The growing integration of renewable energy into electricity markets has introduced significant challenges and opportunities. With the increasing deployment of solar, wind, and other renewable sources, power systems are experiencing a transition from traditional centralized generation to more decentralized and variable power generation. Forecasting approaches have experienced multiple paradigm shifts in recent years. From the dominant position of classical statistics, machine learning (ML) has come into prominence over the last decades, with even more recent advances of deep learning (DL). Each shift in forecasting methodology was driven by a similar motivation, which is the necessity to model the increasing variability and nonlinearity in data [1,2]. Classical methods such as time series models (ARIMA, ETS) are more interpretable, easier to understand and implement, and may be sufficient for short-horizon tasks [3]. However, they usually rely on linear assumptions that can be restrictive for complex, non-stationary signals. ML models (decision trees, random forests, support vector machines (SVM), etc.) can account better for nonlinearities, and DL approaches. In a similar sense, the shift to renewables is essential to reduce carbon emissions and achieve sustainable energy goals, but also creates numerous challenges in terms of grid stability, demand-supply balancing, and electricity price volatility [4].

The efficiency of electricity markets in the future energy world, therefore, also primarily depends on the existence of accurate forecasting models. The forecasting models can forecast energy consumption, production, and market trends, thus enabling grid operators, energy traders, policymakers, and other stakeholders to make informed decisions [3,4]. In the meantime, the volatility and intermittency of renewable energy make this task more difficult and require advanced forecasting techniques beyond those traditionally used [5].

Traditional approaches use statistical techniques such as ARIMA and ETS, wherein the time series are modeled as functions of their lagged values. Though they are utilized today (when short-horizon prediction is necessary), such models basically circumvent the nonlinear character and other complex dependencies, as can be seen among the features of renewable energy generation [6]. For the purposes of evading the quirks of statistical modeling, scientists started to develop ML and DL methods for forecasting because of their inherent capability to fit complex relationships and to harvest rich information from huge data. Various approaches were used to accomplish this, and models such as ANN, SVM, and LSTM networks were reported to be of greater accuracy for predicting energy demand/generation [7]. Even these approaches, however, have drawbacks like high cost of computation, overfitting, and demand for a huge amount of training data.

Hybrid approaches are one of them that have become very popular over time. Hybrid models are simply a collection of a number of forecasting models, which work in conjunction with each other to leverage the strength of each and also enhance forecast accuracy and sensitivity to market patterns. One example is the coupling of statistical forecasting models and deep learning networks (CNNs, LSTMs) that can contribute to better linear and non-linear trend identification for more accurate and robust forecasts [8–12]. Other hybrid methods using ensemble-based techniques are ensemble learning, wavelet transforms, and optimization algorithms [13,14]. They typically work better than traditional methods but also have drawbacks in model interpretability, integration complexity, and computational efficiency.

In this study, we propose FFDEM (Fractional–Fractal Deep Ensemble Model), a hybrid approach that combines fractional long-memory (FARIMA), fractal roughness (MFDFA), and deep learning algorithms (CNN-LSTM/stacking). The dual injection of long-range dependence and multiscale roughness is what sets FFDEM apart from previous hybrids, and what causes the observed error reduction. In the design and development of FFDEM, fractional-order statistical models, fractal-based feature engineering, and deep learning architectures are all designed to work towards improving the forecast accuracy of renewable energy generation. Fractional autoregressive integrated moving average (FARIMA) and fractional exponential smoothing (FETS) models are investigated for the purpose of capturing long-memory dependence in energy time-series, and multifractal detrended fluctuation analysis (MFDFA) is employed for intermittency analysis of renewable energy generation. This paper will present and explore the viability and effectiveness of hybrid forecasting techniques for renewable energy integration in electricity markets. We also benchmark with transformer and decomposition-style long-horizon forecasters (e.g., Informer, Autoformer, FEDformer) as well as recent neural operators such as TimesNet. The key contributions of this research are as follows:

A hybrid framework melding ML/DL with fractional-order statistics and fractal-based feature engineering to enhance forecasting accuracy and efficiency.

Renewable and market forecasting have advanced rapidly in recent years. Comprehensive reviews catalog progress in photovoltaic (PV) forecasting and deep learning pipelines, emphasizing data quality, exogenous drivers, and evaluation protocols [2,9,15]. Concurrently, domain studies continue to motivate high-fidelity forecasting by linking prediction quality to market viability and planning in high-renewable settings [16]. Beyond point accuracy, recent works advocate reliability and robustness through probabilistic treatment and uncertainty quantification (e.g., Gaussian-process and tree-based methods adapted for calibrated predictive distributions) [7,17].

Modern neural approaches extend classical baselines by exploiting temporal hierarchies, exogenous weather, and transfer across locations or markets. Deep hybrids (e.g., CNN/LSTM stacks) and transfer-learning setups have improved short-term market and generation predictions by learning multiscale patterns while reusing representations across datasets [4,9,18]. Deterministic forecasting guided by multi-location meteorology demonstrates additional gains when spatial context is injected at training time [19]. In parallel, demand-side and load-focused studies report accuracy improvements from tailored deep models and careful data engineering, further underscoring the role of inductive bias and exogenous features [14,20]. At the operations interface, recent work links forecast design to market objectives and decision quality. Value-oriented forecasting integrates market structure into the learning target and loss design to improve downstream clearing outcomes.

Reinforcement learning has also been combined with time-series predictors to handle uncertainty and adapt bidding/dispatch actions in the presence of variable renewables [11]. These directions highlight the need for forecasting models that are not only accurate but also decision-aware and robust under distribution shift. Despite these advances, two gaps remain. First, long-memory and intermittency—core properties of renewable generation and price series—are often addressed implicitly via deep architectures rather than through explicit statistical descriptors. Second, hybrid systems rarely fuse such descriptors with learned representations under a leakage-safe ensemble. The framework in this paper targets these gaps by integrating fractional-order statistics and multifractal descriptors with neural sequence models in a unified, interpretable ensemble, complementing the trends above while remaining compatible with probabilistic and decision-aware extensions [13,21].

The proposed FFDEM which fuses fractional long-memory (FARIMA), fractal roughness (MFDFA), and deep learners (CNN–LSTM/stacking). This dual injection of long-range dependence and multi-scale roughness is unique to FFDEM among prior hybrids and underlies the achieved error reductions. Recent transformer and decomposition-style forecasters (Informer, Autoformer, FEDformer) and neural operators such as TimesNet set strong baselines for long-horizon series; we benchmark these models under matched settings. Table 1 summarizes representative techniques, highlighting strengths and limitations.

This comparative review underscores the promise of adaptive, scalable hybrid approaches that balance accuracy with computational efficiency and real-time feasibility. However, scalability, interpretability, and deployment challenges remain [27,28]. Explainability methods integrated into DL architectures could mitigate some of these issues [29]. Building on these observations, our work proposes a hybrid framework combining fractional/fractal models with ML and DL, aiming to deliver accurate and practical solutions for electricity market forecasting.

We develop the Ensemble-Integrated Deep Forecasting Model (FFDEM (Fractional–Fractal Deep Ensemble Model)), a hybrid pipeline that blends statistical, ML, and DL components. Fig. 1 gives a high-level overview.

The fractional and fractal integration were used in which we detail (i) FARIMA with fractional differencing (1−B)d for long-memory; (ii) FETS for fractional smoothing; (iii) MFDFA fluctuation functions Fq(s) as fractal features. These are concatenated with raw/lagged covariates and consumed by CNN–LSTM blocks and a meta-learner as in Fig. 1.

To provide a clearer understanding of the integration of fractional and fractal components in FFDEM. The following annotation steps are for FARIMA, FETS, and MFDFA.

First, the input time series undergoes fractional differencing via FARIMA to preserve long-range memory while stabilizing variance. Second, Fractional Exponential Time Smoothing (FETS) is applied to retain temporal persistence and reduce high-frequency noise. Third, Multifractal Detrended Fluctuation Analysis (MFDFA) is employed to extract non-linear and multi-scale fluctuation features, capturing the intrinsic fractal behavior of renewable energy signals. These extracted statistical and fractal features are concatenated with the raw lagged inputs and passed into the CNN–LSTM backbone, enabling the model to jointly learn from both handcrafted memory-aware features and deep hierarchical representations. Fig. 2 demonstrates the integration of the proposed FFDEM flow from raw data through fractional processing, fractal feature extraction, and deep ensemble learning. This visual illustration helps distinguish our integration from traditional hybrid approaches.

We quantify training and inference cost on a single NVIDIA RTX 3090 (24 GB) with 64 GB RAM, PyTorch 2.1/CUDA 12.1, batch size 64, input window T=12, horizon H=12. Reports seconds/epoch, total minutes (20 epochs), inference latency (ms/step), parameter count (M), and peak GPU memory (GB) averaged over 5 seeds. Relative to the strongest baseline, FFDEM increases train time/epoch by +9.7% (62→68 s), inference latency by +14.3% (2.8→3.2 ms/step), parameters by +13.3% (11.3→12.8 M), and peak memory by +16.0% (5.0→5.8 GB). Given the accuracy gains (e.g., MAE 7.8→6.4; RMSE 12.5→10.8), this overhead is modest and predictable. We follow a fixed protocol: (i) wall-clock with warm-up (3 batches) and synchronized timers, (ii) latency measured on the test loader with autocast disabled, (iii) peak memory via torch.cuda.max_memory_allocated() reset per run.

FFDEM integrates complementary components as follows.

We first select features using recursive feature elimination and correlation screening [3,4]. Each base learner is then trained independently on historical data, allowing it to specialize. Predictions are combined with performance-based weights on a validation split; alternatively, a meta-learner (e.g., XGBoost or a shallow NN) stacks the base outputs to further reduce bias/variance [30]. This strategy exploits model diversity: ARIMA/ETS handle linear/seasonal structure; SVR/RF capture nonlinearities; CNN/LSTM learn complex temporal patterns. The final forecast is a calibrated aggregate that is generally more accurate and stable than any single model. Therefore, the forecasting models’ accuracy and generalization ability are improved by mining the implicit physics information from the selective dataset [31].

Consider a (uni/multi)variate time series 𝒳={x1,…,xt} (e.g., generation, prices, weather). We seek y^t+h=f(Xt;θ) at horizon h: (12)y^t+h=f(Xt;θ).

Linear dynamics use ARIMA (13)Υt=c+∑i=1pϕiYt−i+∑j=1qθjεt−j+εt,with short-term smoothing via ETS (14)y^t+1=αyt+(1−α)y^t,0<α<1.

SVR minimizes (15)minω,b 12‖ω‖2,subject to (16)yt−(ω⊤xt+b)≤ϵ,(ω⊤xt+b)−yt≤ϵ,while Random Forests average M trees: (17)y^t=1M∑m=1Mhm(xt).

DL blocks follow the LSTM update (18)ft=σ(Wf[ht−1,xt]+bf),it=σ(Wi[ht−1,xt]+bi),ot=σ(Wo[ht−1,xt]+bo),ct=ft⊙ct−1+it⊙tanh(Wc[ht−1,xt]+bc),ht=ot⊙tanh⁡(ct),and a CNN mapping (19)ht,j=σ(∑i=0kwixt−i,j+b).

A hybrid CNN–LSTM composes these: (20)ht=LSTM(CNN(Xt)).

Ensembling combines N base predictions (21)y^t=∑i=1Nwiy^i,∑i=1Nwi=1,optionally refined by a meta-learner (22)y^t=g(y^1,…,y^N;θ).

The Informer, Autoformer, FEDformer, TimesNet (and PatchTST/iTransformer where relevant) as benchmarks under identical settings. FFDEM retains its advantage across horizons. We compare CNN–LSTM only vs. +FARIMA, +FETS, +MFDFA, and full FFDEM, demonstrating each module’s incremental contributions. We report paired tests (e.g., Wilcoxon) across rolling windows and 95% CIs for MAE/RMSE; FFDEM’s gains are statistically significant (p<0.05) in most settings. We report SHAP-based global/local attributions and visualize how MFDFA-derived features influence predictions during volatility; attention/gradient maps are included where applicable. The performance of the proposed FFDEM framework was evaluated against multiple baseline models, including ARIMA, SVR, LSTM, and CNN–LSTM, using standard regression metrics (MAE, RMSE, MAPE, and R2). Across all evaluation settings, FFDEM consistently outperformed classical statistical models, standalone machine learning methods, and single deep learning architectures.

The dataset was divided strictly by time to prevent any leakage of future information. The chronological sequence was preserved, with the earliest 70% of samples used for model training, the following 15% for validation, and the most recent 15% for testing. No random shuffling was performed. Hyperparameter tuning was conducted exclusively on the validation set, while the test set remained unseen during model development. A rolling-origin evaluation (time-series cross-validation) was additionally implemented to confirm stability across forecast horizons. Fig. 3 illustrates the temporal segmentation used in the FFDEM framework, ensuring a fair and leak-free comparison among all baseline and hybrid models.

Table 2 presents the robustness evaluation of the proposed method, in which we used a fixed chronological split. We adopt three complementary checks:

(i) Time-aware K-fold CV. We partition the timeline into K=5 consecutive folds; for each fold k, we train on all data strictly before its start and validate/test inside the fold (no shuffling). Metrics are averaged across folds.

(ii) Rolling-origin evaluation (ROE). Starting from an initial window [t0,t1), we iteratively advance the origin by Δ (two weeks for 5-min data; one month for hourly data), retrain on the expanding history, and forecast the next horizon H. We report the mean ± std of MAE/RMSE/MAPE/R2 across all rollouts.

(iii) Out-of-market transfer. We train on source dataset 𝒟S and test on target 𝒟T after aligning feature normalization to 𝒟S and keeping topology fixed (or mapped) when applicable. This probes generalization under distribution shift.

To verify that the gains are not solely due to the CNN–LSTM backbone, we conduct a compact ablation on METR-LA (H=12) under the same protocol (chronological 70/10/20 split, 5 seeds, early stopping, identical optimizer/schedule). We compare: (i) CNN–LSTM only (no fractional/fractal inputs; no meta-learner), (ii) +FARIMA features (d as covariate), (iii) +FETS features (fractional smoothing coefficients), (iv) +MFDFA descriptors (τ(q), H, Fq(s)), (v) FFDEM (full): FARIMA+FETS+MFDFA with the leakage-safe out-of-fold meta-learner g(⋅). For fairness, all variants share data preprocessing, window/horizon, and search ranges. We report mean ± std over 5 seeds and two-sided paired t-tests vs. CNN–LSTM (Holm–Bonferroni across rows) as shown in Table 6.

Fig. 12 shows each fractional/fractal component is found to yield a statistically significant improvement over CNN–LSTM alone, with the multifractal descriptors (MFDFA) providing the most significant single-module gain. Combining FARIMA+FETS+MFDFA is additive, and the full FFDEM (which also employs the out-of-fold meta-learner) achieves the best accuracy (MAE 6.4±0.2, RMSE 10.8±0.3; p<0.001 vs. CNN–LSTM), indicating that the observed gains stem from the fractional/fractal branch and the leakage-safe integration rather than the backbone alone.

The results demonstrate the advantages of integrating fractional–order models, fractal analysis, and deep learning into a unified ensemble. By combining complementary strengths, FFDEM addresses limitations that affect individual approaches: classical statistical models often struggle with nonstationary dynamics, while single deep architectures may overfit or require large datasets. FFDEM overcomes these issues through explicit long-memory and multifractal descriptors in concert with ensembling and leakage-safe meta-learner.

Error distributions reveal that FFDEM achieves both lower and more consistent forecast errors. This property is critical for decision making in electricity markets, where reliability is of the essence. Correlation analyses show, further, that error reduction is consistent with information gain (e.g., in terms of higher R2), which adds confidence in the model outputs.

Benchmarking with recent advanced methods demonstrates the practical impact of FFDEM: from the standpoint of grid operators and market participants, improved accuracy means lower risk, better bidding, and stable grid operation. On the technical side, the framework provides an instance of how fractional and fractal ideas fit in and augment the current wave of neural sequence models, with a principled basis for hybrid modeling in energy analytics. A combined comparison that includes the models featuring explicit fractional and fractal parameters is given in Table 7.

Some limitations remain. The computational cost of training multiple components and the ensemble is higher than that of a single architecture, and broader evaluation on larger and more diverse market datasets would further support generalizability. Future work will streamline computation (e.g., parameter sharing or pruning), expand cross-market studies and seasonal coverage, and deepen operator-facing explainability to increase trust and adoption. In addition, robustness protocols (time–aware cross-validation, rolling-origin evaluation, and out-of-market transfer) indicate that the gains persist under temporal drift and distribution shift. Ablation studies verify that each fractional/fractal component contributes beyond the CNN–LSTM backbone, clarifying where improvements originate. Efficiency measurements quantify a modest increase in latency and memory relative to the strongest baseline, which is often acceptable for day-ahead or 5–15 min dispatch horizons. To support reproducibility, we provide configuration files, seeds, and scripts for data processing and evaluation. Finally, integrating operational constraints (e.g., ramp limits, reserve requirements) and uncertainty quantification into the forecasting pipeline remains a promising direction for deployment at scale.

The paper introduced the Ensemble-Integrated Deep Forecasting Model (FFDEM), a hybrid framework that fuses fractional-order statistics, fractal descriptors, and machine/deep learning within an ensemble. Across MAE, RMSE, MAPE, and R2, FFDEM delivers higher accuracy and more stable forecasts than classical, machine-learning, and deep baselines, translating into practical benefits for market operations, risk management, and grid reliability under high renewable penetration. While this ensemble approach does have the overhead of both training time and inference cost, the gains in accuracy and stability make this a worthwhile approach for day-ahead and short-term scheduling on hardware that can be scaled to handle the increased cost. This work is extended in the future to make the interpretability of the system even stronger (e.g., through explanations of individual predictions that are readily understandable by operators), to integrate reinforcement learning for the purpose of control that can adapt to shifts in the data, and to extend the validation to multiple markets and over multiple seasons. Overall, FFDEM is a strong and practical forecasting approach to aid in the necessary migration to a sustainable power system with renewable integration.