In this study, the data has been collected during the year of 2020 from a wind farm namely Huitengxile wind power plant in Inner Mongolia, Northern China. It is one of the largest wind farms in Asian and it’s located in the suburbs between Chaha’er youyi zhongqi and Ulanqab city. The whole wind farm has multiple wind turbines that are distributed in an open flat grassland which provides rich wind resources. The prevailing wind directions are northwest and southeast which are very stable in recent years. The location and the annual wind rose diagrams has been illustrated in Figure 1 below.

According to Figure 1B, the two prevailing wind directions, around 180° and 315° are visible. The geographic center coordinate is 112°40′E and 41°05′N. It’s annual average wind speed at 10 m height is 7.2 m/s and its annual average wind speed at 40 m height is 8.8 m/s. In the wind farm, the annual average air density is 1.07 kg/m and it contains an effective wind speed of 5–25 m/s with strong stability and high quality.

The data used in this research has been collected by the supervisory control and data acquisition (SCADA) system. Usually, data on more than 100 parameters at 10 s intervals is collected and stored in a SCADA system. The SCADA collected data of individual wind turbines is streamed to a central computer for condition monitoring, performance evaluation, and other forms of analysis.

In this research, the SCADA data collected at 20 wind turbines over the period of 2020 has been analyzed. According to Figure 1B, there are two annual prevailing wind directions and it can be partitioned into four seasons independently as illustrated with the wind roses in Figure 2. In the fall and spring, two prevailing wind directions around 150° and 315°are observed. In the winter and summer, one prevailing wind direction is noted. Since the wind direction data is captured as a discrete angular variable, it needs to be transformed for modeling. A bilinear transformation of the angular wind direction is applied in the next section.

The value of wind direction ranges from 0° to 360°. It is likely that the wind direction may change from the interval, i.e (0°, 10°) to (350°, 360°). Practice shows that bilinear transformation is a better way for transforming discrete wind direction data to continuous data than the sine and cosine transformation (Peng et al., 2020). Geometrically, the two intervals are close to each other and therefore this change would lead to a large prediction error (Bilgili et al., 2007). To avoid such error, transformation of the discrete angular variable into a standardized continuous variable is essential. One option is to use a sine and cosine transformation which is not the best approach due to two variables needed for prediction which enlarges the prediction errors. A better option is to apply a bilinear transformation (Jury 1973).

The bilinear transformation maps the analog plane (s-plane) into the digital plane (z-plane) (Groutage et al., 2003) (see Figure 3). The transformation function, the ratio of two polynomials (Davies 1974), is expressed in Eq. 1. H ( s ) Z = 1 + T 2 s 1 − T 2 s (1) where: s is the original value of angular variable in s-plane; T is the time interval of the transformation. The bilinear transformation expressed the angular variable between 0° and 360° as continuous and normalized. The inverse bilinear transformation is expressed in Eq. 2. s = 1 T 2 − 2 H ( s ) Z − 1 2 + 2 H ( s ) Z − 1 (2) where: s is the inversed value of angular variable in s-plane; and H(s)z is the transformed angular variable.

Since the wind direction data is noisy, a bilinear transformation function acting as a low-pass filter in the continuous-time domain reduces the noise (Davies 1974). A prediction model developed with the transformed data is more accurate than the model based on the discrete time-series angular data.

To integrate the wind direction series with the ICEEMDAN modules, the implementations are introduced as follows (Duan et al., 2021):

Step 1: Compute the local means of realizations using the EMD algorithm described in Eq. 7: x i = x + β 0 E 1 ( w i )   (7) where β 0 = ε 0 σ ( x ) / σ ( E 1 ( w i )   ) ; σ ( ) compute the standard deviation; and ε 0 is the reciprocal of the desired signal-to-noise ratio between the first added noise and the analyzed signal.

Step 2: Compute residual term R ₁ in the first component using Eq. 8: R 1 = M ( x i )   (8)

Step 3: Compute the first mode at the first stage (k = 1) using Eq. 9: d 1 = x − R 1 (9)

Step 4: Estimate the second residue as the average of local means of the realizations R 1 + β 1 E 2 ( w i ) and then define the second mode using Eq. 10 as follows: d 2 = R 1 − M ( R 1 + β 1 E 2 ( w i ) ) (10)

Step 5: For the other terms (k = 3, … ,K) of residuals, they can be computed by Eq. 11: R k = M ( R k − 1 + β k − 1 E k ( w i ) ) (11)

Step 6: Compute the other terms (k = 3, … ,K) of the mode by Eq. 12: d k = R k − 1 − R k (12)

Step 7: Implement step 4 for the next iteration.

For the transformed wind direction series, the IMF components are obtained via the above steps which can be illustrated by the diagram presented in Figure 4.

A major drawback of the classical deep neural networks is that they do not have memory of the past periods. The time series information such as the past clusters of seasonal patterns and seasonal trend may not be reflected (Lee et al. 1993). Introduced by Hochreiter and Schmidhuber (1997) and Gers et al. (2003), the long-short-term memory recurrent neural network (LSTM-RNN) matches the needs and it is used in this paper to predict wind direction.

The long-short-term memory recurrent neural network (LSTM-RNN) contains units called memory blocks composed of memory cells with self-connections storing temporal states. Each memory block includes an input and output gate. The input gate controls the flow of input data into the cell. The output gate controls the output data flow into the rest of the network (Sak et al. 2014). In addition, the LSTM-RNN has peephole connections (Gers et al. 2003) from its internal cells to the gates in the same cell to learn precise timing of the output. The architecture of LSTM-RNN is illustrated in Figure 5.

With a long-short-term memory recurrent neural network (LSTM-RNN) architecture, the mapping from an input to an output layer is iteratively computed from Eqs 13–18 (Gers et al. 2003). i t = s i g ( W i x x t + W i m m t − 1 + W i c c t − 1 + b i ) (13) f t = s i g ( W f x x t + W f m m t − 1 + W i c c t − 1 + b f ) (14) c t = f t ∘ c t − 1 + i t ∘ g ( W c x x t + W c m m t − 1 + b c ) (15) o t = s i g ( W o x x t + W o m m t − 1 + W o c c t + b o ) (16) m t = o t ∘ h ( c t ) (17) y t = ϕ ( W y m m t + b y ) (18) where: W are the weight matrices (i.e., W _ix is the weight matrix from the input to the input layer; W _ic , W _fc , W _oc are diagonal weight matrices of the peephole connections (Gers et al., 2003)); b _i ,b _f ,b _o , and b _c are the bias vectors; m is the cell output activation vector; sig () is the sigmoid function; i, f, o, and c are the input gate, forget gate, output gate, and cell activation vectors, respectively, with all having the same size as the cell output activation vector m; ∘ is the element-wise product of the vectors; and g () and h () are the cell input and cell output activation functions, respectively.

Comparative analysis is performed in this research against the other benchmarking popular deep learning algorithms. All algorithms tested here are using the same ICEEMDAN framework as described in Section “Short-term Wind Direction Forecasting using ICEEMDAN”. The benchmarking deep learning algorithms compared includes deep neural network (DNN) (Xu et al., 2018; Sun et al., 2020; Yi and Xu, 2020), deep belief network (DBN) (Ouyang et al., 2019; Li et al., 2020), kernel-based extreme learning machine (KELM) (Li et al., 2018; Ouyang et al., 2018), and gated recurrent unit network (GRU) (Pan et al., 2019; Tang and Zhang, 2019).

The DNN is a fully connected feedforward network that consists of a cascade of multiple layers and hidden units. It’s structure with multiple processing layers enables it to handle highly nonlinear patterns inside the dataset. The deep temporal representations in the temporal domain can be effectively extracted by DNN.

Similar to DNN, the DBN consists of multiple layers of restricted Boltzmann machines (RBMs). It also contains a supervised regression layer stacked on the top of all RMBs for classification or regression tasks. Inside each RBM, it contains an input layer and a hidden layer with hidden-to-all-visible connections.

The ELM is a single hidden layer feedforward network. Instead of conventional back-propagation, it uses Penn-Moore pseudo-inverse to compute the wights between the hidden layer and output layer. The KELM is the improvement of vanilla ELM which uses the kernel matrix to replace the randomly initialized weights between the input layer and output layer. The most popular applied kernel functions include RBF, linear function, and polynomial function.

The GRU is another type of recurrent neural network other than LSTM-RNN proposed by Cho et al. (2014). In a typical GRU unit, it has one less gate than the LSTM unit and consists of two gates: the reset gate and the update gate. Hence, the GRU is also popular in modeling time-series dataset.

To assess prediction accuracy of the proposed deep learning model, four metrics are computed: the MAE [Mean absolute error (Eq. 19)], the MAPE [Mean absolute percentage error (Eq. 20)], the MSE [Mean square error (Eq. 21)], and the RMSE [Root mean square error (Eq. 22)]. MAE = 1 N ∑ i = 1 N | o j − t j | (19) MAPE = 1 N ∑ i = 1 N | o j − t j t j | (20) MSE = 1 N ∑ j = 1 N o j − t j 2 (21) RMSE = 1 N ∑ j = 1 N o j − t j 2 (22) where: o _j is the jth predicted wind direction; t _j is the jth measured wind direction; and N denotes total number of samples.

To improve the forecasting accuracy, the error correction is implemented in this research. First, after the forecasting outcome produced by each LSTM-RNN, the error series E(t) of the training dataset can be computed by comparing the original transformed wind direction series. The step can be expressed in Eq. 23 as follows: E ( t ) = W L S T M ( t ) − W A c t u a l ( t )   (23) where W L S T M ( t ) is the final forecasted transformed series in the first module; and W A c t u a l ( t ) is the actual transformed wind direction series after bilinear transformation.

The forecasted errors of wind direction series E(t) are oscillatory in the time-series domain (Wasynczuk et al., 1981). The relationship between oscillatory and decaying property of the wind direction errors can be represented by an ARIMA model which predicts the errors. In detail, the ARIMA can be constructed by computing autocorrelation expressed with the autocorrelation factor (ACF) (See. Eq. 24) and the partial autocorrelation factor (PACF)) (See. Eq. 25). Here, Cov() denotes the covariance; Var() denotes the variance; and Corr () denotes the Pearson’s correlation coefficient. ρ k = C o v ( E ( t ) , E ( t − k ) ) V a r ( E ( t ) ) (24) ρ k = C o r r ( E ( t ) , E ( t − k ) | E ( t − 1 ) , … , E ( t − k + 1 ) ) (25)

For each IMF, an ARIMA model is developed and then all outcomes of each ARIMA are integrated to obtain the final error series. Last, as illustrated in Figure 4, the final prediction is achieved by Eq. 26 as follows: W F i n a l ( t ) ^ = W L S T M ( t ) ^ + E ( t ) ^   (26) where W L S T M ( t ) ^ denotes the forecasting results from the LSTMs; E ( t ) ^ denotes the errors forecasted by ARIMA models; and W F i n a l ( t ) ^ is the final forecasting outcomes.

In this section, computational experience with models predicting wind direction is presented. Wind data from four seasons, spring, summer, fall, and winter are used. Prediction of wind direction is conducted using dataset at 10, 20, and 30 s resolution. The prediction horizons are 2, 5, 10 min, and 1 h. The prediction model is expressed in Eqs 27–29. Y = f ( D )   (27) D = [ D t − 1 , D t − 2 , D t − 3 , D t − 4 , D t − 5 , D t − 6 ]   (28) D t − i = ( x t − i , … , x t − i − 359 )   (29) where: f(D) represents the whole framework illustrated in Figure 4; D _t-i is the ith lagged vector containing 1 hour of the historical wind direction data; and x t − i denotes the ith lagged transformed wind direction series.

The wind speed and wind direction are 10 s data. One hour of data (360 data points) is used as the input vector. The six-time lagged vectors containing 1 hour of the historical wind direction data are selected as the input vectors. The wind direction of the 2, 5, 10 min, and 1 h horizon is predicted. The input vector is normalized beforehand and the predicted values are inverse-normalized.

Based on the training strategy stated in Section “Training Strategies”, experiments with the five selected algorithms have been performed. In all experiments, the wind direction has been predicted for the next 2, 5, 10 min, and 1 h. Experiments have been conducted in each of the four seasons of 2020.

The prediction accuracy results in Figure 6 demonstrate that the long-short-term memory recurrent neural network (LSTM-RNN) performs better over short-term horizons than the other algorithms. Since the LSTM-RNN contains long/short term memory, it produces smaller prediction errors than the DNN, DBN, KELM, and GRU. For the short-term horizons (i.e., 2 and 5 min), prediction accuracy of all five algorithms is similar. However, the LSTM-RNN provides more promising results for longer-term predictions (i.e., 10 min and 1 h) of wind direction.

The prediction accuracy in four seasons varies. In the fall and spring season, the prediction errors are larger than the errors in the summer and winter season. This is due to a larger variability of the wind direction over short-term horizons. Hence, training specific prediction models in different seasons is necessary.

Table 1 provide the MAPE for different resolution data (i.e., 10, 20, 30 s) and different prediction horizons (i.e., 2, 5, 10 min, and 1 h) before the error correction. Obviously, the MAPE errors are smaller for the 10 s data than for 20 and 30 s. With the increase of the prediction horizon, the MAPEs increase. The LSTM-RNN algorithm has the smallest MAPE at all resolutions and all prediction horizons. Therefore, it is an effective algorithm for wind direction prediction at short-term horizons.

To correct the errors made by the ICEEMDAN modules, the ARIMA model has been developed to forecast the errors. In the second part of Figure 4, To illustrate this step, the forecasted errors using ARIMA versus the actual errors produced by LSTM-RNNs are visualized in Figure 7. It is obvious that the aggregated results from ARIMAs can represent the temporal trend of the forecasted errors produced from the first component of the proposed framework.

Table 2 provides the MAPE for different resolution data (i.e., 10, 20, 30 s) and different prediction horizons (i.e., 2, 5, 10 min, and 1 h) after the error correction. There exists significant performance for all algorithms tested with respect to the MAPE computed before and after error correction. It validates the effectiveness of implementing error correction in improving the forecasting accuracy of time-series dataset. Meanwhile, the LSTM produces the smallest errors which also demonstrates its superior performance in forecasting wind directions.

The experiments reported in Section “Short-term Predictions” have been conducted using the transformed wind direction data from four seasons. An inverse bilinear transformation, expressed in Eq. 2, is applied to transform the predicted transformed wind direction into the original angular range [0°, 360°]. The actual angular values versus the forecasted angular values by the proposed framework using ICEEMDAN and LSTM-RNN are presented in Figure 8. It can be seen that the majority of the forecasted values fall within a relatively small range with respect to the actual values. It demonstrates the proposed framework can sufficiently provide accurate forecasting performances.

In this section, performance of the ICEEMDAN framework integrated with the long-short-term memory recurrent neural network (LSTM-RNN) for prediction of wind direction at four seasons is discussed. The prediction error of the inverse transformed wind direction at 2, 5, 10 min, and 1 h horizons are presented in Table 3. The mean absolute error (MAE) and mean absolute percentage error (MAPE) of wind direction are smaller in the summer and winter.

The wind direction error shows less variability over short horizons. The changes of a nacelle position are usually made within 5 min and the prediction error should be under 3% (Ouyang et al., 2017). A control chart is applied to monitor the prediction error and facilitate changing the nacelle position. A control chart with lower and upper limits enables monitoring the yaw error. Any prediction error that exceeds the bound (i.e., 3%) may trigger a change of the nacelle position. The final forecasting errors in the angular perspectives are illustrated in Figure 9.

The long-short-term memory recurrent neural network (LSTM-RNN) has been demonstrated to perform better than other algorithms. To validate the effectiveness and robustness of the LSTM-RNN, the data collected from another wind farm located in Shandong Province in the year 2020 has been used. The experiments are conducted following the similar training strategies as described in Section “Data Analysis”. The computational results are presented in Table 4.

The prediction error (see Table 4) in winter and summer seasons of 2020 from the wind farm in Shandong Province produced by the LSTM-RNN is similar to the one based on the 2020 data (see Table 3) in the wind farm in Inner Mongolia. More accurate performance has been observed in the fall and spring seasons with two prevailing wind directions. The favorable prediction error validates the effectiveness and robustness of the LSTM-RNN to predict the nacelle orientation.