Solar energy is expected to reach over 1,500 \( GW_{p}\) of PV capacity by 2030 tripling in just 10 years, which highlights the critical need for accurate forecasting to balance energy supply, optimize grid performance, and decrease dependence on fossil fuels [5]. Real-time grid management and energy markets require accurate short-term predictions (range from about 30 min to 5-6 hours) since errors in these forecasts can result in inefficiencies and increased costs [6, 7]. However, forecasting global solar irradiance presents significant challenges due to the inherent variability and non-stationarity of solar irradiance data. In order to solve these challenges, irradiance data are often normalized. One way to normalize solar irradiance data involves utilizing extraterrestrial irradiance that is defined as solar irradiance at the topmost layer of the Earth’s atmosphere. This extraterrestrial irradiance can be calculated using a formula that accounts for geometric relationship between the Sun and the Earth. Normalizing with extraterrestrial irradiance results in the development of forecasting models utilizing the clearness index [8, 9]. Most of the traditional approaches to solar irradiance forecasting rely on Clearsky model, computing an intermediate stationary form (called Clearsky Index or CSI), as they typically produce better forecasts compared to models using clearness index [10]. Methods employing Clearsky models [11, 12, 13, 14, 15] seek to use solar irradiance in cloud-free (clear sky) conditions and subsequently enhance forecasts by incorporating observed meteorological data characterizing the sky state and content in gas and particles with a seasonal and daily variation. Traditional forecasting methodologies face critical limitations, including inaccurate Clearsky models, elevated computational expenses associated with Numerical Weather Prediction (NWP) systems, and a limited ability to adjust to changing local meteorological conditions [16, 17]. In order to overcome these limitations, this study proposes a framework that is based on a Clearsky-Free, data-driven, methodology that employs Extreme Learning Machines (ELM). This methodology simplifies the forecasting process, enhances adaptability, and provides accurate, real-time predictions that are tailored to modern energy systems by explicitly modeling Global Horizontal Irradiance (GHI) with enriched periodic and probabilistic features.

Clearsky models, such as McClear [1], REST2 [16] and SPARTA [18], are widely used for calculating solar irradiance under cloud-free conditions. These models provide essential baselines using celestial mechanics and interpreting atmospheric effects related to aerosols, water vapor and ozone, with broad applications in climate science, agriculture, solar energy, and air quality monitoring [19, 7, 17]. Despite their versatility, Clearsky-based models (i.e., using the Clearsky model as a stationarization method) exhibit significant limitations, particularly in real-time operational contexts.:

• Synchronization Issues: Mismatched measurements and model outputs, especially at low solar elevations when shading and solar mask can intervene, lead to inaccuracies and reduced reliability [6];
• Dependence on Atmospheric Inputs: Parameters such as aerosol optical depth and water vapor content are widely available thanks to atmospheric composition model or reanalyses such as CAMS or MERRA2. However, the accuracy of these parameters can be poor in some regions leading to unreliable clear sky irradiance estimation citep{gueymard2021solar};
• Gaps in Coverage: CSI derived from these models (as the ratio of GHI to Clearsky irradiance), is undefined during nighttime and prone to significant errors during transitional periods such as dawn and dusk [20, 7];
• Limited Adaptability: Clearsky models struggle to represent diffuse irradiance at low solar elevations and to adapt to localized, dynamic weather conditions, particularly in complex terrains [21];
• Surface Orientation Biases: For forecasting PV power or solar radiation on tilted surface, the irradiance provided by Clearsky model needs to be projected on a inclined plane using so-called transposition model. The limited knowledge of the anisotropy of the diffuse radiance can introduce a bias that propagate through the forecasting pipeline [22].

For photovoltaic systems, these limitations are further compounded by dynamic factors such as inverter efficiency, maintenance schedules, and panel degradation, which Clearsky models cannot capture [23]. As a result, these models, like they are actually used in prediction, could add complexity without significantly improving the accuracy of forecasts. The limitation of Clearsky-Based models underscore the need for alternative approaches where possible. New Clearsky-Free methodologies should demonstrate that it is possible to bypass clearsky model and its limitations by directly leveraging raw irradiance data (e.g., GHI, GTI, BNI, DHI) without relying on theoretical Clearsky baselines [24]. An explanation of the stationary process in solar time series prediction is provided in A, emphasizing that this approach captures the stochastic nature of solar irradiance, enhances robustness and adaptability, and eliminates pre-processing and post-processing steps such as CSI normalization and rescaling, thereby simplifying the forecasting process. These Clearsky-Free methods, powered by advanced machine learning techniques like ELM, represent a significant shift in solar irradiance forecasting. They overcome the key limitations of traditional models for time-series based short-term forecast, providing scalable, accurate, and efficient solutions tailored to modern energy systems [23, 7].

The limitations of CSI-based forecasting methods highlight the need for alternatives that bypass Clearsky models. These Clearsky-independent approaches predict solar irradiance variables directly, without relying on theoretical baselines offers significant advantages with regard to real-time forecasting. Clearsky-Free models avoid the need for atmospheric inputs that are often subject to errors, such as aerosol optical depth and related synchronization issues. Instead, these models rely on endogenous information, such as lagged values of GHI, in order to simplify the forecasting task and avoid cascading errors [25]. They provide continuity for every hour of the day, even at the critical low-light sunrise and sunset hours, therefore improving grid management-energy storage [6]. The hypothesis explored in this paper is that some adaptative statistical model can learn implicitely from past observations the effects of aerosol, ozone and water vapour on the solar irradiance, making the use of a clearsky model unnecessary. Correctly learning this component of atmospheric extinction is not a trivial task, since it is strongly modulated by the effect of clouds, which is several orders of magnitude greater.Not all statistical forecast approaches are capable of learning reliably the effect of aerosol, ozone and water vapour in varying weather conditions. We expect that advanced machine learning techniques, including Extreme Learning Machines, are able to capture complex temporal dependencies scalably and efficiently. Probabilistic extensions [26] enable quantification of uncertainty, allowing for superior operational decision-making [27]. Table 1 provides an overview of the key differences between Clearsky-Based and Clearsky-Free approaches. This comparison underscores how Clearsky-Free methods tackle important challenges, making them especially effective for applications that demand adaptability, scalability, and efficiency. Moreover, such a Clearsky-Free approach have additional benefits: it would be less computationally complex, making them more robust and adaptive.

Despite their utility, Clearsky-Based models face critical limitations in real-time forecasting. These include inaccuracies in atmospheric inputs, reliance on complex pre-processing steps, and limited adaptability in dynamic or data-sparse environments. Addressing these challenges requires an approach that is simpler, more robust, and less dependent on extensive datasets. The objectives of this study are:

• Develop a Clearsky-Free forecasting framework that eliminates the need for Clearsky normalization and minimizes reliance on atmospheric data.
• Leverage historical and endogenous features, such as lagged Global Horizontal Irradiance (GHI) values, to simplify and streamline the forecasting process.
• Design a solution that is adaptable and capable of performing effectively in environments with sparse or incomplete data.

The proposed framework will be evaluated on three criteria: accuracy, to ensure precise solar irradiance forecasts; robustness, to deliver reliable performance across varying climatic conditions and geographic regions; and parsimony, to achieve high forecasting quality with minimal data and low computational complexity. By addressing the shortcomings of traditional Clearsky-Based models, this study positions Clearsky-Free approaches as practical, scalable, and efficient solutions for modern energy systems. The emphasis on simplicity, adaptability, and data efficiency makes this framework particularly suitable for both data-rich and resource-constrained environments. The structure of this paper is as follows: Section 2 describes the methodology, including data collection, quality control, and the modeling framework. Section 3 presents and discusses the results, covering both deterministic and probabilistic forecasting approaches, as well as comparisons to benchmark models. Finally, Section 4 provides the conclusions and outlines future research directions.

This study employs a univariate and endogenous forecasting approach, where Global Horizontal Irradiance (GHI) predicts itself. The dataset is sourced from the SIAR network of agroclimatic weather stations in Spain [28]. It includes high-resolution GHI records from 76 stations over a four-year period, with a 30-min temporal resolution. This dataset offers the fine granularity required for short-term forecasting while encompassing a diverse range of climatic zones, from arid regions to highly variable coastal and mountainous areas, ensuring broad applicability [29]. To ensure data reliability, a strict quality control process proposed by [51] was applied. This included detecting and rectifying anomalies such as sensor malfunctions, extreme weather impacts, or inconsistent measurements. The McClear model, recognized for its accurate estimation of clear-sky irradiance, was used as a reference for validating the GHI data [1]. Since McClear operates in hindcast mode (where required atmospheric inputs are only available two days after the observation date), it provides highly reliable theoretical clear-sky estimates, making it a robust basis for identifying and correcting errors in the historical dataset. However, this operational constraint limits its use in real-time forecasting. Special attention was paid to sunrise and sunset periods, where low solar elevation angles are known to introduce higher uncertainty [30]. To account for temporal dependencies in GHI, lagged input variables were created, enabling dynamic adaptation to the inherent variability of solar irradiance [3]. The dataset was divided into three years for training and validation and one year for testing, ensuring robust evaluation of the model’s generalization capabilities under both stable and dynamic weather scenarios. In this context, the ELM model was trained using a carefully selected set of input features. The primary predictor is historical GHI, using past values to forecast future irradiance. Additionally, the clear-sky index \( k_t\) , defined as the ratio between the measured GHI and its theoretical counterpart from McClear, was included to provide an implicit reference for atmospheric attenuation effects. Temporal features, such as the hour of the day, day of the year, and seasonal indicators, were incorporated to capture diurnal and seasonal variations. Although no direct aerosol measurements are included, their impact on solar radiation is implicitly accounted for through the clear-sky GHI from McClear, which integrates aerosol optical properties from CAMS reanalysis data. By structuring models around historical GHI and clear-sky references, they captures complex dependencies between atmospheric conditions and solar irradiance without requiring explicit AOD measurements. This approach simplifies the forecasting pipeline while maintaining high performance. Furthermore, it allows for a reliable evaluation of the necessity of clear-sky in forecasting accuracy.

This study evaluates a diverse set of models for solar irradiance forecasting, combining deterministic and probabilistic approaches to address the challenges of traditional Clearsky normalization. These models are designed to capture temporal dependencies and the inherent periodic variability of solar irradiance data. Naive predictors serve as benchmarks and include:

• Persistence (P): Assumes future GHI is equal to the most recent observed value [3];
• Clearsky Model (CS): Uses theoretical Clearsky irradiance while ignoring cloud attenuatation [1];
• Smart Persistence (SP): Combines persistence with deviations from theoretical Clearsky irradiance to account for diurnal patterns [6];
• Climatology-Persistence CLIPER: Blends climatological averages with persistence to exploit autocorrelation [31];
• Exponential Smoothing (ES): Applies a weighted average to recent deviations from Clearsky irradiance [32];
• Autoregressive AR(2)-like (ARTU): Incorporates temporal dependencies and accounts for uncertainty [33];
• Combination Model (COMB): Aggregates SP, CLIPER, ES, and ARTU using ensemble methods [33].

The reference model in the deterministic case is an autoregressive AR(p) model computed with classical least square optimisation. In this model, the GHI at time \( y_t \) is expressed as a linear combination of its past values: \( y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} + \epsilon_t \) , where \( \phi_i \) are the coefficients, \( p \) is the model order, and \( \epsilon_t \) represents the error term or residual. The coefficients \( \phi_i \) are estimated by minimizing the sum of squared residuals \( \min_{\phi} \sum_{t=1}^T \left( y_t - \sum_{i=1}^p \phi_i y_{t-i} \right)^2 \) . This AR model serves as a benchmark for assessing the performance of machine learning-based approaches in forecasting. It will be used in two versions: AR (without clear sky) and rAR (with clear sky). See Sections 3.1.1 and 3.1.2 for more details. The Extreme Learning Machine (ELM or EL) model is the central predictive framework in this study, providing a scalable and computationally efficient solution. It relies on a single-hidden-layer neural network architecture with sigmoid and Gaussian activation functions to capture both linear and non-linear relationships in the data. The model configuration includes:

• Input and hidden layer optimization: For each forecast horizon, the number of input neurons and hidden neurons is adjusted from in-sample data (the best nRMSE defines the best configuration). For one particular site with high variability for example, the 1-hour horizon uses in average 78 inputs and 472 hidden neurons, while the 5-hour horizon uses 138 inputs and 450 hidden neurons. The hidden layer size is scaled approximately four times the input size for robust performance [27].
• Training process: Multiple random initializations are used to ensure robustness. Output weights are optimized using Ridge regression, \( \beta = (H^\top H + \lambda I)^{-1} H^\top Y\) , where \( H\) is the hidden layer output matrix, \( \lambda\) is the regularization parameter, and \( Y\) is the target vector [34].
• Model selection: The configuration with the lowest normalized root mean square error (nRMSE) on the validation set is selected for testing.

A detailed ELM definition is given in B. Note that ELM provides a significant advantage over deep learning models in terms of both training and inference speed. Unlike iterative backpropagation-based methods, ELM training is performed through a single matrix inversion step, resulting in high computational efficiency. Furthermore, due to its simple architecture consisting of a single hidden layer, real-time inference is enabled, with predictions being generated in a few seconds. In this study, It was verified that incoming data can be processed for a dataset of approximately 70 sites, with an execution time of less than 30 seconds, confirming the model’s suitability for operational solar forecasting applications where rapid response times are required. Unlike physics-based models that integrate explicit aerosol-related parameters, the ELM operates in a purely data-driven manner. It statistically extracts the most relevant relationships between past observations and future irradiance without requiring a precise estimation of atmospheric attenuation mechanisms. This means that the model indirectly captures the impact of aerosols, clouds, and other atmospheric phenomena based on historical patterns but does not explicitly differentiate their contributions.

Quantile Regression (QR) is employed as a reference probabilistic forecasting method due to its robustness and established performance in uncertainty quantification. QR predicts conditional quantiles of the response variable, allowing for the generation of reliable prediction intervals without assumptions about the underlying data distribution. The method optimizes an asymmetric loss function defined as:

where \( u = y - \hat{y}\) is the residual and \( \tau\) represents the quantile level. By solving this minimization problem using linear programming, QR constructs prediction intervals that align with target coverage levels [35]. In addition to QR, a non-parametric methodology is used to derive prediction intervals directly from residuals of deterministic forecasts [26]. Lookup tables are generated during the training phase, capturing empirical relationships between forecast errors and observed values [36]. This approach avoids predefined assumptions about error distributions, making it particularly robust in dynamic atmospheric conditions. More information are given in C. The methodology used during simulations offers several key advantages. It provides comprehensive benchmarking by incorporating both naive predictors and advanced models, enabling rigorous evaluation. The framework is capable of handling high-dimensional data and varying forecast horizons with minimal computational overhead. Probabilistic forecasting is robustly implemented through QR alongside non-parametric approaches for enhanced flexibility.

The performance of the forecasting models is assessed using both deterministic and probabilistic metrics to capture accuracy, bias, and reliability. This comprehensive evaluation framework is critical for validating the efficacy of Clearsky-Free methodologies in solar irradiance forecasting. For deterministic forecasting, the following metrics are employed:

• Normalized Root Mean Square Error (nRMSE) quantifies the average deviation of predictions from observed values, with a focus on penalizing larger errors, nRMSE \( = {\text{E}[y]}^{-1} \cdot \sqrt{\frac{1}{N} \sum_{i=1}^N \left( y_i - \hat{y}_i \right)^2} \) where \( y_i \) represents the observed values, \( \hat{y}_i \) the predicted values, \( N \) the total number of observations and \( \text{E}[y] \) represents the expected value or mean of the random variable \( y\) [32];
• Normalized Mean Absolute Error (nMAE) provides a robust measure of accuracy, less sensitive to outliers compared to nRMSE, nMAE \( = \text{E}[y]^{-1} \cdot \frac{1}{N} \sum_{i=1}^N \left| y_i - \hat{y}_i \right|;\)
• R-Squared (R\( ^2 \) ) measures the proportion of variance in the observed data explained by the model, providing an indicator of goodness-of-fit, R\( ^2 = 1 - (\sum_{i=1}^N \left( y_i - \hat{y}_i \right)^2)/(\sum_{i=1}^N \left( y_i - \text{E}[y] \right)^2)\) ;
• Normalized Mean Bias Error (nMBE) captures systematic bias in the forecasts [30], nMBE \( = \text{E}[y]^{-1} \cdot \frac{1}{N} \sum_{i=1}^N \left( y_i - \hat{y}_i \right).\)

For probabilistic forecasting, the evaluation framework includes:

• Continuous Ranked Probability Score (CRPS) compares the cumulative distribution functions (CDFs) of predicted and observed values, CRPS\( (F, y) = \int_{-\infty}^{\infty} \left( F(z) - \mathbf{1}_{z \geq y} \right)^2 dz\) , evaluating the entire predictive distribution, where \( F(z) \) is the forecast CDF, and \( y \) is the observed value. This study computes CRPS using centiles, offering finer granularity than quartiles for solar irradiance forecasts [37]. The quantile-based CRPS method used during this study is detailed in D;
• Prediction Interval Coverage Probability (PICP) evaluates the reliability of prediction intervals by measuring the percentage of observed values falling within these intervals, PICP\( = \frac{1}{N} \sum_{i=1}^{N} \mathbf{1}_\underline{{y_i} \leq y_i \leq \overline{y_i}}\) , where \( \underline{y_i} \) and \( \overline{y_i} \) are the lower and upper bounds of the prediction interval [26];
• Mean Interval Length (MIL) assesses the sharpness of prediction intervals, providing an indicator of interval width, \( \text{MIL} = \frac{1}{N} \sum_{i=1}^N \left( \overline{y_i} - \underline{y_i} \right);\)
• Interval Score (IS) evaluates the quality of prediction intervals by balancing two factors: sharpness (ensuring the intervals are as narrow as possible, measured by the MIL) and coverage reliability (Penalizing predictions when observed values fall outside the prediction intervals). The Interval Score is defined as: \( \text{IS} = \text{MIL} + \frac{2}{\alpha} \sum_{i=1}^N \left[ \mathbf{1}_{y_i < \underline{y_i}} \left( \underline{y_i} - y_i \right) + \mathbf{1}_{y_i > \overline{y_i}} \left( y_i - \overline{y_i} \right) \right],\) where \( \alpha \) is the nominal coverage level of the prediction interval, and \( \mathbf{1} \) denotes the indicator function (equals 1 if the condition is true and 0 otherwise) [37].

The evaluation framework uses deterministic metrics like nRMSE and nMAE, along with probabilistic metrics like CRPS and PICP, to assess model reliability, accuracy, and practical relevance. The Mann-Whitney U test [38], a non-parametric method, is used to compare distributions without assuming normality, making it ideal for scenarios where error distributions deviate from Gaussian norms. The test measures shifts in forecast error distributions, with a p-value below the standard threshold of 0.05 indicating a significant difference. For example, Model B outperforms Model A with a p-value of 0.021, indicating superior performance with consistently lower forecast errors.

Point prediction, or deterministic prediction, involves generating a single estimated value for a future variable without accounting for uncertainty or variability. This approach is crucial as it enables precise and timely decision-making in applications such as solar energy management, where accurate forecasts can optimize resource utilization and enhance system efficiency.

3.1.1 Justifying the Use of ELM

Initially, it is essential to validate the use of ELM against a purely linear autoregressive AR(\( p\) ) model, optimizing \( p\) for each site and forecasting horizon using a brute force method comparing in-sample results for \( p=1\) to \( p=96\) (\( p\) median \( =58\) ). The comparison of performances of both models is operated on raw data (without employing Clearsky models). Note that \( p\) can’t be defined from Partial Autocorrelation Function (PACF) while this parameter is periodic with raw data. Performance of the deterministic forecasting, across various metrics and time horizons are presented in Figure 1. The resulting p-values from Mann-Whitney U test are displayed below the respective plots for each time horizon in the figures. The results show that ELM consistently outperforms AR across all error metrics (nRMSE, nMAE, nMBE, R²) and time horizons, with very low p-values confirming the significance of these differences. Therefore, ELM is clearly the better choice for more accurate, reliable, and unbiased predictions.

3.1.2 Validation Against Reference and Naive Predictors

To validate the ELM Clearsky-Free approach, predictions are compared to classical solar irradiance tools, which are defined as reference (rAR) and naive models (SP, P, and CS) derived from the McClear Clearsky model. The order \( p \) of the rAR(\( p \) ) model is optimized using a PACF-based approach: the first minimum of the PACF function determines the maximum lag to consider, which corresponds to \( p \) (\( p\) median \( =6\) ). As shown in Figure 6, ELM consistently outperforms all naive models (SP, P, and CS) across all time horizons (30, 180, and 360 min) with highly significant differences (p < 0.001). While ELM also surpasses rAR at 180 and 360 min (p < 0.001), the difference is not significant for the 30-min horizon (p = 0.239). Among the models, EL ranks as the best, followed by rAR, SP, CS, and finally P.

3.1.3 Benchmark Expansion and ELM’s Dominance

Expanding the benchmark with additional models strengthens the conclusions regarding ELM’s superiority (Figure 7). It consistently outperforms all benchmark models (CLIPER, ES, ARTU, and COMB) across medium (180 min) and long (360 min) horizons, with highly significant improvements (\( p < 10^{-11}\) ). At the short 30-min horizon, the differences are less pronounced, with some comparisons showing non-significant p-values (e.g., CLIPER and ARTU), though EL still performs competitively. Overall, ELM is the most reliable and effective model, particularly excelling in medium and long-term forecasts.

3.1.4 Ranking in Deterministic Case

Including probabilistic forecasts complements deterministic predictions by quantifying uncertainty, which is critical for optimizing energy systems. This approach allows better risk management, improved resource allocation, and enhanced decision-making under variability, leveraging advanced statistical methods to capture forecast distributions and tail risks.

3.2.1 Median Estimation from Probabilistic Case

One advantage of working directly with probabilistic forecasts is that estimating the median, or 0.5 quantile, inherently provides a deterministic prediction. In the case of Quantile Regression (QR), this facilitates seamless integration of probabilistic and deterministic approaches. Similarly, for ELM, the methodology based on lookup tables ensures that both probabilistic and deterministic forecasts are generated concurrently, enhancing efficiency and flexibility. In this section, and particularly in Figure 8, the results comparing the medians obtained from QR and ELM are presented. Overall, ELM outperforms QR Median across most metrics and time horizons, particularly in medium (180 min) and long-term (360 min) forecasts. For nRMSE and nMAE, ELM consistently achieves lower errors, indicating higher predictive accuracy at all horizons. In terms of nMBE, it demonstrates less bias than QR, remaining closer to zero across all horizons. For R², QR shows a slight advantage at the 30-min horizon but declines significantly at longer horizons.

3.2.2 QR Versus ELM

Probabilistic error metrics across ten different forecast horizons are presented in Figure 13. The nMIL metric (Figure 14) evaluates the average width of the prediction intervals, where lower values indicate narrower intervals and thus higher confidence in the forecasts. The results show that both QR and EL maintain consistent interval widths across horizons of 150 min and above, while EL generally provides tighter intervals compared to QR, particularly at longer horizons, indicating higher precision in EL probabilistic forecasts. The PICP metric (Figure 15) reflects the percentage of actual observations captured within the predicted intervals. For this study, the nominal value is set at \( \alpha = 0.2\) , corresponding to an 80% prediction interval. A well-calibrated model should have PICP values close to 80%. The results indicate that the QR model consistently achieves PICP values closer to 80%, suggesting better calibration. In contrast, the EL model’s slightly lower PICP values highlight a trade-off, as its narrower intervals prioritize precision but might compromise calibration in terms of coverage reliability. The CRPS metric (Figure 17) evaluates the alignment between forecast distributions and actual observations, where lower values indicate better performance. The EL model consistently outperforms QR across all horizons, indicating that its forecast distributions are closer to actual observations. Similarly, the nCRPS metric (Figure 16), which normalizes CRPS values, reinforces EL superior performance by highlighting its ability to provide accurate and consistent probabilistic forecasts relative to observed data.

For GHI forecasting over a 30-min horizon (mean concerning all sites), the mean Interval Scores (IS) were 115.9 W/m² for ELM and 131.2 W/m² for QR, as shown in Figure 18. The lower mean IS indicates that ELM is the better model for this horizon. A significance test comparing the IS distributions yielded a p-value of 0.075 for \( \alpha < 0.3\) , indicating that the difference between ELM and QR is not statistically significant at higher confidence levels (\( \geq 70\%\) ). However, as \( \alpha\) increases, the p-value decreases significantly, suggesting that the performance gap becomes statistically significant at lower reliability levels (\( < 70\%\) ). Similar patterns are observed for other forecast horizons, where ELM consistently achieves lower mean IS values than QR. These results, reflected in comparable interval score curves across horizons, confirm that ELM provides superior probabilistic forecasting performance. The statistical significance of the differences, however, varies depending on the reliability level and forecast horizon.

3.2.3 Ranking in Probabilistic Case

The comparison of prediction intervals for QR and ELM models (Figure 19) highlights the superior performance of ELM in probabilistic forecasting. ELM provides consistently narrower prediction intervals across all confidence levels (95%, 80%, 50%, and 20%), demonstrating reduced uncertainty and more confident predictions. In contrast, QR exhibits significantly wider intervals, particularly at upper than 95% confidence level, indicating higher prediction variance and less precise forecasts. Additionally, the median forecasts (solid lines) from ELM align more closely with the observed GHI values (dotted red line), showcasing better calibration and accuracy compared to QR. While QR captures a larger range of possible outcomes due to its wider intervals, this comes at the cost of reduced precision and potentially less actionable predictions. This trade-off between precision and coverage further supports the selection of ELM for applications requiring high-confidence, actionable forecasts in energy systems.

The Clearsky-Free approach demonstrated in this study provides robust results in both deterministic and probabilistic forecasting, but there is substantial potential to further enhance accuracy by integrating the strengths of Clearsky models such as McClear and its forecast-oriented counterpart, CAMS. These models provide high-quality irradiance baselines, while ELM excel at dynamically adapting to local atmospheric variability and temporal dependencies. By coupling these approaches, a hybrid model could outperform traditional methods, leveraging the theoretical precision of Clearsky models and the adaptability of ELM. Clearsky models can also enhance the reliability of input data through quality control, improving the overall robustness of the forecasting system. This is particularly beneficial in regions with sparse or unreliable ground-based measurements, where satellite-derived models like McClear serve as dependable references. A promising future direction involves creating hybrid models that combine the theoretical baselines of McClear with the adaptability of ELM for real-time atmospheric variations. Such models would improve solar irradiance forecasts, particularly in regions with complex dynamics. To enhance robustness and generalizability, training datasets should include diverse climatic conditions, and transfer learning techniques should be integrated to enable global applicability without location-specific historical data. These advancements would provide accurate predictions for any irradiance component (e.g., GHI, DHI, BNI, GTI) and configurations like tilted or azimuthal orientations. Another key research avenue is refining probabilistic forecasting through advanced uncertainty quantification techniques, such as Gaussian Processes or Bayesian Neural Networks, to improve forecast reliability. Additionally, the application of Clearsky-Free methodologies in decision-support systems for energy grid operators must be explored, ensuring scalability and computational efficiency for real-time operational contexts. By pursuing these directions, Clearsky-Free methodologies can evolve into more reliable and flexible tools for solar irradiance forecasting, addressing diverse operational and environmental challenges.

[1] Lefèvre McClear: a new model estimating downwelling solar radiation at ground level in clear-sky conditions Atmospheric Measurement Techniques 2013 6 9 2403-2418 sep 10.5194/amt-6-2403-2013

[2] Abbas Azarpour and Omid Mohammadzadeh and Nima Rezaei and Sohrab Zendehboudi Current status and future prospects of renewable and sustainable energy in North America: Progress and challenges Energy Conversion and Management 2022 269 115945 10.1016/j.enconman.2022.115945

[3] Dazhi Yang and Wenting Wang and Christian A. Gueymard and Tao Hong and Jan Kleissl and Jing Huang and Marc J. Perez and Richard Perez and Jamie M. Bright and Xiang’ao Xia and Dennis van der Meer and Ian Marius Peters A review of solar forecasting, its dependence on atmospheric sciences and implications for grid integration: Towards carbon neutrality Renewable and Sustainable Energy Reviews 2022 161 112348 10.1016/j.rser.2022.112348

[4] Philippe Lauret and Auline Rodler and Marc Muselli and Mathieu David and Hadja Maïmouna and Cyril Voyant A Bayesian model committee approach to forecasting global solar radiation WREF 2012 : World Renewable Energy Forum 2012 1

[5] International Energy Agency (IEA) Renewables 2021: Analysis and Forecasts to 2026 IEA, Paris 2021 Licence: CC BY 4.0

[6] Mbaye Diagne and Mathieu David and Philippe Lauret and John Boland and Nicolas Schmutz Review of solar irradiance forecasting methods and a proposition for small-scale insular grids Renewable and Sustainable Energy Reviews 2013 27 65–76 10.1016/j.rser.2013.06.042

[7] Cyril Voyant and Gilles Notton and Soteris Kalogirou and Marie-Laure Nivet and Christophe Paoli and Fabrice Motte and Aurélie Fouilloy Machine learning methods for solar radiation forecasting: A review Renewable Energy 2017 105 569–582 10.1016/j.renene.2016.12.095

[8] A. Sanfilippo and L. Martin-Pomares and N. Mohandes and D. Perez-Astudillo and D. Bachour An adaptive multi-modeling approach to solar nowcasting Solar Energy 2016 125 77–85 10.1016/J.SOLENER.2015.11.041

[9] E. Akarslan and F.O. Hocaoglu and R. Edizkan Novel short term solar irradiance forecasting models Renew Energy 2018 123 58–66 10.1016/J.RENENE.2018.02.048

[10] P. Lauret and R. Alonso-Suárez and J. Le Gal and L. Salle and M.D.S. David Solar Forecasts Based on the Clear Sky Index or the Clearness Index: Which Is Better? Solar 2022 2 432–444 10.3390/SOLAR2040026

[11] L.S. Hoyos-Gómez and J.F. Ruiz-Muñoz and B.J. Ruiz-Mendoza Short-term forecasting of global solar irradiance in tropical environments with incomplete data Appl Energy 2022 307 118192 10.1016/J.APENERGY.2021.118192

[12] M. David and F. Ramahatana and P.J. Trombe and P. Lauret Probabilistic forecasting of the solar irradiance with recursive ARMA and GARCH models Solar Energy 2016 133 55–72 10.1016/J.SOLENER.2016.03.064

[13] Â. Frimane and J. Munkhammar and D. van der Meer Infinite hidden Markov model for short-term solar irradiance forecasting Solar Energy 2022 244 331–342 10.1016/J.SOLENER.2022.08.041

[14] S. Shan and C. Li and Z. Ding and Y. Wang and K. Zhang and H. Wei Ensemble learning based multi-modal intra-hour irradiance forecasting Energy Convers Manag 2022 270 116206 10.1016/J.ENCONMAN.2022.116206

[15] T. Gneiting and S. Lerch and B. Schulz Probabilistic solar forecasting: Benchmarks, post-processing, verification Solar Energy 2023 252 72–80 10.1016/J.SOLENER.2022.12.054

[16] Christian A. Gueymard REST2: High-performance solar radiation model for cloudless-sky irradiance, illuminance, and photosynthetically active radiation – Validation with a benchmark dataset Solar Energy 2008 82 3 272-285 10.1016/j.solener.2007.04.008

[17] Sobrina Sobri and Sam Koohi-Kamali and Nasrudin Abd. Rahim Solar photovoltaic generation forecasting methods: A review Energy Conversion and Management 2018 156 459-497 10.1016/j.enconman.2017.11.019

[18] José A. Ruiz-Arias SPARTA Renewable and Sustainable Energy Reviews 2023 188 113833 dec 10.1016/j.rser.2023.113833

[19] Richard G Allen and Luis S Pereira and Dirk Raes and Martin Smith Crop evapotranspiration: Guidelines for computing crop water requirements FAO Irrigation and Drainage Paper 1998 56 1–300

[20] Dhivya Sampath Kumar and Gokhan Mert Yagli and Monika Kashyap and Dipti Srinivasan Solar irradiance resource and forecasting: a comprehensive review IET Renewable Power Generation 2020 14 10 1641-1656 10.1049/iet-rpg.2019.1227

[21] M. Santamouris and I. Tselepidaki and N. Dris Evaluation of models to predict solar radiation on tilted surfaces for the Mediterranean region Solar & Wind Technology 1990 7 5 585-589 10.1016/0741-983X(90)90067-C

[22] Yuan Ma and Xuemin Zhang and Shengwei Mei and Zhao Zhen and Rui Gao and Zijie Zhou Ultra-short-term solar power forecasting based on a modified clear sky model 2020 39th Chinese Control Conference (CCC) 2020 5311-5316 10.23919/CCC50068.2020.9189533

[23] Ewa Chodakowska and Joanicjusz Nazarko and Łukasz Nazarko and Hesham S. Rabayah Solar Radiation Forecasting: A Systematic Meta-Review of Current Methods and Emerging Trends Energies 2024 17 13 10.3390/en17133156

[24] José A. Ruiz-Arias and Christian A. Gueymard Worldwide inter-comparison of clear-sky solar radiation models: Consensus-based review of direct and global irradiance components simulated at the earth surface Solar Energy 2018 168 10-29 Advances in Solar Resource Assessment and Forecasting https://doi.org/10.1016/j.solener.2018.02.008

[25] Danny H.W. Li and Siwei Lou and Joseph C. Lam and Ronald H.T. Wu Determining solar irradiance on inclined planes from classified CIE (International Commission on Illumination) standard skies Energy 2016 101 462-470 10.1016/j.energy.2016.02.054

[26] Philippe Lauret and Mathieu David and Hugo T. C. Pedro Probabilistic Solar Forecasting Using Quantile Regression Models Energies 2017 10 10 10.3390/en10101591

[27] Guang-Bin Huang and Qin-Yu Zhu and Chee-Kheong Siew Extreme learning machine: Theory and applications Neurocomputing 2006 70 489–501 10.1016/j.neucom.2005.12.126

[28] Milan Despotovic and Cyril Voyant and Luis Garcia-Gutierrez and Javier Almorox and Gilles Notton Solar irradiance time series forecasting using auto-regressive and extreme learning methods: Influence of transfer learning and clustering Applied Energy 2024 365 123215 10.1016/j.apenergy.2024.123215

[29] S.M. Vicente-Serrano and F. Domínguez-Castro and F. Reig and S. Beguería and M. Tomas-Burguera and B. Latorre and D. Peña-Angulo and I. Noguera and I. Rabanaque and Y. Luna and A. Morata and A. El Kenawy A near real-time drought monitoring system for Spain using automatic weather station network Atmospheric Research 2022 271 106095 10.1016/j.atmosres.2022.106095

[30] Sergio Herrería-Alonso and Andrés Suárez-González and Miguel Rodríguez-Pérez and Raúl F. Rodríguez-Rubio and Cándido López-García A Solar Altitude Angle Model for Efficient Solar Energy Predictions Sensors 2020 20 5 10.3390/s20051391

[31] Dazhi Yang Making reference solar forecasts with climatology, persistence, and their optimal convex combination Solar Energy 2019 193 981-985 10.1016/j.solener.2019.10.006

[32] Spyros Makridakis and Steven C. Wheelwright and Rob J. Hyndman Forecasting methods and applications Wiley 1998 3rd Edition 10.2307/2287014

[33] Cyril Voyant and Gilles Notton and Jean-Laurent Duchaud and Luis Antonio García Gutiérrez and Jamie M. Bright and Dazhi Yang Benchmarks for solar radiation time series forecasting Renewable Energy 2022 113 102–114 10.1016/j.renene.2022.04.065

[34] Yun Zheng and Yisu Ge and Sami Muhsen and Shifeng Wang and Dalia H. Elkamchouchi and Elimam Ali and H. Elhosiny Ali New ridge regression, artificial neural networks and support vector machine for wind speed prediction Advances in Engineering Software 2023 179 103426 10.1016/j.advengsoft.2023.103426

[35] Roger Koenker Quantile Regression Cambridge University Press 2005 10.1017/CBO9780511754098

[36] Cyril Voyant and Philippe Lauret and Gilles Notton and Jean-Laurent Duchaud and Luis Garcia-Gutierrez and Ghjuvan Antone Faggianelli Complex-valued time series based solar irradiance forecast Journal of Renewable and Sustainable Energy 2022 14 6 066502 12 10.1063/5.0128131

[37] Tilmann Gneiting and Adrian E Raftery Strictly Proper Scoring Rules, Prediction, and Estimation Journal of the American Statistical Association 2007 102 477 359–378 10.1198/016214506000001437

[38] H. B. Mann and D. R. Whitney On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other The Annals of Mathematical Statistics 1947 18 1 50 – 60 10.1214/aoms/1177730491

[39] John A Nelder and Roger Mead A simplex method for function minimization The computer journal 1965 7 4 308–313 10.1093/comjnl/7.4.308

[40] Zhiding Liu and Mingyue Cheng and Zhi Li and Zhenya Huang and Qi Liu and Yanhu Xie and Enhong Chen Adaptive Normalization for Non-stationary Time Series Forecasting: A Temporal Slice Perspective Advances in Neural Information Processing Systems 2024 36 10.5555/3666122.3666750

[41] Mijanur Rahman and Mohammad Shakeri and S. Tiong and F. Khatun and N. Amin and J. Pasupuleti and M. K. Hasan Prospective Methodologies in Hybrid Renewable Energy Systems for Energy Prediction Using Artificial Neural Networks Sustainability 2021 13 4 2393 10.3390/SU13042393

[42] Ricardo Petri Silva and B. Zarpelão and Alberto Cano and Sylvio Barbon Junior Time Series Segmentation Based on Stationarity Analysis to Improve New Samples Prediction Sensors (Basel, Switzerland) 2021 21 10.3390/s21217333

[43] Yong Liu Haixu and Wu and Wang Jianmin and Long Mingsheng Non-stationary Transformers: Rethinking the Stationarity in Time Series Forecasting Advances in Neural Information Processing Systems 2022 35 10.48550/arXiv.2205.14415

[44] F. Pan and Haichao Zhao Online sequential extreme learning machine based multilayer perception with output self feedback for time series prediction Journal of Shanghai Jiaotong University (Science) 2013 18 366-375 10.1007/S12204-013-1407-0

[45] Sameer Al-Dahidi and Osama Ayadi and Jehad Adeeb and Mohammad Alrbai and Bashar R. Qawasmeh Extreme Learning Machines for Solar Photovoltaic Power Predictions Energies 2018 11 10 10.3390/en11102725

[46] P. Pinson and H. Nielsen and J. Møller and H. Madsen and G. Kariniotakis Non‐parametric probabilistic forecasts of wind power: required properties and evaluation Wind Energy 2007 10 497-516 10.1002/WE.230

[47] Ke Wang and Yao Zhang and Fan Lin and Jianxue Wang and Morun Zhu Nonparametric Probabilistic Forecasting for Wind Power Generation Using Quadratic Spline Quantile Function and Autoregressive Recurrent Neural Network IEEE Transactions on Sustainable Energy 2022 13 1930-1943 10.1109/TSTE.2022.3175916

[48] S. Perna and Pål Forr Austnes and F. Gerini and Max Chevron and A. Di Fazio and P. De Falco and M. Paolone A Comparative Analysis of Empirical Copula and Quantile Regression Methods for Probabilistic Load Forecasting 2024 18th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS) 2024 1-6 10.1109/PMAPS61648.2024.10667333

[49] J. Bröcker Evaluating raw ensembles with the continuous ranked probability score Quarterly Journal of the Royal Meteorological Society 2012 138 10.1002/qj.1891

[50] M. Zamo and P. Naveau Estimation of the Continuous Ranked Probability Score with Limited Information and Applications to Ensemble Weather Forecasts Mathematical Geosciences 2018 50 209-234 10.1007/s11004-017-9709-7

[51] Luis Garcia-Gutierrez and Cyril Voyant and Gilles Notton and Javier Almorox Evaluation and Comparison of Spatial Clustering for Solar Irradiance Time Series Applied Sciences 2022 12 17 10.3390/app12178529