Research on Physically Constrained VMD-CNN-BiLSTM Wind Power Prediction

Abstract

Accurate forecasting of wind power is crucial for addressing energy demands, promoting sustainable energy practices, and mitigating environmental challenges. In order to improve the prediction accuracy of wind power, a VMD-CNN-BiLSTM hybrid model with physical constraints is proposed in this paper. Initially, the isolation forest algorithm identifies samples that deviate from actual power outputs, and the LightGBM algorithm is used to reconstruct the abnormal samples. Then, leveraging the variational mode decomposition (VMD) approach, the reconstructed data are decomposed into 13 sub-signals. Each sub-signal is trained using a CNN-BiLSTM model, yielding individual prediction results. Finally, the XGBoost algorithm is introduced to add the physical penalty term to the loss function. The predicted value of each sub-signal is taken as the input to get the predicted result of wind power. The hybrid model is applied to the 12 h forecast of a wind farm in Zhangjiakou City, Hebei province. Compared with other hybrid forecasting models, this model has the highest score on five performance indicators and can provide reference for wind farm generation planning, safe grid connection, real-time power dispatching, and practical application of sustainable energy.

1. Introduction

The transition to sustainable energy represents a critical strategy for addressing global environmental challenges while simultaneously fostering economic development [1]. The efficient use and sustainable development of renewable energy sources, such as wind energy, is critical to drive the global transition from non-renewable energy systems to renewable energy systems. However, the large-scale deployment of wind energy also introduces significant challenges to the safe and stable operation of power grids [2,3,4]. Therefore, the accurate prediction of wind power is of great significance to ensure the stable operation of the power system and the sustainable development of wind power. The inherent intermittency, volatility, and randomness of wind energy create substantial difficulties in the wind power forecast. To improve the precision of forecasting, the methodologies for predicting wind power have evolved through three primary stages: beginning with physical calculations, transitioning to statistical models, and currently embracing deep learning techniques [5]. This evolution reflects a concerted effort to improve forecasting reliability in the face of the complexities associated with wind energy generation.

Physical calculation [6] involves predicting wind power based on numerical weather predictions. This method utilizes forecasted weather data and physical information surrounding the wind turbine to calculate wind speed and direction at the hub height, ultimately deriving the predicted wind power through model substitution. While this approach is a well-established method for medium- to long-term wind power forecasting, it entails a complex calculation process that requires substantial computational resources, leading to certain limitations. In order to address these challenges, researchers have increasingly turned to statistical models for wind power forecasting. Unlike physical models, statistical models [7,8,9] do not account for the underlying physical processes, such as variations in wind speed. Instead, they identify relationships between weather conditions and turbine power generation based solely on historical statistical data. By leveraging measured data alongside weather forecasts, these models excel at ultra-short-term and short-term predictions due to their computational efficiency and rapid processing speed. However, they tend to exhibit poor accuracy in predicting gusts and sudden wind changes. Their predictive performance generally deteriorates as the forecast horizon extends. With advancements in artificial intelligence, AI algorithms have emerged as powerful tools for accurately capturing the nonlinear relationships present in wind power data. These algorithms significantly enhance prediction accuracy and have become essential for improving wind power forecasting methodologies [10,11,12,13].

In the initial phases of research on wind power forecasting, attention was focused on the impact of wind speed on power generation. This led to the development of physical prediction models. These models intricately integrate various factors, including wind speed, turbine blade area, terrain characteristics, and air density, to improve the precision of wind power forecasts. Wang L.J. et al. [14] proposed a novel approach that combines cluster analysis and principal component analysis to investigate power prediction and retrogression of wind power generation. This method effectively integrates multiple numerical weather prediction (NWP) datasets surrounding wind farms to improve forecasting accuracy. Similarly, Li L. et al. [15] introduced a wind speed prediction method based on computational fluid dynamics, providing a foundational framework for enhancing wind power forecasting. These advancements illustrate the evolving methodologies in wind power forecast, emphasizing the crucial role of wind speed and environmental factors in generating reliable forecasts.

The physical prediction model comprehensively incorporates meteorological and terrain factors, enhancing its explanatory power for wind power forecasting. However, its reliance on extensive meteorological and geographic data leads to substantial computational demands and low efficiency. This makes it unsuitable for short-term wind power predictions. Consequently, researchers have begun to explore statistical models for wind power forecasting, including time series methods [16] and gray prediction methods [17]. These statistical methods greatly reduce the computational burden and broaden the applicability of prediction models.

While statistical models show high computational efficiency and rapid processing capabilities, they struggle to effectively capture the nonlinear relationships in wind power datasets. In order to address this limitation, researchers have increasingly changed to deep learning methodologies to uncover these nonlinear dynamics and enhance forecasting accuracy. Wang J.D. et al. [18] developed a wind power range forecast model that leverages the multi-output capabilities of back-propagation (BP) neural networks, along with optimization criteria for prediction interval information. Early investigations largely focused on single neural networks; however, the inherent variability of wind energy necessitates a more robust approach. Thus, researchers have proposed a decomposition and integration framework [19,20,21,22], which employs signal decomposition techniques to break the wind power time series into multiple intrinsic sub-signals. Each sub-signal is predicted individually, and the resultant predictions are subsequently aggregated to yield a final forecast. Sun H.B. et al. [20] proposed a CEEMDAN-GWO-BiLSTM wind power prediction model. The GWO algorithm was used to optimize the hyperparameters of the prediction model, an approach which improved the prediction accuracy of wind power. Guo Z.Y. et al. [23] proposed an EMD-CNN-TGNN architecture to improve the adaptability and response to dynamic wind patterns through proportional, integral, and differential control, thereby improving wind power prediction accuracy. However, this prediction method has some defects: in the integration process of each sub-signal prediction result, only from the data themselves, there is a negative power prediction result that does not conform to the physical law (See

Table 1. The summary of the literature review.

Paper	Dataset	Method	Evaluation Metrics
Nicholas J.C. (2011) [6]	The Yambuk and Challicum Hills wind farms owned by Pacific Hydro Pty Ltd., the Bluff Point wind farm owned by Roaring40s Pty Ltd.	Wind farm power curve modeling	Root mean square error
Louka P. (2008) [7]	SKIRON NWP	F-NN, Kalman filtering	Absolute bias, standard deviation of the bias and of the absolute bias, root mean square error
Chen N. (2012) [8]	\	Cross-entropy, ARIMA	Root mean square error
Huang D.G. (2023) [9]	A wind farm in Chongqing	KDE, BDE, SBL	PINAW, CWC
Gu W.G. (2024) [10]	\	OLHS-DBO-BP	Mean square error, mean absolute error, symmetric mean absolute percentage error
Xue Y. (2019) [11]	Supervisory control and data acquisition	GRU, CNN	Mean square error
Han L. (2019) [12]	\	VMD, ILSTM	Mean absolute percentage error
Huang F. (2021) [13]	\	WPF	Mean absolute percentage error, mean square error, mean absolute error
Wang L.J. (2015) [14]	Heilongjiang Yilan wind farm	NWP	Root mean square error, mean absolute error
Li L. (2013) [15]	\	CFD, NWP	\
An X. L. (2012) [16]	Dongtai wind farm	EMD, GM(1,1)	Mean absolute percentage error, root mean square error, mean absolute error
Zhang Q. (2012) [17]	The Changshun wind park in Huade County, Inner Mongolia Autonomous Region, China	LS-SVM	Mean absolute percentage error
Wang J.D. (2017) [18]	Key Laboratory of Smart Grid of Ministry of Education (Tianjin University)	PSO-BP	PINAW, PICP
Sun H.B. (2024) [20]	A wind farm in Gansu Province	CEEMDAN-GWO-Bi-LSTM	Mean absolute percentage error, root mean square error, mean absolute error, R-squared
Guo Z.Y. (2024) [23]	Two wind farms in Xingtai, Hebei	EMD-CNN-TGNN	Root mean square error, mean square error, mean absolute error, R-squared

).

To improve the aforementioned shortcomings and enhance prediction accuracy, this paper introduces a hybrid VMD-CNN-BiLSTM model augmented with physical constraints, built on the decomposition and integration framework. The model can not only predict each sub-signal accurately, but also make the final prediction result conform to the physical law by adding a physical penalty term in the integration process. The key innovations and contributions of this research can be summarized as follows:

(1) To enhance prediction performance, this study introduces a hybrid VMD-CNN-BiLSTM model augmented with physical constraints, built upon the decomposition and integration framework. This innovative model effectively solves the prediction error caused by the fluctuation of historical wind power data, significantly improves the accuracy of wind power prediction, and provides a reference for the sustainable utilization and development of wind power.

(2) To ensure that the prediction results adhere to physical laws, this study proposes an integration strategy that incorporates physical constraints. Specifically, a physical penalty term is integrated into the loss function of the XGBoost ensemble module. In this method, sub-signals are used as inputs and the final predicted result is obtained.

(3) Utilizing the actual wind power dataset, we conduct comparative ablation experiments. The results demonstrate that the VMD-CNN-BiLSTM hybrid model with physical constraints outperforms the other hybrid model in terms of forecast accuracy.

The remainder of this article is organized as follows. Section 2 discusses the key methodologies that form the basis of the model. Section 3 describes the architecture and components of the proposed model. In Section 4, we evaluate the model’s validity through an experimental assessment. Finally, Section 5 concludes the study and suggests possible avenues for future research.

2. Related Methodologies

2.1. VMD

Variational mode decomposition (VMD) [24] is an adaptive signal processing technique designed to decompose complex signals into intrinsic mode functions (IMFs) characterized by specific frequency bandwidths. By formulating a variational problem to identify these IMFs, VMD enhances the robustness and reliability of the decomposition results. The mathematical formulation is described as follows:

Step 1. Define an objective function related to the mode number K:

$$\underset{u_k,w_k}{\min }\left\{{\displaystyle \sum _{k=1}^K{‖{\partial }_t\left[\left(\delta (t)+\frac{j}{\pi t}\right)\ast u_k(t)\right]e^{-j{\omega }_{k}t}‖}_2^2}\right\}$$

(1)

where $u_k(t)$ is the $k$-th mode function and $w_k$ is the corresponding center frequency.

Step 2. The Lagrange multiplier method is introduced, the alternate direction method of multipliers (ADMM) is used to solve the problem, and the original signal is decomposed into several modal functions:

$$u_k^{n+1}=\arg \underset{u_k}{\min }\left\{{‖{\partial }_t\left[\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k(t)\right]e^{-j{\omega }_{k}t}+\lambda \left(f(t)-{\displaystyle \sum _{i=1}^K{u}_i}(t)\right)‖}_2^2\right\}$$

(2)

$${\omega }_k^{n+1}={\displaystyle \frac{{\displaystyle {\int }_0^{\infty }t}{\left|\left[\left(\delta (t)+\frac{j}{\pi t}\right)\ast u_k(t)\right]\right|}^{2}dt}{{\displaystyle {\int }_0^{\infty }{\left|\left[\left(\delta (t)+\frac{j}{\pi t}\right)\ast u_k(t)\right]\right|}^2}dt}}$$

(3)

Step 3. Update iterations $u_k$ and $w_k$ until the convergence conditions are met: ${\displaystyle \sum _{k=1}^K{‖u_k^{n+1}-u_k^n‖}_2^2}<ϵ$, where $ϵ$ is the convergence threshold.

2.2. CNN

The convolutional neural network (CNN) [23,25] is a type of feedforward neural network. It is highly flexible and widely used in various research domains. Key applications include image processing and natural language processing. The architecture of a CNN consists of three primary components: the convolutional layer, the pooling layer, and the fully connected layer. The convolutional layer serves as the core element for extracting intrinsic features from the data. It operates on complex multi-feature inputs using convolutional kernels, enabling efficient feature extraction. Following this, the pooling layer reduces the spatial dimensions of the feature maps while preserving essential features, an approach which decreases computational complexity. Finally, the fully connected layer interprets the extracted features and generates the output results.

Step 1. The formula for calculating the convolution layer is as follows: $x_j^l=\sigma \left({\displaystyle \sum _{i\in M_j}{x}_i^{l-1}}\ast w_{ij}^l+b_j^l\right)$, where $x_j^l$ is the output of the $j$-th convolution kernel of the $l$-th layer and is also the output of the $l-1$-th layer; $\sigma$ is the activation function; $M_j$ is the set of output mapping layers; $w_{ij}^l$ is the weight matrix of the $j$ convolution kernel in layer $l$; $b_j^l$ is the offset term.

Step 2. In the pooling layer, dimensionality reduction is carried out on the output results of the convolution layer by means of maximum pooling and average pooling to facilitate feature extraction.

Step 3. The fully connected layer connects all the output results of the pooling layer and outputs them to the classifier. The forward propagation output of the fully connected layer is $x_j^{l+1}={\displaystyle \sum _{i=1}^n{w}_{ij}^l}{x}_i^l+b_j^l$, where $x_j^{l+1}$ is the value of the $j$-th output neuron of layer $l+1$.

2.3. BiLSTM

While convolutional neural networks (CNNs) excel at automatically extracting spatial features from large datasets, they struggle with processing strongly time-dependent time series data. In contrast, long short-term memory (LSTM) networks [26] effectively address long-term dependency issues through the implementation of gating mechanisms. The combination of CNN and LSTM enhances the performance of both spatial and temporal feature extraction while improving computational efficiency. LSTM utilizes an input gate, forget gate, and output gate to regulate the flow of information, effectively mitigating the challenges of gradient vanishing and explosion. This capability makes LSTM widely applicable for processing time series data.

Among them, the specific calculation formula of the LSTM network is as follows:

$$f_t=\sigma (W_f\cdot [h_{t-1},x_t]+b_f)$$

(4)

$$i_t=\sigma (W_i\cdot [h_{t-1},x_t]+b_i)$$

(5)

$${\stackrel{\sim }{C}}_t=\tanh (W_C\cdot [h_{t-1},x_t]+b_C)$$

(6)

$$C_t=f_t\ast C_{t-1}+i_t\ast {\stackrel{\sim }{C}}_t$$

(7)

$$o_t=\sigma (W_o\cdot [h_{t-1},x_t]+b_o)$$

(8)

$$h_t=o_t\ast \tanh (C_t)$$

(9)

where $f_t$ is the activation value of the forgetting door; $\sigma$ is the activation function; $W_f$ is the weight vector of the forgetting gate; $\ast$ stands for convolution operation; $h_{t-1}$ is the output of time $t-1$ memory unit; $x_t$ is the input of $t$ time memory unit; $b_f$ is the offset of the forgetting gate; ${\stackrel{\sim }{C}}_t$ is the candidate state of memory unit; $W_C$ is the weight vector of the candidate state of the input gate; $b_C$ is the bias of the input gate candidate state; $C_t$ is the state of $t$ moment memory unit; $o_t$ is the activation value of the input gate; $W_o$ is the weight vector of the input gate; $b_o$ is the offset of the output gate.

A bidirectional long short-term memory neural network (BiLSTM) [27,28,29] is composed of forward LSTM and reverse LSTM. $h_0,h_1,\dots ,h_T$ is calculated by the forward LSTM, and $h_T,h_{T-1},\dots ,h_0$ is calculated by the reverse LSTM. Finally, the two are joined together to get a new representation. The formula is as follows:

$$\overrightarrow{h_t}=\mathrm{LSTM}(x_t,\overrightarrow{h_{t-1}})$$

(10)

$$\overleftarrow{h_t}=\mathrm{LSTM}(x_t,\overleftarrow{h_{t+1}})$$

(11)

$$h_t=[\overrightarrow{h_t},\overleftarrow{h_t}]$$

(12)

The network structure of CNN-BiLSTM [30,31] is combined with a convolutional neural network to extract local spatial features and with a bidirectional long short-term memory neural network to capture long-term dependencies in time series and finally get prediction results. The hybrid neural network first inputs one-dimensional time series into the CNN part to obtain multi-feature series, then directly inputs the BiLSTM network through the flattening layer, and finally outputs the prediction results through the fully connected layer.

2.4. XGBoost

XGBoost, developed by Chen T.Q. et al. [32] in 2015, is a machine learning model grounded in the boosting framework. It enhances the predictive performance of traditional gradient boosting decision trees (GBDT) by employing second-order Taylor expansion, regularization, and parallel computation of the loss function. The objective function of XGBoost is defined as follows:

$$P_{obj}={\displaystyle \sum _{i=1}^{n}L(y_i,f_i(x_i))}+{\displaystyle \sum _{i=1}^n\mathrm{\Omega }}$$

(13)

where L is the loss function and $\mathrm{\Omega }$ is the regularization term of all weak learners which is used to prevent overfitting.

3. Proposed Method

3.1. Method Construction Process

In order to accurately forecast wind power, a hybrid VMD-CNN-BiLSTM model with physical constraints is proposed based on the decomposition and integration framework. The model comprises three primary components: data outlier processing, decomposition prediction, and integration. The construction process of the model is shown in detail in Figure 1.

Step 1: input the original wind power time series and establish an outlier detection framework based on the isolation forest algorithm. This framework utilizes random hyperplanes to partition the data space iteratively until only a single sample point remains in each subspace. Subsequently, the abnormality score of each sample point is calculated, allowing for the identification of outliers based on their scores. Finally, the identified outliers are then removed from the original wind power sequence.

Step 2: removing outliers from the original wind power time series results in time discontinuities that adversely affect data quality. To address this issue, we employ the LightGBM algorithm to reconstruct the removed abnormal samples by leveraging the relationship between wind speed and wind power. This process yields a reconstructed wind power time series, denoted as Series $Y=\{y_1,y_2,\cdot \cdot \cdot ,y_n\}$.

Step 3: given the high complexity of the reconstructed sequence data and the inherent characteristics of wind energy, both of which exhibit strong randomness and nonlinearity, we apply the variational mode decomposition (VMD) to facilitate a more structured analysis. We determine the optimal decomposition parameters based on the iterative relationship between the center frequency and the update step length. Consequently, the reconstructed wind power sequence, denoted as Series $Y^{’}=\{y_1^{’},y_2^{’},\cdot \cdot \cdot ,y_n^{’}\}$, is decomposed into multiple sub-signals. This decomposition aims to reduce the randomness, complexity, and nonlinearity present in the original sequence.

Step 4: to mitigate the issue of time series instability during model training, the 13 decomposed sub-signals are standardized to eliminate dimensionality.

Step 5: the standardized sub-signals are input into the CNN-BiLSTM neural network model by using the sliding window technique. Then, the forecast results of each sub-signal are obtained.

Step 6: in order to address the limitations of traditional neural network prediction and integration processes, which primarily focus on data-driven predictions while neglecting real-world physical phenomena, we input the prediction results of 13 sub-signals into the proposed XGBoost ensemble algorithm with physical constraints to obtain the final wind power forecasting outcomes.

3.2. Evaluation Metrics

There are many indexes to assess the effect of the forecast model, and a single evaluation index often has certain limitations. Therefore, this paper selects five model evaluation indicators [33,34,35,36], namely the mean square error (MSE), the root mean square error (RMSE), the mean absolute error (MAE), the symmetric mean absolute error (SMAPE), and the correlation coefficient ($R^2$), as the evaluation indexes of the prediction model. The calculation formula of MSE, RMSE, MAE, SMAPE, and $R^2$, respectively, is as follows:

$$MSE={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{\left(Y_i-\widehat{Y_i}\right)}^2}$$

(14)

$$RMSE=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{\left(Y_i-\widehat{Y_i}\right)}^2}}$$

(15)

$$MAE={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m\left|Y_i-\widehat{Y_i}\right|}$$

(16)

$$SMAPE={\displaystyle \frac{100\%}{m}}{\displaystyle \sum _{i=1}^n\frac{\left|\widehat{Y_i}-Y_i\right|}{(\left|\widehat{Y_i}\right|+\left|Y_i\right|)/2}}$$

(17)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^m{\left(\widehat{Y_i}-Y_i\right)}^2}}{{\displaystyle \sum _{i=1}^m{\left(\overline{Y_i}-Y_i\right)}^2}}}$$

(18)

where m is the length of the test set; $Y_i$ is the actual measured value; $\widehat{Y_i}$ is the predicted value; $\overline{Y_i}$ is the mean value of the actual measurement.

MSE is the square variance of the predicted value and the true value: the smaller the value, the better. The RMSE is the standard deviation between the predicted value and the true value: the smaller the value, the better. MAE is a measure of the average absolute error between the predicted value and the true value, and is a non-negative number: the smaller the value, the better the model effect. SMAPE is a statistical measure of prediction accuracy, and the closer the value is to 0%, the more perfect the model. R² is a measure that explains the difference between the predicted value and the true value, with a value between 0 and 1: the closer the value is to 1, the better the prediction.

4. Experimental Analysis

4.1. Data Description

In a typical wind farm, multiple wind turbines are installed, each operating independently to generate electricity. This paper focuses on the data of fan power generation in one plant area. We collected the dataset from a wind farm in the Hebei Province, China, spanning the period from 1 November 2022 to 27 October 2023, with a time resolution of 15 min. We partitioned the dataset into a training set, comprising 80% of the samples, and a test set, consisting of the remaining 20% (See

Table 2. Partial data.

Date	Original Wind Speed	Original Wind Power
1 November 2022 00:00:00	2.61	15.08
1 November 2022 00:15:00	5.54	18.73
1 November 2022 00:30:00	4.26	18.01
1 November 2022 00:45:00	2.86	18.38
1 November 2022 01:00:00	5.92	19.57
1 November 2022 01:15:00	6.05	16.63
1 November 2022 01:30:00	4.52	16.59
1 November 2022 01:45:00	6.05	19.59
1 November 2022 02:00:00	4.14	17.86
1 November 2022 02:15:00	6.56	19.2
1 November 2022 02:30:00	4.39	24.37
……	……	……
19 February 2023 15:15:00	5.67	14.32
19 February 2023 15:30:00	4.52	13.93
19 February 2023 15:45:00	4.77	10.6
19 February 2023 16:00:00	5.79	10.55
19 February 2023 16:15:00	6.3	13.87
……	……	……
28 July 2023 16:30:00	7.58	54.94
28 July 2023 16:45:00	6.3	32.85
28 July 2023 17:00:00	7.07	37.11
28 July 2023 17:15:00	6.17	35.37
28 July 2023 17:30:00	6.94	31.09
28 July 2023 17:45:00	6.3	38.19
28 July 2023 18:00:00	6.56	33.3
28 July 2023 18:15:00	5.41	33.27
28 July 2023 18:30:00	6.81	26.26
28 July 2023 18:45:00	4.26	11.45
28 July 2023 19:00:00	4.65	4.62
……	……	……

).

4.2. Data Preprocessing

4.2.1. Outlier Repair

Wind speed is the primary factor influencing wind power generation, with output power from wind turbines exhibiting a direct proportionality to wind speed; as wind speed increases, so does the power output. The electricity generated by wind turbines involves two fundamental processes: first, the conversion of aerodynamic forces in the atmosphere into rotational mechanical energy, and second, the transformation of this rotational mechanical energy into electrical energy. The mathematical relationship between wind speed and power output in wind energy generation can be expressed as follows [6]:

$$P=0.5A\cdot \rho \cdot c_P\cdot v^3$$

(19)

where P is the power of the wind turbine; A is the area swept by the fan blade; $\rho$ is the air density at the fan blade; $c_P$ is the utilization rate of wind energy, which is related to factors such as wind speed and slurry distance angle.

The above equation illustrates that wind power is proportional to the cube of the wind speed. To identify the outliers in the dataset, this study employs the isolation forest [37] algorithm, with the number of isolation trees set to 130. Following the application of the isolation forest algorithm, we calculated the anomaly scores, which reveal that more than 2700 samples in the wind power dataset are classified as outliers.

Based on the identification results, we constructed a scatter plot illustrating the distribution of normal and abnormal data within the wind power generation dataset. In this plot, the red scattered points represent abnormal data, while the blue scattered points indicate normal data. The distribution shows that the abnormal data are mainly embodied in the error samples of small wind speed and large power and in the outliers of the power–wind-speed curve. (See Figure 2).

Given the necessity to predict the wind power time series and maintain data continuity, we could not simply remove the abnormal data. Therefore, this study employs the LightGBM model, using wind speed as the input variable, to repair the identified abnormal data.

The model restoration results are shown in Figure 3. The blue scattered points represent normal data. The yellow scattered points indicate the abnormal data identified by the isolation forest algorithm after LightGBM restoration. The results show that the data of the wind power generation dataset are close to the normal data after repair.

In order to verify the influence of the data repair method used in this paper on the model effect, in this paper, two single hybrid neural network models, CNN-LSTM and CNN-BiLSTM, are trained on the datasets before and after repair, and their prediction effects are compared.

As it can be seen from the

Table 3. Comparison of model results before and after data restoration.

Method	Original Wind Power	Adjusted Wind Power
CNN-LSTM	0.939120	0.952621
CNN-BiLSTM	0.943687	0.965677

, the prediction effect of the model improved, overall, on the dataset after outlier processing. Therefore, the data repair method used can not only ensure the actual validity of the data, but also further enhance the quality of the data, so as to improve the prediction effect of the model.

4.2.2. Data Decomposition

In this paper, the obtained reconstruction data are input into the VMD algorithm. The optimal number of decomposition layers was determined by the center frequency method, and the update step size (tau) was determined by the residual index (REI) [38]. The optimal parameter determination process is shown in the following Figure 4.

This paper starts with the number of decomposition layers, and continuously increases the number of decomposition layers. The center frequency was calculated for each input. The optimal number of decomposition layers was determined based on the number of decomposition layers when similar center frequencies occurred. The center frequency table for the number of decomposition levels of order 1 to 15 is shown in

Table 4. Central frequency of each decomposition level.

Decomposition Order	Center Frequency	Decomposition Order	Center Frequency
1	0.001162	9	0.293598
2	0.028151	10	0.335073
3	0.06373	11	0.377056
4	0.090076	12	0.416864
5	0.129276	13	0.444295
6	0.164414	14	0.456638
7	0.210599	15	0.464797
8	0.251984

.

It can be seen from the above table that, when the decomposition order reached 13, the center frequency was not much different from that of the 14-order decomposition; therefore, the optimal decomposition order was 13, and the optimal update step was determined according to the REI index. The REI calculation formula is as follows:

$$REI=\min {\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left[{\displaystyle \sum _{K=1}^K{U}_K-f}\right]}$$

(20)

where $U_K$ is the KTH decomposition mode; f is the original signal; N is the number of signals. The smaller the value, the better.

Since tau is between 0 and 1, the step size was set to 0.001 and the iterative REI was obtained, as shown in Figure 5. It can be seen from the figure that, when tau was greater than 0.18, it tended to converge; therefore, a tau set between 0.18 and 1 is the optimal decomposition parameter.

This paper substitutes the experimental data into the VMD algorithm and sets the decomposition mode number to 13 and the update step as 0.255. The decomposition results are shown in Figure 6.

As it can be seen from Figure 6, the variable mode decomposition algorithm decomposed wind power time series data into IMF1-13 from the perspective of frequency domain. There were multiple sub-signals in the wind power time series dataset which changed with time and had an impact on the wind power prediction model. It can effectively reduce the error caused by random fluctuations.

4.3. Parameter Setting of CNN-BiLSTM Prediction Method

In this paper, Tensorflow2.0 deep learning framework is used to predict each normalized sub-signal as the input of the CNN-BiLSTM model.

After model training and tuning, the main parameters of the CNN-BILSTM prediction model were set as follows: in the CNN module, the number of convolutional nuclei was 64, the number of channels was seven, the ReLU activation function was selected, and the maximum pool window was set to 2; in the BiLSTM module, the number of neurons was 64, the ReLU activation function was still used, and three dense layers were set, two of which were used for output data dimension conversion, and the extracted time features were mapped to the output space. The number of nodes in the two dense layers was set to 32 and 16, respectively, and the third dense layer was used as the output layer with the number of nodes being one. In order to avoid overfitting during model training, the Adam adaptive model optimizer was used to train the model, the learning rate was set to 0.001, the iteration rounds were 32, and the data capture step length was 96.

4.4. Integration

In order to solve the traditional neural network prediction and integration process, it was limited to the prediction of the data themselves, while ignoring the physical phenomenon in reality. This approach utilizes the predicted values of each sub-signal as input and is trained to ensure that these predictions align with physical realities (e.g., actual power remaining greater than zero). In this paper, a physical penalty loss function is designed, and the penalty mechanism is introduced into the loss function. When the model produces a negative prediction, an additional penalty term is imposed, leading the model to avoid negative outputs during training.

The specific implementation can be outlined as follows: (1) define the underlying loss function, specifically the mean square error (MSE), which serves to quantify the discrepancy between the model’s forecasted values and the actual values; (2) introduce a penalty term that will be computed based on the magnitude of negative predictions; (3) employ the ReLU function to determine the average value of negative predictions, which will then be incorporated into the original loss function as an additional loss component.

The loss function is defined as follows:

$$l=l_{base}(y_{true},y_{pred})+\lambda \cdot mean(\mathrm{Re}LU(-y_{pred}))$$

(21)

where $l_{base}$ represents the basic loss function; $\lambda$ is the regularization coefficient, which controls the strength of the penalty term.

4.5. Method Comparison

In order to verify the performance of the VMD-CNN-BiLSTM wind power prediction model with physical constraints, this paper uses the repaired power dataset of the same wind farm. The VMD, ITD, EMD, SSA, and CNN-LSTM models, the CNN-BiLSTM combined model, the CNN-LSTM model, and the CNN-BiLSTM model were respectively used for training, and then the prediction effect of the model on the prediction set was compared and analyzed. In the process of comparison, this paper adopts the control variable method, where the same module uses the same parameter settings, which remain unchanged. Due to the large number of test sets, this paper only shows the prediction effect of part of the experimental samples in the prediction set and the residual distribution of the prediction results.

The comparison between the prediction results of ten wind power prediction models and the original data as well as the residual distribution can be seen from the above figure. By combining Figure 7 and Figure 8 and

Table 5. Residual distribution.

Method	90% Error Interval	100% Error Interval
VMD-CNN-BiLSTM	[−1.1032,2.7440]	[−2.9589,5.3031]
VMD-CNN-LSTM	[−11.4684,8.2913]	[−24.3339,29.1793]
EMD-CNN-BiLSTM	[−4.0295,4.2888]	[−8.6179,7.9375]
EMD-CNN-LSTM	[−4.9226,4.7180]	[−12.7945,11.0244]
ITD-CNN-BiLSTM	[−7.3960,8.5354]	[−26.2526,22.5367]
ITD-CNN-LSTM	[−9.3298,8.8566]	[−28.6323,22.4461]
SSA-CNN-BiLSTM	[−4.1184,5.1928]	[−10.0490,9.1120]
SSA-CNN-LSTM	[−6.2855,2.5584]	[−14.4681,7.8608]
CNN-BiLSTM	[−4.8532,14.6923]	[−24.3245,29.5787]
CNN-LSTM	[−12.5328,12.3496]	[−29.4859,42.3572]

, it can be seen that the wind power prediction curve of CNN-LSTM or CNN-BiLSTM without the decomposition module had the largest deviation from the original wind power data curve, and the residual fluctuation range is obvious. The accuracy and generalization of the prediction results were poor, but the residual distribution of the CNN-BiLSTM prediction model was in the range of −24.3245 to 29.5787, and 90% of the residual distribution was concentrated in the range of −4.8532 to 14.6923. The residual distribution of the CNN-LSTM prediction model was in the range of 42.3572 to −29.4859, and 90% of the residual distribution was concentrated in the range of −12.5328 to 12.3496. It is concluded that the effect of the CNN-BiLSTM prediction model is better than that of the CNN-LSTM prediction model. The prediction curve obtained by the prediction model with the decomposition and physical integration strategy was less deviated from the original wind power data curve than that without decomposition, and the residual error was smaller, so that the prediction effect was relatively good. It can be seen that the residual distribution interval of VMD-CNN-BiLSTM with physical constraints was the smallest when compared with the residual distribution interval of the EMD, ITD, and SSA decomposition methods. In conclusion, the prediction effect of the VMD-CNN-BiLSTM model with physical constraints is significantly better than that of the other models.

In order to further assess the effect of the prediction model, MSE, RMSE, MAE, and SMAPE were selected as evaluation indicators in this paper, and the results are shown in

Table 6. Model comparison prediction error.

Method	R2	MSE	RMSE	MAE	SMAPE
VMD-CNN-BiLSTM	0.997436	1.7924	1.3388	0.8691	16.63%
VMD-CNN-LSTM	0.970595	20.5535	4.5336	2.7648	26.80%
ITD-CNN-BiLSTM	0.980177	13.8562	3.7224	2.2579	24.45%
ITD-CNN-LSTM	0.977234	15.9133	3.9891	2.4284	25.51%
EMD-CNN-BiLSTM	0.993964	4.2189	2.054	1.7184	16.96%
EMD-CNN-LSTM	0.99138	6.0251	2.4546	2.0065	17.94%
SSA-CNN-BiLSTM	0.992849	4.9988	2.2358	1.5743	17.80%
SSA-CNN-LSTM	0.991464	5.9664	2.4426	1.6608	18.28%
CNN-BiLSTM	0.965677	23.9912	4.8981	3.0155	27.66%
CNN-LSTM	0.952621	33.1168	5.7572	3.2669	29.68%
VMD-TCN-Informer [39]	0.994	/	/	/	/
Transformer [40]	0.939	/	/	/	/

and Figure 9.

Among the above compared wind power prediction models, the CNN-BiLSTM model predicted the smallest errors in MSE, RMSE, MAE, and SMAPE, and the largest errors in R². Compared with the CNN-LSTM model’s R², it increased by 0.026841, 0.002943, 0.002584, 0.001385, and 0.013056, respectively, while the MSE decreased by 18.7611, 2.0571, 1.8062, 0.9676, and 9.1256, respectively. The RMSE decreased by 3.1948, 0.2667, 0.4006, 0.2068, and 0.8591, and the MAE decreased by 1.8957, 0.1705, 0.2881, 0.0865, and 0.2514, respectively. The SMAPE decreased by 10.17%, 1.06%, 0.98%, 0.48%, and 2.02%, respectively. Moreover, the wind power prediction model obtained by the decomposition and physical integration strategy was superior to the wind power prediction model without decomposition and physical integration strategy. Among the four decomposition methods, the VMD-CNN-BiLSTM prediction model with physical constraints had the smallest MSE, RMSE, MAE, and SMAPE errors and the largest R² errors. Compared with the other three decomposition methods, R² increased by 0.017259, 0.003472, and 0.004587, the MSE decreased by 12.0638, 2.4265, and 3.2064, and the RMSE decreased by 2.4336, 0.7152, and 0.897, respectively. The MAE decreased by 1.3888, 0.8493, and 0.7052, and the SMAPE decreased by 7.82%, 0.33%, and 1.17%, respectively. It can be seen that the VMD-CNN-BiLSTM prediction model with physical constraints had the best prediction effect and the highest accuracy.

This paper also compares the relatively new wind power prediction models, such as the VMD-CN-Informer and Transformer in the literature, and the results show that their R² is smaller than that of the VMD-CNN-BiLSTM wind power prediction model with physical constraints proposed in this paper. Therefore, compared with self-attention mechanism time series models such as Informer and Transformer, the model proposed in this paper still has greater advantages in prediction accuracy.

In order to make the model proposed in this paper more in line with the background of practical engineering applications, this paper uses 32 vCPU Intel(R) Xeon(R) Platinum 8352 V CPU @ 2.10 GHz with a memory size of 120 GB to analyze the complexity of the model and compare the training time of different models.

The time complexity of the above model was mainly affected by data preprocessing, training model, prediction, and integration; therefore, it is a polynomial time complexity O(n·d + n + E·m·d + t·d + p·q·d·logq), where n represents the number of samples, E represents the training rounds of the model, m represents the number of training samples, d represents the parameters in the model, t represents the number of prediction samples, and p represents the data dimension to be integrated. q indicates the length of the data to be integrated.

It can be seen from the above that time complexity is jointly determined by multiple links. Through calculation, it was found that the number of iterations, data scale, and model parameters (E, m, d) of the training model dominate the time complexity. Despite the increase in the amount of data, the decomposed hybrid model can realize the simultaneous training of sub-signals through massively parallel operation under the existing technical background, so that the time required by the model in the prediction of each sub-signal is only the longest time t_max required for the prediction of a single sub-signal. In this way, the total time required for model training is t_max+ t_merge, where t_merge is the time required for the decomposition and integration of the modules. As it can be seen from Table 6 and

Table 7. Model running schedule.

Data Processing Method	CNN-LSTM	CNN-BiLSTM
VMD	143 s	158 s
ITD	390 s	455 s
EMD	327 s	382 s
SSA	171 s	279 s
Do not Decomposition	127 s	146 s

, the VMD decomposition method adopted in this paper can maintain a short running time with high prediction accuracy. Compared with other decomposition methods, VMD has a more balanced overall performance, especially in relation to the requirements of prediction accuracy and operation efficiency in engineering applications.

By further comparing the two models, i.e., CNN-LSTM and CNN-BiLSTM, it can be found that the VMD decomposition method adopted in this paper has a higher prediction accuracy under the condition of little difference in running time. Considering the high demand for prediction accuracy in engineering applications and the existing hardware development level, it is concluded that the VMD-CNN-BiLSTM wind power prediction model with physical constraints proposed in this paper has a good performance.

5. Conclusions

As renewable energy continues to be developed and used extensively, the penetration of wind energy into the power system is increasing significantly. Wind power prediction is becoming increasingly important to ensure the safe and stable operation of the power grid. Nevertheless, due to the inherent intermittent volatility and randomness of wind energy, accurate wind power prediction is still challenging. In this paper, a VMD-CNN-BiLSTM hybrid model with physical constraints is proposed to improve the performance of wind power prediction.

In this method, the wind power sequence is decomposed into 13 sub-signals, and CNN-BiLSTM is used to predict each sub-signal to ensure that the spatial and temporal characteristics in each sub-signal are preserved. In order to make the final prediction result more consistent with the physical law, a XGBoost integration algorithm with physical constraints is designed. By using the XGBoost algorithm and introducing the physical penalty term into the loss function, the prediction result of wind power is finally obtained. The above hybrid model was applied to the wind power data set of a wind farm in the Hebei Province, with an R² of 0.997436, an MSE of 1.7924, an RMSE of 1.3388, an SMAPE of 16.6277%, and an MAE of 0.8691, indicating that the model has high prediction accuracy.

At the same time, a comparative experiment was carried out to evaluate the performance of the model. The results show that the quality of the data was enhanced after data repair. VMD-CNN-BiLSTM hybrid models with physical constraints have higher prediction accuracy than hybrid neural network models (CNN-LSTM, CNN-BiLSTM) and variant models using other decomposition methods that also use physical constraints.

Although the proposed VMD-CNN-BiLSTM hybrid model with physical constraints based on the decomposition integration framework has certain advantages, it still has some limitations. First of all, the model only considers the influence of wind speed on the prediction results, decomposes the time series of wind power and integrates the physical constraints, and does not consider the impact of other meteorological factors on wind power. Secondly, the decomposition forecast framework used in this paper only considers the characteristics of each sub-signal itself, without considering the relationship between each sub-signal.