A Wind Speed Forecasting Method Using Gaussian Process Regression Model Under Data Uncertainty

Against the backdrop of the current energy crisis and the reduction of traditional fossil energy, the development and utilization of renewable energy has become the hot subject worldwide in the past decade. By the end of 2022, the global installed capacity of renewable energy for electricity generation reached 3372 GW, marking a record increase of $9.6%$ (295 GW) compared to the cumulative total of the previous year. Among them, the installed wind power capacity becomes the second-largest share (⁠$26.89%$⁠) at 906 GW of the total installed renewable energy, with a yearly growth of $9%$ [1]. This provides crucial support for the establishment of a green, low-carbon, clean, and efficient new energy system. However, wind energy is greatly influenced by natural environmental conditions. Specifically, wind speed exhibits characteristics of randomness, variability, and intermittency, which still pose significant technical challenges to ensure the safe and stable operation of wind energy systems and the optimal maintenance plan. Therefore, it is necessary to accurately predict wind speed under different operating conditions to integrate wind energy into existing power grids.

Currently, wind speed forecasting methods are mainly categorized into two types: physics-based and data-driven [2]. The former utilizes meteorological models to convert the data generated by weather forecasting systems into wind speed and direction at the desired location [3]. This type of method also takes into consideration the local effects to convert data from numerical weather prediction into wind speed at the turbine hub. Numerical weather prediction (NWP) models, as a means of wind speed forecasting, have the advantages of high prediction accuracy and interpretability. However, it is difficult to collect enough data, complex to build the model, and computationally time-consuming to model [4–6]. The latter, on the other hand, utilizes statistical correlation methods to forecast wind speed from historical data, such as time series approach, machine learning, and hybrid models [7]. There are different time series models such as moving average (MA), autoregressive (AR), autoregressive moving average (ARMA) and their variants [8]. Mauricio et al. used ARMA to capture the nonstationary characteristics of wind speed data to fit the error series linearly, but its prediction accuracy is limited by the strong nonlinear effects of wind speed [9]. To overcome this drawback, various machine learning methods have been developed for wind speed prediction in recent years. Liang et al. input wind speed data into a Convolutional Neural Network (CNN), extracts the feature information of the data and performs wind speed prediction [10]. It is concluded that the CNN model has a better fitting effect in predicting the nonlinear sequences, but can't handle the strong randomness of data. Wei et al. obtained more accurate prediction results by fully learning the logical relationship between wind speed sequences using long short-term memory (LSTM) networks model [11]. However, LSTM is prone to the problem of information loss when dealing with long sequences of data, resulting in unacceptable prediction accuracy [12].

The aforementioned research mainly focuses on deterministic prediction, which makes it difficult to fully characterize the inherent uncertainty caused by multiple factors in reality, such as data integrity, representativeness, and the inhomogeneity of wind speed. The uncertainty in wind speed can lead to fluctuations and intermittency in wind power output, increasing the difficulty of balancing supply and demand in the power system. Therefore, accurately quantifying and predicting wind speed uncertainty is crucial for optimizing power system scheduling, reducing reserve capacity requirements, and improving the reliability of wind power integration. In recent years, researchers have employed interval prediction methods to quantify the uncertainty in wind speed prediction. Interval prediction provides upper and lower bounds to reflect the possible range of wind speed, thereby more effectively addressing the randomness and intermittency of wind speed. Statistical modeling methods, such as Quantile Regression (QR), Bootstrap, and Kernel Density Estimation (KDE) [13–15], are mainly explored. However, there are some limitations of these methods, for example, the QR method may lead to the quantile crossover problem [16], the computational mechanism of Bootstrap resampling limits its ability to deal with large sample sets, and the KDE method suffers from boundary effects in the boundary region of the density function. Wei et al. proposed the Gaussian process regression (GPR) based on Bayesian theory aims to construct a distribution model of the predicted values, which can effectively yield the distribution of the predicted values and estimate them according to the predefined confidence level. At the same time, the GPR requires fewer parameters, adaptively estimates the hyper-parameters, and obtains the better estimate of the prediction intervals [17].

With the continuous advancement of signal processing techniques, the combination of signal decomposition and various machine learning models has been proven effective in enhancing the accuracy of forecasting complex wind speed sequences. For instance, Lin et al. utilized the Isolation Forest (iForest) method for data preprocessing to identify and remove outliers, thereby improving data quality and enhancing prediction accuracy [18]. iForest works by constructing random isolation trees to isolate data points, where points that are easily isolated are considered outliers. However, this method assumes that data follows independent and homogeneous distributions, making it challenging to capture the temporal correlations in time series data.Similarly, Wang et al. employed the local outlier factor (LOF) method to detect and remove outliers, further enhancing data reliability [19]. LOF identifies outliers by comparing the local density of each data point to its neighbors, but it may face performance issues when dealing with high-dimensional data or datasets with large density variations. Currently, decomposition-based data preprocessing methods have been widely used to improve prediction accuracy in many research cases [20,21]. For example, the wind speed prediction model along with the Variational modal decomposition technique has shown better prediction results than other models [22]. However, Wang et al. pointed out that EMD-based forecasting methods were difficult to be used for practical forecasting because the newly acquired data greatly affected the value of the initial decomposition subsequence, resulting in lower accuracy of the final forecast [23]. To solve this problem, Jiang et al. proposed a denoising method by combining the discrete wavelet packet transform and Bayesian hypothesis testing, which takes into account the noise uncertainty in the original signal, thus lowering the information loss in the traditional wavelet transform method [24,25]. However, to the best knowledge of the authors, no research has been found on the application of Bayesian Discrete Wavelet Packet Transform (BDWPT) method for wind speed prediction. Therefore, in this study we employ the BDWPT method to filter the raw wind speed data in order to improve the predictive accuracy of the proposed model.

The main contributions of this study include: (1) BDWPT is systematically studied for wind speed forecasting for the first time. Specifically, different methods are compared in details to show the denoising performance of the BDWPT method in terms of the forecasting accuracy improvement and (2) A hybrid method combining BDWPT and GPR is proposed to realize both point and interval prediction for wind speed series. The proposed method extracts useful information from the short-term wind speed series, removes interfering factors, handles data uncertainty using implicit functions, and models wind speed behavior under uncertainty.

In the field of wind speed forecasting, the data quality is crucial for building reliable prediction models. However, during the actual collection of wind speed data, there always exist noises resulting from different sources, such as measurement errors, environmental effects, or distortions during data transmission. These noises may make the data vague or inaccurate information, which seriously affects the accuracy of wind speed prediction. In this study, the BDWPT method is chosen to denoise the raw wind speed data to ensure that the prediction model is based on more accurate and reliable data. The BDWPT is mainly composed of three parts: wavelet packet signal decomposition, Bayesian thresholding noise reduction and signal reconstruction. First, the original wind speed data is processed by time-frequency wavelet decomposition using the BDWPT multiscale wavelet packet decomposition method. Then for each decomposition coefficient sequence, the noise reduction process is carried out using the Bayesian hypothesis test, and finally, the signal reconstruction is carried out for the noise reduced coefficients to get the new data after noise reduction.

where $ŝj(k)$ is the denoised scaling coefficient and $ŵj(k)$ is the denoised wavelet coefficient.

Gaussian process regression (GPR) is a machine learning method that combines statistical learning with Bayesian theory. Its input low-dimensional space is mapped to the high-dimensional feature space through the kernel function, which is suitable for dealing with nonlinear complex regression problems. The GPR approach has found its wide applications in many fields such as time series analysis, dynamic system model identification, and so on.

where $K(X,X)=Kn$ the $n×n$ kernel matrix with elements $Kij=k(xi,xj)$⁠,$K(X,X*)=K(X*,X)T$ is the covariance matrix between the test data $X*$ and the input X of the training set; $K(X*,X*)$ is the covariance of $X*$ itself.

The mean vector $f*¯$ is the wind speed mean of GPR model corresponding to the point prediction output, and $σf*2¯=cov(f*)$ is the variance corresponding to $f*¯$ which can be used to obtain wind speed interval prediction in the sense of probability distribution.

This study quantitatively evaluates the effectiveness of denoising methods using signal-to-noise ratio (SNR). Three metrics, namely, the coefficient of determination (⁠$R2$⁠), the root‐mean‐square error (RMSE) and the mean absolute error (MAE), are employed to quantitatively evaluate the point prediction results of the model. and two metrics, namely, the probability of coverage probability (CP), and the mean width percentage (MWP) are utilized to quantitatively evaluate the interval prediction results of the model. These quantitative metrics can be expressed as:

where $sp$ represents the denoised signal, while $sn$ denotes the raw signal. This study illustrates the removed noise by representing the difference between the raw and denoised signals.

where $yi$ denotes the actual value, $y¯$ denotes the mean of the actual value, and $ŷi$ denotes the mean of the predicted value of i samples, n denotes the number of test samples.

where is $Cα$ the number of samples whose observation fall within the prediction interval.

where $upi$ and $downi$ are the lower and upper limits of the prediction interval, respectively.

Figure 1 illustrates the wind speed prediction process based on the fusion of BDWPT and GPR, which consists of eight main parts, summarized as follows:

• Step 1: Data preprocessing. Owing to various reasons such as sensor malfunction, communication failure, turbulence, icing effects, and force majeure incidents, the raw data collected from a wind turbine may contain a lot of imperfections like noise, outliers, non-numeric or missing values. Hence, preprocessing of the raw data is necessary. Initially, a quality check is performed on the original data to remove non-numeric data. Subsequently, missing data within the data series are filled in to ensure data completeness.

• Step 2: Data denoising. This study compares the noise reduction performance of two common methods, LOF and iForest with the previously described BDWPT, using the preprocessed wind turbine speed data. Both the LOF and iForest methods perform noise reduction by identifying the irrational data in the dataset. The former identifies noise points by judging the density difference between a data object and its neighbors, i.e., characterizing the degree of the outlier of the data object, while the latter constructs a binary tree structure called Isolation Tree (iTree), which recursively and randomly isolates each data in the sample from the others by using the dichotomous method, and sparsely fewer noisy data are more easily identified due to the sensitivity to isolation. To better illustrate the noise reduction effect of each method, this study will take the following strategies: a.A sufficient condition assessing a noise reduction method is that it can make the speed prediction after denoising better than that before noise reduction; b.The fundamental requirement for noise reduction is that the denoised speed time series data must remain near the major rising and falling contours of the predenoise ones. If a distinct peak or valley is erased, this noise reduction fails. However, this operation cannot be effectively quantified at present, so visual inspection is first used to determine the denoising intensity; c.The effect of the noise reduction method is quantitatively evaluated by the SNR.

• Step 3: Data normalization. In this study, the mean-variance normalization method is used to normalize the wind speed data after denoising in preparation for subsequent modeling with the following formula:

  $X=X0−μS$

  (16)

  where X is the normalized data which is the input of the forecasting model, $X0$ is the original data, $μ$ is the mean value of the data, and S is the standard deviation of the data.

• Step 4: Modeling wind speed prediction. This study constructs GPR and Back Propagation (BP) models and investigates the impact of different denoising methods on the predictive accuracy of the models. The parameters of these two models are kept consistent to ensure fairness in comparison. Detailed information about the parameters of the models is provided in Table 1.

• Step 5: Deterministic projections. The trained model is utilized to further predict the future wind speed. Both the single-step and multistep predictions of the model are carried out in the case study, aiming to illustrate the superiority of the proposed method by comparing the prediction accuracy of different models.

• Step 6: Uncertainty analysis. Through the utilization of a well-trained GPR model, predictions and corresponding standard deviations were obtained for the test dataset. These forecasted mean values and standard deviations were employed to calculate prediction intervals. In this study, the prediction intervals were constructed by adding and subtracting two standard deviations from the mean, establishing a confidence level of $95%$⁠.This process, based on the predictive probability distribution, provided the most probable values and associated probability information.

• Step 7: Calculation of evaluation metrics. The model evaluation metrics are used to assess noise reduction effectiveness and evaluate model accuracy. The SNR metric assesses the method's noise reduction effectiveness, while the $R2$⁠, MAE, and RMSE metrics are employed to evaluate the accuracy of the model's point predictions. Additionally, the CP and MWP metrics are used for uncertainty analysis of the model.

• Step 8: Selecting the optimal model. Compare the prediction accuracy as well as the reliability of the proposed method with other methods in the wind speed prediction based on the evaluation metrics calculated in step 7.

To demonstrate the effectiveness and reliability of the proposed hybrid prediction model, a set of operational data collected from an offshore wind farm located in the northern Fujian of China, covering the period from Jan. 1, 2017 to Dec. 31, 2018, was used in this case study. The data was acquired by the SCADA system at an interval of 10 min, with the total 49804 sample points. Of them $70%$ was used as the training set with the remainder serving as the testing set. The original time series data of wind speed is plotted in Fig. 2.

According to the process outlined in Fig. 1, the original data undergoes preprocessing to eliminate non-numeric and missing entries. Subsequently, an interpolation regression method is applied to fill in the missing values.

Noise reduction was performed on the raw data using the three noise reduction techniques. Figure 3 illustrates the speed data series before and after noise reduction, clearly showing that the BDWPT and LOF methods retained the peaks and valleys characteristics of the original speed data series while making the curves smoother compared to those before noise reduction. However, the iForest method appears to overly eliminate the data features during the noise reduction process, resulting in significant deviation from the original data. As such, the iForest method is not applicable in this case study.

To further illustrate the effectiveness of the denoising methods, Fig. 4 presents the Power Spectral Density (PSD) plots comparing the original and denoised data. PSD is used to evaluate the denoising effect by analyzing the spectral characteristics of the signal. The PSD curve of the BDWPT method shows that, compared to the original data, the power spectrum retains certain main frequencies in the low-frequency range while significantly decreasing in the high-frequency range. This indicates that BDWPT can effectively remove high-frequency noise while preserving the main frequency components of the signal. In other words, BDWPT not only effectively denoises but also maintains the critical spectral features of the original data, thereby preserving the structure of the signal. This significant retention of key frequencies is an important advantage of BDWPT in denoising. In contrast, the PSD curve of the iForest method shows relatively small differences from the original data after denoising, with limited changes in both low- and high-frequency ranges, indicating a more moderate impact on frequencies. On the other hand, the PSD curve of the LOF method almost overlaps with the original data, suggesting a weaker denoising effect in the frequency domain, as it barely alters the spectral distribution of the original data. This may be because LOF primarily focuses on identifying outliers by examining local density differences between data points, which does not effectively remove noise components and thus is not suitable for this case study. In summary, through the qualitative analysis of Fig. 3 and Fig. 4, we can conclude that the BDWPT method provides superior denoising performance compared to the other two methods.

In addition to this, this study analyzes the effectiveness of several denoising methods from a quantitative perspective. The comparative graph of three denoising methods, BDWPT, iForest, and LOF, is shown in Fig. 3, with SNR values of 18.25, 11.63, and 10.11, respectively. The numerical results indicate that BDWPT exhibits the best denoising performance.

where t is the current moment, $f(Xt,X̂t+T|t)$ is the prediction model, $Xt$ is the measured data before (including the current moment), i.e., the historical wind speed, and $X̂t+T|t$ is the predicted value at the moment $t+T$⁠.

where $yi+m$ is the forecasting output at the $ith$ forecasting time. Hence, the final output vector of wind speed forecasting is $Youtput$⁠: ${y1+m,y2+m,…,yn+m}1×n$⁠.

where c is the forecasting ahead steps (e.g., $c={1,2,3}$ in the experiment herein); $yi+m+(c−1)$ is the forecasting result at the $cth$ step of the $ith$ forecasting experiment; $Xi,c$ is the input vector for the predictor at the $cth$ step of the $ith$ forecasting experiment. Finally, we obtain the multistep prediction result vector by putting them together expressed as $Youtput$⁠: ${y1+m+(c−1),y2+m+(c−1),…,yn+m+(c−1)}1×n$⁠.

To evaluate the performance of different denoising methods and prediction models, this study conducts a series of experiments on the combination of different prediction models and denoising methods. The results of the experiments are shown in Table 2. The following two conclusions can be obtained: (1) Given a forecasting algorithm combined with different denoising methods, the BDWPT approach can achieve higher prediction accuracy compared to the other three denoising methods and the raw data case. For example, when the BP model is used without denoising, the $R2$⁠, RMSE, and MAE values are 0.6348, 0.0632, and 0.0489, respectively; however, after applying the BDWPT denoising, the corresponding $R2$ value is increased to 0.9266, and the RMSE and MAE values are reduced to 0.0226 and 0.0339, respectively. (2) When BDWPT is used as the denoising method, the GPR-based prediction method performs better compared to BP. The $R2$⁠, RMSE, and MAE values of the BDWPT-GPR method reach 0.9771, 0.0208, and 0.0038, respectively. In fact, after comparing the combinations of eight different preprocessing and prediction methods, it is found that the accuracy of the BDWPT-GPR combination is significantly higher than that of the other combination models.

In summary, using BDWPT as a denoising method can improve prediction accuracy, while GPR demonstrates higher prediction accuracy. The BDWPT-GPR combination shows the best performance in the wind speed forecasting.

The numerical results of single-step prediction have demonstrated that BDWPT has a significant improvement on the prediction accuracy. We assume that all input wind speed sequences have been denoised using BDWPT in the preprocessing stage. This experiment aims to explore the performance of the model in comparison with other commonly used multistep wind speed prediction methods. For this purpose, two multistep wind speed forecasting methods are tested with the results presented in Table 3. In these experiments, several evaluation metrics (⁠$R2$⁠, RMSE, and MAE) are used to quantify the prediction accuracy. It was observed that as the prediction step size increases from one to three steps, the difficulty of the prediction task rises, leading to the gradual decrease of prediction accuracy. This trend is reflected in all the prediction models, as shown in Table 3.

Notably, the proposed BDWPT-GPR model outperforms all other models in most cases. Its $R2$ values reach the highest level of 0.9771 (1-step), 0.9769 (2-step), and 0.9758 (3-step), while the smallest RMSE values are 0.0208 (1-step), 0.0209 (2-step), and 0.0214 (3-step), and the smallest MAE values are 0.0038 (1-step), 0.0049 (2-step), and 0.0053 (3-step).

To sum up, it can be concluded that the BDWPT-GPR model demonstrates the superior prediction in terms of adaptation to the dataset and the degree of fitting in both single-step and multistep prediction.

The above analysis focuses on the assessment of the forecasting accuracy. It is clear that given appropriate inputs, BP can only provide point estimates of wind speed. Additionally, in the BP modeling process, there exists bias as it only approximates the true stochastic dynamics of the given dataset. In contrast, GPR is a powerful nonparametric probabilistic approach that can provide the distribution of prediction results, thus addressing the uncertainties in the modeling. As a result, the GPR model can provide information on the most likely predicted values and their corresponding probabilities based on the predictive distributions. In this study, we set the confidence level to $95%$⁠. From Fig. 5, it can be found that most of the actual observations fall within the confidence interval. Specifically, its interval prediction indexes $CP95%$ and $MWP95%$ are 0.9920 and 0.2249, respectively. In comparison to the methods proposed by Zhang et al. [26], the two evaluation indexes show that the BDWPT-GPR model has the best performance in terms of stochastic wind speed prediction.

In this paper, a novel hybrid model named BDWPT-GPR is presented for wind speed forecasting, considering uncertainties in both data and modeling. The BDWPT method is used for noise reduction by integrating discrete wavelet packet transform and Bayesian hypothesis testing, while the GPR model exhibits excellent prediction accuracy for both deterministic and probabilistic forecasting. The combined advantages of BDWPT and GPR construct a hybrid model that outperforms traditional methods.

The proposed BDWPT-GPR method is validated by actual wind turbine data, and the following conclusions are drawn in this paper: (1) In terms of deterministic wind speed prediction, the combination of different prediction models and denoising methods are compared for single-step and multistep prediction. The experimental results show that BDWPT plays a significant role in improving the prediction accuracy, especially when combined with the GPR model, the prediction accuracy of the model is significantly improved to reach the highest $R2$ value of 0.9771, while the RMSE and MAE values are also minimized. (2) For the model uncertainty, this paper also investigates the interval prediction constructed by the GPR model with a confidence interval of $95%$⁠. The results show that most of the actual observations fall within the confidence interval, which confirms that the model can provide accurate wind speed prediction with the corresponding probability information, thus reducing the modeling bias.

Overall, numerical results have demonstrated that the hybrid model by adeptly combining BDWPT with GPR methods can provide excellent forecasting results of wind speed in terms of both single-step and multistep prediction. This study provides a powerful tool for accurately predicting wind speed to estimate the wind power, with ultimate purpose of smart maintenance of wind turbines.

This work was supported in part by the Scientific Innovation Program from DLUT under Grant DUT22 LAB502 and the Dalian Department of Science and Technology under Grant 2022RG10, which is highly appreciated.

• Dalian University of Technology (Award No. 2022RG10; Funder ID: 10.13039/501100002980).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

 $δij$ =

the Kronecker delta function

 $d̂jk$ =

the noiseless coefficients

 $μ$ =

the mean value of the data

 $ψj,k(ti)$ =

the wavelet functions

 $σn2$ =

the noise variance

 $σj$ =

the noise variance of the coefficients of the level decomposition

 $εjk$ =

the noise term

 $φj,k(ti)$ =

the scale functions

 $Cα$ =

the number of samples whose observation fall within the prediction interval

 $djk$ =

the $kth$ coefficient in the $jth$ decomposition

 $f(ti)$ =

the reconstructed time series signal

 $f*$ =

the predicted values

 $H1$ =

signal energy

 $H2$ =

signal-to-noise difference energy

 $In$ =

a unit matrix of dimension $n×n$

 S =

the standard deviation of the data

 $sj(k)$ =

scale coefficients

 $wj(k)$ =

wavelet coefficients

 $X0$ =

the original data