A bi-level mode decomposition framework for multi-step wind power forecasting using deep neural network

Energy Conversion and Management: X 23 (2024) 100650 Available online 22 June 2024 2590-1745/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/bync/4.0/). A bi-level mode decomposition framework for multi-step wind power forecasting using deep neural network ★ Jingxuan Wu a , * , Shuting Li b , ** , Juan C. Vasquez a , Josep M. Guerrero a , b , c a Aalborg University, Pontoppidanstraede 111, Aalborg 9220, Denmark b BarcelonaTech (UPC), Barcelona East School of Engineering (EEBE), Barcelona 08019, Spain c Catalan Institution for Research and Advanced Studies (ICREA), Pg. Lluís Companys 23, Barcelona 08010, Spain ARTICLE INFO Keywords: BiLSTM Deep learning Mode decomposition Wind power prediction ABSTRACT The proportion of wind energy in global energy structure is growing rapidly, promoting the development of wind power forecasting (WPF) technologies to solve the uncertainty and intermittence of wind power generation. However, the nonlinear and stochastic features of wind power time series restrain the accuracy of multi-step prediction performance. A multi-step WPF (MS-WPF) approach based on a time series bi-level empirical mode decomposition (BLEMD) method and BiLSTM neural network is proposed in this paper to improve the WPF accuracy of regional wind power generators. Since the uncertainty is always generated through coupled factors from both wind and weather-to-power conversion, the linearity feature is first introduced as an aspect apart from the frequency in the proposed approach to decompose the wind power time sequence data. The proposed BLEMD introduces Pearson product-moment correlation coefficient to evaluate the linearity of time series and a linearitybased decomposition algorithm is designed accordingly. To further enhance the precision and release computation burdens, a DL-based prediction strategy, including a BiLSTM network, a CNN-BiLSTM network, and a mean weight estimation method are implemented to predict the components separately. The proposed method only relies on local data, greatly reducing the data acquisition and computation cost. The precision of the proposed MS-WPF is verified by a 2.5 kW wind turbine with horizons from 5 s to 30 s, a 1.5 MW wind turbine with horizons from 10 min to 1 h, and a 51 MW wind farm with horizons from 1 h to 6 h. The comparative experimental results with other cutting-edge methods indicated that the proposed MS-WPF has superior prediction accuracy and stable performance for multi-step prediction. 1. Introduction Wind energy has emerged as a highly sought-after form of renewable energy due to the increasing awareness of environmental conservation. The development of wind power generation has become one of the most widely discussed topics around the world. Besides the superiorities of non-pollution and renewability, wind power generation promoting is greatly restricted by the uncertainty and intermittence of wind power. This problem becomes much more critical in some particular small-scale applications like regional Microgrid applications. Different from largescale wind farm scenarios [1], the wind in Microgrids becomes a more complex natural phenomenon. Not only wind oscillation, including speed, direction, and gust but also the mechanical and electrical part of the wind turbines (WTs), for instance, degradation and friction of the generator, energy conversion, and transmission losses, might raise the power oscillation in Microgrid systems [2]. In this sense, wind power forecasting (WPF) becomes an important part of energy management strategies for Microgrids [3,4]. The dynamic characteristics of wind power and wind speed act as chaotic time series with high stochastic features, wind prediction is consequently regarded as a complex regression task. Existing WPF research can be classified into modelbased and data-driven methods. Model-based methods include numerical weather prediction (NWP)- based physical methods and persistence models (PM). NWP can give regional long-term weather forecasts (wind speed, cloud cover, temperature, air pressure, etc.) according to a large amount of weather data ★ This work was supported in part by the China Scholarship Council (CSC). * Corresponding author. ** Principal corresponding author. E-mail addresses: [email protected] (J. Wu), [email protected] (S. Li). Contents lists available at ScienceDirect Energy Conversion and Management: X journal homepage: www.sciencedirect.com/journal/energy-conversion-and-management-x https://doi.org/10.1016/j.ecmx.2024.100650 Energy Conversion and Management: X 23 (2024) 100650 2 [5]. Its performance on WPF is highly restricted by the additional uncertainty caused by the weather-to-power conversion. Besides, the high computation burden of this method also restrains its application on ultra-short-term [6]. The persistence model, on the contrary, is a straightforward method. The prediction value equals the latest observation value. The performance of the persistence model degrades as the time horizon extends. Hence, it provides higher accuracy in the ultrashort-term forecast and the performance is limited in medium-term and long-term prediction tasks [7]. Data-driven methods, using lagged observation as input, get increasing attention due to the popularity of supervisory control and data acquisition (SCADA) systems. Data-driven methods can be divided into conventional statistical and machine learning-based types. The autoregressive moving average (ARMA) model is of great importance in statistical methods [8], and the NWP model is further deployed based on the ARMA model to provide input data [9]. The vector autoregressive (VAR) model with the Least Absolute Shrinkage and Selection Operator is also verified as an effective way to enhance the accuracy [10]. Information fusion is another solution to enhance prediction performance by merging multi-sources data [11]. For wind power interval prediction, fuzzy models [12], Markov chain [13], and Lorenz disturbance sequence [14] can be effective solutions. However, the performance of these statistical methods is highly dependent on the accuracy of statistical models and the non-linearity of the input time series will decrease the prediction accuracy [15]. The machine-learning-based method is developed to solve the nonlinear time-series problem. Thanks to its data mining and feature extraction abilities, it outperforms statistical methods in most cases. Since traditional machine learning-based prediction, including support vector machine, decision tree, or neural network [16], requires lots of experience and skill to develop shallow models, deep learning (DL) [17] shows its advantages in automation and high generalization efficiency [18]. Plenty of research is organized with artificial neural network (ANN) [19], convolutional neural network (CNN) [20], autoencoder [21], recurrent neural network (RNN) [22], etc. In DL approaches, RNN is theoretically designed for time series regression tasks due to its sequence data memory. Long short-term memory (LSTM), bidirectional LSTM (BiLSTM) and gated recurrent units (GRU) [23,24], as variants of RNN, inherit the advantage and solve the exploding or gradient vanishing problem by introducing an extra forget gate to the memory cell, making them the most efficient neural network architectures for time-sequence data forecasting [25]. However, the significant characteristic of wind, uncertainty, is still too stochastic to be fitted. In this sense, decomposition methods are deployed to split timeseries signals into two or more sub-signals and predict them separately. Empirical mode decomposition (EMD), variational mode decomposition (VMD), wavelet decomposition (WD), and Fourier decomposition (FD) are the most common methods for wind data decomposition. EMD splits the time-series signal according to the data feature [26]. A complementary ensemble EMD is proposed to decompose the wind power data into four intrinsic mode functions (IMFs) [27] and build the long-term prediction with the sigma point Kalman filter and short-term prediction with reinforcement learning. EMD can also be used for wind speed prediction [28]. A structuring element is introduced to extract features from the original wind speed signal. However, the main shortcoming of EMD is the lack of mathematic theory, which makes this method highly dependent on experience. VMD decomposes the original signal into several subseries with limited bandwidth in the spectral domain [29], while WD and FD do it according to the frequency feature of the original signal [30]. Based on the assumption that the wind time series contain different frequencies, these methods will lead to an overall improvement in forecasting. While considering the wind is a non-stationary random process, decomposition according to frequency and bandwidth might face the challenge of defining a fundamental wave or wavelet basis. To solve this problem, this paper proposes a multi-step BiLSTMbased WPF (MS-WPF) method, including a Bi-level EMD (BLEMD) with mathematical analysis and cooperated DL-based prediction approaches. The main contributions of this approach can be summarized as follows. 1. A BLEMD considering both linearity and frequency features of time series is proposed to assign the regression algorithm and computation resource. The design of the BLEMD is supported by rigorous mathematical analysis, and a novel linearity evaluation method based on Pearson product-moment correlation coefficient (PCCs) is proposed to extract the linear trend of the time series. 2. A MS-WPF structure is proposed to handle the regression task of time series components. The MS-WPF, including a BiLSTM network, a CNN-BiLSTM network, and a weight mean estimation (WME), provides high-precision forecasting results and reduces the computation burden. 3. The proposed MS-WPF is validated with realistic data from distinct areas. The multi-step prediction performance is verified by comparing it with cutting-edge approaches. The results indicate that the proposed approach can achieve a high prediction accuracy. The rest of the article is organized as follows. The framework of the multi-level WPF method is presented in Section 2. The proposed BLEMD algorithm is discussed in Section 3. The structure and algorithm of the BiLSTM network, CNN-BiLSTM network, and the proposed WME are presented in Section 4. In Section 5, the experimental results with three distinct scenarios are analyzed, and the comparison between the proposed approach and existing methods is studied. A conclusion is drawn in Section 6. 2. Framework of the proposed MS-WPF Due to the data mining ability, DL methods always perform well within long-term or short-term time series prediction. BiLSTM provides satisfying accuracy due to its memory cell and bidirectional training to Fig. 1. Structure of the proposed MS-WPF. J. Wu et al. Energy Conversion and Management: X 23 (2024) 100650 3 improve sequence learning. However, for short-term WPF, the influence of uncertainty might lead to the prediction error being amplified. The MS-WPF approach is proposed to enhance the prediction accuracy. The main structure is shown in Fig. 1. The offline part updates the regression model for online prediction with historical data and the online part conducts the multi-step prediction in real time. Real-time (RT) data is stored in a database for deep neural network training. In the offline training part, the wind power time series will be normalized and decomposed by the BLEMD. The wind power data will be split into three parallel time series according to frequency feature and linearity. Component 1 (C1) describes the trend of wind power. Component 2 (C2) and Component 3 (C3) follow normal distribution around zero with regulated frequency. The offline training process implements a BiLSTM network and a CNN-BiLSTM network for C1 and C2 regression model training. Meanwhile, the distribution of C3 is analyzed to provide the estimation parameters for WME. The Multi-step WPF DL Network matrix is the framework of the regression model, which consists of multiple layers defined by the look-ahead steps. The DL network matrix will be generated by the offline training process and each layer includes two DL networks and a WME algorithm for the three components. The multilevel regression model will be updated in a long-term horizon. In the short-term horizon, RT data would be sampled and stored in a data buffer and then decomposed into three elements as in the offline training process. The multi-level regression model outputs the predicted wind power using the components and wind speed data, finally, the results of the components prediction will be combined into the WPF output. 3. Proposed BLEMD algorithm The uncertainty of wind power and wind speed significantly inhibits the WPF accuracy. In the time series of wind power sampling data, the uncertainty acts like a random error, and the amplitude increases as the time horizon decreases. This Section will introduce the BLEMD algorithm for wind power data and analyze the frequency and linearity features of components. 3.1. Frequency and linearity-based BLEMD BiLSTM network outperforms other DL methods on time sequence regression and prediction, but stochastic components might increase the non-linearity and lead to the performance degradation of BiLSTM. To restrain the influence of stochastic components and allocate the computation resource appropriately, the frequency and linearity-based BLEMD is implemented to divide the time series into three components, the process is shown in Fig. 2. To extract the unpredictable uncertainty from the wind power data, a lowpass filter is deployed to split the high-frequency components. The iteration process of the filter can be described as {LFS(n) = TS ω cP(n) + (1−TS ω c)LFS(n−1); C3(n) = P(n) − LFS(n). (1) in which low-frequency sequence (LFS) is the component extracted by the filter, the P is the wind power time series, Ts represents the sample time, and ω c means passband frequency. The difference between the raw signal and the output LFS is the C3. The ω c must be set as high as possible to limit the amplitude of C3 since its mean and standard deviation are inversely proportional to ω c. Here the ω c is defined according to the look-ahead steps and it should be less than half of the sample rate. After the frequency decomposition, the time series is divided into two sub-signals with specific frequency characteristics, and the residual will be further decomposed according to the linearity features. A Gaussian-weighted moving mean (GWMM)-based data smoother [31] is deployed here to extract the trend of wind power time series, which is C1, and the C2 is the remainder. The window must be right-aligned as ⎧ ⎪ ⎨ ⎪ ⎩ C1(n)=Σn i=n−m+1LFS(i)wi Σn i=n−m+1wi ; C2(n) = LFS(n) − C1(n). (2) where C1(n)donates the output value of GWMM, also the C1 time series, at nth data point, the LFS is the low-frequency sequence defined in (1). The wi is the weight calculated by the average value and standard deviation of the sliding window, and m is the length of the window. The window length is the critical parameter of the GWMM, it affects the linearity of C1 and the correlation between C1 and C2. The prediction of C1 and C2 will benefit from a valid choice of window length m. 3.2. Frequency and linearity analysis The main parameters of the proposed BLEMD are the ω c and the m, which are related to the frequency and linearity separately. The ω c will compress the frequency of C1 and C2 in a limited region, which will greatly benefit the performance of the deep learning-based prediction method. However, the amplitude of C3 grows as the ω c decreases, so the ω c is set as ω c=1 2 α Ts (3) ω c is related to the maximum observation time Ts in the training process and the look-ahead steps α . The α must be smaller than the lookahead step to enhance the periodicity of C1 and C2 with a cycle time equal to the look-ahead horizon. Hence, the prediction performance of C1 and C2 can be improved. Besides, the amplitude of C3 increases as the α decreases, which will generate increasing random errors in the prediction of C3. In this sense, the α should be as large as possible. By jointly considering the effects on the two aspects, the α should be equal to the look-ahead step. Apart from the frequency, linearity is also an essential feature affecting the precision of DL-based prediction methods. However, there are still no widely accepted linearity evaluation methods, so a PCCbased linearity analysis is designed to guide the linearity-based decomposition. It is defined as the PCCs between the raw signal and Fig. 2. wind power data BLEMD process. J. Wu et al. Energy Conversion and Management: X 23 (2024) 100650 4 its 2-order polynomial regression fitting curve in a sliding window. The mathematical procedure of the proposed linearity evaluation includes two steps. Firstly, a sliding window is activated for the local fitting. A 2order polynomial fitting curve in each window is generated through ordinary least squares. As the window moves, a new discrete signal with the same trend as the raw signal is produced as the linearity reference. The second step is calculating the PCCs between the raw data and the reference, which can be denoted as C1= [C1(1),C1(2),…,C1(n)]; (4) and C1ref =[C1ref (1),C1ref (2),…,C1ref (n)]; (5) The linearity l can be calculated by l=Σn i=1(C1(i)−C1)(C1ref (i)−C1ref )  Σn i=1(C1(i) − C1)2Σn i=1(C1ref (i)−C1ref )2 √(6) where l ranges in (0, 1], and the closer l approaches 1, the higher the raw data’s linearity. C2 is defined as the remainder of the LFS after extracting C1. The correlation between them can be evaluated by their correlation coefficient r1−2, r1−2= Σn i=1(C2(i)−C2)(C1(i)−C1)  Σn i=1(C2(i) − C2)2Σn i=1(C1(i) − C1)2 √ (7) The range of r1−2 is (0, 1], and the higher the r1−2, the closer the correlation between the two elements. The BLEMD could be designed based on the linearity evaluation l for C1 and correlation coefficient r1−2 between C1 and C2. It is easier for the DL methods to train an accurate regression model with a linear and highly corresponding input, which means higher l and r1−2. The sliding window length of the smoother is an essential factor to them. The l is rising while the r1−2 is falling with the sliding window length increasing. To balance the l and r1−2, a predictability coefficient ε is defined as, ε =lΣC1+r1−2Σ|C2| ΣC1+r1−2Σ|C2|(8) the ε is a mathematical function of the GWMM window length m, the optimal window length m is the index of the maximum predictability coefficient. m∈ [2,3,4,…]: ε (m)⩾ ε (x),for∀x∈ [2,3,4,…](9) The predictability of the train data will be analyzed and the optimized window length will be figured out for the BLEMD in each training process. 4. Proposed DL-based prediction and error compensation strategy The three components of wind power will be predicted separately. The regression model of C1 will be generated by a BiLSTM network, and the input involves wind speed. The forecasting of C2 is realized via a CNN-BiLSTM network to achieve better accuracy, the input involves wind speed and C1. The C3 is predicted using a WME method to save computation resources. 4.1. BiLSTM-based sequence to sequence DL network The BiLSTM is a further structure based on LSTM, which is an enhanced RNN with a memory cell and a forget gate. The iteration process of LSTM, which is already well-known, can be summarized as four steps. Firstly, the forget gate of kth cycle is figured by, ft(k)= σ (Wf⋅[h(k−1),xt(k)]+bf)(10) In (10), the xt is the input time sequence, the h is the hidden layer status of LSTM, σ is the passing rate within [0, 1], then Wf will decide which part of the memory and input sequence should be forgotten. Secondly, the memory gate is calculated, it(k) = σ (Wi⋅[h(k−1),xt(k)] + bi)(11) where it is the memory gate. The  Ct is the temporary cell status, and the cell status Ct could be updated in the third step as,  Ct(k)=tanh(Wc⋅[h(k−1),xt(k)]+bc)(12) and Ct(k)=ft(k)⋅Ct(k−1)+it(k)⋅ Ct(k)(13) After the cell status is updated, the output gate ot is calculated, and then the hidden layer is updated as, ot(k) = σ (Wo⋅[h(k−1),xt(k)] + bo)(14) and h(k) = ot(k)⋅tanh(Ct(k)) (15) The historical input and cell status will be conditionally considered during the iteration. However, there is still one shortcoming: the effect of the next cycle is not considered. As a result, BiLSTM is developed to solve this problem. The proposed sequence-to-sequence BiLSTM network is established based on the LSTM, as shown in Fig. 3. The input sequence of the C1 network includes two dimensions which are wind Fig. 3. Architecture of the proposed BiLSTM training network. J. Wu et al. Energy Conversion and Management: X 23 (2024) 100650 5 speed and its historical sequence, while the input sequence of the C2 network contains one more dimension: C1. The BiLSTM layer has a forward LSTM and a backward one, the hidden layer of the two LSTM marked as hf and hb will be combined to achieve the improved hidden layer sequence ht. Then the drop layer will delete some cells randomly to prevent overfitting. The fully connected layer is employed to fetch the feature and after the activation of the tanh Layer, another is deployed to achieve the mapping between the hidden layer and responses. 4.2. CNN-BiLSTM-based DL network The C2 sequence is predicted through a CNN-BiLSTM network since it contains more mutations than C1. The C2 prediction process involves C1, C2, and wind speed sequences, a convolution filter will scan the input matrix and detect the lead and lag of different sequences. The result is transmitted to the BiLSTM network for time sequence data mining. The C2 sequence has limited frequency and follows normal distribution, an essential challenge to its prediction task is avoiding over-fitting. Apart from employing drop layers in the DL network structures, the learning rate is set to decrease piecewise and a validation process is established to prevent over-fitting problems. 4.3. High-frequency component estimation methods Since the C3 is stochastic and unpredictable, an estimation method is designed instead to compensate for the random error in the WPF. Based on the two features of the C3: 1) The frequency of the C3 has limited above the pass-band frequency; 2) The distribution of C3 follows the normal distribution around zero; the WME is designed. The procedure has three steps. Step 1: fetch a data window of historical data with the length of nWME and store it in matrix X (Sort by time in descending order), the estimation elements matrix  E will be calculated by, { E= [e(1),e(2),…,e(nWME −1)]; e(k)=(k+1) μ −Σk i=1X(i). (16) where μ represents the long horizon mean. Step 2: The matrix E is the absolute estimation error of each element in the previous cycle, and the weight matrix W is defined by the accuracy of each element in the last iteration, {E= [e(1),e(2),…,e(nWME −1)]; e(k)= |X(1)−(k+1) μ +Σk+1 i=2X(i)|. (17) ⎧ ⎪ ⎨ ⎪ ⎩ W= [w(1),w(2),…,w(nWME −1)]; w(k)=ΣE−E(k) (nWME −2)ΣE. (18) Step 3: calculate the WME output as, WME =WET(19) Both the long horizon and short horizon mean are calculated in  E, and the weight matrix W will adjust the dynamic response of WME accordingly. The accuracy of the proposed WME relies on the distribution of C3. 5. Case study Computational experiments are implemented to validate the performance of the proposed MS-WPF on different look-ahead steps and horizons against existing methods. Three datasets with divergent scenarios, listed in Table 1, are utilized to test the advance of the proposed method. Data from scenario 1 comes from the real-time data acquisition system of the IoT laboratory, Aalborg University, Denmark, in which a 2.5 kW WT is established and analyzed in this scenario. Scenarios 2 and 3 are extracted from a public dataset from KDD Cup 2022 [9], the data comes from the SCADA system of a wind farm in China. Scenario 2 is the output power of one single WT. Scenario 3 uses the total wind farm output power and average wind speed data of 34 WTs. The horizons of the scenarios are distinct, from ultra-short-term (5s) to long-term (1 h). The persistence model is deployed as one of the baselines for prediction methodology due to its high performance in short-term and ultra-shortterm prediction. The other baselines include ARIMA [28,32], GRU [24], CNN-BiLSTM [20,23] and WD-BiLSTM [30]. For all the DL-based methods, 80% of data is allocated for training, 1% for validation during the training and 19% for testing. Mean absolute percent error (MAPE) and nominal root mean squared error (nRMSE) are the two critical parameters for WPF accuracy evaluation. MAPE =1 NΣN i=1 yi−yi yi(20) in which yi is the actual wind power and yi is the wind power prediction. The nRMSE is defined by, nRMSE = ΣN i=1(yi−yi)2 ΣN i=1(yi−y)2 √(21) y is the mean of actual value. MAPE is a very intuitive interpretation of relative error, and nRMSE is more sensitive to outliers. Additionally, the coefficient of determination, R2, is deployed to evaluate the regression performance, which is defined as, R2=1−ΣN i=1(yi−yi)2 ΣN i=1(yi−y)2(22) when the R2 approaches 1, the prediction is positively correlated to the original signal. If the R2 approaches −1, the prediction is negatively correlated to the original signal. R2=0 indicates that y and y are not related. Table 1 Configuration of the experimental datasets. Parameters Scenario 1 Scenario 2 Scenario 3 Horizon 6*5 s 6*10 min 6*1 h Nominal power 2.5 kW 1.5 MW 1.5 MW*34 Location Denmark China China Fig. 4. The wind power time series and output components of BLEMD in 5-s, 10-min, and 1-h scenarios. J. Wu et al. Energy Conversion and Management: X 23 (2024) 100650 6 5.1. 6-step prediction results of the proposed MS-WPF The performance of the MS-WPF is verified in three different scenarios with different scales and horizons. An ablation experiment is conducted to validate the effectiveness of the proposed BLEMD. A conventional EMD using the detrend function, which also decomposes the original signal into three components, is deployed to cooperate with the same DL framework. The wind power time series raw data should be preprocessed through the BLEMD. The output is three components with different linearity and complementary frequency features. The output of BLEMD for MS-WPF of the three scenarios is presented in Fig. 4. The parameters of the BLEMD are optimized by the methodology proposed in Section 3. The sliding window length is 39 for scenario 1, 5 for scenario 2, and 35 for scenario 3, as Fig. 4 shows, C1 has the same trend as wind power, C2 and C3 oscillate around 0. The three components are transmitted to the parallel DL networks to train the regression models, the hyperparameters are presented in Table 2. The 1-step prediction results of the MS-WPF and the EMDBiLSTM for the three components are presented in Fig. 5. The prediction results presented by the dots are compared with the actual wind power value in the same figure, and the distance between the dots and the reference line represents the prediction error. It is evident that the BiLSTM and the CNN-BiLSTM network have satisfying accuracy with the prediction task of C1 and C2. In Fig. 5(g–i), the proposed WME method also shows promising performance. Table 2 Hyper parameters of the proposed MS-WPF Layers BiLSTM CNN-BiLSTM Convolution Layer NA C[4,1] – F[256] BiLSTM B[128] B[256] Drop Layer 0.1 0.1 Fully connect Layer 1 F[256] F[512] Learn rate/Drop factor 0.005/0.5 0.005/0.5 Fig. 5. 1-step components prediction results of 5-s scenario, 10-min scenario and 1-h scenario. J. Wu et al. Energy Conversion and Management: X 23 (2024) 100650 7 The distribution of black dots and blue dots indicates the results of the MS-WPF and the EMD-BiLSTM. In both decomposition methods, the general trend of the wind power time series is extracted as C1. The proposed BiLSTM and CNN-BiLSTM can effectively estimate the C1 and C2. However, conventional EMD cannot ensure the frequency and linearity features of the C1 and C2, so the accuracy is restrained. The proposed BLEMD decomposes the wind power time series into three components with specific characteristics: C1 (high linearity, low frequency), C2 (low linearity, low frequency), and C3 (low linearity, high frequency). The accuracy of the wind power trend is greatly enhanced first by predicting the C1. The prediction accuracy of C2 benefits from the regulated frequency of C2. The amplitude of the C3 is restrained by the BLEMD and will have a limited effect on overall performance. The 6-step prediction results of the proposed MS-WPF evaluated through MAPE, nRMSE, and R2 are shown in Fig. 6. The prediction precision degrades as the look-ahead step grows as expected, and the MAPE is restrained by the proposed methodology. The MAPE of 1-step prediction is below 10 %, which greatly benefits from the high precision of C1 and C2 prediction, as shown in Fig. 5. The influence of C3 prediction error is restrained by the low amplitude of C3. As the lookahead step grows, the ω c for frequency decomposition must increase to achieve better C1 and C2 prediction results. Hence the amplitude of Fig. 6. Statistical prediction result evaluation of a) 5-s scenario, b) 10-min scenario, and c) 1-h scenario. Fig. 7. Multi-step prediction results of the proposed MS-WPF and existing approaches. J. Wu et al. Energy Conversion and Management: X 23 (2024) 100650 8 C3 is rising, which leads to the reduction of forecasting accuracy. The R2 indicates a good regression performance. The prediction results track the realistic wind power well. The prediction accuracy for the 10-min scenario is the best among the three cases. Scenario 1 represents the ultra-short-term prediction which involves dramatic wind dynamics. The dynamics gather in the C3 time series and directly affect the precision of the MS-WPF. Scenario 3 is a long-term prediction case that indicates a long-term horizon. The 1step prediction in scenario 3 reaches an accurate level (MAPE = 0.056,nRMSE =0.059,R2=0,996). However, the precision degrades exponentially as the horizon grows due to the stochastic of wind. In summary, the proposed MS-WPF provides high-performance prediction for wind power in different scenarios and horizons. The 1Table 3 Hyper parameters of the benchmarks Benchmarks Hyper parameters PM NA GRU G[1024] – Drop[0.1] – F[1024] Learn rate/Drop factor – [0.001/0.2] CNN-BiLSTM C[4,32] – B[512] – Drop[0.1] – F[512] Learn rate/Drop factor – [0.001/0.5] WD-BiLSTM WD[‘db3’, 9] B[128, 128] – Drop [0.3, 0.7] Learn rate/Drop factor – [0.01/0.5] ARIMA P[4] – D[0] – Q[5] Table 4 Multi-step prediction results evaluation of proposed MS-WPF and existing approaches. Steps Step-1 Step-2 Step-3 Step-4 Step-5 Step-6 Case 1 (5s) MAPE MS-WPF 0.067 0.126 0.187 0.232 0.256 0.332 PM 0.281 0.590 0.708 0.781 0.875 0.961 GRU 0.108 0.185 0.376 0.578 0.683 0.744 CNN-BiLSTM 0.152 0.300 0.547 0.734 0.790 0.801 WD-BiLSTM 0.307 0.425 0.547 0.638 0.685 0.740 ARIMA 0.234 0.430 0.712 0.928 1.072 1.169 nRMSE MS-WPF 0.110 0.215 0.312 0.361 0.407 0.451 PM 0.546 0.941 1.039 1.113 1.177 1.236 GRU 0.191 0.288 0.524 0.755 0.843 0.865 CNN-BiLSTM 0.240 0.465 0.766 0.905 0.929 0.936 WD-BiLSTM 0.488 0.630 0.719 0.781 0.819 0.848 ARIMA 0.468 0.905 1.361 1.625 1.802 1.884 R2 MS-WPF 0.987 0.953 0.902 0.869 0.834 0.797 PM 0.702 0.114 −0.079 −0.238 −0.385 −0.528 GRU 0.963 0.917 0.725 0.430 0.289 0.252 CNN-BiLSTM 0.942 0.783 0.413 0.181 0.137 0.123 WD-BiLSTM 0.762 0.602 0.482 0.389 0.329 0.280 ARIMA 0.780 0.180 −0.851 −1.639 −2.245 −2.548 Case 2 (10 min) MAPE MS-WPF 0.067 0.179 0.196 0.223 0.235 0.263 PM 0.291 0.462 0.592 0.664 0.716 0.798 GRU 0.150 0.216 0.344 0.487 0.587 0.642 CNN-BiLSTM 0.149 0.244 0.416 0.570 0.653 0.702 WD-BiLSTM 0.332 0.381 0.450 0.490 0.548 0.589 ARIMA 0.140 0.355 0.504 0.583 0.693 0.823 nRMSE MS-WPF 0.058 0.150 0.154 0.195 0.215 0.237 PM 0.272 0.445 0.561 0.644 0.711 0.767 GRU 0.108 0.163 0.283 0.401 0.468 0.501 CNN-BiLSTM 0.121 0.205 0.355 0.466 0.511 0.541 WD-BiLSTM 0.242 0.309 0.365 0.412 0.458 0.498 ARIMA 0.148 0.352 0.494 0.603 0.697 0.793 R2 MS-WPF 0.996 0.977 0.976 0.962 0.953 0.943 PM 0.925 0.801 0.684 0.584 0.493 0.411 GRU 0.988 0.973 0.919 0.839 0.780 0.749 CNN-BiLSTM 0.985 0.958 0.873 0.782 0.738 0.707 WD-BiLSTM 0.941 0.904 0.867 0.830 0.790 0.752 ARIMA 0.978 0.876 0.755 0.636 0.514 0.371 Case 3 (1 h) MAPE MS-WPF 0.056 0.120 0.163 0.204 0.276 0.437 PM 0.261 0.578 0.828 0.977 1.026 1.022 GRU 0.105 0.181 0.325 0.464 0.564 0.611 CNN-BiLSTM 0.188 0.374 0.544 0.645 0.758 0.852 WD-BiLSTM 0.232 0.330 0.416 0.501 0.574 0.637 ARIMA 0.341 0.676 0.812 0.707 0.670 0.796 nRMSE MS-WPF 0.059 0.116 0.158 0.185 0.249 0.378 PM 0.301 0.595 0.782 0.885 0.942 0.975 GRU 0.097 0.188 0.350 0.504 0.597 0.649 CNN-BiLSTM 0.184 0.353 0.493 0.565 0.619 0.676 WD-BiLSTM 0.238 0.348 0.438 0.516 0.587 0.646 ARIMA 0.300 0.602 0.708 0.636 0.613 0.674 R2 MS-WPF 0.996 0.986 0.975 0.965 0.937 0.856 PM 0.909 0.645 0.388 0.215 0.111 0.049 GRU 0.991 0.964 0.877 0.745 0.643 0.578 CNN-BiLSTM 0.966 0.875 0.756 0.679 0.616 0.541 WD-BiLSTM 0.942 0.878 0.808 0.733 0.655 0.582 ARIMA 0.909 0.637 0.498 0.595 0.623 0.544 J. Wu et al. Energy Conversion and Management: X 23 (2024) 100650 9 step prediction error is suppressed within 7%, and the accuracy degradation as the horizon grows is effectively restrained by the proposed MSWPF approach. The proposed BLEMD, which decomposes the complex wind power time series according to linearity and frequency, has a significant effect on enhancing the accuracy of the MS-WPF. 5.2. Comparison between MS-WPF and existing approaches The superiority of the proposed MS-WPF method is validated through comparison with cutting-edge approaches in three scenarios. The prediction results of the proposed MS-WPF and existing approaches with different prediction steps are shown in Fig. 7. The hyperparameters of the DL-based prediction method in the proposed MS-WPF are presented in Table 2 and hyperparameters of existing approaches are listed in Table 3. The 1-step ahead prediction results of two scenarios are shown in Fig. 7(a), (d), (g), while the 3-step ahead results are shown in (b), (e), (h) and the 6-step ahead in (c), (f), (i). The blue curve representing the results of the proposed method always tracks the actual value well, the tracking of wind power trend is precise, and the response to wind power corners is fast. The accuracy of the proposed MS-WPF degrades in 3-step and 6-step predictions due to the decreased sensitivity to the wind power dynamics. However, the overall trend of the wind power change can still be effectively tracked. The PM result describes the characteristics of the wind power dataset, which is nonlinear, stochastic, and hard to forecast. Most of the methods have promising results on 1-step prediction, but the precision degrades dramatically. For 6-step prediction, most of the existing approaches already lose the trend of wind power data. The proposed MS-WPF outperforms all the existing approaches in effectively capturing the trend of wind power. Hence, the overall accuracy of MSWPF for multi-step prediction is the highest among the selected methods. From the figures, it can be found that DL-based methods handle nonlinear prediction tasks better than conventional mathematical algorithms. GRU network can efficiently predict wind power even without any decomposition approaches. If comparing the 1-step prediction in the three scenarios, it can be concluded that the prediction task can be divided into two subobjectives: 1) follow the trend, and 2) detect the turning point. RNNbased DL approaches can follow the trend very well while conventional mathematical methods and CNN-based DL algorithms are sensitive to turning points. The proposed MS-WPF can effectively consider two sub-objectives jointly since the BLEMD has decoupled the trend and oscillation components. The trend is predicted by a BiLSTM network, the turning point is detected by a CNN network, and the dramatic oscillation is compensated by a mathematical approach. In this way, the interaction of components with different linearity and frequency in the DL network training process is avoided. The objectives of the DL networks are simplified and concretized. The MAPE, nRMSE, and R2 of the multi-step prediction methods with three scenarios are presented in Table 4. Generally, DL-based methods have higher accuracy and regression performance compared to conventional mathematical approaches. The proposed MS-WPF shows the highest accuracy, and other DL-based approaches also show good results with one or two-step-ahead prediction. However, with more look-ahead steps, most existing methods will meet a precision degradation due to the over-fitting problem. The stochastic and nonlinear components in the time series will affect the prediction of linear components, leading to the algorithm losing the overall trend of wind power, further decreasing the overall WPF accuracy. This problem is significantly suppressed in the proposed MS-WPF, the interaction of the linear and nonlinear components in the prediction algorithm is decoupled by the linearity-frequency BLEMD. The accurate prediction of linear components (C1) ensures the stable performance of multi-step forecasting. 6. Conclusion In this paper, an MS-WPF is proposed to enhance the prediction accuracy of wind power prediction and improve the performance as the look-ahead step grows. The performance of the proposed method is verified through three cross-experiments. The advancement of the proposed MS-WPF is proved by comparing it to five conventional methods. The main works of the paper can be summarized in the following aspects. 1. An improved BLEMD with rigorous mathematical analysis is designed for wind power time series decomposition. Apart from frequency features, a linearity feature evaluated by PCCs is proposed to decompose the time series legitimately. The proposed BLEMD effectively decouples the nonlinear and linear components in the wind power time series and restrains their interactions, further enhancing the accuracy and stability of multi-step forecasting. 2. A multi-level prediction structure, including a BiLSTM-based neural network, a CNN-BiLSTM-based neural network, and a WME method, is established to solve the regression task of stochastic time series while efficiently allocating the computation resource. The C1, high linearity and low-frequency component, is predicted with less computation resource; the C2, low linearity and low-frequency component, is solved by a more complex neural network to achieve better results; the C3, low linearity and high-frequency component, is compensated through the proposed WME to release computation burden. 3. The proposed MS-WPF method outperforms the existing approaches in this paper. The precision reaches 93% and remains around 70 % after 6-step-ahead forecasting in all cases. The proposed BLEMD and DL-based multi-step prediction method provide an essential reference for forecasting tasks of time series with a high proportion of uncertainty. CRediT authorship contribution statement Jingxuan Wu: Conceptualization, Formal analysis, Data curation, Software, Investigation, Validation, Writing – original draft. Shuting Li: Methodology, Visualization, Writing – review & editing. Juan C. Vasquez: Resources, Writing – review & editing. Josep M. Guerrero: Project administration, Supervision. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. References [1] Li Menglin, Yang Ming, Yu Yixiao, Lee Wei Jen. A wind speed correction method based on modified hidden markov model for enhancing wind power forecast. IEEE Trans Ind Appl 2022;58:656–66. [2] Wang Jianzhou, Qian Yuansheng, Zhang Linyue, Wang Kang, Zhang Haipeng. A novel wind power forecasting system integrating time series refining, nonlinear multi-objective optimized deep learning and linear error correction. Energy Convers. Manage. 2024;299:117818. [3] Khodayar Mahdi, Wang Jianhui. Spatio-temporal graph deep neural network for short-term wind speed forecasting. IEEE Trans Sustain Energy 2019;10:670–81. 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