# A New Approach for Satellite-Based Probabilistic Solar Forecasting with Cloud Motion Vectors

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## Abstract

**:**

## 1. Introduction

## 2. Proposed CMV-Based Probabilistic Approach

- calculation of the clear-sky index ${k}_{c}$ for each pixel of a given satellite sub-image, centered at the location of interest with a radius of typically 50 km to 100 km; n this paper, a radius of roughly 50 km is used;
- estimation of the Cloud Motion Vectors (CMV) using an approach of Optical Flow (OF);
- identification of the pixels that are likely to advect to the location of interest for different time horizons (named hereinafter upcoming or converging ${k}_{c}$ pixels or values); and
- building of the Probability Distribution Function (PDF) of the upcoming ${k}_{c}$ based on the candidate pixels.

#### 2.1. Calculation of the Clear-Sky Index ${k}_{c}$

#### 2.2. Identification of the Cloud Motion Vectors

#### 2.3. Identification of the Candidate Pixels

- a standard deviation of 2 km/h for the Gaussian noise added to the cloud speed (${\sigma}_{r}$), which is roughly one order of magnitude lower than the typical wind speed in the lower atmosphere (a few dozen km/h);
- a standard deviation of $\pi /12$ radians for the Gaussian noise added to the cloud direction (${\sigma}_{theta}$); significantly higher values were considered, as to represent the difficulty in accurately estimating a CMV and the fact that it varies over time (which the Eulerian approach here used disregards), but larger values resulted in the loss of all information regarding trajectories (i.e., considerably worse sharpness values); and
- a monitoring radius of 1 km, intentionally lower than the 3 km satellite resolution; larger values increase the number of selected candidates (with the new ones being farther from the sensor location), which would then describe a variability that is less similar to that of a point sensor.

#### 2.4. Building of the Empirical Distribution of the Clear-Sky Index ${k}_{c}$

## 3. Performance Assessment

- The proposed probabilistic CMV approach is noted Pr-CMV.
- We can also provide deterministic forecasts using the Pr-CMV, by taking the median of the forecast distribution as the deterministic forecast. This method of obtaining deterministic forecasts is noted Pr-CMV-Det.

- A standard CMV approach with no Monte-Carlo procedure noted St-CMV. With this method, the motion vectors are propagated assuming an Eulerian approach to estimate the upcoming GHI map for the next time step.
- Two baseline models that make no use of satellite information: the Smart Persistence model noted persistence, for deterministic evaluation, and the Complete-History Persistence Ensemble noted CH-PeEn, for probabilistic evaluation. These models are defined in Section 3.2.

#### 3.1. Metrics

#### 3.2. Baseline Models

#### 3.3. Data

#### 3.4. Deterministic Performance

#### 3.5. Probabilistic Performance

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDF | Cumulative Distribution Function |

CMV | Cloud Motion Vectors |

CRPS | Continuous Ranked Probability Score |

GHI | Global Horizontal Irradiance |

MPINAW | Mean Prediction Interval Normalized Average Width |

MRD | Mean Reliability Deviation |

nBIAS | Normalized bias |

nMAE | Normalized Mean Absolute Error |

nRMSE | Normalized Root Mean Square Error |

OF | Optical Flow |

PINAW | Prediction Interval Normalized Average Width |

Pr-CMV | The proposed probabilistic CMV-based solar forecasting |

Det-Pr-CMV | The deterministic mode of the Pr-CMV |

PV | Photovoltaic |

SSI | Surface Solar Irradiance |

St-CMV | Standard CMV-based solar forecasting |

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**Figure 1.**Example of the CMV-based probabilistic forecast on Carpentras, 11 July 2016 at 12;00 UT, for 1-h ahead every 5 min. (

**a**) Clear-sky index map at 12:00 UT, along with CMV (small red arrows), with a radius of 50 km around the location of interest. (

**b**) Contour map of the clear-sky index map with 0.5 as a threshold, along with dots identifying pixels converging to the monitored perimeter of interest. The color of the dots represents the estimated time of intersection. (

**c**) The gray area represents the reference measured GHI at 15-min time step. The blue line represents the corresponding estimation from satellite. The black-dash line represents the corresponding clear-sky GHI. The probabilistic forecasting issued at 12:00 UT (vertical red dash line) up to 2-h ahead (vertical black dash line) is represented by the ensemble of potential converging clear-sky index ${k}_{c}$ in black dots. For illustration purposes, the 10th, 50th, and 90th percentiles of this ensemble are represented, respectively, by the red, green, and yellow lines.

**Figure 3.**Reliability of the probabilistic forecast models CH-PeEn and Pr-CMV for the two sites, Carpentras (

**a**) and Signes (

**b**).

**Figure 4.**Sharpness of the probabilistic forecast models CH-PeEn and Pr-CMV for the two sites, Carpentras (

**a**) and Signes (

**b**).

**Figure 5.**CRPS of the probabilistic forecast models CH-PeEn and Pr-CMV for the two sites, Carpentras (

**a**) and Signes (

**b**).

**Table 1.**Summary of the sites location and description of the yearly dataset of measured 15-min GHI used for training and for evaluation.

Location | Latitude (°) | Longitude (°) | Elevation (m) | Training Period | Evaluation Period |
---|---|---|---|---|---|

Carpentras | 44.0830 | 5.0590 | 100 | 2015: 14,573 data | 2016: 13,995 data |

Signes | 43.25551 | 5.8 | 441 | 2015: 14,686 data | 2016: 13,693 data |

${\mathit{\sigma}}_{\mathit{r}}$ (km/h) | ${\mathit{\sigma}}_{\mathit{\theta}}$ (radians) | ${\mathit{R}}_{\mathit{m}}$ (km) | ${\mathit{N}}_{\mathit{mc}}$ |
---|---|---|---|

2 | $\pi $/12 | 1 | 5000 |

Deterministic Evaluation (Carpentras) | |||||||||||||||

Category (reference value) | Low variability days 217 days (540.3 W/m${}^{2}$) | High variability days 148 days (441.3 W/m${}^{2}$) | All days 365 days (500.2 W/m${}^{2}$) | ||||||||||||

Horizon (minutes) | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All |

nBIAS - Persistence (%) | 0 | −0.1 | −0.2 | −0.4 | −0.2 | 0.1 | 0.1 | −0.1 | −0.3 | 0 | 0 | 0 | −0.2 | −0.4 | −0.1 |

nBIAS - St-CMV (%) | 0.2 | 0 | −0.3 | −0.8 | −0.2 | 0 | −0.2 | −0.4 | -1 | −0.4 | 0.1 | −0.1 | −0.4 | −0.9 | −0.3 |

nBIAS - Det-Pr-CMV (%) | 0.3 | 0 | −0.4 | −0.9 | −0.2 | 0.9 | 0.2 | −0.6 | −1.7 | −0.3 | 0.5 | 0.1 | −0.5 | −1.2 | −0.3 |

nMAE - Persistence (%) | 2.6 | 3.5 | 4.2 | 4.9 | 3.8 | 16.1 | 21.7 | 25 | 27.2 | 22.6 | 7.4 | 10.1 | 11.7 | 12.9 | 10.5 |

nMAE - St-CMV (%) | 4 | 4.4 | 4.9 | 5.6 | 4.7 | 17.2 | 19.5 | 22.2 | 24.7 | 20.9 | 8.7 | 9.8 | 11.1 | 12.5 | 10.5 |

nMAE - Det-Pr-CMV (%) | 4 | 4.3 | 4.9 | 5.6 | 4.7 | 17 | 18.3 | 19.9 | 21.9 | 19.3 | 8.6 | 9.3 | 10.3 | 11.4 | 9.9 |

nRMSE - Persistence (%) | 6.2 | 8 | 9.3 | 10.5 | 8.6 | 24.8 | 32.1 | 36.2 | 38.9 | 33.5 | 14.9 | 19.2 | 21.8 | 23.6 | 20.1 |

nRMSE - St-CMV (%) | 7.2 | 8.2 | 9.2 | 10.4 | 8.8 | 24.5 | 27.6 | 31.2 | 34.3 | 29.7 | 15 | 16.9 | 19.1 | 21.2 | 18.2 |

nRMSE - Det-Pr-CMV (%) | 7.1 | 7.7 | 8.8 | 9.9 | 8.4 | 24.3 | 25.7 | 27.7 | 30.2 | 27.1 | 14.9 | 15.8 | 17.2 | 18.8 | 16.7 |

Probabilistic Evaluation (Carpentras) | |||||||||||||||

Category (reference value) | Low variability days 217 days (540.3 W/m${}^{2}$) | High variability days 148 days (441.3 W/m${}^{2}$) | All days 365 days (500.2 W/m${}^{2}$) | ||||||||||||

Horizon (minutes) | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All |

MRD - CH-PeEn (%) | 8.4 | 8.4 | 8.4 | 8.4 | 8.4 | 12.2 | 12.2 | 12.2 | 12.2 | 12.2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |

MRD - Pr-CMV (%) | 5.2 | 4.8 | 5.2 | 6.5 | 4.7 | 5.3 | 5.9 | 6.5 | 7.3 | 6.3 | 2.2 | 3.5 | 4.8 | 6 | 4.1 |

MPINAW - CH-PeEn (%) | 46.1 | 46.1 | 46.1 | 46.1 | 46.1 | 57.1 | 57.1 | 57.1 | 57.1 | 57.1 | 50.2 | 50.2 | 50.2 | 50.2 | 50.2 |

MPINAW - Pr-CMV (%) | 11.3 | 11.7 | 12.0 | 12.6 | 11.8 | 22.9 | 25.0 | 27.4 | 29.1 | 26.1 | 15.4 | 16.4 | 17.6 | 18.6 | 17.0 |

CRPS - CH-PeEn (%) | 13.8 | 13.8 | 13.8 | 13.8 | 13.8 | 25.2 | 25.2 | 25.2 | 25.2 | 25.2 | 17.8 | 17.8 | 17.8 | 17.8 | 17.8 |

CRPS - Pr-CMV (%) | 3.2 | 3.5 | 3.9 | 4.4 | 3.7 | 12.8 | 13.6 | 14.8 | 16.2 | 14.3 | 6.6 | 7.1 | 7.8 | 8.6 | 7.5 |

Deterministic Evaluation (Signes) | |||||||||||||||

Category (reference value) | Low variability days 178 days (599.2 W/m${}^{2}$) | High variability days 187 days (452.1 W/m${}^{2}$) | All days (530.7 W/m${}^{2}$) | ||||||||||||

Horizon (minutes) | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All |

nBIAS - Persistence (%) | 0.5 | 0.8 | 1 | 0.9 | 0.8 | 0.4 | 0.7 | 0.9 | 0.8 | 0.7 | 0.5 | 0.8 | 0.9 | 0.8 | 0.7 |

nBIAS - St-CMV (%) | 0.7 | 0.5 | 0.2 | −0.4 | 0.3 | 1.7 | 1.6 | 1.3 | 0.6 | 1.3 | 1.1 | 0.9 | 0.6 | 0 | 0.7 |

nBIAS - Det-Pr-CMV (%) | 0.7 | 0.5 | 0 | −0.8 | 0.1 | 1.4 | 1.4 | 0.7 | −0.1 | 0.8 | 0.9 | 0.9 | 0.3 | −0.5 | 0.4 |

nMAE - Persistence (%) | 2.6 | 3.8 | 4.8 | 5.8 | 4.2 | 17.8 | 23.5 | 26.3 | 28.6 | 24.1 | 8.6 | 11.6 | 13.3 | 14.8 | 12.1 |

nMAE - St-CMV (%) | 6.5 | 6.8 | 7.3 | 8.3 | 7.2 | 19.9 | 22.6 | 24.9 | 27 | 23.6 | 11.8 | 13.1 | 14.3 | 15.5 | 13.7 |

nMAE - Det-Pr-CMV (%) | 6.5 | 6.9 | 7.5 | 8.2 | 7.3 | 19.6 | 21.7 | 23.2 | 25.1 | 22.4 | 11.7 | 12.8 | 13.7 | 14.9 | 13.3 |

nRMSE - Persistence (%) | 5.7 | 7.6 | 8.9 | 10 | 8.2 | 27.2 | 34.7 | 38 | 40.5 | 35.4 | 16.5 | 21.1 | 23.2 | 24.9 | 21.7 |

nRMSE - St-CMV (%) | 8.9 | 9.5 | 10.2 | 11.3 | 10 | 28 | 31.6 | 34.8 | 37.4 | 33.1 | 17.8 | 20 | 21.9 | 23.6 | 21 |

nRMSE - Det-Pr-CMV (%) | 8.8 | 9.5 | 10.3 | 11.6 | 10.1 | 27.4 | 30.2 | 32.4 | 34.7 | 31.3 | 17.5 | 19.2 | 20.6 | 22.3 | 20 |

Probabilistic Evaluation (Signes) | |||||||||||||||

Category (reference value) | Low variability days 178 days (599.2 W/m${}^{2}$) | High variability days 187 days (452.1 W/m${}^{2}$) | All days (530.7 W/m${}^{2}$) | ||||||||||||

Horizon (minutes) | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All | 15 | 30 | 45 | 60 | All |

MRD - CH-PeEn (%) | 10 | 10 | 10 | 10 | 10 | 11 | 11 | 11 | 11 | 11 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |

MRD - Pr-CMV (%) | 2 | 1.5 | 2.1 | 3 | 2.1 | 432 | 4.7 | 4.7 | 5.1 | 4.6 | 1.3 | 1.7 | 2 | 2.8 | 1.8 |

MPINAW - CH-PeEn (%) | 45.6 | 45.6 | 45.6 | 45.6 | 45.6 | 58.0 | 58.0 | 58.0 | 58.0 | 58.0 | 50.5 | 50.5 | 50.5 | 50.5 | 50.5 |

MPINAW - Pr-CMV (%) | 16.3 | 16.7 | 17.0 | 17.5 | 16.9 | 28.0 | 29.9 | 32.2 | 33.9 | 31.0 | 20.9 | 21.9 | 23.1 | 24.0 | 22.5 |

CRPS - CH-PeEn (%) | 12.8 | 12.8 | 12.8 | 12.8 | 12.8 | 25.0 | 25.0 | 25.0 | 25.0 | 25.0 | 17.6 | 17.6 | 17.6 | 17.6 | 17.6 |

CRPS - Pr-CMV (%) | 4.9 | 5.2 | 5.6 | 6.2 | 5.5 | 14.4 | 15.7 | 17.0 | 18.4 | 16.4 | 8.6 | 9.4 | 10.1 | 11.0 | 9.8 |

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Carrière, T.; Amaro e Silva, R.; Zhuang, F.; Saint-Drenan, Y.-M.; Blanc, P.
A New Approach for Satellite-Based Probabilistic Solar Forecasting with Cloud Motion Vectors. *Energies* **2021**, *14*, 4951.
https://doi.org/10.3390/en14164951

**AMA Style**

Carrière T, Amaro e Silva R, Zhuang F, Saint-Drenan Y-M, Blanc P.
A New Approach for Satellite-Based Probabilistic Solar Forecasting with Cloud Motion Vectors. *Energies*. 2021; 14(16):4951.
https://doi.org/10.3390/en14164951

**Chicago/Turabian Style**

Carrière, Thomas, Rodrigo Amaro e Silva, Fuqiang Zhuang, Yves-Marie Saint-Drenan, and Philippe Blanc.
2021. "A New Approach for Satellite-Based Probabilistic Solar Forecasting with Cloud Motion Vectors" *Energies* 14, no. 16: 4951.
https://doi.org/10.3390/en14164951