# Ensemble Learning Approach for Probabilistic Forecasting of Solar Power Generation

^{*}

## Abstract

**:**

## 1. Introduction

#### Our Contributions

## 2. Related Work

#### 2.1. Point Forecasting

#### 2.1.1. Statistical Methods

#### 2.1.2. Machine Learning Methods

#### 2.2. Probabilistic Forecasting

## 3. Dataset and Problem Formulation

- Zone 1: altitude = 595 m; panel type = Solarfun SF160-24-1M195; No. of panels = 8; nominal power = 1560 W; panel orientation = 38°clockwise from north; panel tilt = 36°.
- Zone 2: altitude = 602 m; panel type = Suntech STP190S-24/Ad+; No. of panels = 26; nominal power = 4940 W; panel orientation = 327°clockwise from north; panel tilt = 35°.
- Zone 3: altitude = 951 m; panel type = Suntech STP200-18/ud; No. of panels = 20; nominal power = 4000 W; panel orientation = 31°clockwise from north; panel tilt = 21°.

- tclw: Total column liquid water, vertical integral of cloud liquid water content. Unit of measurement: kg/m
^{2}. - tciw: Total column ice water, vertical integral of cloud ice water content. Unit: kg/m
^{2}. - SP: Surface pressure. Unit: Pa.
- r: Relative humidity at 1000 mbar, defined with respect to saturation over ice below −23 °C and over water above 0 °C. For the period in between, a quadratic interpolation is applied. Unit: %.
- TCC: Total cloud cover. Unit: zero to one.
- 10u: 10-meter Uwind component. Unit: m/s.
- 10v: 10-meter Vwind component. Unit: m/s.
- 2T: two-meter temperature. Unit: K.
- SSRD: Surface solar radiation down. Unit: J/m
^{2}. - STRD: Surface thermal radiation down. Unit: J/m
^{2}. - TSR: Top net solar radiation, net solar radiation at the top of the atmosphere. Unit: J/m
^{2}. - TP: Sum of convective precipitation and stratiform precipitation. Unit: m.

## 4. Proposed Method

#### 4.1. Grouping of Data

#### 4.2. Generating Point Forecasts

- Decision tree regressor: A model is fitted using each of the input variables. For each of the individual variables, the mean squared error is used to determine the best split. The maximum number of features to be considered at each split is set to the total number of features [28].
- Gradient boosting: An ensemble model that uses decision trees as weak learners and builds the model in a stage-wise manner by optimizing the loss function [29].
- KNN regressor (uniform): The output is predicted using the values from the k-nearest neighbors (KNNs) [30]. In the uniform model, all of the neighbors are given an equal weight. Five nearest neighbors are used in this model, i.e., $k=5$. The “Minkowski” distance metric is used in finding the neighbors.
- KNN regressor (distance): In this variant of KNN, the neighbors closer to the target are given higher weights. The choice of k and the distance metric are the same as above.
- Lasso regression: A variation of linear regression that uses the shrinkage and selection method. The sum of squares error is minimized, but with a constraint on the absolute value of the coefficients [31].
- Random forest regressor: An ensemble approach that works on the principle that a group of weak learners when combined would give a strong learner. The weak learners used in random forest are decision trees. Breiman’s bagger, in which at each split all of the variables are taken into consideration, is used [32].
- Ridge regression: It penalizes the use of a large number of dimensions in the dataset using linear least squares to minimize the error [33].

#### 4.3. Generating Probabilistic Forecasts

#### 4.3.1. Method I: Linear Method

#### 4.3.2. Method II: Normal Distribution Method

#### 4.3.3. Method III: Normal Distribution Method with Additional Features

## 5. Experimental Setup

#### 5.1. Training and Testing Datasets

#### 5.2. Evaluation Metrics

#### 5.3. Benchmark Models

- ARIMA: The autoregressive integrated moving average (ARIMA) model is one of the most widely-used techniques in time series forecasting. The function
`auto.arima()`from the forecast package [37] in R is used. It automatically detects the best parameters to fit the data. - Naive: In this method, all of the forecasts are set to the last observed value. Surprisingly enough, this model works well for many economic and financial time series problems [39].
- Seasonal naive: This method is similar to the naive method, but the forecasts are set to the last observed value from the same season [39].

## 6. Experimental Results

#### 6.1. Benchmark Models

#### 6.2. Individual Machine Learning Models

#### 6.3. Ensemble Models

#### 6.3.1. Ensemble Method III

## 7. Conclusions

- Does combining the results from different models improve the performance?
- Does grouping the data from each hour and running separate models on them give a better performance?

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Average RMSE values of benchmark models’ point forecasts before and after grouping of data by hours of the day.

**Figure 3.**Average MAE values of benchmark models’ point forecasts before and after grouping of data by hours of the day.

**Figure 4.**Average pinball loss scores of benchmark models’ point forecasts before and after grouping of data by hours of the day.

**Figure 5.**Average RMSE values of seven individual machine learning models’ point forecasts before and after grouping of data by hours of the day. Results for: (

**a**) decision tree; (

**b**) gradient boosting; (

**c**) KNN (distance); (

**d**) KNN (uniform); (

**e**) lasso regression; (

**f**) random forests; and (

**g**) ridge regression.

**Figure 6.**Average MAE values of seven individual machine learning models’ point forecasts before and after grouping of data by hours of the day. Results for: (

**a**) decision tree; (

**b**) gradient boosting; (

**c**) KNN (distance); (

**d**) KNN (uniform); (

**e**) lasso regression; (

**f**) random forests; and (

**g**) ridge regression.

**Figure 7.**Pinball loss scores of seven individual machine learning models’ point forecasts before and after grouping of data by hours of the day. Results for: (

**a**) decision tree; (

**b**) gradient boosting; (

**c**) KNN (distance); (

**d**) KNN (uniform); (

**e**) lasso regression; (

**f**) random forests; and (

**g**) ridge regression.

**Figure 8.**Pinball loss scores of three ensemble models’ probabilistic forecasts before and after grouping of data by hours of the day.

**Figure 9.**Average pinball loss scores for different hours. (Note: the hours shown on the X-axis are just nominal and not the real wall-clock hours. The offset between these two is not disclosed by the Global Energy Forecasting Competition (GEFCOM) 2014 organizers.)

**Figure 12.**Example of probabilistic forecasting by Method III. First, 50th and 99th percentile forecasted values are shown along with actual solar power generated for the 72-h period (25 May, 0 h to 27 May, 23 h in year 2013) in Zone 1.

Model | Before Grouping | After Grouping | ||||
---|---|---|---|---|---|---|

RMSE | MAE | Pinball Loss | RMSE | MAE | Pinball Loss | |

Benchmark | ||||||

ARIMA | 0.33363 | 0.26691 | 0.08418 | 0.13988 | 0.07454 | 0.02318 |

Naive | 0.40756 | 0.35748 | 0.08526 | 0.16410 | 0.08433 | 0.03518 |

Seasonal Naive | 0.36894 | 0.25997 | 0.08535 | 0.17405 | 0.08829 | 0.02873 |

Machine Learning | ||||||

Decision Tree | 0.12973 | 0.06211 | 0.03954 | 0.11190 | 0.04999 | 0.02483 |

Gradient Boosting | 0.10105 | 0.05719 | 0.07159 | 0.08284 | 0.03784 | 0.02164 |

KNN (Distance) | 0.14537 | 0.08109 | 0.04055 | 0.09790 | 0.04519 | 0.02259 |

KNN (Uniform) | 0.14406 | 0.08072 | 0.03633 | 0.09696 | 0.04501 | 0.01891 |

Lasso Regression | 0.17546 | 0.13690 | 0.07108 | 0.08826 | 0.04329 | 0.02028 |

Random Forest | 0.09801 | 0.04886 | 0.04036 | 0.08312 | 0.03798 | 0.02251 |

Ridge Regression | 0.17349 | 0.13471 | 0.03185 | 0.08320 | 0.04056 | 0.01936 |

Ensemble | ||||||

Method I | 0.02775 | 0.01544 | ||||

Method II | 0.02934 | 0.01503 | ||||

Method III | 0.03105 | 0.01457 |

**Table 2.**Performances of three individual sets and any combinations thereof of Method III (in terms of the average pinball loss score after grouping by hours of the day).

Contributor | Pinball Loss |
---|---|

original set only (i.e., Method II) | 0.01503 |

1st additional set only | 0.01516 |

2nd additional set only | 0.01794 |

original set + 1st additional set | 0.01510 |

original set + 2nd additional set | 0.01498 |

1st additional set + 2nd additional set | 0.01483 |

original set + 1st additional set + 2nd additional set (i.e., Method III) | 0.01457 |

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**MDPI and ACS Style**

Ahmed Mohammed, A.; Aung, Z. Ensemble Learning Approach for Probabilistic Forecasting of Solar Power Generation. *Energies* **2016**, *9*, 1017.
https://doi.org/10.3390/en9121017

**AMA Style**

Ahmed Mohammed A, Aung Z. Ensemble Learning Approach for Probabilistic Forecasting of Solar Power Generation. *Energies*. 2016; 9(12):1017.
https://doi.org/10.3390/en9121017

**Chicago/Turabian Style**

Ahmed Mohammed, Azhar, and Zeyar Aung. 2016. "Ensemble Learning Approach for Probabilistic Forecasting of Solar Power Generation" *Energies* 9, no. 12: 1017.
https://doi.org/10.3390/en9121017