# Quantile Regression Post-Processing of Weather Forecast for Short-Term Solar Power Probabilistic Forecasting

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## Abstract

**:**

## 1. Introduction

- The Quantile Regression (QR) uses a GBRT methodology and generates an ensemble of forecasts based on the IFS forecasts of the ECMWF model and the plant power output measured previously under similar conditions.
- The Ensemble Prediction System (EPS) uses the results of the EPS weather forecast ensemble of the ECMWF model, processed through the regression model used in deterministic prediction, and no calibration or ensemble post-processing technique.
- The Variance Deficit (VD) applies the variance deficit calibration technique to the results of the EPS model.
- The Quantile Ensemble (QE) still uses the EPS forecasting ensemble and the plant model, but applies a post-processing procedure based on quantile regression with the GBRT methodology to the predictions obtained.
- The Quantile Ensemble Plus (QE+) is similar to the previous one, but uses both the IFS and EPS forecasts of the ECMWF model as features in quantile regression, in order to make the most of the information that can be extracted from both weather models.

- the use of the power index to efficiently forecast solar power generation even with temporally-coarse weather forecasts;
- the effectiveness of the gradient-boosted quantile regression trees for the post-processing of the ensemble of predictions from the EPS;
- the relative importance of the EPS and IFS forecasts of the ECMWF depending on the forecast horizon considered;
- the combined use of the IFS and EPS forecast of ECMWF for an accurate probabilistic and deterministic PV power forecast.

## 2. Methods

#### 2.1. Gradient-Boosted Regression Trees

#### 2.2. Deterministic Forecasting

#### 2.3. Quantile Regression

#### 2.4. Ensemble Probabilistic Forecasting

#### 2.5. VD Calibration

#### 2.6. Quantile Ensemble

- The deterministic model is applied to the ECMWF EPS forecasts, generating an ensemble of 51 power forecasts for the plant
- For each time step, 9 quantiles [0.1, 0.2, …, 0.9] are used to describe the distribution of the ensemble and constitute a set of new features.
- Fifty one GBRT quantile regressors are trained, choosing the quantile of the target function, in the quantile range [1/52, 2/52, …, 51/52], using as input features the 9 quantiles with which the distribution of power forecasts for the training period is represented and, as the training variable, the power actually measured.

#### 2.7. Quantile Ensemble Plus

#### 2.8. Implementation

## 3. Forecast Verification

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PF | Probabilistic Forecasting procedures |

GBRT | Gradient-Boosted Regression Trees |

PV | Photo-Voltaic |

ECMWF | European Centre for Medium-Range Weather Forecasts |

IFS | Integrated Forecasting System |

EPS | Ensemble Prediction System |

QR | Quantile Regression |

VD | Variance Deficit |

NWP | Numerical Weather Predictions |

AnEn | Analog Ensemble |

PeEn | Persistence Ensemble |

WRF | Weather Research and Forecasting regional model |

EMOS | Ensemble Model Output Statistics |

QE | Quantile Ensemble |

QE+ | Quantile Ensemble Plus |

DNI | Direct Normal Irradiation |

DHI | Diffuse Horizontal Irradiation |

GHI | Global Horizontal Irradiation |

POA | Plane Of Array |

RTM | Radiative Transfer Models |

CF | Control Forecast |

ME | Mean Error |

MAE | Mean Absolute Error |

RMSE | Root Mean Square Error |

nME | normalized Mean Error |

nMAE | normalized Mean Absolute Error |

nRMSE | normalized Root Mean Square Error |

SS | Skill Score |

ROC | Receiver Operating Characteristic |

AUC | Area Under the ROC Curve |

BS | Brier Score |

UNC | Uncertainty |

REL | Reliability |

RES | Resolution |

CRPS | Continuous Ranked Probability Score |

CDF | Cumulative Distribution Function |

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**Figure 1.**Flowchart showing the steps required to realize the QE and QE+ forecast starting from the EPS and IFS power forecasts.

**Figure 2.**Rank histograms for forecast ensembles produced by the five methods examined for the 24–48-h forecast horizon: (

**a**) Quantile regression; (

**b**) Uncalibrated EPS (full scale); (

**c**) Uncalibrated EPS; (

**d**) EPS with variance deficit calibration; (

**e**) Quantile ensemble with EPS forecast; (

**f**) Quantile ensemble with IFS and EPS forecasts. The calibration of the EPS method is not satisfactory, and the VD method has a better performance. The QR and QE methods show a good calibration.

**Figure 3.**ROC curves for a threshold equal to half the clear sky power and for a forecast interval of 24–48 h. In the legend within the figure, the values of the area under the curve are also shown.

**Figure 4.**CRPS results as the lead time forecast varies for the probabilistic forecast methods examined: (

**a**) CRPS values for times of forecast in 0–24-h time range; (

**b**) same as before for 24–48-h; (

**c**) same for 48–72-h.

Jan–Feb | Mar–Apr | May–Jun | Jul–Aug | Sep–Oct | Nov–Dec | Total | |
---|---|---|---|---|---|---|---|

Samples (-) | 5664 | 5856 | 5856 | 5952 | 5856 | 5856 | 35,040 |

mean value (kW) | 68.4 | 204.2 | 261.6 | 276.5 | 150.3 | 49.1 | 169.2 |

std. deviation (kW) | 161.8 | 309.1 | 344.0 | 361.3 | 262.2 | 128.2 | 289.9 |

max value (kW) | 957.4 | 1187.9 | 1326.0 | 1276.2 | 1111.5 | 824.0 | 1326.0 |

Parameter | Search Values | Deterministic | QE | QE+ |
---|---|---|---|---|

learning rate | 0.05, 0.02, 0.01 | 0.05 | 0.05 | 0.01 |

maximum depth of trees | 3, 4, 5 | 3 | 3 | 3 |

minimum number of samples in leaf | 3, 5, 7 | 7 | 7 | 5 |

number of estimators | 100, 200, 500 | 100 | 200 | 500 |

**Table 3.**Feature importance as the relative occurrence of the feature in the GBRT regression trees for the prediction of ${k}_{pv}$.

Feature | Relative Importance |
---|---|

k | 56.9% |

Precipitation | 3.9% |

Temperature 2 m | 11.3% |

Zenith | 9.5 % |

Azimuth | 18.4 % |

Forecast (h) | R2 (-) | ME (kW) | MAE (kW) | RMSE (kW) | nMAE (%) | nRMSE (%) | SS(%) |
---|---|---|---|---|---|---|---|

0–24 | 0.785 | $-3.3$ | 93.3 | 146.4 | 7.18 | 11.26 | 43.61 |

24–48 | 0.687 | 6.9 | 114.3 | 176.5 | 8.80 | 13.57 | 35.62 |

48–72 | 0.636 | 7.5 | 124.8 | 190.4 | 9.60 | 14.65 | 27.78 |

**Table 5.**Probabilistic error measures with different forecast lead times. The Brier Score (BS), its components reliability (REL), Resolution (RES) and Uncertainty (UNC) and the Area Under the ROC Curve (AUC) are calculated for a threshold of $0.5\xb7{p}_{cs}$. The cumulative ranked probability score is the mean value for the test set in the daylight hours only.

Forecast (h) | Model | BS (-) | REL (-) | RES (-) | UNC (-) | AUC (-) | CRPS (kW) |
---|---|---|---|---|---|---|---|

0–24 | QR | 0.140 | 0.008 | 0.117 | 0.249 | 0.882 | 65.7 |

EPS | 0.204 | 0.043 | 0.089 | 0.249 | 0.814 | 87.5 | |

VD | 0.179 | 0.014 | 0.085 | 0.249 | 0.808 | 82.3 | |

QE | 0.162 | 0.006 | 0.094 | 0.249 | 0.845 | 74.9 | |

QE+ | 0.143 | 0.006 | 0.113 | 0.249 | 0.876 | 67.7 | |

24–48 | QR | 0.170 | 0.008 | 0.087 | 0.249 | 0.827 | 79.2 |

EPS | 0.199 | 0.027 | 0.077 | 0.249 | 0.803 | 86.3 | |

VD | 0.188 | 0.012 | 0.074 | 0.249 | 0.794 | 85.3 | |

QE | 0.173 | 0.007 | 0.083 | 0.249 | 0.821 | 78.4 | |

QE+ | 0.165 | 0.007 | 0.091 | 0.249 | 0.836 | 76.8 | |

48–72 | QR | 0.184 | 0.007 | 0.073 | 0.249 | 0.800 | 85.9 |

EPS | 0.199 | 0.026 | 0.077 | 0.249 | 0.796 | 86.2 | |

VD | 0.191 | 0.014 | 0.072 | 0.249 | 0.789 | 86.3 | |

QE | 0.176 | 0.006 | 0.079 | 0.249 | 0.815 | 81.2 | |

QE+ | 0.172 | 0.008 | 0.086 | 0.249 | 0.825 | 80.7 |

**Table 6.**Error measures for the deterministic forecast based on the median of the quantile ensemble probabilistic model with the IFS and EPS forecasts.

Forecast (h) | R2 (-) | ME (kW) | MAE (kW) | RMSE (kW) | nMAE (%) | nRMSE (%) | SS (%) |
---|---|---|---|---|---|---|---|

0–24 | 0.784 | −5.1 | 93.5 | 146.8 | 7.19 | 11.29 | 43.46 |

24–48 | 0.721 | 0.6 | 110.0 | 166.7 | 8.46 | 12.82 | 39.18 |

48–72 | 0.704 | 0.7 | 117.1 | 171.6 | 9.00 | 13.20 | 34.92 |

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## Share and Cite

**MDPI and ACS Style**

Massidda, L.; Marrocu, M. Quantile Regression Post-Processing of Weather Forecast for Short-Term Solar Power Probabilistic Forecasting. *Energies* **2018**, *11*, 1763.
https://doi.org/10.3390/en11071763

**AMA Style**

Massidda L, Marrocu M. Quantile Regression Post-Processing of Weather Forecast for Short-Term Solar Power Probabilistic Forecasting. *Energies*. 2018; 11(7):1763.
https://doi.org/10.3390/en11071763

**Chicago/Turabian Style**

Massidda, Luca, and Marino Marrocu. 2018. "Quantile Regression Post-Processing of Weather Forecast for Short-Term Solar Power Probabilistic Forecasting" *Energies* 11, no. 7: 1763.
https://doi.org/10.3390/en11071763