# Short-Term Deterministic Solar Irradiance Forecasting Considering a Heuristics-Based, Operational Approach

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## Abstract

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## 1. Introduction

#### 1.1. Statistical Forecasting Methods: Introduction and Challenges

#### 1.2. Emergence and Challenges of Artificial Intelligence as Forecasting Tool

#### 1.3. Current State of the Art in Solar Power Forecasting Performance Assessment

#### 1.4. Present Work and Scientific Contributions

- A novel solar radiation forecasting method was developed based on pattern identification and classification, probability and heuristic methodology, considering operational needs of decision-making parties. The heuristic method developed for this application presents an intuitive, explainable, interpretable and effective way to forecast solar irradiance, by relying on concepts of probability, possibility and human reasoning, overcoming the limitation of complex mathematical abstraction and black-box characteristics of advanced state-of-the-art statistical and artificial intelligence methods.
- A generalized explicit irradiance pattern classification scheme was employed for performance assessment and forecasting, by classifying irradiance patterns through an analytical expression that yields similar results to clustering techniques, with the advantage of easy implementation across studies.
- A comprehensive performance assessment framework was developed to analyze not only how forecasting performance changes as a function of forecast horizon and lead time, but to evaluate the effect of data aggregation into the knowledge base has on forecasting skill, how quality control and/or data gaps affect performance assessment and how the forecaster performs under different objectively defined day types.

## 2. Data Sources and Methodology

_{t}-K-QS value to a certain irradiance class and Figure 6 presents the workflow for each variable, using QS as example. First, the variable is evaluated using Gaussian membership functions defined for each of the 20 defined classes presented in Figure 3 to determine the most similar class or origin state (Figure 6a). Then, the transition probability distribution corresponding to the origin state (Figure 6c) is extracted from the general transition matrix (Figure 6b). This distribution represents the probability of transition from the current origin state to the destination (or forecasted) state. Equation (8) presents the Gaussian membership function in which x refers to the variable to be tested, ${\mu}_{x}$ and ${\sigma}_{x}$ represent the mean and standard deviation of the set, respectively.

_{t}-K-QS space exists for a given $V{S}_{cdf}$ (Figure 7a) and the forecast output must belong to it. Since $V{S}_{cdf}$ is used to describe the intrahourly variability, has a wider distribution of values and has the least effect in QS compared to ${K}_{t}$ and K [33], for this work it has been assumed that the intrahourly variability is preserved across hours. With this $V{S}_{cdf}$ value, the K

_{t}-K-QS space is constructed (Figure 7b) and using the previously determined QS likelihood function (Figure 7c) the K

_{t}-K-QS space can be mapped into the QS likelihood function in the K

_{t}-K space (Figure 7d).

_{t}-K combinations violate physical irradiance limits, for which no forecast can be produced. Hence, there exists a binary plausibility space (Figure 7e) representing this (1 for plausible and 0 for implausible values), with ${K}_{t}$ and K limits being derived from the irradiance limits for GHI, DHI and direct normal irradiance (DNI) established by [47,48] calculated at the forecast’s corresponding α and ${I}_{o}$. The last space to consider is constructed from the likelihood functions of ${K}_{t}$ and K (Figure 7f). Finally, the product operator is used to aggregate the QS likelihood space, the combined ${K}_{t}$-K likelihood and the plausibility space, to obtain the overall combined likelihood in the ${K}_{t}$-K-QS space, representing the overall likelihood for the combination of variables. To obtain the crisp numerical ${K}_{t}$, K and QS values, a centroid defuzzification has been chosen, as this criterion considers the information of all the space and not just the maximum value.

## 3. Assessment Methodology

^{2}≅ 25 °C) and plants vary in capacity, it is reasonable to estimate electric power production as the product of the nominal production capacity by the irradiance on the module’s surface and reference the error metric to the nominal plant capacity ${C}_{nominal}$, following [32]. Therefore, the errors are calculated as follows: ${e}_{irr}$ for irradiance, while ${e}_{power}$ for power production at tilted plane cases. Only valid data points that passed quality control are considered for error calculation.

## 4. Results and Discussion

#### 4.1. Data Aggregation Effect on Forecasting Performance

#### 4.2. Effect of Forecast Horizon and Lead Time in Forecasting Performance

#### 4.3. Forecast Performance Assessment as a Function of Day Type

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Error metrics for ${K}_{t}$(

**top row**) and K (

**bottom row**) as a function of day class. Dashed line represents yearly performance. Cumulative distribution of day types presented in secondary axis.

**Figure A2.**Error metrics for GHI (

**top row**), DHI (

**middle row**) and DNI (

**bottom row**) as a function of day class. Dashed line represents yearly performance. Cumulative distribution of day types presented in secondary axis.

**Figure A3.**MAPE for irradiance forecasting as a function of day class. Cumulative distribution of day types presented in secondary axis.

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**Figure 2.**Application of Box–Jenkins methodology to different irradiance patterns from overcast to clear sky. One-minute clear sky irradiance hourly patterns (

**top left**), with most representative time series of each set (

**top right**). Lower subfigures present autocorrelation (

**bottom left**) and partial autocorrelation functions (

**bottom right**) for each representative time series. Color bar in top-left subfigure represents probability of occurrence.

**Figure 3.**Typical one-minute clear sky clearness index time series for a representative hour for each irradiance class, with its corresponding ${K}_{t}$ and K value ranges, based on [33].

**Figure 4.**Workflow to construct transition probability matrixes after irradiance pattern classification has been performed.

**Figure 6.**Workflow to determine the likelihood function for QS on a test case with origin state: ${K}_{t}$ = 0.667, K = 0.333 and QS = 0.667. Membership function for QS (

**a**), state transition probability matrix (

**b**), future state probability of occurrence (

**c**), QS ranges for each destination state (

**d**), probability-weighed likelihood function for QS at each possible destination state (

**e**), likelihood function for QS (

**f**).

**Figure 7.**Workflow for the inference and de-fuzzification stages. Case with origin state: ${K}_{t}$ = 0.667, K = 0.333 and QS = 0.667. QS in the K

_{t}-K space at the likely $V{S}_{cdf}$ value (

**a**), QS likelihood function (

**b**), QS likelihood mapped to the K

_{t}-K space (

**c**), plausibility space (

**d**), combined K

_{t}-K likelihood (

**e**), overall likelihood function for a K

_{t}-K-QS triad, with the forecasted value at the centroid (

**f**).

**Figure 9.**Example forecasting time series for the proposed method, showcasing completely clear (first), clear (second) and highly variable (third) days in the Santiago 2017 dataset: clearness index and diffuse fraction (

**a**), horizontally referenced irradiance (

**b**) and tilted plane irradiance (

**c**).

**Figure 10.**Forecasting skill performance for different forecast horizons and lead times (FLT), time units in hours: clearness index and diffuse fraction in the first row, horizontally referenced irradiance middle row and tilted plane irradiance bottom row.

**Figure 11.**Taylor diagrams for Kt (

**left**) and K (

**right**) forecasting. Hours in the chart indicate forecast horizon. Standard deviation and RMSD are in percent.

**Figure 12.**Taylor diagrams for irradiance forecasting. Hours in chart indicate forecast horizon, standard deviation and RMSD are in percent for tilted irradiance and in W/m

^{2}for horizontally referenced irradiance. Mean GHI = 502.62 W/m

^{2}Mean DNI = 503.74 W/m

^{2}and Mean DHI = 148.35 W/m

^{2}.

**Figure 14.**Skill score distributions for ${K}_{t}$ and K as a function of day class. Dashed line represents yearly performance. Cumulative distribution of day types presented in secondary axis.

**Figure 15.**Skill score distributions for irradiance as a function of day class. Dashed line represents yearly performance. Cumulative distribution of day types presented in secondary axis.

Source | Method | Dataset |
---|---|---|

Lan et al. [13] | Combination of frequency analysis to identify irradiance patterns, principal component analysis to identify characteristic features and neural networks are used to forecast future features of irradiance that are translated back to an irradiance forecast. | One year of data |

Theocharides et al. [14] | Several neural networks were developed to produce GHI forecasts, which are then clusterized and finally, through statistical processing, the final forecast is produced as a linear combination of the clusters. | 210 days |

Du Plessis et al. [15] | Several neural networks were developed for meteorological data to forecast a photovoltaic plant output at the subunit level and then scaled up to plant-wide production. | Two years training and 1 year of validation data. |

Source | Issue or Concern |
---|---|

Wang et al. [16] and Sethi and Kantardzic [17] | Neural networks/deep learning approaches have not seen sufficient adoption, despite growing interest in them, due to their complex black-box nature and lack of explainability and interpretability |

Wang. et al. [18] | These approaches are prone to model overfitting and insufficient generalization ability, being hyperspecific. |

Wang. et al. [16] | Explainability is of great importance, therefore, proposed a new approach through direct explainable neural networks that can provide further insights in the input–output relationship to assist in result interpretation and model explanation. |

Ahmed et al. [19] | Appropriate weather classification is important for solar photovoltaic power forecasting assessment, and there are challenges to overcome in these classifications, presenting that most authors employ four or less classes. |

Wang et al. [20] | Separate forecast models for each weather class should improve forecasting performance; therefore, having a higher number of classes would be beneficial. |

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**MDPI and ACS Style**

Castillejo-Cuberos, A.; Boland, J.; Escobar, R.
Short-Term Deterministic Solar Irradiance Forecasting Considering a Heuristics-Based, Operational Approach. *Energies* **2021**, *14*, 6005.
https://doi.org/10.3390/en14186005

**AMA Style**

Castillejo-Cuberos A, Boland J, Escobar R.
Short-Term Deterministic Solar Irradiance Forecasting Considering a Heuristics-Based, Operational Approach. *Energies*. 2021; 14(18):6005.
https://doi.org/10.3390/en14186005

**Chicago/Turabian Style**

Castillejo-Cuberos, Armando, John Boland, and Rodrigo Escobar.
2021. "Short-Term Deterministic Solar Irradiance Forecasting Considering a Heuristics-Based, Operational Approach" *Energies* 14, no. 18: 6005.
https://doi.org/10.3390/en14186005