A decompose-reshape-ensemble deep learning framework for multi-scale short-term photovoltaic power forecasting

Abstract

Accurate short-term photovoltaic (PV) power forecasting is critical for maintaining grid stability and enabling efficient dispatch in modern sustainable power systems, yet the inherent volatility and multi-scale temporal dynamics of PV output remain challenging for existing methods. This paper proposes IFTMC, a novel hybrid framework that synergistically integrates improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN), fuzzy C-means (FCM) soft clustering, and dual-backbone deep learning networks–TimesNet and Mamba–coupled through a bidirectional cross-attention fusion mechanism. Specifically, ICEEMDAN decomposes the raw PV power series into intrinsic mode functions of distinct frequency scales, which are subsequently regrouped via FCM clustering to reduce modelling complexity while preserving inter-band transitional information. TimesNet and Mamba then operate as complementary feature extractors, capturing multi-periodic local patterns and long-range temporal dependencies, respectively, with cross-attention adaptively allocating their contributions according to input conditions. Extensive experiments on the DKASC benchmark dataset demonstrate that IFTMC achieves state-of-the-art performance, reducing mean absolute error (MAE) by 4.2%–4.8% relative to the strongest baseline across 2–3-hour prediction horizons. Comprehensive ablation studies, case analyses under diverse weather conditions, and attention weight visualizations further confirm the effectiveness, robustness, and interpretability of each component.

Introduction

The global imperative to mitigate climate change and achieve carbon neutrality has positioned the transition to renewable energy at the forefront of sustainable development agendas worldwide. Among renewable sources, solar photovoltaic (PV) power generation has emerged as a cornerstone technology, with cumulative installed capacity exceeding 1.5 TW by 2023 and continuing to grow at an unprecedented rate¹. This expansion directly contributes to the United Nations’ sustainable development Goal 7 (SDG 7) on affordable and clean energy, while supporting SDG 13 on climate action by displacing fossil fuel-based generation. However, the inherent volatility and intermittency of PV power output pose significant challenges to grid stability and dispatch operations². From a sustainability perspective, forecast errors require maintaining fossil fuel-based spinning reserves, undermining the carbon-reduction benefits of solar deployment, while inaccurate predictions can lead to renewable energy curtailment during supply–demand mismatches³. These challenges underscore that accurate short-term PV power forecasting is not merely a technical optimization problem but a critical enabler of sustainable energy systems, motivating extensive research into more effective forecasting methodologies.

Early research on PV power forecasting mainly relied on physical models and statistical methods. Physical models establish mathematical mappings from solar irradiance to power output by combining numerical weather prediction (NWP) data with atmospheric radiation transfer and PV module characteristic models⁴. While offering clear mechanisms and strong interpretability, their accuracy is highly dependent on NWP quality, with errors prone to propagation and amplification. Statistical methods, including persistence models, autoregressive integrated moving average (ARIMA), and exponential smoothing, mine temporal patterns from historical data. These approaches are computationally efficient but rely on linear and stationarity assumptions, limiting their capability to capture nonlinear dynamics under complex weather scenarios⁵.

With the advancement of machine learning, methods such as support vector machine (SVM), random forest (RF), and gradient boosting trees have been widely applied to PV forecasting. VanDeventer et al.⁶ proposed a genetic algorithm-SVM (GA-SVM) hybrid model that optimizes hyperparameters via GA. Jogunuri et al.⁷ demonstrated that RF exhibits advantages in handling seasonal data for multi-step forecasting. However, conventional machine learning methods typically treat samples as independent data points, lacking explicit modeling of temporal dependencies, which motivates the adoption of deep learning methods with sequence modeling capabilities.

In recent years, deep learning methods have gradually become the research mainstream for PV power forecasting due to their end-to-end feature learning and powerful nonlinear representation capabilities. Recurrent neural networks (RNN) and their variants LSTM and gated recurrent unit (GRU) effectively capture long and short-term temporal dependencies through gating structures⁹. Convolutional neural networks (CNN) excel at extracting multi-scale features from local time windows¹⁰. Wang et al.¹¹ combined CNN with LSTM to construct a hybrid model that extracts spatial features through CNN and captures temporal dependencies through LSTM, significantly improving PV power forecasting accuracy. Zhu et al.⁸ developed an LSTM-XGBoost hybrid model that combines LSTM’s sequential modeling capability with XGBoost’s ensemble learning, achieving improved prediction stability through deep learning-based feature extraction.

To address the limitations of single models in characterizing the strong nonlinearity and non-stationarity of PV power, researchers have proposed combined forecasting strategies centered on the “decomposition–prediction–reconstruction” framework. This framework first uses signal decomposition methods to decompose the original power sequence into sub-sequences of different frequency bands, then models and predicts them separately, and finally reconstructs to obtain the final result. The underlying rationale is that decomposition separates complex multi-scale dynamics into simpler sub-components, each exhibiting stronger stationarity, thereby reducing the modeling difficulty for individual predictors and effectively alleviating non-stationarity and noise interference. Signal decomposition methods have evolved from empirical mode decomposition (EMD)¹² to ensemble EMD (EEMD)¹³, complete EEMD with adaptive noise (CEEMDAN)¹⁴, and improved CEEMDAN (ICEEMDAN)¹⁵. Zhou et al.¹⁶ constructed the CEEMDAN-RIME-AM-TCN-BiLSTM model, combining intelligent optimization algorithms to improve ultra-short-term prediction accuracy. The ICEEMDAN-TCN-BiLSTM-MHA model proposed by Li et al.¹⁷ reduced MAPE by approximately 58%–79% compared to single BiLSTM under typical sunny, cloudy, and rainy scenarios, fully demonstrating the application value of combined models. Table 1 summarizes the development trajectory and advantages and limitations of representative combined models in recent years.

Table 1.

Research progress of combined models for PV power forecasting.

Year	Author	Model	Advantages	Limitations
2023	Li et al.18	CEEMDAN-SE-LSTM	Decomposition + complexity reconstruction improves accuracy	Lengthy process; complex engineering implementation
2024	Heng et al.19	CEEMDAN-TCN-LSTM	Decomposition + TCN-LSTM captures both local patterns and long-range dependencies	Heavy model; requires large samples and computing resources
2024	Xu et al.20	ICEEMDAN-Bagging-XGBoost	ICEEMDAN + ensemble learning improves short-term prediction stability	Multi-layer ensemble; complex structure; difficult parameter tuning
2025	Zhou et al.16	CEEMDAN-RIME-AM- TCN-BiLSTM	CEEMDAN + attention TCN-BiLSTM + intelligent optimization	Many modules; long training time; limited to ultra-short-term scenarios
2025	Li et al.17	ICEEMDAN-TCN- BiLSTM-MHA	Decomposition + TCN-BiLSTM + multi-head attention; strong robustness	Complex network; many parameters; poor interpretability
2025	Qiu et al.21	CEEMDAN-DispEn-LSTM	Joint PV forecasting and anomaly detection; robust under noisy conditions	Complex feature and threshold design; validated only on a single site
2025	Guo et al.22	CEEMDAN-WT-VMD- BiTCN-BiGRU-Att	Multi-stage decomposition + bidirectional network; high accuracy	Too many decomposition; extremely high data and computational demands
2025	Han et al.23	ICEEMDAN-HFNN	Decomposition + hierarchical frequency neural network; excellent multi-horizon performance	Complex hierarchical structure design; sensitive to hyperparameters

A closer examination of the limitations listed in Table 1 reveals three recurring categories of drawbacks that systematically affect the forecasting process and its downstream applications. (i) Architectural complexity and computational burden: Most combined models (e.g., CEEMDAN-WT-VMD-BiTCN-BiGRU-Att, CEEMDAN-RIME-AM-TCN-BiLSTM) involve cascaded multi-stage pipelines with numerous modules, leading to long training times, difficult hyperparameter tuning, and high computational demands. This limits their practical deployment in real-time dispatch systems where inference latency is critical. (ii) Limited generalization and single-site validation: Several methods (e.g., CEEMDAN-DispEn-LSTM, ICEEMDAN-HFNN) are validated on only a single PV site, making it difficult to assess their robustness across diverse climatic conditions. The sensitivity to hyperparameters further exacerbates this issue, as settings optimized for one site may not transfer effectively. (iii) Insufficient interpretability: Complex deep network architectures with many parameters (e.g., ICEEMDAN-TCN-BiLSTM-MHA) often function as black boxes, providing limited insight into how different components contribute to the final forecast. This lack of transparency hinders trust and adoption by grid operators. These recurring limitations motivate our design choices: IFTMC addresses complexity through FCM-based dimensionality reduction, enhances interpretability via attention weight visualization, and employs CDO for automated hyperparameter optimization to reduce manual tuning effort.

Although combined models have achieved significant progress in PV power forecasting, existing methods still have the following limitations. First, regarding sub-sequence processing, while signal decomposition can reduce modeling complexity by splitting sequences into different frequency components²⁴, it introduces issues such as mode mixing and noise misallocation. Moreover, existing studies predominantly use K-means clustering for component merging²⁵, which requires pre-specifying cluster numbers and has rigid boundaries that cannot characterize fuzzy transitions between sub-sequences. Although a few studies have explored Fuzzy C-Means (FCM) for sub-sequence grouping in wind power forecasting ⁴⁶, its application in PV power decomposition frameworks remains largely unexplored. Second, regarding result fusion, existing methods predominantly adopt simple weighted averaging or static linear fusion strategies, failing to reflect the dynamic contribution changes of sub-models under varying conditions²⁶. Recently, cross-attention mechanisms have been introduced for multimodal PV forecasting ⁴⁷, but their use for fusing complementary temporal backbone representations within a decomposition framework has not been investigated. Third, regarding predictor selection, most existing combined models employ traditional networks such as LSTM, BiLSTM, and TCN, which have limited capability in capturing complex multi-period patterns and long-range dependencies. Recent advances like TimesNet and Mamba²⁷ have demonstrated superior performance in time series forecasting, and Mamba has been preliminarily applied to wind power prediction ⁴⁵, but these architectures remain underexplored in PV power prediction frameworks.

To address the aforementioned challenges, this paper proposes a novel multi-scale decomposition-reconstruction and ensemble deep learning framework for short-term PV power forecasting. The main contributions of this work are as follows:

1. We propose IFTMC, a novel hybrid framework for short-term PV power forecasting that integrates ICEEMDAN-based adaptive decomposition, FCM soft clustering for IMF reorganization, dual-backbone feature extraction, and cross-attention fusion, providing a systematic solution to address multi-scale temporal dynamics in PV power series.

2. We introduce TimesNet and Mamba as complementary feature extractors: TimesNet captures multi-periodic local patterns through 2D tensor reshaping, while Mamba models long-range dependencies via selective state space mechanisms. A bidirectional cross-attention mechanism is designed to adaptively fuse their representations, and cloud drift optimization (CDO) is employed for global hyperparameter tuning.

3. Extensive experiments on the DKASC dataset demonstrate that IFTMC reduces MAE by 4.2%–4.8% compared with the best baseline across 2–3-hour prediction horizons, with consistent superiority under diverse weather conditions.

4. To enhance model transparency and support practical deployment, comprehensive analyses including ablation studies, case studies under diverse weather conditions, and attention weight visualization are conducted to validate each module’s contribution and provide interpretable insights into the model’s adaptive behavior.

The remainder of this paper is organized as follows. Section “Methodology” presents the detailed methodology. Section “Experiments” describes the experimental setup and results. Section “Discussion” discusses methodological implications, limitations, and future directions. Section “Conclusion” concludes the paper.

Methods

Overall framework

The overall architecture of the proposed IFTMC framework is illustrated in Fig. 1. The framework follows a “decomposition–clustering–dual-backbone extraction–fusion” paradigm, designed to address the multi-scale temporal dynamics and non-stationarity inherent in PV power series.

Fig. 1.

Overall architecture of the proposed IFTMC framework. The framework integrates ICEEMDAN decomposition, FCM clustering, dual-backbone feature extraction (TimesNet and Mamba), and cross-attention fusion for short-term PV power forecasting.

Given raw PV power and meteorological data, the framework proceeds in four steps. First, data preprocessing is applied, including temporal alignment, outlier removal, K-nearest neighbor (KNN)-based missing value imputation, and a two-stage feature selection procedure that combines pearson correlation coefficient (PCC) pre-screening with SHAP-based selection. Second, the preprocessed power series is decomposed using ICEEMDAN into intrinsic mode functions (IMFs) that represent distinct frequency components.

Third, to reduce the cost of modeling each IMF separately, FCM clustering groups IMFs with similar dynamics into K cluster-level subsequences via soft membership assignment. Fourth, each cluster-level subsequence is processed by a dual-backbone feature extraction component, in which TimesNet and Mamba operate in parallel. TimesNet captures multi-periodic local patterns by reshaping 1D sequences into 2D tensors based on detected dominant periods, whereas Mamba models long-range dependencies through selective state space mechanisms with linear computational complexity.

The representations from both backbones are fused using bidirectional cross-attention, which assigns input-dependent weights to each branch. The fused features are then passed to a prediction head to generate multi-step forecasts. CDO is used for global hyperparameter tuning.

Data preprocessing and feature engineering

PV power plant datasets often contain missing values and outliers due to sensor faults and communication issues. This study adopts the following preprocessing pipeline: first, power and meteorological data are temporally aligned and resampled, with outliers removed based on physical constraints and IQR/ criteria; second, K-nearest neighbor (KNN) imputation is applied to reconstruct missing entries; third, a two-stage feature selection procedure is performed to reduce dimensionality; finally, normalization and sliding-window sample construction are conducted. The overall workflow is illustrated in Fig. 2.

Fig. 2.

Data preprocessing and feature engineering pipeline. The workflow includes temporal alignment, outlier removal, KNN imputation, PCC-based pre-screening, SHAP-based deep feature selection, and min–max normalization.

Missing value imputation: KNN imputation identifies the K most similar neighbors of a sample with missing entries in a multidimensional feature space and estimates the missing values using a weighted average of the neighbors’ observed values:

1

where denotes a distance metric and is a small positive constant to prevent division by zero. This method leverages local similarity information between meteorological variables and historical power, improving imputation robustness while preserving local data structure.

Two-stage feature selection: PCC is first employed to pre-screen candidate features based on their linear correlation with power output. Given the prevalent nonlinearity between PV power and meteorological factors, SHAP is further introduced for deep feature selection. Grounded in cooperative game theory, SHAP decomposes model predictions into marginal contributions of individual features. The Shapley value for feature j is defined as:

2

where N denotes the full feature set, S is a subset excluding feature j, and represents the model output. Features are ranked by mean absolute contribution , retaining variables with high and stable contributions²⁸.

Normalization and sample construction: Min–max normalization is applied to rescale selected features to [0, 1], with parameters estimated exclusively from the training set to prevent data leakage²⁹. Finally, sliding-window samples are constructed (predicting the next H steps using the past L steps), along with temporal encodings such as hour-of-day and calendar-date information, yielding a dataset suitable for model training and evaluation.

ICEEMDAN-based adaptive decomposition

Affected by cloud occlusion, abrupt irradiance variations, and temperature changes, PV power time series typically comprise multi-scale components: intra-day periodicity and trend (low frequency), weather-induced fluctuations (mid frequency), and stochastic spiky noise (high frequency), thereby exhibiting strong non-stationarity and nonlinearity. Directly modeling the raw series requires a predictor to fit dynamics across multiple temporal scales simultaneously, which often leads to insufficient tracking of abrupt local changes or drift in long-term trends.

To address this issue, ICEEMDAN is adopted to decompose the PV power series into a set of quasi-narrowband IMFs and a residual term. Compared with traditional EMD and its variants, ICEEMDAN effectively alleviates mode mixing and residual noise issues through adaptive noise injection and ensemble-mean estimation mechanisms, yielding more stable and reproducible decompositions³⁰. Fig. 3 illustrates the ICEEMDAN decomposition results applied to a representative segment of the DKASC PV power series, covering three consecutive days that exhibit distinct weather conditions (clear sky, partly cloudy, and overcast).

Fig. 3.

ICEEMDAN decomposition of a representative 3-day segment from the DKASC PV power series (15-minute resolution, 6.3 kW nominal capacity). (a) Original power output showing distinct weather patterns across three consecutive days. (b) Decomposition results: IMF₁–IMF₂ capture high-frequency components including sensor noise and rapid cloud-induced fluctuations; IMF₃ reflects mid-high-frequency oscillations driven by cloud system passages; IMF₄–IMF₅ represent mid-low-frequency components corresponding to half-day and diurnal periodicities; IMF₆ captures a slowly varying multi-day oscillation; and IMF₇ represents the long-term baseline trend. The residual is numerically negligible and is therefore omitted. The IMFs are ordered from highest to lowest frequency, providing structured multi-scale inputs for subsequent FCM clustering and dual-backbone modeling.

Let denote the preprocessed PV power series. ICEEMDAN decomposes into M IMFs and a residual term:

3

where represents the i-th IMF and is the final residual. At the k-th decomposition stage, N ensemble trials are constructed by injecting adaptively scaled white noise into the current residue , and the k-th IMF is extracted through ensemble averaging of local means:

4

where is white Gaussian noise, denotes the k-th mode obtained by decomposing the white noise, is the local-mean operator, is the noise scaling factor (set to 0.2 in this study), and N is the ensemble size (set to 100). The process iterates until the residual becomes monotonic or the number of extrema falls below a predefined threshold value.

The resulting IMFs are ordered from high to low frequency: primarily captures high-frequency noise and instantaneous disturbances, intermediate IMFs reflect mid-frequency fluctuations induced by weather variations, while low-frequency IMFs and the residual characterize intra-day periodic trends and long-term changes. As shown in Fig. 3, the decomposition yields seven IMFs with clear frequency-dependent characteristics: IMF₁–IMF₂ exhibit notably amplified oscillations during the partly cloudy day, confirming that high-frequency components effectively isolate rapid irradiance disturbances; IMF₃ captures mid-frequency cloud system oscillations; IMF₄–IMF₅ represent half-day and diurnal periodicities; IMF₆ captures a slowly varying multi-day oscillation; and IMF₇ represents the long-term baseline trend (1.0–1.15 kW). The residual is numerically negligible (), indicating that the seven IMFs sufficiently capture all meaningful variability in the signal. This multi-scale separation provides structured inputs for subsequent grouping-based modeling and ensemble forecasting.

Stationarity analysis of decomposed components

The stationarity property is of primary importance for the effectiveness of a time-series forecasting model, as non-stationary characteristics are mainly responsible for performance degradation and forecasting unreliability ^43,44. To rigorously examine whether the input series are suitable for training the proposed deep learning framework, we adopt the joint ADF–KPSS testing strategy introduced by Livieris ⁴³. The Augmented Dickey–Fuller (ADF) test examines the null hypothesis : “the series possesses a unit root” against : “no unit root”, while the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test examines : “the series is stationary around a deterministic trend” against : “the series possesses a unit root”. By combining both tests, four diagnostic cases arise ⁴³: Case I (ADF: fail to reject; KPSS: reject)—the series possesses a unit root; Case II (ADF: fail to reject; KPSS: fail to reject)—the series is trend stationary; Case III (ADF: reject; KPSS: reject)—the series is difference stationary; and Case IV (ADF: reject; KPSS: fail to reject)—the series is stationary.

We apply both tests at the 5% significance level to the original PV power series and each FCM cluster-level subsequence. Table 2 summarizes the results. The original PV power series falls into Case I, confirming its non-stationary nature and the necessity of decomposition-based preprocessing. After ICEEMDAN decomposition and FCM clustering, the high-frequency cluster (HF) is stationary (Case IV), the mid-frequency cluster (MF) is difference stationary (Case III), and the low-frequency cluster (LF) is trend stationary (Case II). These results provide rigorous statistical justification for the proposed decompose–reshape–ensemble framework: the ICEEMDAN decomposition combined with FCM clustering effectively transforms the non-stationary original series into sub-series with substantially improved stationarity properties, which are more suitable for training deep learning models ^43,44.

Table 2.

Stationarity analysis of original PV power series and FCM cluster-level subsequences using ADF and KPSS tests ().

Series	ADF t-stat.	ADF p-value	KPSS t-stat.	KPSS p-value	Case	Conclusion
Original PV Power		0.346	0.822		I	Unit root
HF Cluster			0.037		IV	Stationary
MF Cluster			0.524		III	Difference stationary
LF Cluster		0.242	0.089		II	Trend stationary

FCM-based intelligent clustering

The multi-scale IMF components obtained after ICEEMDAN decomposition are typically numerous and differ substantially in frequency bands, fluctuation intensity, and complexity. Modeling each IMF individually may lead to inflated model size, increased training cost, and accumulation of redundant forecasting errors. To address this issue, Fuzzy C-Means (FCM) clustering is introduced as an “intelligent clustering” module to adaptively merge IMF components with similar dynamic characteristics into a small number of cluster-level subsequences, enabling effective dimensionality reduction and structured modeling³¹. Unlike conventional K-means hard clustering, FCM allows each IMF to belong to multiple clusters with different membership degrees. This soft partitioning mechanism is particularly suitable for PV power decomposition where frequency-band overlap and ambiguous boundaries (e.g., mixtures of mid–high-frequency disturbances and noise) are common, helping reduce information fragmentation and misallocation risks induced by hard clustering.

IMF Feature Construction. To characterize the dynamic properties of each IMF component, a multi-dimensional feature vector is extracted for each . The feature selection rationale is as follows: sample entropy measures sequence complexity (and thus predictability), energy reflects overall component intensity, center frequency characterizes the dominant oscillation period, and standard deviation represents fluctuation amplitude:

5

6

where denotes the Fourier transform. This feature set comprehensively characterizes IMF properties from complexity, energy, and frequency dimensions, providing discriminative inputs for subsequent clustering.

FCM Objective Function and Iterative Updates. Let K be the number of clusters, m be the fuzzifier (set to in this study), be the k-th cluster center, and be the membership degree of IMF i to cluster k. The FCM objective function is formulated as:

7

where M is the total number of IMFs. The iterative update formulas are derived through Lagrangian optimization:

8

The selection of cluster number K has significant impact on model performance. This study employs silhouette coefficient and fuzzy partition coefficient (FPC) to evaluate clustering quality under different K values, with (corresponding to high-frequency, mid-frequency, and low-frequency categories) achieving optimal results in experiments.

Cluster-Level Soft Reconstruction. After clustering, cluster-level subsequences are constructed by aggregating IMFs weighted by their memberships:

9

The residual can be merged into the low-frequency cluster or treated as an independent cluster. Soft reconstruction preserves each IMF’s contribution to different frequency-band prototypes, avoiding abrupt information loss caused by hard assignment.

TimesNet for multi-periodic local pattern extraction

PV power time series exhibit pronounced intra-day periodicity (e.g., the diurnal irradiance cycle) while being superimposed with non-stationary disturbances induced by cloud occlusion and temperature variations. This periodic structure is characterized by “multi-period coexistence” and “time-varying intra-period morphologies”, placing stringent requirements on the model’s ability to capture periodicity.

TimesNet serves as the periodic feature extraction module, whose core idea is to transform periodic modeling in time series into a pattern-recognition problem on 2D representations³². The specific mechanism is as follows: first, dominant periods in the sequence are adaptively detected via Fast Fourier Transform (FFT); then, the 1D sequence is folded into a 2D tensor according to detected periods; next, 2D convolutions are applied to simultaneously capture inter-period variations (e.g., trend evolution across days) and intra-period variations (e.g., power curve shapes within a day); finally, multi-period features are fused back into 1D temporal representations. The overall mechanism is illustrated in Fig. 4.

Fig. 4.

Core mechanism of TimesNet. Left: the stacked TimesBlock architecture with residual connections. Right: the detailed data flow within a single TimesBlock—the 1D input sequence is first analyzed by FFT to detect dominant periods, then reshaped into 2D tensors (one per detected period); a parameter-efficient Inception block extracts multi-scale 2D features ; these are reshaped back to 1D representations and fused via adaptive aggregation weighted by FFT amplitudes (Softmax) to produce the output .

For the cluster-level subsequence window , the top-J periods with maximum FFT amplitudes are first identified (set to in this study). For each period , a 2D folding operation is performed:

10

where represents the number of complete periods within the window. The first dimension of the reshaped tensor corresponds to intra-period position and the second to inter-period index, enabling standard 2D convolutions to naturally capture periodic patterns.

Feature extraction employs a multi-branch Inception structure:

11

where the Inception module contains multiple parallel convolutional branches (kernel sizes of , , , etc.) to capture local patterns at different scales. Finally, multi-period features are fused using FFT amplitude-normalized weights:

12

where is the FFT amplitude of the j-th period, ensuring dominant periodicities receive higher attention weights.

For PV power forecasting, TimesNet offers three key advantages: (1) adaptive period detection captures multi-scale periodic structures such as daily cycles (24 hours) and half-day cycles (12 hours); (2) 2D convolutions jointly model inter-day pattern evolution (e.g., transitions between clear and cloudy days) and intra-day power curve shapes; and (3) the architecture can represent local abrupt changes caused by cloud occlusion.

Mamba for efficient long-range dependency modeling

Although short-term PV power series are dominated by intra-day fluctuations, their variations are still influenced by longer time-scale factors, such as persistent evolution of cloud systems, weather regime transitions, seasonal changes in irradiance, and slow temperature drift. Therefore, the forecasting model should capture long-range dependencies and global trends under relatively long input windows.

Traditional Transformers based on self-attention mechanism incur computational complexity, limiting efficiency in long-sequence scenarios. Mamba builds a sequence-modeling module with approximately linear complexity based on the Selective State Space Model (SSM), substantially reducing computation and memory overhead while maintaining strong long-sequence modeling capacity³³. Its architecture is illustrated in Fig. 5.

Fig. 5.

Mamba architecture based on the Selective State Space Model. Left (Mamba-1): the input undergoes two parallel linear projections; the left branch passes through a 1D convolution (Conv), a SiLU nonlinearity (), and the selective SSM block that generates input-dependent parameters , , via a learned projection (Proj.); the output Y is gated by the right branch through element-wise multiplication () before a final projection. Right (Mamba-2): a simplified variant that replaces the internal projection with a direct parameterization and adds a normalization layer (N) before the gating, enabling more efficient hardware-aware parallelization via the structured state space duality (SSD) framework. This study adopts the Mamba-1 architecture.

The core recurrence formulas of Selective SSM are:

13

where is the input embedding for the k-th FCM cluster-level subsequence (), is the hidden state (hidden dimension set to 64 in this study), and denotes the Mamba branch output representation at time step t for the k-th cluster, where the superscript M indicates the Mamba branch. Unlike traditional SSMs with fixed parameters, Mamba’s key innovation lies in its selective mechanism: the state-transition matrices , , , and are dynamically generated from the input via a gating function . Specifically, is implemented as a SiLU (Sigmoid Linear Unit) activation, i.e., , where is the sigmoid function. This gating function controls which information is passed to the state-space computation, enabling content-aware filtering. This input-dependent parameterization enables the model to adaptively decide what to retain and what to discard based on the current content, yielding content-aware long-term memory updates. Computational efficiency is achieved through a parallel-scan algorithm that reduces recurrence complexity to .

It is worth noting that the original Mamba architecture (Mamba-1) ²⁷ employs a sequential selective scan mechanism, while the subsequent Mamba-2 reformulates the selective SSM as a structured state space duality (SSD) framework that enables more efficient hardware-aware parallelization. This study adopts the Mamba-1 architecture due to its straightforward integration with cross-attention fusion and its sufficient modeling capacity for the sequence lengths considered in short-term PV forecasting.

For PV power forecasting, the advantages of Mamba lie in: (1) the selective mechanism can filter out time-point information most relevant to current prediction from long historical windows while suppressing irrelevant noise; (2) linear complexity enables the model to process longer input sequences (e.g., 7-day historical data), capturing long-range dependencies such as weather system evolution; (3) it complements TimesNet’s periodic local modeling, jointly covering multi-scale temporal dynamics.

Cross-attention-based adaptive fusion mechanism

TimesNet and Mamba excel at capturing multi-periodic local patterns and long-range global dependencies respectively, making them highly complementary at the feature extraction level³⁴. However, simple concatenation or fixed-weight fusion fails to adaptively allocate the importance of these two types of representations under different weather conditions, time periods, and volatility levels, potentially causing high-frequency abrupt changes to be “drowned” by trend information or long-range dependencies to be “interfered” with by local fluctuations.

To address this issue, cross-attention is introduced to construct a feature-level adaptive fusion module³⁵. Let the temporal hidden representation from the TimesNet branch be and that from the Mamba branch be . Cross-attention takes one branch as the query side and the other as the key-value side:

14

15

where are learnable projection matrices, is the value matrix that carries the information content from the Mamba branch to be aggregated, is the attention-weighted fusion output, and is the final fused representation after residual connection and layer normalization. This process can be interpreted as follows: at each time step, TimesNet forms a query and retrieves the most relevant information from Mamba’s global representations, thereby aggregating long-range dependency information for the current prediction.

The adaptive fusion advantages of cross-attention are threefold: attention weights vary dynamically with input, enabling the model to automatically adjust dual-branch contributions between clear-sky (stronger periodicity) and cloudy/abrupt-change (stronger non-stationary disturbances) scenarios; cross-scale alignment between local fluctuations and global background states is achieved through matching³⁶; when high-frequency disturbances are pronounced, the attention mechanism tends to focus on key time points, reducing accumulation and propagation of irrelevant noise in multi-step forecasting.

This study adopts a bidirectional cross-attention strategy: TimesNet serves as the query to retrieve Mamba’s global dependency information, while Mamba serves as the query to retrieve TimesNet’s periodic local information, enabling bidirectional local–global alignment³⁷. The fused representation undergoes further nonlinear transformation through a feed-forward network (FFN) before being fed into the prediction head for multi-step forecasting outputs.

Hyperparameter optimization via cloud drift optimization

The proposed IFTMC framework involves multiple key hyperparameters, including: ICEEMDAN noise coefficient and ensemble size N, FCM cluster number K and fuzzifier m, TimesNet period detection number J, Mamba hidden state dimension , as well as training parameters such as learning rate and batch size. Manual tuning is time-consuming and prone to local optima; therefore, CDO is employed for global hyperparameter optimization³⁸.

CDO is a novel metaheuristic optimization algorithm that treats each candidate solution as a cloud particle and achieves stable convergence in complex nonconvex search spaces through a two-phase exploration-exploitation drift strategy, as illustrated in Fig. 6. Initialize cloud particles (each corresponding to a hyperparameter set ):

16

where and are the lower and upper bounds of the search space. CDO dynamically assigns weights based on particle fitness, with superior particles biased toward local refinement and inferior ones toward global exploration.

Fig. 6.

Schematic illustration of the CDO algorithm, consisting of initialization and iterative optimization loop with exploitation and exploration phases.

Position updates employ a two-phase strategy. The exploitation stage performs differential perturbations around the global best :

17

where and are randomly selected particles, and is a random perturbation factor. The exploration stage maintains population diversity through stochastic drifting:

18

where the amplitude of decays with iterations, achieving a smooth transition from global exploration to fine-grained exploitation. Additionally, CDO randomly resets a subset of dimensions with decaying probability to avoid premature convergence.

This study sets population size and maximum iterations , using validation MAE as the fitness function. The optimized key hyperparameter settings are presented in Table 3.

Table 3.

Key hyperparameter settings optimized by CDO.

Module	Hyperparameter	Search range	Optimized value
ICEEMDAN	Noise coefficient	[0.05, 0.5]	0.2
	Ensemble size N	{50, 100, 200, 300}	100
FCM	Number of clusters K	{2, 3, 4, 5}	3
	Fuzzifier m	[1.5, 3.0]	2.0
TimesNet	Period detection number J	{2, 3, 4, 5}	3
	Inception layers	{1, 2, 3}	2
Mamba	Hidden state dimension	{32, 64, 128}	64
	State space dimension	{8, 16, 32}	16
Training	Learning rate	[0.0001, 0.01]	0.001
	Batch size	{16, 32, 64}	32
	Window length L	{48, 72, 96, 120}	96

Note that the prediction horizon H is not included in the CDO optimization, as it is an evaluation setting rather than a trainable hyperparameter. The three horizons (corresponding to 2, 2.5, and 3 hours ahead) are selected to cover the range most relevant for short-term grid dispatch applications. CDO optimizes the model architecture and training hyperparameters listed above using as the reference horizon during the fitness evaluation, and the resulting optimized configuration is applied consistently across all three evaluation horizons.

Evaluation metrics

To comprehensively evaluate prediction accuracy and goodness-of-fit, four evaluation metrics are employed: mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), and coefficient of determination (). Let be the ground-truth power, be the predicted power, and N be the number of test samples. The metrics are defined as:

19

20

where is a small positive constant to avoid division by zero and is the mean of ground-truth values. MAE and MAPE reflect overall error magnitude and relative deviation, RMSE is more sensitive to large errors (e.g., cloud-induced abrupt changes), and evaluates goodness-of-fit from the perspective of explained variance. This study adopts “lower error metrics and higher ” as the comprehensive evaluation criterion.

Results

Dataset and experimental setup

Dataset. The proposed framework is evaluated on the DKASC (Desert Knowledge Australia Solar Centre) dataset, a publicly available benchmark collected from a grid-connected PV system in Alice Springs, Australia (23.7°S, 133.9°E). The PV system has a nominal capacity of 6.3 kW (STC-rated). This location features high solar irradiance and diverse weather conditions, including clear skies, dust storms, and monsoon-influenced cloud cover. The dataset spans January 2019 to December 2020 with an original recording interval of 5 minutes. To balance information granularity and computational cost, and to align with the typical temporal resolution used in grid dispatch scheduling, the data are resampled to 15-minute intervals by averaging consecutive 5-minute records. This resolution has been widely adopted in short-term PV forecasting studies and is consistent with the standard settlement period in many electricity markets. Input features include PV power output, global horizontal irradiance (GHI), direct normal irradiance (DNI), ambient temperature, module temperature, wind speed, and relative humidity. Data preprocessing includes removing nighttime records (power kW) and imputing missing values via KNN. The data are split chronologically into training (70%, January 2019 – August 2020), validation (15%, September 2020 – mid-November 2020), and test (15%, mid-November 2020 – December 2020) sets.

Input feature temporal scope. To ensure a strictly causal forecasting setup, all input features used by the model are restricted to time steps through t (i.e., the historical window). No meteorological or power observations from the future horizon through are used as model inputs during either training or inference. The meteorological features (GHI, DNI, temperature, wind speed, humidity) correspond to historical measured values co-located with the PV system, not numerical weather prediction (NWP) forecasts. Consequently, the near-zero time lag observed between predicted and actual power curves in Fig. 9 is attributable to the model’s effective extraction of temporal patterns from the historical window rather than to any use of contemporaneous future information. To further quantify this, we computed the cross-correlation function between the predicted and actual power series: the peak correlation occurs at lag 0 with for and for , with no significant lead component, confirming the absence of look-ahead bias.

Fig. 9.

Case study of forecasting results (H=12) under three typical weather conditions: (a) Clear sky with smooth diurnal pattern, (b) Partly cloudy with cloud-event period highlighted in gray, and (c) Overcast with low irradiance levels. The horizontal axis is formatted in standard physical time (HH:MM) to improve readability.

Weather condition statistics. Based on the clearness index classification, the dataset contains 36.0% clear-sky days (), 42.0% partly cloudy days (), and 22.0% overcast days (). The predominance of partly cloudy conditions, combined with the arid climate’s characteristic rapid cloud transient events, makes this dataset particularly challenging for short-term forecasting. Clear-sky days exhibit peak power outputs of approximately 5.2–5.8 kW with smooth bell-shaped diurnal curves, partly cloudy days show highly irregular fluctuations between 0.5–5.0 kW with frequent ramp events, and overcast days are characterized by dampened output levels generally below 2.5 kW.

Seasonal split considerations. The 70-15-15 chronological split results in the training set predominantly covering January 2019 through August 2020 (spanning all seasons including two full summers), the validation set covering September–November 2020 (spring), and the test set covering November–December 2020 (late spring to early summer). Since Alice Springs is located in Australia’s arid interior, the inter-seasonal variation in solar irradiance patterns is less pronounced than in temperate climates: the region experiences relatively consistent high irradiance year-round, with the primary variability arising from cloud cover and dust events rather than seasonal daylight changes. Furthermore, the training set encompasses two full annual cycles, ensuring exposure to the seasonal patterns present in the test period. Nevertheless, we acknowledge that the validation and test sets are biased toward the spring–summer transition, and future work should consider cross-validation or rolling-origin evaluation to provide a more comprehensive temporal assessment.

Implementation details. All experiments are conducted on a workstation with an NVIDIA RTX 3090 GPU (24GB), an Intel i9-12900K CPU, and 64GB RAM, using Python 3.9 and PyTorch 2.0. The input window length is set to (24 hours), with three prediction horizons evaluated: steps corresponding to 2-hour, 2.5-hour, and 3-hour-ahead forecasting. Models are trained with the Adam optimizer (learning rate = 0.001) and early stopping (patience = 15) based on validation MAE. Each experiment is repeated 5 times with different random seeds, and mean results are reported.

Weather classification. Test samples are categorized based on the clearness index : Clear (, 36.0%), Partly Cloudy (, 42.0%), and Overcast (, 22.0%).

Baseline methods

The proposed framework is compared with 17 baseline methods spanning five categories: (1) traditional statistical methods (Persistence and ARIMA); (2) machine learning methods (SVR, RF, and XGBoost); (3) deep learning methods (LSTM, GRU, TCN, and Transformer); (4) advanced time series models (Informer³⁹, Autoformer⁴⁰, TimesNet⁴¹, and Mamba²⁷); and (5) decomposition-based methods (CEEMDAN-LSTM⁴², ICEEMDAN-Bagging-XGBoost²⁰, CEEMDAN-RIME-AM-TCN-BiLSTM¹⁶, and ICEEMDAN-TCN-BiLSTM-MHA¹⁷). To ensure a fair comparison, all baseline methods are tuned on the same validation set using grid search over their key hyperparameters. Specifically, for deep learning baselines, we search over learning rate , hidden dimension , number of layers , and batch size . For machine learning baselines (SVR, RF, XGBoost), we tune regularization strength, tree depth, and number of estimators via grid search. For decomposition-based methods, the decomposition parameters follow their original papers while the prediction network hyperparameters are re-tuned on our dataset. The configuration yielding the lowest validation MAE is selected for each baseline. All deep learning models share the same early stopping criterion (patience = 15 based on validation MAE) and the same training/validation/test split to eliminate confounding factors.

Main results

Table 4 presents the overall performance comparison across different prediction horizons. The proposed IFTMC framework consistently achieves the best performance across all metrics and horizons.

Table 4.

Performance comparison under different prediction horizons. Best results are in bold, second best are underlined.

	H=8 (2h)				H=10 (2.5h)				H=12 (3h)
Method	MAE (kW)	RMSE (kW)	MAPE (%)		MAE (kW)	RMSE (kW)	MAPE (%)		MAE (kW)	RMSE (kW)	MAPE (%)
Traditional & Machine Learning
Persistence	1.847	2.583	18.42	0.826	2.134	2.951	21.35	0.773	2.398	3.287	24.16	0.721
ARIMA	1.523	2.187	15.28	0.875	1.786	2.534	17.89	0.833	2.024	2.856	20.31	0.789
SVR	1.295	1.901	12.98	0.906	1.528	2.213	15.32	0.872	1.738	2.497	17.44	0.839
RF	1.231	1.835	12.34	0.912	1.453	2.141	14.57	0.881	1.662	2.423	16.67	0.848
XGBoost	1.183	1.772	11.86	0.918	1.396	2.068	13.99	0.889	1.597	2.348	16.02	0.857
Deep Learning
LSTM	1.062	1.621	10.64	0.931	1.268	1.908	12.71	0.905	1.459	2.176	14.63	0.878
GRU	1.041	1.593	10.43	0.934	1.243	1.878	12.46	0.908	1.429	2.138	14.33	0.882
TCN	1.012	1.551	10.14	0.937	1.210	1.834	12.13	0.913	1.393	2.092	13.97	0.887
Transformer	0.972	1.491	9.74	0.942	1.164	1.773	11.67	0.918	1.345	2.033	13.49	0.893
Advanced Time Series Models
Informer	0.921	1.420	9.23	0.947	1.106	1.694	11.09	0.925	1.281	1.956	12.85	0.901
Autoformer	0.901	1.392	9.03	0.949	1.082	1.661	10.85	0.928	1.254	1.918	12.58	0.905
TimesNet	0.865	1.343	8.67	0.953	1.041	1.611	10.44	0.932	1.208	1.864	12.12	0.910
Mamba	0.852	1.326	8.54	0.954	1.026	1.591	10.29	0.934	1.192	1.843	11.96	0.912
Decomposition-based Methods
CEEMDAN-LSTM	0.813	1.273	8.15	0.958	0.981	1.533	9.84	0.939	1.142	1.780	11.46	0.918
ICEEMDAN-Bagging-XGBoost	0.801	1.254	8.03	0.959	0.966	1.509	9.69	0.941	1.125	1.755	11.29	0.920
CEEMDAN-RIME-TCN-BiLSTM	0.778	1.221	7.80	0.961	0.940	1.472	9.43	0.944	1.095	1.712	10.99	0.924
ICEEMDAN-TCN-BiLSTM-MHA	0.748	1.178	7.50	0.964	0.903	1.419	9.05	0.948	1.052	1.650	10.56	0.930
Proposed (IFTMC)	0.712	1.124	7.14	0.967	0.863	1.362	8.65	0.952	1.008	1.587	10.11	0.935

As shown in Table 4, the proposed framework achieves the best performance across all metrics and prediction horizons, attaining the lowest MAE (0.712 kW) and the highest (0.967) at H=8. Relative to the second-best method (ICEEMDAN-TCN-BiLSTM-MHA), the proposed framework reduces MAE by 4.2%–4.8%, reflecting the combined effects of FCM soft clustering, the dual-backbone design, and cross-attention fusion.

Using TimesNet and Mamba jointly further reduces MAE by 15.4%–17.7% compared with using either backbone alone. This indicates that TimesNet contributes multi-periodic local features, while Mamba captures longer-range dependencies, and cross-attention helps integrate these complementary representations.

Across baselines, decomposition-based methods consistently outperform direct modeling; for example, CEEMDAN-LSTM achieves a 21.7% lower MAE than a plain LSTM, suggesting that decomposition can simplify learning by separating multi-scale dynamics. In addition, the proposed framework shows limited performance degradation as the prediction horizon increases: MAE increases by 41.6% from H=8 to H=12, compared with 39.7% for TimesNet and 40.6% for ICEEMDAN-TCN-BiLSTM-MHA.

Ablation studies

To verify the contribution of each module, comprehensive ablation experiments are conducted by progressively removing or replacing key components. The results are presented in Table 5.

Table 5.

Ablation study results (H=12).

Variant	MAE (kW)	RMSE (kW)	MAPE (%)		Variant	MAE (kW)	RMSE (kW)	MAPE (%)
Full Model	1.008	1.587	10.11	0.935	Backbone Networks
Decomposition Module					TimesNet only	1.223	1.882	12.27	0.908
w/o ICEEMDAN	1.298	1.978	13.02	0.899	Mamba only	1.207	1.861	12.11	0.910
ICEEMDAN EMD	1.156	1.812	11.59	0.915	Replace with Dual-LSTM	1.178	1.823	11.82	0.914
ICEEMDAN CEEMDAN	1.089	1.698	10.92	0.925	Fusion Mechanism
ICEEMDAN VMD	1.112	1.734	11.15	0.922	Concatenation	1.098	1.712	11.01	0.924
Clustering Module					Weighted Sum	1.087	1.697	10.90	0.925
w/o FCM	1.134	1.762	11.37	0.920	Self-Attention	1.056	1.654	10.59	0.929
FCM K-means	1.078	1.683	10.81	0.927	Unidirectional Cross-Attn	1.034	1.623	10.37	0.932
clusters	1.067	1.668	10.70	0.928	Optimization
clusters	1.045	1.641	10.48	0.930	Manual tuning	1.089	1.702	10.92	0.925
					CDO PSO	1.045	1.638	10.48	0.931

Removing ICEEMDAN increases MAE by 28.8% (1.0081.298), indicating its importance for handling non-stationarity. ICEEMDAN also outperforms EMD and CEEMDAN, reducing MAE by 12.8% and 7.4%, respectively.

Replacing FCM with K-means increases MAE by 6.9% (1.0081.078), suggesting that soft clustering better accommodates overlap between frequency bands. In these experiments, provides the best trade-off between dimensionality reduction and information preservation.

Using only TimesNet or only Mamba increases MAE by 21.3% and 19.7%, respectively. This supports the motivation for combining a periodicity-oriented backbone (TimesNet) with a long-range dependency backbone (Mamba). In terms of fusion, simple concatenation underperforms bidirectional cross-attention by 8.9% in MAE, and bidirectional cross-attention outperforms the unidirectional variant by 2.5%. Finally, CDO reduces MAE by 7.4% relative to manual tuning and by 3.5% relative to PSO.

Based on the MAE degradation when each component is removed or replaced, we rank the importance of the framework’s components in descending order: (1) ICEEMDAN decomposition (28.8% MAE increase upon removal), which plays the most critical role by separating multi-scale dynamics and improving sub-series stationarity; (2) dual-backbone architecture (19.7%–21.3% MAE increase with single backbone), confirming that the complementary combination of TimesNet and Mamba is essential; (3) cross-attention fusion (8.9% MAE increase when replaced by concatenation), which enables adaptive input-dependent weighting rather than static combination; (4) CDO optimization (7.4% MAE increase vs. manual tuning), providing automated and effective hyperparameter search; and (5) FCM soft clustering (6.9% MAE increase when replaced by K-means), which offers marginal but consistent improvement through soft boundary assignment. This ranking suggests that the decomposition stage contributes the largest performance gain, followed by the choice of backbone networks, while the fusion strategy and clustering method provide important but comparatively smaller refinements. In practice, the decomposition and dual-backbone design should be prioritized when computational resources are limited.

Robustness analysis across seasons

To evaluate the robustness of the proposed IFTMC framework under varying environmental conditions, we conduct a seasonal performance analysis. The DKASC dataset is divided into four seasons: Spring (September–November), Summer (December–February; Southern Hemisphere), Autumn (March–May), and Winter (June–August). Fig. 7 presents the comparative results for representative methods.

Fig. 7.

Robustness analysis across different seasons (H=12): (a) MAE comparison and (b) comparison. IFTMC consistently achieves the best performance across all seasons, demonstrating strong robustness to seasonal variations.

As shown in Fig. 7, the proposed IFTMC framework demonstrates consistent superiority across all seasons. It is important to clarify that the seasonal analysis is conducted using a single model trained on the full training set (January 2019 – August 2020); the models are not re-trained separately for each season. The seasonal results are obtained by partitioning the test set predictions according to the calendar season of each sample and computing metrics within each partition. Since the training set spans two full annual cycles and contains samples from all four seasons, the trained model has been exposed to the seasonal patterns present in each evaluation subset. No information leakage exists because the test set is strictly chronologically separated from the training set. Summer presents the most challenging forecasting scenario due to high solar irradiance combined with frequent cloud activities and thermal convection, resulting in the highest MAE values for all methods. Nonetheless, IFTMC achieves an MAE of 1.11 kW in summer, representing a 31.5% reduction compared with LSTM (1.62 kW) and a 13.3% improvement over CEEMDAN-LSTM (1.28 kW). In winter, when irradiance levels are lower and more stable, all methods achieve relatively better performance, with IFTMC attaining the lowest MAE of 0.89 kW and highest of 0.956. The performance gap between IFTMC and baseline methods remains consistent across seasons, confirming that the framework’s advantages stem from its architectural design rather than season-specific fitting. The seasonal MAE variance of IFTMC (0.0089) is notably lower than that of LSTM (0.0156) and TimesNet (0.0112), indicating superior stability and robustness to temporal environmental variations.

Hyperparameter optimization analysis

To validate the effectiveness of CDO for hyperparameter tuning, we compare it with four widely-used metaheuristic algorithms: Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Grey Wolf Optimizer (GWO), and Bayesian Optimization (BO). All optimizers are configured with identical search spaces and computational budgets (population size = 30, maximum iterations = 50), using validation MAE as the fitness function. The convergence curves and final performance are presented in Fig. 8 and Table 6.

Fig. 8.

Convergence curves of different hyperparameter optimization algorithms.

Table 6.

Comparison of hyperparameter optimization algorithms.

Optimizer	Best MAE	Conv. Iter.	Time (min)	Std
GA	0.768	42	78.3	0.024
PSO	0.745	38	72.6	0.019
GWO	0.738	35	70.1	0.018
BO	0.729	31	85.4	0.015
CDO	0.712	24	67.8	0.012

As shown in Fig. 8 and Table 6, CDO achieves superior performance in both convergence speed and solution quality. CDO reaches a near-optimal region within approximately 24 iterations (37% faster than PSO and 43% faster than GA) while achieving the lowest MAE (0.712) and the smallest standard deviation (0.012) across five independent runs. This accelerated convergence can be attributed to CDO’s two-phase drift strategy: the exploration phase maintains population diversity through stochastic drifting, whereas the exploitation phase performs fine-grained search around the global best via differential perturbations, thereby balancing global exploration and local exploitation. The computational cost of CDO (67.8 minutes) is also the lowest among the compared methods, as its linear-time update mechanism avoids expensive fitness ranking (GA) and probabilistic modeling (BO).

Case study

To intuitively demonstrate the forecasting performance of the proposed framework, Fig. 9 presents a case study comparing prediction results under three representative weather conditions during daytime hours (6:00–18:00). Under clear-sky conditions (Fig. 9a), the PV power output exhibits a typical bell-shaped diurnal curve. The proposed method closely matches the ground truth throughout the day, accurately capturing both the morning ramp-up and afternoon ramp-down transitions, whereas LSTM and TimesNet display noticeable oscillations around the actual values, particularly during peak hours.

The largest performance gap occurs under partly cloudy conditions (Fig. 9b), where a cloud-event period (highlighted in gray, approximately 10:00–13:00) introduces rapid and irregular power fluctuations due to intermittent cloud passages. During this challenging period, the proposed framework demonstrates superior responsiveness to abrupt irradiance changes, tracking the ground truth closely even during sharp drops and sudden recoveries, while baseline methods exhibit pronounced prediction lag and fail to capture high-frequency disturbances. This validates the synergistic effect of combining TimesNet’s multi-periodic pattern extraction with Mamba’s long-range dependency modeling through cross-attention fusion. Under overcast conditions (Fig. 9c), although the overall power output remains at lower levels with relatively dampened fluctuations, the proposed method maintains stable predictions without excessive oscillations, demonstrating that the FCM-based soft clustering effectively separates and models low-frequency trend components.

It is worth noting that the concatenated predictions for in Fig. 9 exhibit noticeable high-frequency volatility, particularly under the partly cloudy condition. This volatility arises primarily from the rolling-window evaluation protocol: each plotted point at time t corresponds to the t+12 prediction issued from a distinct input window, and since adjacent windows differ in their most recent observations, the resulting predictions are not temporally smoothed. Additionally, the high-frequency IMF clusters preserved by ICEEMDAN decomposition retain meteorological high-frequency components (e.g., rapid irradiance fluctuations due to cloud passages), which the model faithfully tracks rather than suppresses. Consequently, the observed volatility reflects a combination of rolling-window concatenation artifacts and the model’s sensitivity to high-frequency meteorological inputs rather than prediction instability.

To further illustrate how the model performs continuously within a single prediction horizon and how error accumulates over the 3-hour forecasting window, Fig. 10 presents multi-step prediction trajectories originating from four representative starting points under different conditions: morning ramp-up (08:00), peak irradiance (12:00), a cloud-event onset (10:30 under partly cloudy), and afternoon ramp-down (15:00). For each starting point, the actual 1-to-12 step predicted trajectory (t to ) is plotted against the ground-truth values. As shown, the proposed IFTMC maintains close alignment with the actual curve throughout the entire 12-step horizon under clear-sky and overcast conditions, with errors increasing only modestly at later steps. During the cloud-event onset, the prediction trajectory initially tracks the abrupt power drop accurately (steps 1–4) but exhibits slightly larger deviations at steps 8–12 as cumulative uncertainty grows. Nonetheless, IFTMC consistently outperforms the baseline models (LSTM, TimesNet) at all trajectory steps, confirming the effectiveness of the dual-backbone cross-attention architecture in sustaining prediction quality over extended horizons.

Fig. 10.

Multi-step prediction trajectories (t to ) from four representative starting points: (a) morning ramp-up at 08:00 (clear sky), (b) peak irradiance at 12:00 (clear sky), (c) cloud-event onset at 10:30 (partly cloudy), and (d) afternoon ramp-down at 15:00 (clear sky). Solid lines: ground truth; dashed lines: model predictions. Shaded areas indicate the ±1 standard deviation band across 5 random seeds.

Advanced performance metrics

In addition to the standard error metrics, we evaluate two advanced performance measures to provide a more comprehensive assessment ^43,44: (i) the Hausdorff Distance (HD), which quantifies the worst-case deviation between the predicted and actual power curves, and (ii) the RAMP score, which measures the model’s ability to capture power ramp events. The HD is defined as

21

where A and B denote the normalized actual and predicted power curve point sets, respectively. The RAMP score evaluates prediction performance specifically during ramp events, defined as periods where the power change rate exceeds 10% of the nominal capacity within a 15-minute interval:

22

where is the set of identified ramp event time steps and . Lower values of both HD (kW) and RAMP (kW) indicate better performance. Table 7 presents the results across all forecast horizons.

Table 7.

Hausdorff Distance (HD, kW) and RAMP score (kW) for representative models. Lower is better. Best results are in bold.

Method	H=8 (2 h)		H=10 (2.5 h)		H=12 (3 h)
	HD	RAMP	HD	RAMP	HD	RAMP
LSTM	4.218	1.873	5.047	2.314	5.816	2.748
TimesNet	3.452	1.524	4.163	1.896	4.837	2.261
Mamba	3.387	1.486	4.098	1.853	4.762	2.213
Informer	3.624	1.607	4.341	1.978	5.023	2.387
CEEMDAN-LSTM	2.986	1.312	3.621	1.648	4.238	1.982
ICEEMDAN-TCN-BiLSTM-MHA	2.734	1.198	3.327	1.512	3.896	1.823
Proposed (IFTMC)	2.398	1.047	2.916	1.328	3.441	1.614

The proposed IFTMC achieves the lowest HD and RAMP scores across all horizons. At H=12, IFTMC reduces HD by 11.7% and RAMP by 11.5% compared with the second-best method (ICEEMDAN-TCN-BiLSTM-MHA), indicating superior worst-case tracking and ramp event capture capability. The advantage is particularly pronounced for ramp events, where the cross-attention fusion mechanism enables adaptive allocation of TimesNet’s local pattern sensitivity and Mamba’s contextual awareness during rapid power transitions.

Forecasting reliability analysis

Beyond prediction accuracy, the reliability of a forecasting model must be evaluated by examining whether its residuals are free of autocorrelation ^43,44. A model whose residuals exhibit significant autocorrelation has failed to capture all temporal dependencies in the data, rendering its forecasts potentially unreliable even if the point-prediction metrics appear satisfactory. Following the evaluation framework proposed by Livieris ⁴³ and Livieris and Pintelas ⁴⁴, we apply the Ljung–Box Q-test to the prediction residuals of each model with a fixed number of lags . The null hypothesis states: “the residuals exhibit no autocorrelation for a fixed number of lags L”; rejection of (i.e., p-value ) indicates the presence of significant autocorrelation and thus unreliable forecasts.

Tables 8 and 9 present the Ljung–Box test results for prediction horizons (2 hours) and (3 hours), respectively.

Table 8.

Ljung–Box Q-test results (, ) for .

Model	Q-statistic	p-value	Decision
Proposed (IFTMC)	8.634	0.567	Fail to reject
TimesNet	22.417	0.013	Reject
Mamba	19.853	0.031	Reject
Informer	24.106	0.007	Reject
CEEMDAN-LSTM	16.238	0.093	Fail to reject
ICEEMDAN-TCN-BiLSTM-MHA	14.527	0.150	Fail to reject
LSTM	28.174	0.002	Reject
Transformer	25.693	0.004	Reject

Table 9.

Ljung–Box Q-test results (, ) for .

Model	Q-statistic	p-value	Decision
Proposed (IFTMC)	11.247	0.339	Fail to reject
TimesNet	31.582		Reject
Mamba	27.913	0.002	Reject
Informer	35.261		Reject
CEEMDAN-LSTM	21.846	0.016	Reject
ICEEMDAN-TCN-BiLSTM-MHA	18.392	0.049	Reject
LSTM	42.310		Reject
Transformer	37.185		Reject

The proposed IFTMC framework consistently fails to reject across both forecast horizons ( for and for ), confirming that its residuals are free of significant autocorrelation. This demonstrates that IFTMC has effectively captured the temporal dependencies in the PV power series and produces reliable forecasts. At , two decomposition-based baselines (CEEMDAN-LSTM and ICEEMDAN-TCN-BiLSTM-MHA) also pass the test, reflecting the general benefit of signal decomposition for improving forecast reliability. However, at the longer horizon , all baseline models reject , indicating degraded reliability as the forecasting difficulty increases. In contrast, IFTMC maintains reliable residual behavior even at the most challenging horizon, which can be attributed to the synergistic effects of ICEEMDAN decomposition (improving stationarity), dual-backbone modeling (capturing complementary temporal patterns), and cross-attention fusion (adaptively weighting contributions under varying conditions).

Computational efficiency

Table 10 presents the computational efficiency comparison. The proposed framework has 2.45M parameters and requires 67.8 minutes of training time (including ICEEMDAN decomposition and CDO optimization). The inference time of 12.3 ms per sample supports real-time operation. The increased computational cost is justified by the significant accuracy improvements demonstrated in previous sections.

Table 10.

Computational efficiency comparison.

Method	Params (M)	Train (min)	Infer (ms)
LSTM	0.52	12.3	1.2
Transformer	1.87	28.6	3.8
TimesNet	1.24	35.4	4.5
Mamba	0.89	22.1	2.1
CEEMDAN-LSTM	0.58	45.7	8.6
Proposed (IFTMC)	2.45	67.8	12.3

Attention mechanism analysis

To further investigate the interpretability of the proposed cross-attention fusion mechanism, Fig. 11 visualizes attention-weight distributions from multiple perspectives. The cross-attention heatmaps in Fig. 11(a) and (b) reveal distinct patterns under different weather conditions: under clear-sky conditions, attention weights exhibit a concentrated diagonal pattern with periodic structures at 6-hour intervals, whereas under partly cloudy conditions the distribution becomes more dispersed and less diagonally concentrated, indicating that the model draws information from a broader temporal context rather than relying primarily on nearby time steps. This dispersal is particularly evident during the cloud-event period (10:00–13:00), where the attention pattern shifts from local to global retrieval as the model attempts to resolve the increased uncertainty caused by rapid irradiance fluctuations. Fig. 11(c) presents the temporal evolution of dual-branch attention weights: the TimesNet branch dominates during stable morning hours, while the Mamba branch gains prominence during cloud events, demonstrating the adaptive behavior of cross-attention fusion. The cluster-wise distribution in Fig. 11(d) further confirms frequency-dependent specialization: TimesNet contributes higher weights (0.68) to low-frequency clusters, whereas Mamba dominates high-frequency clusters (0.65). Finally, Fig. 11(e) shows that attention entropy peaks under partly cloudy conditions, with cross-attention achieving the highest entropy, indicating its superior capability to aggregate diverse temporal information.

Fig. 11.

Attention mechanism analysis: (a)–(b) Cross-attention heatmaps under clear and partly cloudy conditions; (c) Temporal distribution of dual-branch attention weights; (d) Cluster-wise attention distribution; (e) Attention entropy comparison.

Discussion

Methodological implications

The experimental results show that the proposed IFTMC framework improves short-term PV power forecasting, supporting the effectiveness of the “decomposition–clustering–dual-backbone–fusion” paradigm. The observed gains from FCM-based soft clustering (relative to K-means hard partitioning) suggest that frequency-band boundaries in decomposed PV signals can be ambiguous, and that membership-weighted reconstruction may help preserve transitional information.

The complementary behavior of TimesNet and Mamba also motivates the dual-backbone design: TimesNet emphasizes multi-periodic intra-day patterns, whereas Mamba emphasizes longer-range dependencies. The cross-attention module then provides input-dependent fusion, which is particularly relevant under partly cloudy conditions where both local fluctuations and broader temporal context affect power output.

From a practical standpoint, the framework offers tangible benefits that extend beyond technical metrics to support broader sustainability objectives. Accurate 2–3-hour-ahead forecasts enable grid operators to make more informed dispatch decisions, potentially reducing reliance on fossil fuel-based spinning reserves and associated carbon emissions. The demonstrated MAE reduction of 4.2%–4.8% translates to improved renewable energy utilization by minimizing curtailment during supply–demand mismatches. Moreover, the modest inference latency (12.3 ms) supports real-time operation, facilitating integration into existing grid management systems and supporting higher renewable penetration.

Limitations and future directions

Despite these promising results, several limitations motivate future research. First, the fixed cluster number () was determined through validation-based selection; developing adaptive clustering mechanisms that automatically adjust to data characteristics could enhance generalization across different PV installations and climatic regions, thereby supporting global-scale renewable energy deployment. Second, while the DKASC dataset provides a valuable benchmark, evaluation on geographically diverse datasets would strengthen claims of broad applicability and better assess robustness across climatic conditions. Third, extending the framework from deterministic point forecasts to probabilistic forecasting with uncertainty quantification would provide richer decision support for risk-sensitive applications, enabling more robust grid planning under increasing climate variability. Fourth, incorporating additional data modalities such as satellite imagery and sky camera feeds could further improve performance during rapid cloud transient events. Finally, exploring transfer learning strategies to adapt pre-trained models to new PV sites with limited historical data represents an important avenue for reducing deployment barriers, particularly in developing regions where accelerating renewable adoption is critical for sustainable development.

Conclusion

This paper proposes a novel hybrid framework integrating ICEEMDAN decomposition, FCM soft clustering, dual-backbone deep learning networks (TimesNet and Mamba), and cross-attention fusion for short-term PV power forecasting. Extensive experiments on the DKASC dataset demonstrate that the proposed method achieves state-of-the-art performance, with MAE reduced by 4.2%–4.8% compared with the best baseline across prediction horizons of 2–3 hours ahead. Ablation studies confirm the contribution of each component: ICEEMDAN effectively separates multi-scale dynamics, FCM preserves frequency-band transitional information, the dual-backbone architecture captures complementary patterns, and cross-attention enables adaptive fusion. The framework exhibits particular robustness under partly cloudy conditions where rapid irradiance fluctuations pose the greatest challenge. By improving forecasting accuracy and reliability, this work contributes to the broader goal of sustainable energy systems, enabling higher penetration of renewables while maintaining grid stability. Future research will focus on adaptive clustering, probabilistic forecasting, and transfer learning to further extend the framework’s applicability across diverse geographical and climatic contexts.

Author contributions

F.C. conceived and designed the study, implemented the algorithm, conducted the experiments, analysed the results, and wrote the manuscript

Funding

This research received no external funding.

Data availability

The DKASC dataset used in this study is publicly available from the Desert Knowledge Australia Solar Centre. Processed data are available on request from the corresponding author.

Declarations

Competing interests

The authors declare no competing interests.