

**right**), respectively.

**Figure 2.**Wind power (MV) compared with wind direction (degrees) and speed (m/s) for the

**Rokas**farm. The panels represent the empirical power curves of the farm, being subject to wind direction (

**left**) and speed (

**right**), respectively.

**Figure 3.**Wind roses providing a view of how wind speed and direction are distributed at the Aeolos (

**left**) and Rokas (

**right**) wind farms.

**Figure 4.**Weibull distributions of the wind speed at the Aeolos (

**left**) and Rokas (

**right**) wind farms.

**Figure 5.**Forecasting performance in terms of wind power

**point forecasts**(mean absolute error (MAE), measured in MW) of the statistical and machine learning (ML) methods considered in this study for the

**Aeolos**wind farm when different types of noise and noise intensities are considered. The three panels on the top display the accuracy of the methods for the case of the Gaussian noise of various intensities, $\sigma $, applied to wind speed forecasts, wind direction forecast, and both, respectively. The three panels at the bottom display the same results for the case of the uniform noise of various intensities s.

**Figure 6.**Forecasting performance in terms of wind power

**point forecasts**(MAE, measured in MW) of the statistical and ML methods considered in this study for the

**Rokas**wind farm when different types of noise and noise intensities are considered. The three panels on the top display the accuracy of the methods for the case of the Gaussian noise of various intensities, $\sigma $, applied to wind speed forecasts, wind direction forecast, and both, respectively. The three panels at the bottom display the same results for the case of the uniform noise of various intensities s.

**Figure 7.**Forecasting performance in terms of wind power

**prediction intervals**(Mean Interval Score (MIS), measured in MW) of the statistical and ML methods considered in this study for the

**Aeolos**wind farm when different types of noise and noise intensities are considered. The three panels on the top display the accuracy of the methods for the case of the Gaussian noise of various intensities, $\sigma $, applied to wind speed forecasts, wind direction forecast, and both, respectively. The three panels at the bottom display the same results for the case of the uniform noise of various intensities s.

**Figure 8.**Forecasting performance in terms of wind power

**prediction intervals**(MIS, measured in MW) of the statistical and ML methods considered in this study for the

**Rokas**wind farm when different types of noise and noise intensities are considered. The three panels on the top display the accuracy of the methods for the case of the Gaussian noise of various intensities, $\sigma $, applied to wind speed forecasts, wind direction forecast, and both, respectively. The three panels at the bottom display the same results for the case of the uniform noise of various intensities s.

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**MDPI and ACS Style**

Spiliotis, E.; Petropoulos, F.; Nikolopoulos, K. The Impact of Imperfect Weather Forecasts on Wind Power Forecasting Performance: Evidence from Two Wind Farms in Greece. *Energies* **2020**, *13*, 1880.
https://doi.org/10.3390/en13081880

**AMA Style**

Spiliotis E, Petropoulos F, Nikolopoulos K. The Impact of Imperfect Weather Forecasts on Wind Power Forecasting Performance: Evidence from Two Wind Farms in Greece. *Energies*. 2020; 13(8):1880.
https://doi.org/10.3390/en13081880

**Chicago/Turabian Style**

Spiliotis, Evangelos, Fotios Petropoulos, and Konstantinos Nikolopoulos. 2020. "The Impact of Imperfect Weather Forecasts on Wind Power Forecasting Performance: Evidence from Two Wind Farms in Greece" *Energies* 13, no. 8: 1880.
https://doi.org/10.3390/en13081880