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# Energy-Consistent Multi-Input Multi-Output Multi-Horizon Extreme Learning Machines with Embedded Reconciliation for Multi-Source Power Forecasting

Elena Magliaro, Cyril Voyant, Yoan Jheelan, Andrea Michiorri, Alan Julien, Yehia Eissa, Jamie Bright

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Elena Magliaro, Cyril Voyant, Yoan Jheelan, Andrea Michiorri, Alan Julien, et al.. Energy-Consistent Multi-Input Multi-Output Multi-Horizon Extreme Learning Machines with Embedded Reconciliation for Multi-Source Power Forecasting. 2025. hal-05360624

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# Energy-Consistent Multi-Input Multi-Output Multi-Horizon Extreme Learning Machines with Embedded Reconciliation for Multi-Source Power Forecasting

Elena Magliaro,¹,* Cyril Voyant,¹,† Yoan Jheelan,¹,²,‡ Andrea

Michiorri,³,§ Alan Julien,¹,¶ Yehia Eissa,¹,** and Jamie Bright,⁵,††

¹Mines Paris, PSL University, O.I.E., 06904 Sophia Antipolis, France

²Université Paris-Saclay, France

³Mines Paris, PSL University, PERSEE, France

⁴Distribution System Operator, UK Power Networks, London, United Kingdom

(Dated: November 12, 2025)

Forecasting energy production from multiple sources is essential for balancing supply and demand, especially in island electricity grids with limited interconnections and no nuclear baseload capacity. This work introduces a multi-input, multi-output, multi-horizon (MIMO-MH) framework that combines Extreme Learning Machines (ELM) with forecast reconciliation. The ELM core couples the robustness of linear closed-form training with the expressiveness of random feature mappings, enabling deep-learning-like accuracy at negligible computational cost. The proposed MIMO-MH jointly predicts thermal, hydropower, solar PV, wind, and import generation over 1–24 h horizons, ensuring built-in coherence across both sources and lead times. Validation on seven years of Corsican grid data (2016–2022, hourly resolution) shows: (i) consistent forecasts across all horizons, (ii) 30–45% error reduction versus persistence, and outperforming NeuralProphet and TimeGPT-1, and (iii) sub-minute training times compatible with operational deployment. Because MIMO-MH forecasts are inherently reconciled, they naturally preserve physical energy balances without any post-processing. This effectively enforces the principle of energy conservation across sources and horizons, making forecast models directly actionable tools for dispatch, reserve sizing, and market bidding. By linking statistical coherence to operational reliability, this framework bridges predictive modeling and grid management, addressing a long-standing limitation of renewable integration in non-interconnected systems.

**Keywords:** Multi-source forecasting; MIMO-MH architecture; Forecast reconciliation; Renewable energy integration; Non-interconnected power systems; Dispatch and market operations

## Significance Statement

This study demonstrates that embedded reconciliation in MIMO-MH forecasting ensures physical coherence required by grids—enabling dispatch-ready forecasts without post-processing—achieving up to 45% error reduction over baselines.

## Highlights:

* Introduces a multi-input, multi-output, multi-horizon (MIMO-MH) framework for joint forecasting of heterogeneous energy sources.

* Embeds forecast reconciliation within the ELM structure, producing inherently coherent multi-source and multi-horizon predictions.

* Achieves 30–45% lower errors than persistence, outperforming NeuralProphet and TimeGPT-1 on seven years of Corsican grid data.

* Ensures sub-minute training times, enabling real-time retraining and operational deployment.

## I. INTRODUCTION

The global energy transition increasingly relies on variable renewable energy sources (RES) such as wind and solar, whose intermittency challenges the stability of electrical systems [1, 2]. Accurate short-term forecasting has therefore become a key operational tool to balance demand and supply, optimize reserves, and limit fossil dependence. This need is particularly acute in non-interconnected zones (NIZ), where limited interconnections [3] and a lack of a nuclear baseload capacity amplify imbalance risks [4].

Corsica represents a typical NIZ and operates under EDF SEI (Systèmes Énergétiques Insulaires), which acts simultaneously as transmission system operator (TSO), distribution system operator (DSO), and sole electricity supplier [5]. This vertically integrated structure implements centralized dispatch within a regulated framework, without day-ahead market (DAM) or intraday market (IDM) [6, 7]. All production sources (thermal, hydropower, solar, wind, and imports via a single subma-

<footnote>* elena.magliaro@minesparis.psl.eu</footnote>

<footnote>† cyril.voyant@minesparis.psl.eu</footnote>

<footnote>‡ yoan.jheelan@universite-paris-saclay.fr</footnote>

<footnote>§ andrea.michiorri@minesparis.psl.eu</footnote>

<footnote>¶ alan.julien@minesparis.psl.eu</footnote>

<footnote>** yehia.eissa@minesparis.psl.eu</footnote>

<footnote>†† jamie.bright@ukpowernetworks.co.uk</footnote>
rine link to Italy i.e. Sardigna) are scheduled directly by EDF SEI. In this configuration, forecasting errors cannot be offset by market mechanisms and translate directly into operational imbalances, requiring immediate reserve activation and costly thermal generation adjustments [8].

The energy mix is structurally fragile: it relies on limited hydropower storage, small thermal units burning heavy- and light-fuel oil, and a single interconnection often constrained by technical or weather conditions. Fuel delivery and storage limitations frequently constrain thermal availability, especially during summer when hydropower reserves are depleted and photovoltaic (PV) generation dominates. In such conditions, reliable multi-horizon forecasting (1–24 h ahead) becomes essential [9]: short horizons (1–6 h) support real-time reserve activation and thermal dispatch, while longer horizons (6–24 h) guide storage management and import scheduling [10]. Without coherent multi-source forecasts, EDF SEI faces higher ancillary services costs, increased fossil dependency, and growing exposure to supply security risks.

Current operational practice relies predominantly on single-input single-output (SISO) models that forecast each energy source independently for a single horizon [11]. While such approaches are computationally tractable, they systematically overlook interdependencies among sources and across time horizons [12]. These interactions are operationally significant: thermal generation frequently substitutes hydropower during dry periods, while imports and PV exhibit anticorrelation due to daily irradiance patterns. More critically, SISO forecasts are typically not reconciled, meaning that individual source predictions do not sum to the total system load. For a vertically integrated operator managing unit commitment (UC [13]), reserve allocation, and import contracts, such incoherence creates direct operational inefficiencies: dispatchers must manually adjust forecasts or face imbalanced schedules that trigger costly corrective actions. The problem is real and acute in NIZ environments where operational margins are tight and corrective resources are limited.

To address this gap, forecasts must be jointly modeled and reconciled to be operationally useful. Multi-input multi-output (MIMO) architectures naturally capture cross-source dependencies by predicting all energy components simultaneously, while reconciliation ensures that disaggregated forecasts remain mathematically consistent with system totals. It is important to keep in mind that in multi-source forecasting systems, the aggregation constraints require forecast coherence: preliminary independent forecasts often violate these, resulting in physically inconsistent and operationally unusable predictions. Forecast reconciliation [14] methods adjust those base forecasts to enforce constraint coherence, improving overall accuracy and ensuring system-wide consistency. This principle is central to hierarchical and grouped time series forecasting and is well formalized in the literature [15]. Extending this approach

to multiple horizons (MIMO-MH) further exploits temporal dependencies, yielding forecasts that are coherent both across sources and across lead times. Unlike conventional forecasting pipelines where reconciliation is applied as an external post-processing step, the proposed MIMO-MH framework embeds hierarchical consistency directly within its architecture: joint training across all sources and horizons naturally enforces energy conservation. This intrinsic coherence eliminates the need for costly iterative adjustments and ensures that forecasts are immediately usable for dispatch without manual intervention. However, deploying such frameworks in real-time operations still requires IT efficiency. Extreme Learning Machines (ELM) (single-layer neural networks with randomly assigned input weights and closed-form output training [16]) offer sub-minute model updates, making MIMO-MH forecasting practically feasible for operational scheduling.

In this paper, a unified framework is introduced: a multi-input multi-output, multi-horizon Extreme Learning Machine (MIMO-MH) enhanced with forecast reconciliation. The framework is evaluated on seven years of hourly electricity generation data from the Corsican power system: a representative NIZ characterized by high renewable penetration, fossil-fuel imports, and strong seasonal variability. It is benchmarked against state-of-the-art models, including NeuralProphet, TimeGPT-1, and LSTM. The ELM core leverages ridge-regularized closed-form output estimation [16], combining computational efficiency with robustness. The main contributions of this study are:

1. Demonstrating the added value of MIMO modeling for capturing interdependencies across energy sources;

2. Extending the framework to MIMO-MH forecasting, ensuring temporal consistency across lead times;

3. Designing an ELM-based architecture that achieves fast and interpretable training through ridge regularization;

4. Embedding reconciliation within the model to guarantee physically coherent multi-source forecasts and improve operational reliability.

Altogether, these contributions deliver a unified, operationally deployable forecasting framework for non-interconnected power systems. The remainder of the article is organized as follows: Section II presents the dataset and methodology, Section III reports the results, Section IV discusses implications and limitations, and Section V presents the conclusions and prospects.

## II. DATA AND METHODS

The section presents the Corsican multi-source power dataset (2016–2022, hourly resolution) and the forecasting framework combining MIMO-MH & ELM architectures
with forecast reconciliation. It specifies data preprocessing steps, model configuration, and benchmarking protocols across 1–24 h horizons, designed to evaluate both predictive accuracy and operational coherence under non-interconnected grid conditions.

## A. Power System Dataset

The analysis relies on hourly electricity production data from EDF SEI covering 7 years (2016–2022). The dataset describes generation in Corsica by source: thermal, hydropower, micro-hydro, solar PV, wind, bioenergy, and imports, together with total production (similar to electrical demand plus or minus the losses incurred) and average generation cost. Corsica constitutes an prototype of Non-Interconnected Zones (NIZ) within the French system. Despite a moderate monthly demand of 0.3–0.4 TWh, the island operates almost autonomously, with no continental interconnection beyond limited AC/DC links. Its energy mix is hybrid and structurally constrained: thermal generation provides around 40% of supply, imports 25–30%, and renewables 30–35%. This configuration results from both climatic and socio-economic drivers. Hydropower varies strongly with Mediterranean rainfall and reservoir management, while a pronounced summer demand peak arises from tourism and air conditioning loads. Solar production is subject to high inter-annual variability, due to the diurnal cycle and the variability of clouds and atmospheric aerosols that attenuate the solar radiation as it traverses through the atmosphere. Wind and biomass remain marginal, limiting balancing flexibility. These conditions make Corsica a unique testbed for short-term forecasting: the system is small enough for full observability yet diverse enough to reproduce the dynamics of larger isolated grids. The open availability of the EDF SEI data offers a rare opportunity to benchmark forecasting frameworks on an island system with transparent operational context and multi-year coverage (https://opendata-corse.edf.fr/explore/). Summary statistics are provided in Table I, and the average seasonal energy mix is illustrated in Fig. 1. Thermal generation supplies the baseload, hydropower and solar alternate seasonally, and imports serve as the residual balancing lever. This interplay between local variability and external support typifies the operating constraints of non-interconnected systems. For model training, all data from January 2016 to December 2020 were used for fitting, while the two most recent years (2021–2022) were reserved for testing. No separate validation set was used; instead, the full training period served for parameter estimation, and out-of-sample evaluation was performed on the consecutive test years to preserve temporal integrity.

<figure><img src="image_2.png" /><figcaption>FIG. 1. Seasonal Electricity Generation Mix in Corsica (2016–2022) by Energy Source.</figcaption></figure>

<table><caption>TABLE I. Summary Statistics of the Corsican Dataset (2016–2022).</caption><thead><tr><th>Variable</th><th>Mean</th><th>Std</th><th>Max</th></tr></thead><tbody><tr><td>Thermal (MWh)</td><td>106.14</td><td>46.57</td><td>311.02</td></tr><tr><td>Hydropower (MWh)</td><td>47.08</td><td>49.61</td><td>196.58</td></tr><tr><td>Micro-hydro (MWh)</td><td>6.25</td><td>5.55</td><td>21.15</td></tr><tr><td>Solar PV (MWh)</td><td>24.53</td><td>34.15</td><td>146.32</td></tr><tr><td>Wind (MWh)</td><td>1.79</td><td>2.70</td><td>17.71</td></tr><tr><td>Bioenergy (MWh)</td><td>0.75</td><td>0.40</td><td>1.91</td></tr><tr><td>Imports (MWh)</td><td>74.26</td><td>36.38</td><td>154.56</td></tr><tr><td>Total production (MWh)</td><td>260.81</td><td>62.17</td><td>510.94</td></tr><tr><td>Prod. cost (€/MWh)</td><td>202.74</td><td>73.37</td><td>480.02</td></tr></tbody></table>

## B. Forecasting Methodologies

The proposed forecasting framework combines recent advances in machine learning with structured multi-source modeling. Figure 2 provides an overview of the end-to-end pipeline, from raw data to benchmarking, highlighting the integration of MIMO and MIMO-MH extensions [17, 18] with ELM [16, 19] and reconciliation [15]. At the core of this framework lies the Extreme Learning Machine, adapted to different forecasting structures and complemented by forecast reconciliation. Benchmark models are included for fair comparison.

### 1. ELM-Based Models

Three forecasting structures based on ELM are evaluated:

* single-input single-output (SISO): Baseline configuration in which each energy source is predicted independently;

* MIMO: Joint learning that predicts all sources si-
<figure><img src="image_3.png" /><figcaption>FIG. 2. End-to-end Forecasting Pipeline. Historical Multi-Source Data are Transformed Into Temporal Features and Processed by an ELM-Based MIMO-MH Architecture with Embedded Reconciliation.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3. Comparison of Three ELM-based Forecasting Architectures on the Variables (S) × Horizons (H) Grid. SISO: One Model per Source and Horizon (fully decoupled). MIMO: One Model per Horizon, Jointly Predicting All Sources. MIMO-MH: a Single Model Spanning All Sources and Horizons.</figcaption></figure>

multaneously to capture cross-dependencies (e.g., substitution between thermal and hydropower, or anticorrelation between solar and imports);

* **MIMO-MH:** Extension in which all horizons (1-24 h) are forecast jointly, promoting temporal consistency and reducing contradictions across successive horizons.

Fig. 3 compares the three variants with a ELM core using: a single-hidden-layer neural network with randomly initialized input weights and analytically computed output weights using ridge regression [16]. This design provides rapid training and reproducibility while maintaining competitive accuracy [20]. The formal definitions of the three variants are given below, using the notation of Fig. 3. In the SISO configuration (Fig. 3), $S \times H$ independent models are trained, one per source and forecast horizon ($x_t \in \mathbb{R}^p \rightarrow \hat{y}_{t+h}^{(s)} \in \mathbb{R}$). The MIMO approach trains one model per horizon, each jointly predicting all sources ($x_t \in \mathbb{R}^{p \times S} \rightarrow \hat{y}_{t+h} \in \mathbb{R}^S$). As a result, in SISO, $x_t$ contains features for one source; in MIMO, $X_t$ stacks

features for all S sources. Finally, the MIMO-MH configuration uses a single model spanning all sources and horizons ($X_t \in \mathbb{R}^{p \times S} \rightarrow \hat{Y}_{t+1:t+H} \in \mathbb{R}^{S \times H}$), thereby ensuring intrinsic coherence across both dimensions. Unless otherwise specified, ELM-based configurations use $L=1000$ hidden neurons, ReLU activation, and Ridge regularization with $\lambda \in [10^{-6}, 10^{-2}]$, selected by in-sample data mining to balance bias and variance. These hyperparameters were fixed across SISO, MIMO, and MIMO-MH (and recompiled versions) to ensure consistent comparison. For more details on the ELM framework, technical configuration, and notation, see Appendix A.

## 2. Benchmark Models

ELM-based configurations are benchmarked against representative forecasting paradigms, following recent work emphasizing the need for balanced comparisons between transparent statistical baselines and complex machine-learning models [21]; see Appendix B for implementation details.

* **Persistences P & P\_cyclo**: Naïve baseline widely used in energy forecasting [22];

* **NeuralProphet**: Seasonal-trend decomposition model designed for robust and interpretable forecasts [23-25], implemented using v0.8.0[26].

* **TimeGPT-1**: Pre-trained transformer providing zero-shot probabilistic forecasts without local training [23-25];

* **LSTM**: Deep recurrent neural network capable of capturing nonlinear dynamics at the cost of higher computational demand [27].

The objective of this benchmarking is not to inflate accuracy through ad hoc tuning, but to quantify the structural gain introduced by the proposed architecture. Comparing MIMO-MH against both simple and trainless models isolates the contribution of multi-source and multi-horizon coupling, rather than that of parameter optimization. This approach follows the principle that scientific progress in forecasting stems from methodological understanding (*why* and *how* a model performs) not from overfitted competition metrics. In this sense, benchmarking serves here as a diagnostic instrument to evaluate the impact of model design on coherence, robustness, and computational efficiency.

## 3. Forecast Reconciliation

Forecast reconciliation adjusts base forecasts so that all aggregation constraints are satisfied, a necessity for physical and operational consistency in multi-source energy systems. Multiple approaches exist (including bottom-up, top-down, and middle-out methods) but the optimal
approach exploits the covariance structure of forecast errors via weighted least squares (optimal reconciliation using WLS optimization), systematically minimizing overall error while enforcing coherence [15, 28]. Formally, given base forecasts $\hat{\mathbf{y}}$, reconciled forecasts in WLS sense, are obtained as $\hat{\mathbf{y}} = P\hat{\mathbf{y}}$ using:

$$P = S(S^{\top} W^{-1} S)^{-1} S^{\top} W^{-1},$$

where S is the summing matrix defining aggregation constraints. The weighting matrix W plays an important role: it ideally corresponds to the covariance of forecast errors, which leads to the so-called Minimum Trace (MinT) estimator that minimizes the total variance of the reconciled forecasts [29]. As an illustrative two-level case ($K$ sources → total), let $\mathbf{y}_t = [P_t^{(1)}, \dots, P_t^{(K)}, C_t]^{\top}$ denote the stacked vector of productions, with $\mathbf{b}_t = [P_t^{(1)}, \dots, P_t^{(K)}]^{\top}$ the base series of sources, $P_t^{(k)}$ the production of source $k$, and $C_t$ the total generation (approximately equal to total consumption up to network losses). Then, $\mathbf{y}_t = \mathbf{S}\mathbf{b}_t$, and $\mathbf{S} = [I_K, \mathbf{1}_{K \times 1}]^{\top}$, where $I_K$ is the $K \times K$ identity matrix and $\mathbf{1}_{K \times 1}$ is a column vector of ones. Intuitively, reconciliation projects the base forecasts onto the subspace where aggregation rules hold, ensuring mathematical coherence while minimizing squared error under the chosen weighting scheme. In the next, an asterisk appended to a model name (e.g., $\square^*$) denotes the WLS-reconciled version.

### C. Comparison Metrics

All ELM-based and benchmark models are evaluated on the Corsican dataset for horizons from 1–24 h. Both predictive accuracy and computational efficiency are assessed. Following standard practice in energy forecasting [17, 22], the main evaluation metrics are listed in Table II. Normalized forms of the classical metrics are also computed to ensure comparability across sources of different magnitudes:

$$\text{nRMSE} = \frac{\text{nRMSE}}{\bar{\mathbf{y}}}, \text{nMAE} = \frac{\text{MAE}}{\bar{\mathbf{y}}}, \text{nMBE} = \frac{\text{MBE}}{\bar{\mathbf{y}}}$$

where $\bar{\mathbf{y}}$ is the mean observed generation. A model is considered physically consistent when $|\text{nMBE}| < 5\%$, since larger biases imply a structural energy imbalance even if other errors appear small. The NICE$^k$ family [30] extends classical metrics by embedding a persistence-based normalization (P):

$$\text{NICE}^k = \frac{\| \hat{\mathbf{y}} - \mathbf{y} \|_k}{\| \mathbf{P} - \mathbf{y} \|_k}, \quad (1)$$

with $\| \cdot \|_k$ the $L^k$ norm. NICE$^k = 0$ denotes a perfect model, NICE$^k = 1$ corresponds to persistence (worst one). The aggregated index $\text{NICE}^\Sigma = \frac{1}{3}(\text{NICE}^1 + \text{NICE}^2 + \text{NICE}^3)$ combines $L^1, L^2$, and $L^3$ norms to evaluate, respectively, median accuracy, global variance, and

<table><caption>TABLE II. Evaluation Metrics Used for Model Comparison.</caption><thead><tr><th>Metric</th><th>Description</th></tr></thead><tbody><tr><td>RMSE</td><td>Root Mean Square Error; emphasizes large deviations, relevant for reserve sizing and peak-load estimation.</td></tr><tr><td>MAE</td><td>Mean Absolute Error; measures global accuracy, robust to outliers.</td></tr><tr><td>MBE</td><td>Mean Bias Error; captures systematic over- or underestimation. Persistence models have MBE = 0 by construction.</td></tr><tr><td>R<sup>2</sup></td><td>Coefficient of determination; proportion of variance explained.</td></tr><tr><td>NICE<sup>k</sup></td><td>Normalized Informed Comparison of Errors; relative L<sup>k</sup> norm of model error to persistence. Lower values indicate higher informational gain.</td></tr><tr><td>Computation time</td><td>Total training and inference time, indicating operational feasibility.</td></tr><tr><td>CR</td><td>Consistency Ratio quantifies structural incoherence between aggregated and disaggregated forecasts.</td></tr></tbody></table>

sensitivity to extremes. Forecast validation in power systems must be multidimensional, a model must be accurate (nRMSE, nMAE), unbiased (nMBE ≈ 0), and stable across regimes (NICE$^\Sigma$). MBE acts as a physical constraint rather than a ranking criterion, while the NICE family quantifies informational gain beyond persistence, thus connecting statistical accuracy to operational reliability. Structural coherence between sources must also be verified, since low individual errors may still yield an unbalanced system total. Several reconciliation performance diagnostics have been proposed in the literature, including structural coherence and trace-based efficiency measures [14, 15, 31]. To remain consistent with these frameworks while keeping implementation straightforward, a normalized $L^1$-based formulation referred to as the Coherence Ratio (CR), is adopted and defined as:

$$\text{CR} = \frac{\sum_t |\sum_s \hat{y}_{s,t} - \hat{y}_{\text{total},t}|}{\sum_t \sum_s \hat{y}_{s,t}}, \quad (2)$$

where $\hat{y}_{s,t}$ and $\hat{y}_{\text{total},t}$ denote, respectively, the predicted power for source $s$ and the total generation. CR = 0 indicates perfect additivity; larger values reveal structural inconsistency. While nMBE measures global energy bias, CR captures instantaneous deviation from energy conservation, a prerequisite for operational validity.

## III. EVALUATION AND RESULTS

The proposed framework is first evaluated through an analysis of statistical dependencies between energy sources, which motivates the use of MIMO-MH formulations. This preliminary assessment quantifies inter-source correlations and
substitution effects before benchmarking forecasting performance. All the simulations are available in https://github.com/YJheelan/MIMO-MH-ELM and https://github.com/YJheelan/Program_SISO-Prophet_Time_gpt_MIMO_MIMO-MH_Reconciliation.

## A. Inter-Source Statistical Dependencies

Prior to forecasting, cross-dependencies among production means are analyzed using the Spearman rank correlation (Fig. 4), a robust nonparametric measure widely applied in energy systems [7, 11]. This choice avoids linearity assumptions and captures monotonic relationships between sources, which are typical in non-Gaussian and seasonally driven datasets. Three dominant interaction patterns emerge: substitution (or negative correlation) between thermal and hydropower, $ρ = -0.54$, synergy (or positive correlation) between hydropower and micro-hydro, $ρ = 0.77$, and opposition (also negative correlation) between solar and imports, $ρ = -0.40$. These patterns reflect both physical complementarity and dispatching strategies applied by EDF SEI, thermal units are mobilized when hydropower is scarce, while imports compensate solar intermittency, consistent with operational reports on insular grids [32]. Beyond pairwise relations, the structured pattern of positive and negative dependencies suggests that certain combinations of sources co-evolve under shared constraints or seasonal storage, hydraulic availability, and operational rules. The proposed MIMO-MH architecture naturally embeds these relationships across both sources and forecast horizons: the multi-input structure captures cross-source couplings in the inputs, while the multi-horizon output enforces temporal consistency across lead times. In practice, this design allows the model to internalize not only physical interactions but also the behavioral imprint of dispatch decisions, bridging statistical learning and operator behavior. Spearman analysis thus provides empirical justification for the adoption of a joint MIMO-MH approach, although its full validation (i.e., whether these learned dependencies translate into improved coherence and accuracy) must ultimately rely on the forecasting experiments presented in the next sections.

## B. Forecasting Performance on Test Sampling

Forecasting accuracy was evaluated over horizons from 1–24 h using normalized errors (nRMSE, nMAE, nMBE) and a persistence-normalized information index from the NICE$^k$ family [30]. At the aggregated model level, $L^2$ component is used, denoted here by NICE$^2$, and instantiated as the ratio of the model nRMSE to that of the simple persistence baseline P. Figures 6 illustrates the horizon-wise evolution, while Fig. 5 summarizes mean performance across all sources and horizons. The proposed MIMO-MH* achieves the lowest normalized

errors among all tested models, with average nRMSE values 30–45% lower than the persistence baseline and systematically below those of NeuralProphet [33] and TimeGPT-1 [24, 25]. Error growth with horizon remains nearly linear for SISO but stabilizes beyond 6 h for MIMO-MH, confirming its temporal regularization effect [17, 18]. From an informational standpoint, the NICE$^2$ values in Fig. 5 reinforce this hierarchy: MIMO-MH* and SISO*/MIMO* achieve the lowest ratios (NICE$^2 \approx 0.6$), $P_{cyclo}$ is intermediate ($\approx 0.77$), and the simple persistence P sits at 1 by construction. The simple persistence baseline (P) performs reasonably only at very short horizons ($h<3$ h), reflecting the strong short-term autocorrelation of the series, whereas the daily-cyclostationary variant ($P_{cyclo}$) exhibits nearly constant errors across all horizons, indicating that while it captures the mean diurnal pattern, it fails to reproduce intra-day variability and weather-driven deviations characteristic of insular grids [7]. This contrast highlights the added value of multi-horizon coupling: by learning both inter-hour dependencies and cross-source adjustments, MIMO-MH maintains coherence across lead times and avoids the discontinuities and bias accumulation observed in recursive or naïve formulations [34]. Operationally, MIMO-MH* remains within the physical consistency threshold (|nMBE| < 5%), ensuring balanced energy totals and realistic substitution between thermal, hydropower and imports, an essential condition for dispatch and reserve sizing in isolated systems [32].

<figure><img src="image_5.png" /><figcaption>FIG. 4. Spearman Correlation Between Energy Sources.</figcaption></figure>

## C. Reconciliation Effects

Forecast reconciliation [15, 28] enforces additivity across hierarchical components by projecting unconstrained forecasts onto the coherent subspace defined
<figure><img src="image_6.png" /><figcaption>FIG. 5. Aggregate Benchmark Across all Sources and Horizons. Cells Report Mean (nRMSE, nMAE, nMBE & NICE²), While Color Encodes Performance (Lower is Better, and NICE²=1 for Simple Persistence).</figcaption></figure>

by the summation matrix *S*. In the present framework, reconciliation is applied as a weighted least-squares projection (MinT/WLS) to all base models (*SISO*, *MIMO*, *MIMO-MH*, *NeuralProphet*, *TimeGPT-1*), yielding reconciled versions denoted by an asterisk (*e.g.* *MIMO-MH*\*). In line with the metrics defined in Sec. II C, the impact of reconciliation along two dimensions is assessed: structural coherence, measured by the Consistency Ratio CR (Eq. 2), and predictive accuracy, measured by the persistence-normalized index NICEΣ (Eq. 1). The ratio CR quantifies the normalized deviation between the predicted total and the sum of disaggregated forecasts (CR=0 indicates perfect additivity), whereas NICEΣ compares the *L*¹, *L*², and *L*³ norms of the model error to those of the simple persistence baseline *P* (NICEΣ=1 for persistence; smaller values indicate informational gain beyond *P*). Table III summarizes mean values over all sources and horizons on the test set. From a structural standpoint, the raw CR values reveal marked differences between models before reconciliation: *MIMO* and *MIMO-MH* exhibit the smallest incoherence (CR ≈ 0.004), indicating that joint training across components already captures much of the aggregation structure. In contrast, *SISO*, *NeuralProphet*, and especially *TimeGPT-1* show larger deviations between the sum of sources and the total (raw CR up to 0.0418), consistent with their independently generated component forecasts. After reconciliation, all models become perfectly coherent by construction, and CR is exactly zero for every configuration. From an accuracy standpoint, reconciliation has a more modest effect. The NICEΣ values, which measure accuracy relative to the persistence baseline, decrease slightly for all models. These changes represent consistent but small gains and do not alter the overall ranking: *MIMO-MH* and *MIMO* remain the most accurate models, followed by *NeuralProphet* and *SISO*, with *TimeGPT-1* slightly worse on this dataset. Figure 7 complements this information-theoretic view with a more classical error

<figure><img src="image_7.png" /><figcaption>FIG. 6. Normalized RMSE (nRMSE) by Source Across Models.</figcaption></figure>

metric by plotting the average reduction in normalized RMSE ($\Delta$nRMSE = nRMSE<sub>raw</sub> - nRMSE<sub>rec</sub>) by model. Larger $\Delta$nRMSE for *SISO* and *TimeGPT-1* confirm that reconciliation mainly corrects models whose base forecasts are structurally less coherent. Conversely, the very small gains for *MIMO-MH* indicate that most of the coherence is already embedded in the architecture, and reconciliation behaves as a light correction rather than a fundamental overhaul. Altogether, these results show that reconciliation is structurally important (it eliminates incoherence for all models) while its impact on pointwise accuracy remains limited once the base model already internalizes the hierarchical structure, as in *MIMO-MH*.

## D. Source-specific Dynamics

Forecast accuracy differs markedly across sources, reflecting intrinsic variability and the degree of cross-dependency captured by the model. Figure 6 reports the normalized RMSE versus horizon (1–24 h) for each energy variable and model. Thermal and imports achieve the lowest errors under *MIMO-MH*\*, consistent with their higher inertia and stronger coupling with the rest of the
<table><caption>TABLE III. Effect of Reconciliation on Structural Coherence and Accuracy. Mean Values Over All Sources and Horizons (Test Set). CR is the Structural Coherence Ratio (CR=0 Indicates Perfect Additivity). NICE$^{Σ}$ is the Persistence-Normalized Accuracy Index (NICE$^{Σ}$=1 for Simple Persistence; Smaller is Better).</caption><thead><tr><th rowspan="2">Model</th><th colspan="2">NICE<sup>Σ</sup></th><th colspan="2">CR</th></tr><tr><th>Raw</th><th>Reconciled</th><th>Raw</th><th>Reconciled</th></tr></thead><tbody><tr><td>SISO</td><td>0.383</td><td>0.379</td><td>0.023</td><td>&lt; 10<sup>-3</sup></td></tr><tr><td>MIMO</td><td>0.376</td><td>0.376</td><td>0.004</td><td>&lt; 10<sup>-3</sup></td></tr><tr><td>MIMO-MH</td><td>0.375</td><td>0.375</td><td>0.004</td><td>&lt; 10<sup>-3</sup></td></tr><tr><td>NeuralProphet</td><td>0.372</td><td>0.370</td><td>0.022</td><td>&lt; 10<sup>-3</sup></td></tr><tr><td>TimeGPT-1</td><td>0.396</td><td>0.389</td><td>0.042</td><td>&lt; 10<sup>-3</sup></td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>FIG. 7. Average Improvement in Normalized RMSE After Reconciliation by Model.</figcaption></figure>

mix. Hydropower and micro-hydro also benefit from joint learning (MIMO/MIMO-MH), indicating that inter-basin correlations and diurnal patterns are effectively captured. In Fig. 8, the SISO configuration fails to detect a cloud event at 14:00, producing a ~35 MW imbalance driven by coupled errors in solar, thermal, and imports. The MIMO-MH model correctly anticipates the solar drop and adjusts dispatchable sources, limiting the deviation to <8 MW. Cross-source learning therefore enforces physical balance and prevents operationally costly forecast divergence. For most sources, MIMO-MH* achieves the lowest errors beyond 6 h; however, for Solar PV, NeuralProphet remains best across all horizons. Wind and bioenergy remain the least predictable due to low installed capacity and sparse data [35], yielding larger dispersion and stronger horizon sensitivity. Overall, MIMO-MH* reduces normalized errors by approximately 30-45% relative to the persistence baseline across most sources and horizons, while maintaining a more controlled error growth with lead time. This stability is operationally critical, as it limits the accumulation of horizon-dependent errors typical of recursive or horizon-decoupled approaches. Although LSTM-based architectures [27] can reach similar accuracy, their computational cost is substantially higher (see Sec. III E)).

## E. Execution Time

Computational efficiency is an important criterion for operational deployment in power systems, where models must be retrained daily or hourly as new measurements arrive. Table IV and Figure 9 summarize the total training and inference times together with the effective model size. ELM-based architectures (SISO, MIMO, and MIMO-MH) complete training within one minute on a standard CPU, enabling near real-time retraining without the need for GPU. This rapid convergence stems from the closed-form least-squares solution of the output layer [16], which avoids iterative backpropagation. MIMO-MH further reduces complexity by jointly learning all sources and horizons in a single run, decreasing the parameter count by one order of magnitude relative to SISO. In contrast, recurrent models such as LSTM require more than 5 minutes of training for equivalent input size, and are therefore less compatible with frequent model updates required in solar and hybrid-grid forecasting [20]. Although TimeGPT-1 relies on a large transformer (~10⁸ parameters), inference is served server-side; in the batched cross-validation it achieved among the lowest client-side latencies (0.21 min; Table IV). NeuralProphet shows intermediate runtime in the setup; as the number of series and exogenous regressors increases, wall-clock time rises markedly compared with ELM and server-side TimeGPT-1 (Table IV). The reconciliation step, involving the estimation and inversion of a covariance matrix of dimension S×S (here S=8), adds a constant computational overhead of less than 0.5 min, negligible compared to model training. Overall, ELM-based MIMO-MH* achieves the optimal trade-off between speed, coherence, and accuracy, allowing deployment at sub-hourly refresh rates in operational energy-management systems.

## IV. DISCUSSION

Three findings emerge from the analysis, consistent with the physical structure of non-interconnected power systems.

* Only renewable sources require explicit forecasting. In islanded systems, the truly variable components are solar, wind, and, to a lesser extent, hydropower. Thermal generation, imports, and self-consumption behave as balancing variables: their values are determined by the residual demand once renewable output and storage levels are known. The observed variability of hydropower is mainly seasonal, constrained by reservoir stocks that are reduced in summer;

* Global modeling ensures physical and statistical coherence. The proposed MIMO-MH framework jointly predicts all sources within a consistent energy balance, integrating renewables, dispatchable units,
<figure><img src="image_9.png" /><figcaption>FIG. 8. SISO vs MIMO-MH: Cross-Source Learning Eliminates Solar-Driven Imbalances and Stabilizes Net-Load Forecast Errors.</figcaption></figure>

<table><caption>TABLE IV. Training and Inference Times (24 h) and Total Number of Parameters.</caption><thead><tr><th>Model</th><th>Training (min)</th><th>Inference (min)</th><th># Parameters<sup>a</sup></th></tr></thead><tbody><tr><td>Persistence</td><td>—</td><td>&lt; 0.01</td><td>0</td></tr><tr><td>NeuralProphet</td><td>—</td><td>—</td><td>~ 10<sup>3</sup>–10<sup>4</sup></td></tr><tr><td>TimeGPT-1 (Transformer)</td><td>—</td><td>0.21<sup>b</sup></td><td>~ 10<sup>8</sup> (server-side, pre-trained; no local training)</td></tr><tr><td>SISO (ELM)</td><td>—</td><td>—</td><td>4,992,192</td></tr><tr><td>MIMO (ELM)</td><td>0.8</td><td>0.1</td><td>4,824,192</td></tr><tr><td>MIMO--MH (ELM)</td><td>0.9</td><td>0.1</td><td>385,192</td></tr><tr><td>LSTM (2 layers)</td><td>5.3</td><td>0.8</td><td>68,288</td></tr><tr><td>Reconciliation<sup>c</sup></td><td>0.4</td><td>0.05</td><td>O(S<sup>2</sup>) ≈ 100</td></tr></tbody></table>

and residual demand in a single system. Auto-consumption, which effectively appears as a decrease in demand, is implicitly captured in this joint representation. Unlike independent forecasts, this global approach preserves additivity and temporal smoothness without external reconciliation, reproducing the operational coupling between PV, wind, hydro, and imports;

* ELM-based implementation enables real-time operation. Closed-form training allows retraining on standard CPU in less than one minute, while maintaining comparable accuracy to recurrent or transformer-based models [27, 36, 37]. This computational efficiency makes MIMO-MH* directly compatible with the short update cycles required for energy management in isolated grids [16, 20].

Forecast skill remains weaker for marginal or poorly instrumented sources such as wind and bioenergy, where signal-to-noise ratios are low [35]. The proposed framework is purely deterministic: it provides point forecasts

without uncertainty bounds. While reconciliation enforces aggregate consistency [15], the absence of probabilistic modeling limits its applicability in stochastic or reserve-optimization settings [38]. This design choice favors real-time deployment and operational simplicity, at the expense of uncertainty quantification and risk-aware decision-making. Therefore, in practice, reconciliation aligns statistical coherence with operational feasibility, so that dispatch schedules derived from forecasts remain consistent with physical energy balances and reservoir constraints.

V. CONCLUSION

This work presents a unified forecasting framework combining multi-input, multi-output modeling (MIMO), multi-horizon learning (MIMO-MH), Extreme Learning Machines (ELM), and forecast reconciliation. Applied to the Corsican power system (NIZ), the framework consistently

<footnote>a Assumptions: $S=8$ sources, $H=24$ horizons, $p=24$ input features, $L=1000$ hidden neurons (ELM), and $h=64$ hidden units (LSTM). For TimeGPT-1, the underlying Transformer has $O(10^8)$ pre-trained weights, but no local parameters are updated during inference.</footnote>

<footnote>b Client-side wall-clock for one batched cross-validation call (8 series, 720 effective windows, $h=24 \Rightarrow 138,240$ forecasts): API 12.612s, total 12.771s; includes network latency.</footnote>

<footnote>c Includes estimation of covariance $W$ and matrix inversion for $S=8$; same for all models.</footnote>
<figure><img src="image_10.png" /><figcaption>FIG. 9. Execution Time per Model (other models omitted for clarity).</figcaption></figure>

outperforms classical and deep-learning baselines such as NeuralProphet [33], TimeGPT-1 [24], and LSTM [27], achieving coherent, accurate, and computationally efficient forecasts across all energy sources.

The reconciliation stage plays a central role: it enforces physical consistency between disaggregated and aggregated forecasts, ensuring that the sum of individual sources equals total system generation at every horizon. This property is not only statistically desirable but essential for operational feasibility, as imbalances directly translate into reserve activation or curtailment. By embedding reconciliation within a MIMO-MH structure, coherence becomes intrinsic rather than ex-post, allowing forecasts to evolve smoothly across both sources and horizons. Such stability is particularly valuable for non-interconnected grids, where balancing flexibility is limited and dispatch decisions must rely on internally consistent trajectories of renewable, thermal, and import components.

Operationally, the proposed framework provides a robust basis for integrating forecasting into energy-management workflows. Although the Corsican system operates under centralized dispatch rather than a market structure, the same principles apply to scheduling, reserve allocation, and UC optimization. Coherent multi-horizon forecasts prevent contradictory decisions between consecutive time blocks, improving production planning and storage management. In systems operating under market logic (such as DAM or IDM exchanges) the same coherence would enable consistent bidding and real-time adjustment of positions. By jointly modeling renewables, imports, and residual demand, the approach links technical forecast reliability to economic and operational efficiency, providing operators and aggregators with actionable signals for dispatch and flexibility planning. While demonstrated on an isolated system, the framework remains transferable to regional or microgrid contexts within larger networks, especially for modeling distributed generation and self-consumption. However, its

direct application to continental systems dominated by nuclear or large baseload units would require additional constraints on dispatchability and market coupling.

Future work will extend this deterministic framework toward probabilistic forecasting [38], hybrid ELM-Transformer architectures [36], and ensemble meteorological inputs [39]. Such developments will enhance robustness and quantify uncertainty, paving the way for market-coupled forecasting systems capable of supporting both operational reliability and optimal renewable valorization.

## ACKNOWLEDGMENTS

This research was carried out under the SciDoSol Chair (Sciences of Data applied to Solar Energy) hosted at the O.I.E. Laboratory, Mines Paris-PSL, with support from its founding sponsors (TotalEnergies, RTE, Somfy, TSE and CalibSun) and Mines Paris Foundation.

## Appendix A: MIMO Framework and ELM Architecture Details

Historical energy production and consumption data (solar, wind, hydropower, thermal, imports, bioenergy) are preprocessed through a sliding window and encoded with temporal auto-indicators with $T = 24$ h such as $sin(2\pi t/T)$ and $cos(2\pi t/T)$, capturing diurnal seasonality. These features are then propagated through an ELM network that jointly predicts all energy sources in a multi-output configuration. The outputs represent coherent generation forecasts over 1-24 h horizons, while the evaluation stage aggregates classical and informational metrics (nRMSE, nMAE, $R^2$, and NICE$^\Sigma$). This global pipeline corresponds to the MIMO-MH (multi-input, multi-output, multi-horizon) setting, in which all sources and horizons are learned simultaneously, ensuring both inter-source and inter-temporal consistency. Figure 10 details the internal structure of the ELM used within the MIMO framework. Each input vector aggregates: (i) normalized historical observations of all sources over a fixed window (here 48 time steps), (ii) exogenous temporal features (hour, day, season, $sin(2\pi t/T)$, $cos(2\pi t/T)$), and (iii) operational indices such as total demand and imports. The hidden layer is composed of $N = 1000$ neurons with randomly initialized weights $W$ and biases $b$. These parameters remain fixed after initialization; only the output weights $W_{out}$ are estimated analytically using Moore-Penrose pseudo-inversion $W_{out} = H^\dagger Y$, where $H = \max(0, XW + b)$ denotes the ReLU-activated hidden matrix. This closed-form training ensures high computational efficiency and numerical stability, making it suitable for frequent retraining in operational forecasting environments. Outputs correspond to the predicted energy trajectories $\hat{Y} = [\hat{P}_{\text{Tot}}, \hat{P}_{\text{T}}, \hat{P}_{\text{H}}, \hat{P}_{\text{S}}, \hat{P}_{\text{W}}, \hat{P}_{\text{B}}, \hat{P}_{\text{I}}]$ for
all sources and horizons, consistent by design under the MIMO-MH formulation.

## Appendix B: Model Descriptions

This appendix summarizes the models compared in this study. Data sources, feature construction, and reconciliation procedures are detailed in Section II.

### Notation

All models predict energy production for S sources over H forecast horizons using a look-back window of length W.

$\mathbf{X}_t \in \mathbb{R}^{W \times p}$: input W past steps with p features,
$\hat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{S \times H}$: matrix for S sources & H horizons,
$s \in \{1, \dots, S\}$: index of an individual energy source,
$h \in \{1, \dots, H\}$: forecast horizon index (hours ahead).

The operator $\mathrm{vec}(\cdot)$ stacks the columns of a matrix into a single column vector. All models use the same dataset, feature set, and evaluation protocol described in the main text.

## 1. ELM-based Models

All ELM variants share the same single-hidden-layer architecture with random input weights and a closed-form ridge-regularized output layer [16]. For an input $\mathbf{x}_t \in \mathbb{R}^p$ and L hidden units with activation function $g(\cdot)$, the hidden representation is

$$ \mathbf{h}_t = [g(\mathbf{w}_1^\top \mathbf{x}_t + b_1), \dots, g(\mathbf{w}_L^\top \mathbf{x}_t + b_L)]^\top \in \mathbb{R}^L, $$

and predictions are obtained by a single ridge-regularized least-squares solve for the output weights.

### a. Single Input & Single Output (SISO)

One model per source and per horizon (independent fits):

$$ \hat{y}_{t+h}^{(s)} = \sum_{j=1}^{L} \beta_{j}^{(s,h)} g(\mathbf{w}_{j}^{\top} \mathbf{x}_{t}^{(s)} + b_{j}). $$

Training uses L = 1000 hidden units, ReLU activation, and a ridge penalty selected on a validation set. This configuration is extremely fast and robust but ignores cross-source correlations; reconciliation is recommended when aggregate coherence is required.

### b. Multiple Inputs & Multiple Outputs (MIMO)

Jointly predicts all sources for a given horizon h, capturing cross-source dependencies:

$$ \hat{\mathbf{y}}_{t+h} = \mathbf{H}_t \mathbf{B}^{(h)}, \quad \hat{\mathbf{y}}_{t+h} \in \mathbb{R}^S, $$

where $\mathbf{H}_t = [\mathbf{h}_t^{(1)}, \dots, \mathbf{h}_t^{(S)}]^\top$ stacks hidden responses. The same hyperparameters as in SISO are used, with one output matrix $\mathbf{B}^{(h)}$ per horizon. This structure improves accuracy and inter-source coherence; reconciliation brings only marginal gains.

### c. Multiple Inputs & Multiple Outputs with Multiple Horizons (MIMO-MH)

Produces all sources and all horizons in a single forward pass, ensuring built-in temporal consistency:

$$ \mathrm{vec}(\hat{\mathbf{Y}}_{t+1:t+H}) = \mathbf{H}_t \mathbf{B}, \quad \hat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{S \times H}. $$

Here $\mathbf{B} \in \mathbb{R}^{L \times (SH)}$ maps hidden features to all outputs simultaneously. Training uses the same hidden layer and activation as above, with a single ridge-regularized solve. This design reduces error accumulation versus recursive one-steps and yields smooth horizon profiles; reconciliation is optional depending on operational constraints.

## 2. Benchmark Models

All benchmarks follow the same notation as the ELM-based models for input $\mathbf{X}_t$ and output $\hat{\mathbf{Y}}_{t+1:t+H}$.

### a. Persistences

Two naïve baselines are considered: (i) simple persistence, P, which repeats the last observed value, and (ii) daily persistence, or cyclo-persistence ($P_{cyclo}$), which exploits the cyclostationary nature of energy time series by repeating the value from the same hour of the previous day (24 h seasonal naïve). This variant assumes that the underlying process is cyclostationary, i.e., its statistical properties repeat periodically with a 24 h cycle, a realistic assumption for solar and demand-related variables [40]. Both are applied independently to each energy source, with $h \in \{1, \dots, 24\}$ denoting the forecast horizon.

(i) Simple persistence (P): $\hat{y}_{t+h}^{(s)} = y_t^{(s)}$,

(ii) Daily persistence (24 h, $P_{cyclo}$): $\hat{y}_{t+h}^{(s)} = y_{t+h-24}^{(s)}$.

No training is required: the simple variant uses the latest observation, whereas the daily variant requires at least one day of history. Simple persistence remains competitive at very short horizons for highly autocorrelated series, while the daily form leverages diurnal periodicity
<figure><img src="image_11.png" /><figcaption>FIG. 10. Technical Implementation of Extreme Learning Machine (ELM) Used for Time Series Forecasting.</figcaption></figure>

and generally performs better beyond 3–4 h. Either variant may be post-processed via reconciliation (□*) to enforce coherence between individual sources and the aggregated system total.

## b. NeuralProphet

NeuralProphet extends Facebook Prophet [33] by integrating neural components for automatic feature extraction and nonlinear autoregression [41]. It retains Prophet additive and interpretable structure (trend, seasonality, and event effects) while embedding autoregressive (AR) terms and lagged covariates within a PyTorch implementation [42]:

$$y(t) = T(t) + S(t) + AR(t) + \mathbf{X}(t) + E(t) + \varepsilon(t),$$

where $T(t)$ denotes the trend component, $S(t)$ represents multiple seasonality (via Fourier terms), $AR(t)$ models lag dependencies, $\mathbf{X}(t)$ gathers exogenous regressors, and $E(t)$ encodes event or holiday effects. Training uses mini-batch stochastic gradient descent with a Huber loss and early stopping based on validation error. NeuralProphet thus combines the interpretability of

statistical decomposition with the flexibility of neural architectures, making it suitable for time series dominated by trend and strong periodic components at short horizons. In all experiments, we used the Python implementation NeuralProphet v0.8.0 (MIT-licensed, available at https://github.com/ourownstory/neural_prophet).

## c. TimeGPT

TimeGPT-1 [24, 25] is a large pre-trained transformer model developed by Nixla for general-purpose time-series forecasting. It adopts an encoder-decoder architecture trained on millions of heterogeneous series, enabling zero-shot and few-shot forecasting without task-specific fine-tuning. Input sequences $\mathbf{X}_t$ are embedded with positional encodings and processed through multi-head self- and cross-attention layers, while decoder hidden states are projected to the forecast horizon via a linear output head:

$$\mathrm{vec}(\hat{\mathbf{Y}}_{t+1:t+h}) = \mathbf{W}_o \mathrm{vec}(\mathbf{H}^{\mathrm{dec}}) + \mathbf{b}_o \in \mathbb{R}^{Sh}.$$

In this study, forecasts are generated via the standardized Nixla API using fixed input window $W$ and forecast horizon $h$, without local re-training. TimeGPT-1
offers strong cross-domain generalization and negligible deployment cost, but its accuracy depends on the representativeness of the target data with respect to the pre-training corpus.

d. Long Short-Term Memory networks

LSTM [27] are recurrent architectures designed to capture long-range temporal dependencies through gated cell states, mitigating vanishing-gradient effects that limit

standard recurrent neural networks. In this work, a two-layer LSTM encoder processes the input sequence $X_t$; the final hidden state feeds a multi-horizon output layer:

$$ \mathrm{vec}(\hat{\mathbf{Y}}_{t+1:t+h}) = \mathbf{W}_{\mathrm{out}}\mathbf{h}_{t}^{(\mathrm{top})} + \mathbf{b}_{\mathrm{out}} \in \mathbb{R}^{S h}. $$

The architecture uses two stacked layers with 64 hidden units each, a dropout rate of 0.2, ReLU activation on dense heads, and Adam optimization ($10^{-3}$) with early stopping on validation loss. LSTM models capture nonlinear and regime-dependent dynamics effectively, but their higher computational cost limits real-time retraining compared with ELM-based approaches.

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