Short-Term Multi-Step Wind Direction Prediction Based on OVMD Quadratic Decomposition and LSTM

Abstract

Accurate and reliable wind direction prediction is important not only for enhancing the efficiency of wind power conversion and ensuring safe operation, but also for promoting sustainable development. Wind direction forecasting is a challenging task due to the random, intermittent and unstable nature of wind direction. This paper proposes a short-term wind direction prediction model based on quadratic decomposition and long short-term memory (LSTM) to improve the accuracy and efficiency of wind direction prediction. Firstly, the model adopts a seasonal-trend decomposition procedure based on the loess (STL) method to divide the wind direction series into three subsequences: trend, seasonality and the remainder, which reduces the impact of the original sequence’s complexity and non-stationarity on the prediction performance. Then, the remainder subsequence is decomposed by the optimal variational mode decomposition (OVMD) method to further explore the potential characteristics of the wind direction sequence. Next, all the subsequences are separately input into the LSTM model, and the prediction results of each subsequence from the model are superimposed to obtain the predicted value. The practical wind direction data from a wind farm were used to evaluate the model. The experimental results indicate that the proposed model has superior performance in the accuracy and stability of wind direction prediction, which also provides support for the efficient operation of wind turbines. By developing advanced wind prediction technologies and methods, we can not only enhance the efficiency of wind power conversion, but also ensure a sustainable and reliable supply of renewable energy.

1. Introduction

1.1. Research Background

In the pursuit of carbon neutrality, the focus of the power industry has shifted to the development of renewable energy sources. As an environmentally friendly and adaptable renewable energy source, wind energy shows great potential in the development and application of renewable energy and plays a crucial role in achieving sustainable development goals. China is a large energy-consuming country, and the rapid development of industry is hungry for more and more energy. Therefore, it is urgent to develop renewable energy to relieve the pressure on the energy supply [1,2]. As an environment-friendly renewable energy, wind energy shows good adaptability in development and application [3]. Additionally, China possesses abundant wind energy resources, and the total exploitable wind energy is estimated to be 1000 GW [4]. According to the Global Wind Energy Council, Chinese-installed wind power capacity reached 395.6 GW by the end of 2022, which not only accounts for 43.7% of the world’s total installed wind power capacity, but also drives the transformation of our energy towards green and low-carbon [5]. In addition to reducing carbon emissions and mitigating climate change, vigorously developing wind energy can also contribute to the promotion of the sustainable development of the energy industry and improve the overall wellbeing of society.

With the development of wind power industry, how to utilize wind energy efficiently has become a prominent topic. The conversion efficiency of the wind energy conversion system and the performance of the wind turbine are closely related to the wind direction [6,7]. Adjusting the wind turbine orientation and blade angle according to the wind direction can maximize the efficiency of wind energy capture and conversion. Therefore, an accurate prediction of the wind direction can optimize the operation and power generation efficiency of wind turbines and improve the utilization efficiency of wind energy. In the process of wind power generation, a yaw system usually relies on historical wind direction data to adjust its direction. However, the adjustment of the yaw system often lags behind the current wind direction data. An accurate prediction of the wind direction can realize efficient regulation of the yaw system [8,9] and prevent machine damage caused by inaccurate yaw regulation. In order to actualize the efficient control of wind turbines and promote the efficiency of wind power conversion, it is necessary to study a short-term wind direction prediction model for guiding wind direction regulation [10].

1.2. Relevance of Wind Direction Prediction Research and Sustainability

Wind prediction research has an important relationship between wind energy and sustainability. As a renewable energy source, wind energy plays an important role in the field of sustainable energy, and an accurate prediction of the wind direction is crucial for the planning, operation and management of wind power generation systems.

First, wind prediction research can improve the efficiency of wind power generation systems. Accurately predicting the wind direction can help optimize the layout and scheduling of wind power generation equipment, so that the wind energy system can better capture and utilize wind energy resources. Through accurate wind direction prediction, the steering angle and blade angle of the wind turbine can be reasonably arranged to capture the wind energy to the greatest extent and improve the power generation efficiency and capacity utilization.

Secondly, wind direction prediction research can improve the stability and reliability of a wind power generation system. An accurate prediction of the wind direction can help predict the output power fluctuation of a wind power generation system, so as to improve system scheduling, operation and maintenance. A reliable wind direction prediction can reduce the risk and uncertainty of the wind energy system, improve the reliability and stability of the system, and ensure that the system continues to provide a stable power output.

In addition, wind direction prediction research is also of great significance for site selection and the planning of wind power generation sites. By accurately predicting the wind direction, it is possible to assess the potential of wind energy resources in different locations, select areas suitable for the construction of wind farms, and carry out reasonable planning and layout. This helps to maximize the use of wind energy resources and improves the overall sustainability of the wind power generation system.

In summary, wind direction prediction research has important implications for wind energy and sustainability. By accurately predicting the wind direction, the efficiency, stability and reliability of the wind power generation system can be improved, the development and utilization of sustainable energy can be promoted, and the sustainable energy transition can be promoted.

1.3. Literature Review

In the context of wind data prediction, several factors significantly impact the accuracy and reliability of the predictions. These factors include data quality, selection of the prediction model and modeling method, feature selection, optimization of model parameters and determination of the prediction range. Therefore, it is crucial to construct and optimize a wind direction prediction model from these aspects.

Numerous domestic and international scholars have carried out relevant research on forecasting wind direction. In a previous study based on the single prediction model, Hu et al. improved the deep belief network (DBN) in the data input and training process to achieve a better forecast effect [11]. Zhen et al. combined bidirectional long short-term memory (BiLSTM) [12] with the convolutional neural network (CNN) to mine the bidirectional temporal and spatial characteristics of input sequences, respectively, to improve the prediction performance. However, due to the influence of the terrain, climate and other environmental factors, wind direction data under different environmental conditions will show their differences. These discrepancies increase the complexity of wind direction data, making it challenging for a single model to handle complex data. The prediction accuracy will be reduced if the complex data is not well processed before being input into a single model. Therefore, researchers proposed a decomposition ensemble model in an effort to make up for the shortcomings of the single model [13]. This model decomposes the original series by a certain method and predicts each subsequence independently. Finally, the prediction results of each subsequence are integrated as the original sequence prediction result. Hu [14] et al. decomposed the original data by variational mode decomposition (VMD), then forecasted each subsequence by the echo state network model (ESN) optimized with the differential evolution (DE) and obtained the final prediction result through integrating the prediction results of each subsequence. Guo et al. [15] adopted a two-stage data processing strategy consisting of variable mode decomposition and sparse autoencoder (VMD-SAE) to extract the original wind speed characteristics and adopted high-order fuzzy cognitive mapping (HFCM) neural network modeling and a batch gradient descent optimization algorithm to make up for its shortcomings. The two-stage data processing strategy smooths the original data and extracts the feature information. Gao et al. [16] used the sparrow search algorithm (SSA) to optimize the variational mode decomposition (VMD) parameters, which has the advantage of adaptive data decomposition. However, the SSA algorithm increases the computation and complexity of the model, and the algorithm itself is easy to fall into local optimization. Lu et al. [17] adopted the K value of VMD optimized by MCC and used the optimized VMD for decomposition prediction; however, they also faced the same problem as SSA. In addition, Xiaozhi et al. [18] combined wavelet change and transformer models for decomposition prediction, but the adaptability of the wavelet transform to wind direction data is insufficient and the amount of calculation required in the transformer model is large. In addition, Yıldırım and Kamil [19] used a variety of methods to produce wind energy forecasts.

The most commonly used data decomposition methods include empirical mode decomposition (EMD) [20], ensemble empirical mode decomposition (EEMD) [21], complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) [22], wavelet transform (WT) [23] and others. The OVMD adopted in this paper is based on the evolution of VMD. Therefore, OVMD was selected to compare with mainstream time-domain decomposition algorithms, both of which are adaptive decomposition algorithms. EMD is an empirical mode decomposition based on an iterative algorithm that decomposes the signal into a series of vibration modes, each with a specific frequency range and amplitude characteristics, based on the local properties and adaptability of the signal. However, there are some problems with the decomposition of EMD, such as sequence discontinuity and noise interference. To solve these problems, integrated empirical mode decomposition (EEMD) has been proposed. The problem of discontinuity in sequence decomposition is solved by adding noise to the sequence to be decomposed. Then, adaptive noise adjustment technology was introduced on the basis of EEMD, and the CEEMDAN method was proposed, which can automatically adjust the noise level according to the characteristics of the signal to improve the decomposition effect. The method adaptively estimates the noise level of the signal and introduces it into the signal decomposition process as auxiliary data, which improves the stability and reliability of the signal decomposition. Studies have demonstrated that this decomposition method can effectively reduce the complexity of series data, particularly sequence data with strong mutability and high complexity.

1.4. Contribution to the Current Literature

In order to solve the problem of wind direction data being difficult to predict accurately, we are committed to developing a wind direction prediction model that combines wind direction data processing, feature extraction and the prediction of future values to achieve accurate wind direction prediction.

The innovation of this paper is that a wind direction decomposition ensemble prediction model based on a quadratic decomposition strategy is proposed. Firstly, according to the strong randomness, high fluctuation and nonlinear characteristics of the original wind direction data, the correlation stationarity analysis and data pre-processing were carried out. Then, the original wind direction sequence data was decomposed by STL. It separates out three sub-sequences: seasonality, trend, and residual. Then, to solve the problem of the residual sequence containing unpresented information features, an OVMD decomposition method based on center frequency observation and residual index minimization was proposed, and the decomposition parameters of VMD were optimized to achieve the optimal decomposition of the residual sequence. The complexity of a wind direction sequence can be effectively reduced by these steps. Finally, LSTM was used to predict all the subsequences, and the prediction results of each subsequence were linearly integrated to obtain the prediction results of the original wind direction data. Extensive experiments results denote that in the error evaluation indexes of the root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE), the quadratic decomposition model are respectively improved by 81.7%, 73% and 75% compared with the LSTM model, and increased by 17.3%, 33% and 31.3%, respectively, compared with the single decomposition model OVMD–LSTM, which demonstrates the potential of our method for wind direction prediction.

In the field of wind energy prediction, different from traditional wind speed research, this paper mainly focuses on wind direction research. In the past, for complex nonlinear sequences, the decomposition results were divided according to the frequency of sequence data. In this paper, the wind direction itself has periodic periodicity and trend change.

The rest of this article is organized as follows. Section 2 introduces the model framework and methods involved in this paper in detail. In Section 3, the experimental cases and prediction results are elaborated, the validity of the proposed framework is validated and the experimental results are discussed. See Section 4 for the conclusions and prospects for future work.

2. Methodology

2.1. Trend and Seasonality Feature Extraction Based on STL

Aiming at problems such as high complexity, strong mutability and multi-feature fusion of the original wind direction data, this study firstly employed STL to filter out the basic features of trend and seasonality in the original wind direction sequence. In contrast to conventional decomposition techniques, the STL method presents stronger robustness in processing time series with outliers [24].

The STL decomposes the original wind direction sequence into three additive subsequences: trend subsequence, seasonality subsequence and the remainder subsequence [25]. This relationship can be expressed as follows:

$$Y_t=T_t+S_t+R_t,t=1,2,\dots N$$

(1)

where $Y_t$ is the original sequence, $T_t$ is the trend, $S_t$ is the seasonality, $R_t$ is the remainder and N is the length of the sequence.

The STL method decomposes time series data by combining internal and external cycles. The internal cycle generates the trend and seasonality, and the external cycle is used for the calculation of the remainder. After each internal cycle, the trend and seasonality are updated once, then the external cycle will calculate the robust weight based on the internal cycle results to reduce the impact of the remainder on the seasonality and trend updates in the next internal cycle. The STL method decomposes the time series step by step, resulting in the corresponding seasonality subsequence, trend subsequence and remainder subsequence.

2.2. Optimal Variational Mode Decomposition

The VMD is an adaptive data decomposition method that is suitable for handling non-stationary sequence data [26,27]. It decomposes a sequence into a specific number of subsequences, and each subsequence holds its own center frequency and finite bandwidth.

To obtain all the decomposed subsequences, a constraint variational model needs to be constructed. The subsequence is represented as an intrinsic mode function (IMF) in the model. There are two conditions in modeling: first, the sum of the bandwidths of each IMF is required to be the minimum; second, the sum of each IMF must be equal to the original sequence f(t), which is the constraint condition of the model [28]. According to these two conditions, the following constraint variational model was constructed:

$$\{\begin{array}{c}\begin{array}{c}\min \\ u_{{}_k},{\omega }_{{}_k}\end{array}\left\{{\displaystyle \sum _{k=1}^K\left|{\partial }_t[\delta (t)+\frac{j}{\pi t})\ast u_{{}_k}(t)]\exp (-j{\omega }_{{}_k}t)\right|}\right\};\\ \mathrm{s}.\mathrm{t}.{\displaystyle \sum _{k=1}^K{u}_{{}_k}(t)=f(t).}\end{array}$$

(2)

where $u_k$ is the kth IMF, ${\omega }_k$ is the kth center frequency of the mode function and K is the number of IMF. $\delta \left(t\right)$ is the Dirac distribution, where j represents a fixed constant in the transformation process and π is the constant. $(\delta \left(t\right)+\frac{j}{\pi t}){\ast u}_k(t)$ is the unilateral spectrum of each mode, which is then multiplied with the operator to obtain its corresponding fundamental frequency band:$[(\delta \left(t\right)+\frac{j}{\pi t}){\ast u}_k(t)]\mathrm{e}\mathrm{x}\mathrm{p}(-j{\omega }_{\mathrm{k}}t)$. Then, the derivative of time variable t is calculated for the fundamental frequency band, the square norm L of the demodulation gradient is calculated, and the estimated bandwidth of each component is obtained. Finally, the minimum value of the sum of the bandwidths of each component is obtained. The second formula of Equation (2) is the objective function, which requires the sum of the components to be equal to the original sequence.

In order to solve the constraint problem, penalty factors $\alpha$ and Lagrange multipliers $\lambda$ are added to transform the constrained variational model into an unconstrained variational model. The Lagrange function expression is constructed as follows:

$$\begin{array}{cc}\hfill L(u_k,{\omega }_k,\lambda )=& \alpha {\displaystyle \sum _{i=1}^K\left|{\partial }_t[\delta (t)+\frac{j}{\pi t})\ast u_k(t)]\exp (-j{\omega }_{k}t)\right|}+\hfill \\ & \left|f(t)-{\displaystyle \sum _{k=1}^K{u}_k(t)}\right|+\langle \lambda ,f(t)-{\displaystyle \sum _{k=1}^K{u}_k(t)}\rangle .\hfill \end{array}$$

(3)

Then, using the alternating multiplier direction method, the optimal solution of the Lagrange function is solved alternately iteratively according to the following formula.

$${\widehat{u}}_i^{n+1}(\omega )=\frac{\widehat{f}(\omega )-{\displaystyle \sum _{i=1}^K{\widehat{u}}_i(\omega )}+\frac{\widehat{\theta }(\omega )}{2}}{1+2\alpha {(\omega -{\omega }_{{}_i})}^2}$$

(4)

$${\omega }_i^{n+1}=\frac{{{\displaystyle {\int }_0^{\infty }\omega |{\widehat{u}}_i(\omega )|}}^{2}d\omega }{{{\displaystyle {\int }_0^{\infty }|{\widehat{u}}_i(\omega )|}}^{2}d\omega }$$

(5)

$${\widehat{\theta }}_i^{n+1}(\omega )={\widehat{\theta }}^n(\omega )+\tau [\widehat{f}(\omega )-{\displaystyle \sum _{i=1}^K{\widehat{u}}_i^{n+1}(\omega )}]$$

(6)

where $\tau$ is the Lagrange multiplier update step. Finally, the decomposition is complete and the subsequences are obtained.

Several studies have demonstrated that the performance of the VMD is heavily influenced by the values of its two key parameters, K and $\tau$, when applied to decompose the remainder subsequences [29]. Specifically, if the K is set too large, the center frequencies of each mode tend to converge or even overlap. When K is too small, part of the modes will be assigned to similar modes or abandoned. Similarly, different IMFs will also be generated when $\tau$ takes a different step size. Therefore, the appropriate setting of these two parameters is crucial for achieving optimal decomposition results. In this study, the OVMD was selected to decompose the remainder subsequence, which employs the center frequency observation method to settle K and determine the value of $\tau$ by the minimization of the residual error index (REI). The REI is calculated as follows: the optimal value of τ can be obtained by identifying the value that corresponds to the minimum REI.

$$REI=\min \frac{1}{N}{\displaystyle \sum _{i=1}^N{\left|{\displaystyle \sum _{k=1}^K{u}_k-f(t)}\right|}_i}$$

(7)

where N is the length of the decomposition sequence and the other parameters are the same as in the VMD.

In comparison with decomposition methods such as EMD, EEMD and CEEMDAN, VMD offers distinct advantages for analyzing nonlinear and non-stationary signals. It not only effectively avoids issues such as over-envelope, under-envelope and end effects, but also exhibits strong data decomposition and noise reduction capabilities [30]. Building upon these strengths, the OVMD method optimizes important parameters in VMD decomposition to reach better a decomposition effect of the remainder subsequence [31].

2.3. Long Short-Term Memory Neural Network

The LSTM model is a significant advancement in deep learning compared to the traditional recurrent neural network (RNN) model [32]. The RNN model often encounters the issue of gradient disappearance or explosion as the training depth increases, making it unsuitable for long-term time series learning and training. Compared to RNN, LSTM introduces gated self-circulation to solve these problems [33]. This innovation reduces dependence on information length [34] and realizes long-term tracking of time series data [35,36,37].

The memory cell structure of the LSTM model is depicted in Figure 1. The memory cells are controlled by three gates: the forgetting gate, input gate and output gate. These three gates, respectively, carry out weighted learning of cell state information $c_{t-1}$ at time t − 1, newly entered cell information ${\tilde{c}}_t$ at time t, and cell state information $c_t$ at time t. This process accomplishes the memory function of historical sequence data.

Firstly, the value $f_t$ of the forgetting gate is obtained by substituting the output $h_{t-1}$ of the hidden layer at time t − 1 and the new input $x_t$ at the current time into Equation (9).

$$f_t=\sigma (W_{fx}{x}_t+W_{fh}{h}_{t-1}+b_f)$$

(8)

where $\sigma$ is the sigmoid function, whose value range is (0, 1); $W_{fx}$, $W_{fh}$ is the weight of the forgetting gate; and $b_f$ is the bias of the forgetting gate.

Then, calculate the value of the input gate $i_t$ and obtain the candidate state information ${\tilde{c}}_t$ of the cells at time t:

$$i_t=\sigma (W_{ix}{x}_t+W_{ih}{h}_{t-1}+b_i)$$

(9)

$${\tilde{c}}_t=\tanh (W_{cx}{x}_t+W_{ch}{h}_{t-1}+b_c)$$

(10)

where $W_{cx}$, $W_{ch}$ is the weight of cell state, $b_c$ is the bias of cell state, $W_{ix}$, $W_{ih}$ is the weight of the input gate and $b_i$ is the bias of the input gate.

Then, we obtain the cell state information $c_t$ at time t:

$$c_t=f_t\otimes c_{t-1}+i_t\otimes c_t$$

(11)

where $\otimes$ denotes the bit-by-bit dot product operation. Under the action of the forgetting gate value $f_t$, the invalid information in the cell state at t − 1 moment can be “forgotten”; at the same time, under the action of the input gate value $i_t$, the newly input effective information at time t can be retained, so as to combine the two results into the cell state information $c_t$ at time t.

Finally, the output gate value $o_t$ is calculated:

$$o_t=\sigma (W_{ox}{x}_t+W_{oh}{h}_{t-1}+b_o)$$

(12)

where $\sigma$ is the sigmoid function, whose value range is (0,1); $W_{ox}$, $W_{oh}$ is the weight of the output gate; and $b_o$ is the bias of the output gate.

Finally, the output of the whole memory cell at time t is obtained:

$$h_t=o_t\otimes \tanh (c_t)$$

(13)

2.4. Overall Structure of STL–OVMD–LSTM Model

In this paper, a decomposition ensemble prediction model based on STL–OVMD–LSTM is proposed for the prediction of short-term wind direction sequence data. The model aims to extract the most influential features in the wind direction sequence, and then mine the residual information from the remainder. The model mainly includes the quadratic decomposition stage and the forecasting stage, and the basic procedure is illustrated in Figure 2. There are two main stages:

(1)

The quadratic decomposition stage. According to the seasonal and trend characteristics of wind direction data, the original series data are decomposed by the STL method, and the seasonal and trend sub-series are separated, as well as the remainder sub-series containing the remainder characteristic information. However, there is still some effective information of the original sequence data in the remainder subsequence; therefore, the OVMD method is utilized to decompose the remainder subsequence to further mine the potential features, and several different subsequences are obtained.

(2)

The forecasting stage. Each component obtained after the quadratic decomposition is input into the LSTM model for prediction, and then the predicted results of each component are linearly added to obtain the final predicted value of the original sequence data of wind direction.

3. Cases Studies

3.1. Analysis of Wind Direction Sequence Data

Figure 3 displays the quarterly wind direction in a rose chart according to the practical data from a wind farm for a whole year. The concentric circles in the rose diagram represent the frequency of the wind direction; the longer the rays at the corresponding angle, the higher the frequency of wind direction in that particular area. Each season has a dominant wind direction that represents the wind direction with the highest frequency during that season. For this wind farm, the main wind direction of the four seasons is (255°, 315°), (240°, 285°), (150°, 195°), and (255°, 300°), respectively. In each region, there is a corresponding main wind direction, but the other wind directions also have a certain frequency. These characteristics make wind direction random and increase the difficulty of wind direction prediction.

The establishment of the wind direction prediction model relies on historical wind direction data information, and the autocorrelation analysis was performed to obtain its relevant data information, such as whether it has autocorrelation and stationarity. Figure 4 shows the autocorrelation diagram of the wind direction data. The vertical axis represents the autocorrelation coefficient, while the horizontal axis represents the lag number. The symmetrical straight line placed above and below the horizontal axis indicates the confidence interval of the autocorrelation coefficient of the wind direction sequence [−0.0527, 0.0527]. The vertical lines on the graph that fall within the confidence interval indicate an autocorrelation coefficient of 0. As can be seen from Figure 4, the autocorrelation coefficient drops from 1 to the confidence interval after 164 sampling lags and exceeds the confidence interval after 180 sampling lags. The autocorrelation coefficient changes periodically and does not exhibit stationary time series features, such as truncation and trailing, which demonstrates that the wind direction sequence has an obvious non-stationary property.

The aforementioned outcome provides further confirmation of the random and unstable nature of wind direction sequence data, highlighting the need to preprocess such data. To solve the randomness and instability of the original wind direction sequence data, the quadratic decomposition strategy is proposed. This strategy filters out invalid information from the original wind direction sequence, reserving a sequence containing only effective information.

3.2. Experimental Design

To validate the performance of STL–OVMD–LSTM, wind direction data were collected from a wind farm for the experiments, and three parts of the experiments were setup: a single model, a single decomposed model and a quadratic decomposed model. The prediction effect of each model is judged from the prediction accuracy and the stability of the multi-step prediction errors. Among them, the single decomposition aims to extract the main features of the original sequence, then the method with higher decomposition accuracy should be selected for the secondary decomposition.

The feature extraction capability of the STL and OVMD methods was tested in the experiment. The comparative decomposition methods applied in the experiment include empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD) and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). For the sake of generality, the LSTM network was exploited in all the prediction models.

The LSTM model consists of an input layer, a hidden layer and an output layer. The hyperparameters of this model are specified as follows: the time window steps are 1-step, 3-step and 5-step. The number of neurons in the hidden layer was 200; the optimizer used is ADAM; the maximum number of iterations was set to 200; and the gradient threshold was set to 1. The initial learning rates of the trend subsequence prediction model, periodic subsequence prediction model and remainder subsequence prediction model were 0.005, 0.01 and 0.007, respectively. The three sequences have different data characteristics. According to the experience of previous experiments or similar problems, we chose the learning rate of the sequence with similar characteristics in the model as the initial learning rate of the corresponding model. The number of iterations of the learning rate decline is 125 and the attenuation factor is 0.2.

3.3. Evaluation Criteria

In this paper, RMSE, MAE and MAPE were used as evaluation indexes to evaluate the predictive performance of the model. The respective calculation formulas are as follows:

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{(y{’}_i-y_i)}^2}}$$

(14)

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

(15)

$$MAPE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\frac{{\mathrm{y}\prime }_i{-\mathrm{y}}_i}{{\mathrm{y}}_i}\right|}\ast 100\%$$

(16)

where $y_i$ is the real data, ${y’}_i$ is the forecast data, n is the number of data, RMSE reflects the error fluctuation, MAPE represents the overall accuracy, and the smaller values of the three evaluation indexes indicate the higher prediction accuracy of the model.

3.4. Experimental Results

3.4.1. Data Decomposition Result

The experimental data consist of 1440 wind direction measurements taken at 10 min intervals between 1 January and 10 January. This data was subjected to STL decomposition, resulting in three subsequences, as plotted in Figure 5. The first subsequence represents the original wind direction data. The second subsequence captures the overall trend in the wind direction data, while the third subsequence represents the periodicity of changes in wind direction, also known as seasonality. The fourth subsequence is known as the remainder subsequence, which contains complex fluctuations, such as multiple mutations and weak regularity.

On this basis, the OVMD method was used for decomposing the remainder subsequence. To achieve the best possible decomposition results, the center frequency observation was employed to compute the center frequency of each mode component for a different set of K. As is illustrated in

Table 1. Center frequency when K takes different values.

Mode Number	1	2	3	4	5	6	7
K = 2	0.0216	0.1728
K = 3	0.0201	0.1146	0.3098
K = 4	0.0195	0.1073	0.2013	0.3692
K = 5	0.0191	0.104	0.1764	0.2904	0.3863
K = 6	0.0158	0.0999	0.1208	0.1971	0.3227	0.3993
K = 7	0.0184	0.0865	0.1136	0.1789	0.2724	0.3575	0.4249

, when K equals 6, IMFs with similar frequencies appear, which result from the over-decomposition phenomenon. Thus, in this study, the optimal number of K was set to 5.

The parameter τ is determined based on the minimum value of the REI. In order to increase the accuracy of the subsequent predictions, the REI must be minimized. When $\tau$ is changed in the interval [0, 1] with a step size of 0.01, there are changes in the REI resulting from OVMD decomposition, which are shown in Figure 6. When $\tau$ is equal to 0.98, the value of the REI reaches the minimum; therefore, in this study, the optimal value of $\tau$ was set at 0.98.

Based on the parameters K and $\tau$ settled by the aforementioned process, OVMD decomposition was used to obtain five intrinsic mode functions (IMFs) with different frequencies and amplitudes. As seen in Figure 7, the first row shows the remainder subsequences, while rows 2–6 reflect the five IMFs obtained by decomposition, each of which captures different details of the random fluctuations. Compared with the original sequence, the sequence after the quadratic decomposition can be more stable and smooth, which provides a foundation for the precise prediction of subsequent models.

3.4.2. Comparison Model and Prediction Result Analysis

To verify the effectiveness of the proposed method, the single prediction model and the single decomposition model were designed as comparison models, including the LSTM, EMD–LSTM, CEEMDAN–LSTM, STL–LSTM and OVMD–LSTM models. This paper utilized the RMSE, MAE and MAPE to evaluate the model prediction effect. The one-step and multi-step prediction error of each model are detailed in

Table 2. One-step, three-step and five-step prediction results of each model.

Model	One-Step			Three-Step			Five-Step
	RMSE	MAE	MAPE	RMSE	MAE	MAPE	RMSE	MAE	MAPE
LSTM	0.0679	0.0356	0.0469	0.4617	0.4598	1.1665	0.6335	0.6314	2.8592
EMD-LSTM	0.0543	0.0242	0.0750	2.6828	2.5946	1.0398	4.4542	3.7371	3.7675
CEEMDAN-LSTM	0.1380	0.1271	0.2073	1.8222	1.6466	1.7563	2.5248	2.1232	1.2991
STL-LSTM	0.0616	0.0276	0.0344	0.4545	0.4389	1.1521	0.8900	0.8189	3.3971
OVMD-LSTM	0.0150	0.0144	0.0169	1.3774	1.1473	1.5867	1.6681	1.3498	1.5519
STL-OVMD-LSTM	0.0124	0.0096	0.0116	0.0617	0.0327	0.0613	0.3925	0.3065	0.9162

, and the visual effect is plotted in Figure 8. The one-step prediction fitting effects of each model are shown in Figure 9. From the experimental results, the following conclusions can be found:

(1)

The original wind direction data manifests obvious mutability and complexity, and the seasonality and trend characteristics are not readily apparent. Comparing the error values of the prediction results from each model reveals that the decomposed prediction model outperformed the single model. Given that wind direction data is influenced by various inherent characteristics, an appropriate decomposition method can effectively improve the prediction effect of the model on the complex original data. In the one-step prediction, the error values of all the decomposed prediction models are smaller than the single prediction model. In the multi-step prediction, each error value of the decomposed prediction model based on STL is still smaller than that of all the single prediction models. This not only proves that the beneficial application of the decomposition methods for enhancing the prediction performance of the model in one-step prediction, but also highlights the advantage of the STL method on various decomposition methods. These findings have significant implications for the accuracy and reliability of wind direction prediction models.

(2)

The quadratic decomposition technique extracts additional meaningful information from the wind direction data, thereby enhancing the prediction accuracy of the model. Despite the STL decomposition, the remainder subsequence retains some valuable information that requires further processing. From the one-step prediction results of the single decomposition prediction model, it is evident that the OVMD–LSTM method has the least error. Consequently, for one-step prediction accuracy alone, the OVMD technique is the most effective decomposition method. However, once the trend and seasonality attributes of the original sequence are removed, the remainder subsequence becomes more complex and unpredictable. Therefore, a powerful decomposition method for complex sequences is necessary. Accordingly, the OVMD method was selected as the secondary decomposition method.

(3)

Furthermore, it can be observed that although OVMD-LSTM and the proposed STL–OVMD–LSTM have similar errors in one-step prediction, STL–OVMD–LSTM exhibits superior performance in subsequent 3-step and 5-step predictions, with significantly smaller growth rates and absolute error values compared to OVMD–LSTM. It can be seen in Figure 8 that in the multi-step prediction, STL–OVMD–LSTM consistently achieves the lowest prediction error among all the models, while the errors of the other models increase with the number of prediction steps. Even in the OVMD–LSTM, which has the most similar error to STL–OVMDLSTM in one-step prediction, shows weaker performance in multi-step prediction. It could be concluded that STL–OVMD–LSTM combines the high accuracy of OVMD in single-step prediction and the stability of STL in multi-step prediction, leading to an overall improved prediction performance. These results provide strong evidence supporting the superiority of STL–OVMD–LSTM over other models in multi-step prediction tasks.

(4)

In addition, we performed a comparative experiment to determine the mode number by the center frequency in OVMD. As previously established, the optimal value of K is 5. Therefore, K was varied from 3 to 7 to decompose the remainder into different subsequence groups. Then, the subsequence groups were tested under the same prediction model, STL–OVMD–LSTM. The prediction results of the different mode numbers are displayed in Figure 10. As K increased from 3, the prediction error decreased continuously. When K climbed to 5, the error dropped to the minimum, and then the error started to become larger when K continued to rise. Therefore, when K = 5, the RMSE, MAE and MAPE are the smallest, thereby validating the theory presented in Section 2.2.

(5)

Finally, based on the different decomposition numbers and LSTM parameters in OVMD, the findings can be summarized as follows: In the process of OVMD decomposition, based on the center frequency observation method, it was also observed that the mode separation effect is better when the decomposition number is 10 to 12; however, considering the algorithm complexity and prediction accuracy, 5, which has the most obvious mode separation characteristics, was chosen as the best decomposition number. This is because the complexity and amount of computation required by the algorithm when the decomposition number is 5 is much smaller than when the decomposition number is 12, and the prediction accuracy is higher. In addition, for the selection of LSTM parameters for three different sequences, we have attempted to develop three models that use the same learning rate, but the trend sequence is easy to overfit, and the periodic sequence is easy to underfit. This is due to the different characteristics of the different sequences; if the appropriate parameters are not selected, not only can the amount of calculation required by the model increase, but it is also difficult to achieve better prediction accuracy.

The findings reveal that the proposed prediction model, STL–OVMD–LSTM, which utilizes quadratic decomposition, combines the stability of STL and the accuracy of OVMD for decomposing sequence data, while benefiting from the LSTM model for predicting multi-feature complex sequences. In the practical wind direction data from a wind farm, the STL–OVMD–LSTM model outperformed other models in terms of its prediction accuracy and overall fitting effect. Compared with the model using only one decomposition method, the quadratic decomposition employed by our model can fully extract the features from complex wind direction sequence data, thus diminishing the difficulty of forecasting complex series and significantly boosting the prediction effect of the wind direction time series.

4. Conclusions and Future Studies

From the perspective of data feature-driven models and based on the idea of “divide and rule”, this paper constructs a decomposition-integrated prediction model for wind direction data. The model considers the inherent development law of wind direction sequence data. Firstly, it extracts the trend and season characteristics of the series to preliminarily study the changes of the series. After that, it adopts the quadratic decomposition strategy to filter through the remainder of the results of the first decomposition, and inputs all the subsequence data obtained from the two decompositions into the prediction model. Finally, the prediction results of each subsequence are integrated to obtain the final comprehensive prediction results. From the results of the experiment, the following conclusions were obtained.

• In the error evaluation indexes of RMSE, MAE and MAPE, OVMD–LSTM has the highest prediction accuracy in the first decomposition model of single-step prediction, followed by STL–LSTM and EMD–LSTM. However, when it comes to the subsequent multi-step prediction, STL–LSTM is the most accurate in the first decomposition model. This shows that STL–LSTM is the best model for comprehensive accuracy and stability.
• The quadratic decomposition model is 81.7%, 73% and 75% higher than the LSTM model, and 17.3%, 33% and 31.3% higher than the single decomposition model OVMD–LSTM, respectively. It can be seen that the accuracy of the subsequent prediction model can be effectively improved after data decomposition. At the same time, choosing an appropriate method for the second decomposition of the first time series can further improve the accuracy of the prediction model.
• To solve the problem of high complexity and mutability of the original wind direction sequence data, the STL method was used to decompose the original wind direction sequence, which has the obvious effect of stabilizing the sequence and can effectively separate the trend and period in the original sequence. Moreover, in multi-step prediction, STL decomposition of the sequence data can maintain a stable prediction accuracy.
• OVMD based on the center frequency observation method and REI formula method can determine important parameters k and t quickly and efficiently and has the highest accuracy in single-step prediction of a decomposition model. OVMD can effectively improve the prediction accuracy of the model.

The conclusions above can provide a reference for wind direction forecasting and other fields. However, there are still many challenges in this area. According to the current research, data decomposition methods have been applied to many wind energy prediction models, but they have certain limitations. For example, in STL decomposition, the parameter sliding window size and the length of the seasonal cycle are usually fixed, while the wind direction data change has a unique cycle. From this point of view, we can try to innovate an adaptive STL method so that the trend and cycle can be separated according to the characteristics of the data itself and the accurate wind direction. For example, it is a challenge to explore whether there is a correlation between wind speed and wind direction changes at the same time, and to combine the two for wind energy prediction. In addition, by using the wind direction prediction results to control the direction and blade angle of the wind turbine, an intelligent control system could be added to the wind turbine. In the system, the wind energy prediction model can be used to predict the historical data in memory in real time, and the wind turbine direction and blade angle can be adjusted according to the prediction results. Based on the long- and short-term trends of wind direction forecast, the intelligent control algorithm is applied to optimize the operation strategy of a wind power system. This includes the control of wind turbine start–stop cycles, capacity adjustment and utilization of energy storage systems designed to maximize the use of wind energy and economic benefits.