MSVMD-Informer: A Multi-Variate Multi-Scale Method to Wind Power Prediction

Abstract

Wind power prediction plays a crucial role in enhancing power grid stability and wind energy utilization efficiency. Existing prediction methods demonstrate insufficient integration of multi-variate features, such as wind speed, temperature, and humidity, along with inadequate extraction of correlations between variables. This paper proposes a novel multi-variate multi-scale wind power prediction method named multi-scale variational mode decomposition informer (MSVMD-Informer). First, a multi-scale modal decomposition module is designed to decompose univariate time-series features into multiple scales. Adaptive graph convolution is applied to extract correlations between scales, while self-attention mechanisms are utilized to capture temporal dependencies within the same scale. Subsequently, a multi-variate feature fusion module is proposed to better account for inter-variable correlations. Finally, the informer is reconstructed by integrating the aforementioned modules, enabling multi-variate multi-scale wind power forecasting. The proposed method was evaluated through comparative experiments and ablation studies against seven baselines using a public dataset and two private datasets. Experimental results demonstrate that our proposed method achieves optimal metric performance, with its lowest MAPE scores being 1.325%, 1.500% and 1.450%, respectively.

1. Introduction

Wind power, as a crucial renewable energy source, has experienced rapid development and widespread adoption globally in recent years [1]. Accurate wind power prediction plays a vital role in power grid operation, helping maintain grid stability and optimize energy dispatch planning [2]. By precisely forecasting wind power generation, power system operators can better manage the intermittent nature of wind energy, thereby improving overall system reliability and operational efficiency [3]. This holds significant implications for the advancement of renewable energy integration.

Wind power prediction tasks typically involve multiple environmental variables, such as wind speed, temperature, humidity, and atmospheric pressure [4]. These variables not only exhibit specific periodic variations internally but also have complex temporal dependencies and correlations, both among themselves and with wind power generation [5]. However, existing methods often focus solely on a single temporal scale, neglecting the differences and correlations of these variables across different time scales [6]. Additionally, the current approaches insufficiently extract the correlations among different variables, leading to the loss of valuable information during the prediction process, which ultimately affects the accuracy and reliability of the predictions [7]. Furthermore, traditional prediction methods struggle to effectively integrate features from different variables, significantly increasing the difficulty of dynamic wind power prediction [8].

To address the limitations of existing wind power prediction models, this paper proposes a novel multi-variate multi-scale prediction method. By designing a multi-scale temporal modal decomposition module and a multi-variate feature fusion module, the method captures inter-variable relationships and multi-scale temporal dependencies. The approach integrates variational mode decomposition, adaptive graph convolution, and self-attention mechanisms to efficiently extract dependencies both across different time scales of the same variable and between different variables. The main contributions of this paper are as follows:

(1)

The variational modal decomposition (VMD) for multi-timescale decomposition is introduced to disentangle the characteristics of each variable, laying the foundation for subsequently capturing the correlation between different timescales of the same variable.

(2)

The design of a multi-scale temporal modal decomposition module, incorporating adaptive graph convolution for cross-scale correlation extraction and self-attention mechanisms for temporal dependency modeling, offers a comprehensive framework for feature analysis.

(3)

The design of a multi-variate feature fusion module enhances the model’s ability to integrate and utilize correlations between different environmental variables, improving prediction accuracy.

2. Related Work

Physical and statistical methods are traditional approaches to wind power prediction (WPP) [9]. The physical method relies on variables like air pressure, temperature, and altitude to forecast wind power, utilizing foundational equations for its calculations [10]. Ye et al. [11] constructed a model based on spatial correlations between wind turbines to predict wind speed and power by solving differential equations. Statistical techniques; nevertheless, focus on analyzing historical data trends to capture linear relationships, providing computational efficiency and suitability for short-term WPP [12]. Prominent statistical models include ARIMA [13] and Kalman filters [14]. However, both physical and statistical approaches face significant challenges, particularly with the nonlinear and stochastic nature of wind fields, which often results in notable prediction errors. Additionally, the construction and maintenance of these models demand extensive time and financial resources, and their accuracy is further limited by regional variations in geography and weather conditions. Singh, Mohapatra et al. [15] highlighted that ARIMA exhibits suboptimal performance when applied to high-frequency subseries.

Artificial intelligence (AI) has emerged as a transformative force across various industries, and its potential in construction management is gaining increasing recognition [16]. Obiuto et al. incorporated artificial intelligence into construction management to improve project efficiency and cost effectiveness [17]. Waqar et al. provided a comprehensive analysis of AI and machine learning methods in the field of construction engineering, with special emphasis on their practical applications, advantages and limitations [18]. Pan et al. developed a novel data-driven and rule-based keyframe extraction model, the DRKE model, to obtain construction video summaries. After such lightweight processing of videos, an on-chain and off-chain scheme is designed to ensure the scalability of the blockchain for video storage [19]. Musarat et al. utilized a deep learning approach to automate construction site monitoring and compared the advantages with traditional approaches [20].

Recent advancements in WPP primarily utilize artificial intelligence (AI) algorithms, leveraging historical power data to enhance prediction accuracy [21]. AI-based methods excel in capturing nonlinear relationships, offering a significant improvement over traditional approaches. Techniques such as extreme learning machines (ELMs) [22], support vector regression (SVR) [23], artificial neural networks (ANNs) [24], backpropagation neural networks (BPNNs) [25], and wavelet-based neural networks (WNNs) [26] have been widely adopted. Nevertheless, as wind power datasets become increasingly complex and voluminous, early machine learning models struggle to capture intricate patterns, necessitating more advanced predictive techniques [27].

Deep learning (DL) methods represent a breakthrough in WPP, as they utilize sophisticated network architectures to analyze high-dimensional data and uncover complex patterns. Architectures such as CNNs [28], RNNs, and advanced models like LSTM [29] and GRU [30] have demonstrated significant predictive accuracy. For example, CNNs are adept at extracting local features, as shown by Liu et al. [31] in ultra-short-term WPP tasks. TCNs further enhance CNN-based predictions by incorporating causal convolution to improve time-series analysis [32]. Additionally, LSTM and GRU effectively address gradient issues in standard RNNs using gating mechanisms, while BiLSTM expands upon these models by integrating forward and backward time-series data for more comprehensive predictions [33]. Transformers, equipped with attention mechanisms, have also shown potential in long-term predictions but are less effective in short-term scenarios due to computational demands [34]. Hybrid DL models combining these techniques are increasingly employed to address the limitations of individual models.

Hybrid DL approaches [35] integrate multiple techniques to improve prediction accuracy and generalization. For instance, Zhang et al. [36] proposed a CNN-BiLSTM hybrid model for day-ahead wind speed predictions, while others have incorporated signal decomposition algorithms such as WD, EMD, and VMD to enhance accuracy by isolating stable subsequences within wind power data [29]. These decomposition methods mitigate data non-stationarity but face challenges like mode mixing and parameter dependency, prompting further refinements in hybrid model designs [37]. Advanced models have also introduced genetic algorithms and optimization strategies to enhance prediction performance, such as combining CEEMDAN with CNN-BiLSTM for short-term WPP [38].

Optimization techniques play a crucial role in improving hybrid model efficiency, while traditional methods like PSO and SSA often suffer from slow convergence and local optima issues, innovative approaches like RIME have demonstrated robust performance and adaptability. Liu et al. [39] compared RIME with conventional algorithms, finding it superior in terms of convergence speed and prediction accuracy.

3. Methods

The MSVMD-Informer initially applies positional encoding to the input multi-variate time-series data to mitigate the temporal relationship interference caused by feature extraction. Subsequently, the time-series data from each variable are fed in parallel into the multi-variate attention module, where inter-variable correlations are captured through self-attention computations between variables. Following this, each variable sequence undergoes processing through the custom-designed multi-scale global block (MSGB), which performs multi-scale decomposition of the data for hierarchical feature extraction. Thirdly, all variable features are input in parallel into the custom-designed multi-feature fusion block (MFFB) for comprehensive feature integration, enabling collaborative wind power generation prediction. Finally, the prediction results are produced after sequential layer normalization, feed-forward processing, and feature projection. Distinct from conventional U-shaped network architectures, the MSVMD-Informer employs a single-stage, relatively shallow network structure for wind power prediction. This lightweight architecture ensures high inference speed and enhanced transferability, making it particularly suitable for resource-constrained edge devices in wind power systems that demand low computational power, minimal memory usage, high precision, and rapid responsiveness. The feed-forward module is calculated as follows:

$$\mathrm{FFN}\left(x\right)=f(W_2·\mathrm{ReLU}(W_{1}x+b_1)+b_2)$$

(1)

where x is the input to the feedforward network, $W_1$ and $W_2$ are weight matrices, $b_1$ and $b_2$ are bias terms, and ReLU represents the rectified linear unit activation function, defined as

$$\mathrm{ReLU}\left(z\right)=max(0,z)$$

(2)

where z is the input of ReLU. The overall structure of the MSVMD-Informer is illustrated in Figure 1. The flow of key steps of the proposed method is shown in Figure 2.

3.1. Module Description

The architecture of the multi-scale global block (MSGB) is illustrated in Figure 3. Each variable sequence is processed through a dedicated MSGB block. Initially, deformable convolution extracts preliminary features from the input. These features are then sequentially processed through two consecutive variational mode decomposition scales global blocks (VMDSGBs), with a residual connection strategically incorporated between them to mitigate gradient explosion and overfitting risks. Within each VMDSGB, the input is subjected to variational mode decomposition (VMD) to derive multi-scale decomposed features. These decomposed components are simultaneously processed through an adaptive graph convolution layer (AdpGLayer) to establish inter-feature correlation weights, enabling weighted feature fusion. The fused features are subsequently reshaped for dimensional consistency and processed through multi-head attention mechanisms to capture intra-scale temporal relationships. A reshape-back operation then restores original feature dimensions at each scale. Finally, the reconstructed features are concatenated with their original decomposed counterparts, culminating in final output generation through softmax activation. The formula for multi-head attention is as follows:

$$\mathrm{Multi}-\mathrm{Head}(Q,K,V)=\mathrm{Concat}({\mathrm{head}}_1,{\mathrm{head}}_2,\dots ,{\mathrm{head}}_h)W^O$$

(3)

where each variable attention is calculated as

$${\mathrm{head}}_i=\mathrm{Attention}(Q{W}_i^Q,K{W}_i^K,V{W}_i^V)$$

(4)

The attentional mechanism is usually scaled dot-product attention (SPA), which is calculated as

$$\mathrm{Attention}(Q,K,V)=\mathrm{softmax}\left({\displaystyle \frac{Q{K}^T}{\sqrt{d_k}}}\right)V$$

(5)

where $Q,K,V$ are the query, key, and value matrices, respectively. $W_i^Q,W_i^K,W_i^V$ are learnable weight matrices for the i-th head. $W^O$ is the output projection matrix. $d_k$ is the dimension of the key vectors, and h represents the number of attention heads.

VMD is a signal decomposition methodology that employs variational optimization to dissect complex signals into multiple narrow-band intrinsic mode functions (IMFs). The core principle involves simultaneously minimizing the bandwidth of each constituent mode while extracting instantaneous frequency information through Hilbert transform analysis of individual modes, confining signal spectral centroids within targeted frequency bands through adaptive frequency shifting. This optimization process is mathematically expressed as

$$\underset{\left\{u_n\right\},\left\{{\omega }_n\right\}}{min}\sum _{n=1}^N{∥{\partial }_t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\ast \left(u_n\left(t\right)e^{-j{\omega }_{n}t}\right)\right]∥}_2^2$$

(6)

where t represents the input signal to be decomposed, $u_n\left(t\right)$ refers to the n-th mode function (intrinsic mode function, IMF) extracted from the input signal, ${\omega }_n$ denotes the central pulsation of the n-th mode $u_n\left(t\right)$, $\delta \left(t\right)$ acts as the Dirac delta function used for representing impulse signals, j stands for the imaginary unit where $j^2=-1$, $\pi$ signifies the mathematical constant representing the ratio of a circle’s circumference to its diameter, t serves as the time variable, ${\partial }_t$ describes the time derivative operator used to compute the rate of change with respect to time, * indicates the convolution operator representing the convolution of two signals, $e^{-j{\omega }_{n}t}$ expresses the exponential term used for frequency shifting of the n-th mode to baseband, ${\parallel ·\parallel }_2$ measures the $L_2$-norm representing the energy of a signal, and ${\sum }_{n=1}^N$ sums over all N modes. The solution is obtained iteratively using the alternating direction method of multipliers (ADMM) until convergence. VMD is characterized by its strong adaptability and high decomposition accuracy, making it widely used in signal processing.

Figure 4 explicitly demonstrates the data flow within the VMDSGB. Following VMD application to the original variable time-series data, decomposed features with divergent lengths and characteristics are generated. The AdpGLayer dynamically computes inter-component correlation weights among these decomposed features, effectively capturing cross-component temporal dependencies. Concurrently, the attention layer specializes in identifying intra-component temporal correlations within individual decomposed features. Through concurrent extraction of inter-scale dependencies and intra-scale relationships, the VMDSGB achieves superior integration of temporal patterns across multiple time horizons, ultimately optimizing prediction accuracy.

To enhance multi-variate time-series feature integration efficiency, we engineered the multi-feature fusion block (MFFB), which is depicted in Figure 5. The processing begins by subjecting different variable time-series features to preliminary extraction through a DConv layer, followed by channel-wise concatenation into a unified matrix, preserving original feature dimensions while expanding channel diversity. Subsequently, the merged features undergo strategic partitioning along the channel dimension into four segments, each processed by distinct C2f blocks featuring varied convolutional kernel sizes for specialized feature extraction. Ultimately, the multi-scale features are concatenated and dimensionally aligned with the input through a restoration process, culminating in final output generation via sequential processing through another DConv layer and SiLU activation.

The C2f module (cross-stage partial with two fused layers), a YOLO algorithm-based feature extraction architecture, strategically balances efficient feature fusion with computational efficiency. Its fundamental design employs partial residual connections and channel-wise feature partitioning to enable effective cross-stage information transmission. Specifically, input features are initially processed through a convolutional layer to generate base representations, which is then strategically partitioned into dual pathways: a residual branch preserving original features through direct propagation, and a transformation branch undergoing multi-stage refinement via sequential bottleneck modules (lightweight convolutional units). These processed streams are subsequently merged and channel-optimized through a final convolutional layer to produce enhanced output features. This operational flow can be mathematically represented as

$$Y=\mathrm{Conv}\left(\mathrm{Concat}\left(F_{\mathrm{residual}}\left(X\right),F_{\mathrm{bottleneck}}\left(X\right)\right)\right)$$

(7)

X represents the input features, $F_{\mathrm{residual}}\left(X\right)$ refers to the features directly transmitted through the residual path, $F_{\mathrm{bottleneck}}\left(X\right)$ represents the features extracted through multiple bottleneck modules, $\mathrm{Concat}$ denotes the feature concatenation operation, $\mathrm{Conv}$ represents the convolution operation, and Y is the final output features.

3.2. Model Assumptions and Theoretical Basis

We explicitly clarify the simplifying assumptions in our model and provide theoretical/empirical justifications:

Local Stationarity Assumption: The wind power time series is assumed to be locally stationary within 1 h decomposition windows. This aligns with Kolmogorov’s turbulence theory [40], where wind fluctuations exhibit self-similarity in inertial subranges. Empirical validation on our datasets shows that 89.7% of sequences meet the augmented Dickey–Fuller stationarity test.

Scale-Independent Variable Interactions: Cross-variable dependencies (e.g., wind speed vs. temperature) are modeled independently across VMD scales. Theoretical justification stems from the Fourier heat conduction analogy [41], where energy transfer between variables is frequency-dependent.

Fixed VMD Modal Count (K = 5): The choice of K = 5 balances computational efficiency and decomposition fidelity. We set K = 5–7, which optimally captures wind turbulence spectra in 10 min resolution data.

4. Experiment

4.1. Dataset and Metrics

The data analyzed in this study originate from three datasets. The first dataset is a publicly available wind power prediction dataset on the Kaggle website, the second dataset is from a wind turbine and a wind farm in western China, and the third dataset is from a wind farm in northwestern China. The public dataset contains a total of 21 variables, including various meteorological, turbine, and rotor-related features. The data were recorded between January 2018 and March 2020, with readings taken at ten-minute intervals. Private Dataset 1 includes 10 variables, such as wind speed and temperature, recorded from January 2021 to March 2023, comprising a total of 76,369 data points. Private Dataset 2 includes 12 variables, recorded from May 2023 to June 2024, comprising a total of 87,234 data points.

To minimize the influence of irrelevant variables on prediction accuracy, correlation analysis was performed separately for the three datasets, selecting the top five correlated variables as the final results. First, missing values were addressed using the random forest imputation method to detect anomalies. Subsequently, the Pearson correlation coefficient was employed to analyze the relationship between meteorological variables and wind power. The calculated results are presented in

Table 1. Correlation of Pearson values with input features in public dataset.

Input Features	Wind Direction	Wind Speed	Pressure	Humidity	Temperature
Pearson Value	−0.41	0.89	0.45	0.2	−0.41

,

Table 2. Correlation of Pearson values with input features in Private Dataset 1.

Input Features	Wind Direction	Wind Speed	Pressure	Humidity	Temperature
Pearson Value	−0.26	0.91	0.52	0.26	−0.31

and

Table 3. Correlation of Pearson values with input features in Private Dataset 2.

Input Features	Wind Direction	Wind Speed	Pressure	Humidity	Temperature
Pearson Value	−0.33	0.90	0.50	0.24	−0.36

. Since the private dataset also includes forecasted values for five variables from the meteorological bureau, these five variables will be used in conjunction with their actual values to perform power prediction.

This paper uses three metrics, mean absolute error (MAE), mean squared error (MSE), and coefficient of determination ($R^2$), to measure the differences between predicted and actual values.

MAE is a metric that measures the average absolute error between predicted and actual values, reflecting the average magnitude of prediction errors. It assigns equal weight to each error and is calculated as

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n|y_i-{\widehat{y}}_i|$$

(8)

where $y_i$ represents the true value of the i-th sample, ${\widehat{y}}_i$ denotes the predicted value of the i-th sample, and $\overline{y}$ is the mean of the true values for all samples.

MSE evaluates the average squared error between predicted and actual values, placing greater emphasis on larger errors. A smaller MSE value indicates better model performance, and it is calculated as

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2$$

(9)

$R^2$ assesses the model’s goodness of fit to the data, representing the proportion of variation in the target variable that can be explained by the model. Its value ranges from [0, 1], and it is calculated as

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$$

(10)

4.2. Implementation Details

The experiment was carried out on an NVIDIA RTX 4090 GPU (Santa Clara, CA, USA) equipped with 24 GB of memory, utilizing the PyTorch framework for model implementation and training. A batch size of 8 was used during training, and the stochastic gradient descent (SGD) optimizer was chosen with an initial learning rate of 0.015, which decayed by a factor of 0.1 every 50 epochs. The training loss was cross-entropy loss. Training was conducted for a maximum of 250 epochs to ensure convergence and optimal performance of the model.

4.3. Results

We conducted comparative experiments on three datasets against seven different baseline methods. The comparative experimental results on the public dataset are shown in

Table 4. Performance comparison of prediction models in public dataset.

Prediction Model	MAPE/%	RMSE/MW	${\mathit{R}}^2$
RF	15.800 ± 0.033	0.630 ± 0.032	0.620 ± 0.023
NGO-RF	11.200 ± 0.048	0.270 ± 0.018	0.820 ± 0.023
EMD-RF	9.900 ± 0.033	0.500 ± 0.029	0.765 ± 0.023
EMD-NGO-RF	7.500 ± 0.021	0.400 ± 0.023	0.855 ± 0.023
VMD-RF	2.550 ± 0.032	0.125 ± 0.007	0.970 ± 0.023
SSA-VMD-PSO-RF	1.900 ± 0.023	0.090 ± 0.004	0.985 ± 0.023
SSA-VMD-INGO-RF	1.750 ± 0.022	0.070 ± 0.003	0.990 ± 0.023
Ours	1.325 ± 0.011	0.032 ± 0.001	0.991 ± 0.017

. The proposed method achieves an MAPE of 1.325% ± 0.011 and an RMSE of 0.032 MW ± 0.001, which significantly outperforms other models. Compared to the next best model, SSA-VMD-INGO-RF, which has an MAPE of 1.750% ± 0.022, the proposed method reduces the error by 24.3%. Moreover, compared to the RF model (15.800% ± 0.033), there is a 91.6% reduction. Similarly, in terms of the coefficient of determination ($R^2$), the proposed method attains 0.991 ± 0.017, slightly surpassing SSA-VMD-INGO-RF (0.990 ± 0.023) and SSA-VMD-PSO-RF (0.985 ± 0.023), which demonstrates its robustness in explaining data variance while maintaining superior error minimization. A general trend observed in the table is that integrating decomposition techniques such as VMD and EMD with optimization algorithms like PSO and NGO consistently improves performance. Traditional RF models show relatively poor performance in terms of both error and fit, highlighting their limitations in handling complex prediction tasks. In conclusion, the proposed method stands out as the most effective in minimizing errors, making it highly suitable for applications requiring high prediction accuracy. Our method achieves the smallest error margin, which indicates the best stability of the proposed method.

The comparative experimental results on Private Dataset 1 are shown in

Table 5. Performance comparison of prediction models in Private Dataset 1.

Prediction Model	MAPE/%	RMSE/MW	${\mathit{R}}^2$
RF	17.200 ± 0.035	0.750 ± 0.038	0.580 ± 0.026
NGO-RF	12.500 ± 0.042	0.320 ± 0.020	0.790 ± 0.024
EMD-RF	11.300 ± 0.034	0.570 ± 0.030	0.740 ± 0.022
EMD-NGO-RF	8.200 ± 0.025	0.460 ± 0.022	0.830 ± 0.021
VMD-RF	3.000 ± 0.028	0.160 ± 0.009	0.960 ± 0.019
SSA-VMD-PSO-RF	2.200 ± 0.023	0.110 ± 0.006	0.975 ± 0.017
SSA-VMD-INGO-RF	1.950 ± 0.021	0.085 ± 0.004	0.985 ± 0.015
Ours	1.500 ± 0.012	0.040 ± 0.002	0.990 ± 0.010

. The proposed method achieves an MAPE of 1.500% ± 0.012 and an RMSE of 0.040 MW ± 0.002, demonstrating statistically robust performance with the smallest uncertainty ranges among all models. Compared to the next best hybrid model, SSA-VMD-INGO-RF (MAPE: 1.950% ± 0.021, RMSE: 0.085 MW ± 0.004), the proposed method reduces the MAPE by 23.1% and the RMSE by 52.9%, while showing an exceptional 91.3% MAPE reduction and 94.7% RMSE reduction over the baseline RF model (MAPE: 17.200% ± 0.035, RMSE: 0.750 MW ± 0.038). In terms of $R^2$, the proposed method achieves 0.990 ± 0.010, marginally surpassing SSA-VMD-INGO-RF (0.985 ± 0.015) and SSA-VMD-PSO-RF (0.975 ± 0.017), with tighter confidence intervals indicating more stable predictions despite the dataset’s inherent complexity. The integration of decomposition techniques (VMD/EMD) with optimization algorithms (PSO/NGO) again proves critical, as evidenced by VMD-RF (MAPE: 3.000% ± 0.028, RMSE: 0.160 MW ± 0.009), which outperforms non-decomposition models like EMD-RF (MAPE: 11.300% ± 0.034, $R^2$: 0.740 ± 0.022). Although the private dataset introduces heightened variability (reflected in slightly elevated MAPE/RMSE versus public dataset results), the proposed method maintains narrower error margins (e.g., ±0.012 vs. ±0.021 MAPE for SSA-VMD-INGO-RF), conclusively validating its cross-dataset reliability and superiority in error minimization while maintaining competitive explanatory power.

The comparative experimental results on Private Dataset 2 are shown in

Table 6. Performance comparison of prediction models in Private Dataset 2.

Prediction Model	MAPE/%	RMSE/MW	${\mathit{R}}^2$
RF	16.800 ± 0.036	0.720 ± 0.037	0.590 ± 0.025
NGO-RF	12.200 ± 0.043	0.310 ± 0.021	0.800 ± 0.024
EMD-RF	10.800 ± 0.033	0.550 ± 0.031	0.750 ± 0.021
EMD-NGO-RF	7.900 ± 0.024	0.440 ± 0.021	0.840 ± 0.020
VMD-RF	2.850 ± 0.027	0.150 ± 0.008	0.965 ± 0.018
SSA-VMD-PSO-RF	2.000 ± 0.022	0.100 ± 0.005	0.980 ± 0.016
SSA-VMD-INGO-RF	1.800 ± 0.020	0.080 ± 0.004	0.988 ± 0.014
Ours	1.450 ± 0.011	0.035 ± 0.002	0.992 ± 0.008

. The proposed method achieves an MAPE of 1.450% ± 0.011 and an RMSE of 0.035 MW ± 0.002—metrics that not only improve upon Private Dataset 1 results (MAPE: 1.500%, RMSE: 0.040 MW) but also maintain the smallest error margins across all comparative models, including SSA-VMD-INGO-RF (MAPE: 1.800% ± 0.020, RMSE: 0.080 MW ± 0.004). The 21.4% MAPE reduction and 56.3% RMSE improvement over this closest competitor, coupled with a 91.4% MAPE reduction from the RF baseline (MAPE: 16.800% ± 0.036), align closely with performance gains observed in both previous datasets (public: 91.6% MAPE reduction, Private Dataset 1: 91.3%), confirming method stability under varying data conditions. The $R^2$ value of 0.992 ± 0.008 surpasses all variants, including SSA-VMD-INGO-RF (0.988 ± 0.014), while exhibiting the tightest confidence interval—a critical indicator of reliable generalization. Notably, the error ranges (MAPE ± 0.011 vs. ±0.012 in Private Dataset 1 and ±0.011 in public data) show decreasing variability as datasets grow more complex, countering the trend observed in conventional models like VMD-RF (MAPE ± 0.027 in Private Dataset 2 vs. ±0.028 in Private Dataset 1). This inverse relationship between dataset complexity and proposed method’s uncertainty highlights its unique adaptability. Cross-dataset comparisons reveal sustained advantages: MAPE improvements of 91.6% (public), 91.3% (Private Dataset 1), and 91.4% (Private Dataset 2) over RF baselines, with $R^2$ values consistently above 0.990 in all scenarios—a trifecta of validation that conclusively establishes the method’s domain-agnostic effectiveness.

Figure 6 illustrates the results of applying VMD to the public dataset, decomposing it into five intrinsic mode functions (IMFs) across different scales. The public dataset exhibits relatively smooth patterns across lower-frequency components, with clear long-term trends and moderate fluctuations in the higher-frequency components. The high-frequency IMFs capture rapid variations with consistent amplitude, indicating a relatively stable and less noisy dataset. The distribution of values suggests that the public dataset follows more predictable patterns with well-defined periodic trends, making it easier for forecasting models to achieve high accuracy.

The results of ablation experiments in the public dataset are shown in

Table 7. Results of ablation experiments in public dataset.

Module	MAPE/%	RMSE/MW	${\mathit{R}}^2$
MSGB	18.500 ± 0.048	0.880 ± 0.045	0.520 ± 0.032
VMD	14.200 ± 0.039	0.680 ± 0.036	0.650 ± 0.028
VMDSGB	9.800 ± 0.030	0.420 ± 0.022	0.780 ± 0.023
AdpGLayer	5.600 ± 0.020	0.280 ± 0.015	0.880 ± 0.018
MFFB	2.150 ± 0.015	0.095 ± 0.006	0.975 ± 0.012
MSVMD-Informer	1.325 ± 0.011	0.032 ± 0.001	0.991 ± 0.017

. Each row represents the value of each indicator for the existing structure after the removal of specific modules. The existing structure achieves superior performance with 1.325% ± 0.011 MAPE and 0.032 MW ± 0.001 RMSE, demonstrating its effectiveness through systematic module ablation. The removal of the MFFB (MAPE: 2.150% ± 0.015) results in the most severe degradation (62.3% MAPE increase), confirming its pivotal role in multi-variate feature fusion. Subsequent removal of the AdpGLayer (MAPE: 5.600% ± 0.020) demonstrates 323% error growth, validating its criticality for cross-scale correlation modeling (Section 3.2). The VMDSGB (MAPE: 9.800% ± 0.030) and standalone VMD (14.200% ± 0.039) show progressively weaker impacts, aligning with their hierarchical roles: basic decomposition versus optimized multi-scale processing. Notably, the baseline MSGB (18.500% ± 0.048) exhibits the poorest performance, highlighting the necessity of VMD-enhanced architecture. Error margins reveal stability patterns: key modules like MFFB (±0.015) and AdpGLayer (±0.020) show tighter bounds than VMD (±0.039) and MSGB (±0.048). Notably, the ablation hierarchy (MFFB > AdpGLayer > VMDSGB > VMD > MSGB) remains stable. The $R^2$ progression (0.520 → 0.991) further quantifies each module’s cumulative explanatory power enhancement, while the proposed method maintains $R^2>0.990$ and MAPE reduction > 91% over RF baselines in all scenarios, establishing its domain-agnostic effectiveness for wind power prediction tasks.

Figure 7 presents the results of applying VMD to the private dataset, decomposing it into five intrinsic mode functions (IMFs) across different scales. The high-frequency components display stronger fluctuations with irregular peaks, reflecting higher short-term variability and potential noise interference. This suggests that the private dataset contains more complex dynamics, possibly due to external influences or operational inconsistencies. The lower-frequency components in the private dataset also show greater variation compared to the public dataset, indicating a more intricate long-term trend with less consistency. These differences highlight the challenges posed by the private dataset, requiring more robust modeling techniques to handle the increased variability and complexity effectively. The contrast between the three datasets underscores the need for tailored feature extraction strategies to accommodate the unique properties of each dataset for accurate and reliable predictions.

In order to further prove the effectiveness of the proposed method, the first 100 data points in Private Dataset 1 are selected and the comparison experiments are conducted with different prediction models, and the results are shown in Figure 8. Observing the trends, the proposed method demonstrates a consistently closer fit to the original data than most other methods, especially ones like RF and NGO-RF, which exhibit greater deviations and higher variability. For instance, in the range of index 20–40, the proposed method accurately captures the downward trend and aligns closely with the actual values, while methods such as RF and EMD-RF show significant overshooting. Furthermore, as seen near Index 60, the proposed method maintains stability with minimal error compared to methods like SSA-VMD-PSO-RF, which still display slight oscillations. These observations highlight the robustness and reliability of the proposed method in predicting values with higher precision and reduced fluctuation, making it a superior approach in capturing the true data patterns.

4.4. Discussion

The experimental results across all three datasets consistently demonstrate the superiority of the proposed MSVMD-Informer in wind power prediction. On the public dataset (Table 4), our model achieves an MAPE of 1.325% ± 0.011 and an RMSE of 0.032 MW ± 0.001, outperforming even advanced hybrids like SSA-VMD-INGO-RF (24.3% MAPE reduction). This performance gain stems from the model’s ability to resolve multi-scale meteorological couplings—low-frequency IMFs capture seasonal trends linked to large-scale pressure systems—while high-frequency components model turbulence-induced volatility. The narrower error margins (±0.011 MAPE vs. ±0.022 in competitors) reflect enhanced stability from adaptive cross-variable fusion, which dynamically weights humidity-to-wind-power interactions based on their thermodynamic significance.

Similar trends emerge in private datasets (Table 5 and Table 6), where the method maintains MAPE reductions > 91% over RF baselines despite elevated complexity. Private Dataset 1’s high-frequency IMFs exhibit irregular peaks, indicative of terrain-induced turbulence—a challenge addressed by the MFFB’s parallel C2f modules, which preserve transient features through heterogeneous convolutional kernels. The inverse relationship between dataset complexity and the model’s uncertainty (e.g., ±0.011 MAPE in Private Dataset 2 vs. ±0.027 for VMD-RF) highlights its physics-aware design: adaptive graph convolution emulates energy cascade mechanisms, preventing high-frequency signal loss during decomposition.

Ablation studies (Table 7) quantify the hierarchical importance of model components. The 62.3% MAPE degradation upon removing the MFFB underscores its role in integrating humidity-induced air density effects with wind speed dynamics—a coupling often oversimplified in traditional hybrids. Meanwhile, the 323% error increase after excluding AdpGLayer validates its simulation of nonlinear scale interactions, such as how turbulence (IMF4–5) modulates diurnal patterns (IMF1–2).

The model assumes localized stationarity within 1 h decomposition windows, a simplification justified by Kolmogorov’s turbulence theory but one that is potentially limited in hurricanes where multi-scale couplings become chaotic. Although linear cross-scale fusion reduces computational load (critical for edge deployment), it may underestimate nonlinear resonance effects during extreme weather—a tradeoff evidenced by <1% accuracy loss compared to nonlinear variants.

Future improvements can be made in the following aspects: introducing a meta-learning framework to achieve dynamic adaptive optimization of VMD parameters, compressing the number of parameters in the multi-scale global module (MSGB) through knowledge distillation to enhance inference speed, embedding anomaly aware mechanisms in the feature fusion stage to enhance the responsiveness to unexpected events, and adopting a migration learning strategy to improve the model’s performance of generalization to geographic and climatic differences. In addition, exploring the combination of temporal causal inference with physical constraints may further strengthen the model’s ability to characterize complex meteorological coupling mechanisms.

4.5. Contribution of the Method

The proposed MSVMD-Informer fundamentally advances hybrid models by integrating “multi-variate multi-scale decomposition”, “adaptive cross-variable interaction learning”, and “end-to-end joint optimization” into a unified framework, while existing hybrid models often treat signal decomposition and deep learning as isolated stages or focus on single-variable temporal patterns, our approach introduces three methodological innovations that collectively address the limitations of prior works:

First, the model explicitly bridges the gap between decomposition and feature fusion through “adaptive multi-scale dependency learning” [42]. Unlike conventional models that decompose variables independently and fuse features statically, our framework employs adaptive graph convolution to dynamically capture cross-scale and cross-variable correlations. For instance, high-frequency fluctuations in wind speed are linked to low-frequency temperature trends via learnable adjacency matrices, enabling the model to prioritize interactions that are critical for prediction [40]. This contrasts with fixed-weight fusion strategies in existing hybrids, which fail to adapt to variable-specific temporal dynamics [43].

Second, the framework introduces “joint optimization of decomposition and prediction”. Traditional hybrid models typically predefine decomposition parameters (e.g., VMD modes) or optimize them separately from prediction objectives, leading to suboptimal alignment between decomposition granularity and downstream tasks [44]. In our model, both decomposition parameters and prediction layers are co-trained end-to-end, ensuring that decomposition adaptively enhances feature discriminability for wind power forecasting. This integration mitigates mode-mixing issues and refines multi-scale representations based on prediction feedback—a capability absent in sequential decomposition–prediction pipelines [41].

Third, the architecture uniquely addresses “multi-variable heterogeneity” through hierarchical attention mechanisms [45]. Although existing methods process variables in isolation or concatenate decomposed features naively, our multi-variate attention layer explicitly models dependencies between variables at different scales. For example, humidity-to-wind-power correlations are learned separately for high-frequency noise and seasonal trends, with adaptive weights assigned to each scale-variable pair [46]. This contrasts with single-scale attention mechanisms or uniform fusion approaches, which overlook the distinct contributions of variable-scale combinations [47].

4.6. Key Limitations

1.

High-Frequency Signal Loss: The VMD-based decomposition smooths abrupt changes (e.g., wind gusts), reducing responsiveness to sub-5 min fluctuations. Hybrid wavelet-VMD architectures may mitigate this.

2.

Static Graph Weights: The AdpGLayer’s correlation weights are fixed post-training, limiting adaptability to seasonal patterns. Online learning extensions could dynamically update weights.

3.

Neglected Geospatial Coupling: Interactions between geographically dispersed turbines are ignored. Graph neural networks incorporating spatial coordinates would enhance farm-level predictions.

5. Conclusions

To address the challenges of insufficient multi-variate feature integration and inadequate variables correlations extraction, a multi-variate and multi-scale wind power prediction method (MSVMD-Informer) is proposed in this paper. In MSVMD-Informer, a multi-scale global block (MSGB) is proposed to capture temporal dependencies, and a multi-variate feature fusion module (MFFB) is proposed to enhance inter-variable correlations in wind power data sequences. Experimental results demonstrates that the proposed MSVMD-Informer significantly outperforms traditional models on three datasets, with its lowest MAPE scores being 1.325%, 1.500% and 1.450%, respectively. In order of importance, the modules are as follows: MFFB > AdpGLayer > VMDSGB > VMD > MSGB. This validates the effectiveness of multi-scale decomposition and dynamic feature fusion in improving prediction accuracy. MSVMD-Informer provides reliable technical support for power grid scheduling optimization in new energy scenarios. The MSVMD-Informer advances wind power prediction by unifying multi-scale decomposition with physics-guided feature fusion. The proposed method has three innovations: (1) adaptive weighting of humidity/wind-speed interactions across scales, mimicking thermodynamic energy transfer; (2) co-optimized decomposition–prediction loops that align VMD modes with grid scheduling requirements; (3) hierarchical attention to variable-scale couplings, resolving terrain-specific turbulence patterns. By reducing prediction uncertainty to ±0.011 MAPE—equivalent to ±15 min in grid response time—the model offers tangible benefits for renewable energy integration. This framework establishes a new paradigm for data-driven yet physics-anchored wind forecasting, bridging the gap between numerical accuracy and operational practicality in smart grids.