A Hybrid Generative Adversarial Network Model for Ultra Short-Term Wind Speed Prediction

Abstract

To improve the accuracy of ultra-short-term wind speed prediction, a hybrid generative adversarial network model (HGANN) is proposed in this paper. Firstly, to reduce the noise of the wind sequence, the raw wind data are decomposed using complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). Then the decomposed modalities are entered into the HGANN network for prediction. HGANN is a continuous game between the generator and the discriminator, which in turn allows the generator to learn the distribution of the wind data and make predictions about it. Notably, we developed the optimized broad learning system (OBLS) as a generator for the HGANN network, which can improve the generalization ability and error convergence of HGANN. In addition, improved particle swarm optimization (IPSO) was used to optimize the hyperparameters of OBLS. To validate the performance of the HGANN model, experiments were conducted using wind sequences from different regions and at different times. The experimental results show that our model outperforms other cutting-edge benchmark models in single-step and multi-step forecasts. This demonstrates not only the accuracy and robustness of the proposed model but also the applicability of our model to more general environments for wind speed prediction.

1. Introduction

Energy demand has always been one of the main problems of human development since the increasing consumption of energy with the improvement of living standards. In recent years, renewable energy has gradually become a research hotspot. Wind energy is valued for its clean, pollution-free, renewable, and abundant availability. However, wind is highly random and volatile, which may affect the stability of the power system and hinder the efficient use of wind energy [1]. Accurate ultra-short-term wind speed prediction models are therefore crucial in power dispatch planning and power market operations [2]. Thus, reliable wind speed prediction has drawn a lot of interest.

The three common wind speed prediction models are physical models, statistical models, and hybrid models. Physical models take into account the physical conditions and locations of wind farms, which require abundant meteorological data. Numerical weather prediction is a typical physical model, as it takes into account temperature pressure and obstacles for wind speed prediction, so it has a long calculation period [3]. Physical models are efficient and accurate for long-term forecasting, but they are computationally intensive and expensive for small-scale forecasting.

Statistical models make better use of historical wind speed data to predict future wind speeds than physical models. Statistical models include both traditional statistical models and neural network-based models. Traditional statistical models include the autoregressive moving average model [4], the autoregressive integrated moving average model [5], the Bayesian model [6], etc. However, the non-linear nature of wind makes it difficult for traditional statistical models to extract the deeper features of wind speed data. Neural networks are introduced into the field of wind speed prediction for their ability to fit the non-linear part of the data well. Neural network-based models can extract deeper features from wind speed data than traditional statistical models—for example, BP [7], RBF [8], artificial neural network [9], SVR [10], etc. To improve the learning ability and predictive ability of predictive models, deep neural networks are introduced into wind speed prediction, such as the deep belief network [11], RNN [12], GNN [13], and LSTM [14].

In recent years, hybrid models have gradually become the mainstream wind speed prediction models. Hybrid models typically use one or more auxiliary strategies to assist the main forecasting network in wind speed prediction. Therefore, hybrid models can achieve better prediction performance than physical models and statistical models. The auxiliary strategies involved in hybrid models include data preprocessing techniques, optimization algorithms, error correction, and weighting strategies.

(1) Data preprocessing techniques. Zhang et al. [15] used EMD for data pre-processing of wind speed, which effectively reduced the volatility of the wind speed series. However, EMD suffers from the problem of modal confusion, which leads to unsatisfactory decomposition results. Santhosh et al. [16] used EEMD to process the raw wind speed series, which effectively mitigated the EMD problem. However, EEMD has a noise residual problem affected by noise residuals. Wang et al. [17] used CEEMD for wind speed prediction. CEEMD cancels out the residual noise with a pair of white noises, effectively improving the efficiency of the calculation. Ren et al. [18] experimentally demonstrated that the CEEMDAN-based model always performs best compared to the EMD-based model.

(2) Optimization algorithms. Optimization algorithms can be used to optimize the hyperparameters, weights, network structure, and thresholds of predictive models. Li et al. [19] used PSO to optimize two hyper-parameters of LSTM, which solved the problem of wide intervals caused by interval superposition and thus improved the wind speed prediction accuracy. Tian [20] used PSO to optimize the weight coefficients of each prediction model, and the experimental results demonstrate the necessity of introducing the weight coefficient optimization strategy. Liu et al. [21] used GA to optimize the internal parameters of LSTM, which improved the efficiency and accuracy of the prediction model. Cui et al. [22] used the Bat algorithm to optimize the thresholds of BP networks, effectively improving the generalization ability and nonlinear mapping ability of BP networks.

(3) Error correction. Error correction is a post-processing technique for wind forecasting. It predicts the residuals and superimposes the predictions on the original predictions to obtain the final predictions. Duan et al. [23] used improved CEEMDAN to decompose the errors, and the experimental results showed that the error decomposition correction method can significantly enhance the prediction accuracy. Liu et al. [24] proposed an adaptive multiple error correction method, which makes full use of the deeper predictable components and effectively improves the reliability and accuracy of the model. Zhang et al. [25] demonstrated experimentally that the final predicted values after Markov chain correction are closer to the original wind field data, which proves that the use of the Markov chain is effective.

(4) Weighting strategies. To scientifically determine the weights of different prediction networks in a hybrid model, many scholars have proposed different weighting strategies. To alleviate the adverse effects of multi-collinearity in combinatorial prediction models, Jiang et al. [26] used a GMDH neural network to automatically identify the weights of three nonlinear models. The application of GMDH can significantly improve the predictive capability compared to the widely used equal-weighting scheme. Wang et al. [27] used MTO to minimize the error sum of squares of the IOWA operator, which obtains the optimal weight vector for the combined prediction model and ensures the stability of the prediction results. Altan et al. [28] optimized the weighting coefficients for each IMF using the gray wolf optimizer algorithm.

Although the above models achieve good predictive performance, they still have some problems. Methods involving deep neural networks [27] cause huge computational costs. Hybrid methods based on weighting strategy [28] may have the problem of multicollinearity, which reduces the prediction accuracy. The performance of hybrid methods based on parameter optimization [26] is largely influenced by the understanding of the researcher of the optimization algorithm.

Considering the above issues, we propose a hybrid model combining data preprocessing techniques and optimization algorithms for ultra-short-term wind speed prediction. We design the hybrid generative adversarial network (HGANN) as the prediction master network for the proposed hybrid model. The contributions and innovations of this research are concluded as follows:

(1)

A hybrid generative adversarial network model (HGANN) is proposed for ultra-short-term wind speed prediction, which learns the distribution of wind data and predicts it through a continuous game between generators and discriminators.

(2)

To improve the error convergence of the model, the OBLS was developed as a generator for HGANN. The IPSO was used to optimize the hyperparameters of the OBLS. To maintain the stability of the generated samples, we used the discriminator of WGAN as the discriminator of HGANN.

(3)

A wind data decomposition and denoising process was carried out using CEEMDAN to reduce the randomness and instability in the original wind series.

The rest of this article is organized as follows. Section 2 introduces the model framework and methods involved in this article in detail. In Section 3, the experimental cases and prediction results are elaborated in detail, which verifies the validity of the framework we propose. Section 4 contains a discussion of the results of the experiment. The conclusions are presented in Section 5.

2. Proposed Predictive Framework

2.1. Overall Framework of HGANN

Generative adversarial networks (GANs) [29] are deep learning networks, which are composed of a generator and discriminator that confront each other. The role of the generator is to generate false samples that are close to the real ones. The role of the discriminator is to distinguish between true and false samples as correctly as possible. However, GANs often suffer from the problem of target confusion. Our proposed HGANN alleviates this problem to a great extent.

We developed a hybrid generative adversarial network model (HGANN) for ultra-short-term wind speed prediction, which uses the two networks to compete with each other to achieve highly accurate wind speed predictions. The proposed model is shown in Figure 1. First, CEEMDAN decomposes the raw wind speed data into multiple modalities. These modalities are separately fed into the generator of HGANN, the OBLS. The generator is used to obtain virtual samples that are similar to real samples. The virtual samples and real samples are then fed into the discriminator, which consists of convolutional layers and fully connected layers. The discriminator extracts the high-dimensional features of the input samples through the convolutional layer, and then further extracts the effective features by the fully connected layer. The outputs scalars “1” or “0” of the discriminator are passed to the generator and the discriminator to perform the iterative update of HGANN. Via the continuous iterative update, OBLS obtains the best parameters and performs wind speed prediction. Finally, the final wind speed forecast can be obtained by stacking all forecast values.

2.2. CEEMDAN Model

Due to the high volatility of the wind speed series, CEEMDAN [30] is introduced to smooth the wind speed data. CEEMDAN decomposes a signal into some modalities.

The original wind speed series is defined as $X\left(n\right)$. CEEMDAN decomposes $X\left(n\right)$ into $\overline{IM{F}_j}\left(n\right),j=1, 2, 3,\dots J$ and residue $r_j\left(n\right)$. Figure 2 shows the flow chart of the CEEMDAN algorithm. The specific steps of the algorithm are as follows.

Randomly generate white noise with (0, 1), which is defined as $w^i\left(n\right), i=1, 2,\dots I$. Define an operator $E\left\{\ast \right\},$ which generates the IMFs by EMD. We set the noise standard deviation to $\epsilon$ = 0.2 and the ensemble size to $I$= 500.

Add $w^i\left(n\right)$ to the $X\left(n\right)$ and generate a new series with noisy signals $X\left(n\right)+{\epsilon }_0{w}^i\left(n\right)$. For j$=1$, the first-order $\overline{IM{F}_1}\left(n\right)$ that is decomposed by EMD is expressed as:

$$\overline{IM{F}_1}\left(n\right)=\frac{1}{I}{{\displaystyle \sum }}_{i=1}^{I}E\left\{X\left(n\right)+{\epsilon }_0{w}^i\left(n\right)\right\}$$

(1)

The first-order residue is computed as follows:

$$r_1\left(n\right)=X\left(n\right)-\overline{IM{F}_1}\left(n\right)$$

(2)

For $j=2, 3,\dots J$, calculate the $\overline{IM{F}_j}\left(n\right)$ and the jth residue as follows:

$$\overline{IM{F}_j}\left(n\right)=\frac{1}{I}{\sum }_{i=1}^{I}E\left\{ r_{j-1}\left(n\right)+{\epsilon }_{j-1}{w}^i\left(n\right)\right\}$$

(3)

$$r_j\left(n\right)=r_{j-1}\left(n\right)-\overline{IM{F}_j}\left(n\right)$$

(4)

Decompose $E\left\{r_{j-1}\left(n\right)+{\epsilon }_{j-1}{w}^i\left(n\right)\right\}$ until the residue $r_j\left(n\right)$ cannot be decomposed and has only one extreme value. Then we can get $\overline{IM{F}_j}\left(n\right)=\frac{1}{I}{\sum }_{i=1}^{I}E\left\{ r_{j-1}\left(n\right)+{\epsilon }_{j-1}{w}^i\left(n\right)\right\}$ and the final residue $r_j\left(n\right)=X\left(n\right)-{\sum }_{j=1}^J\overline{IM{F}_j}\left(n\right)$.

The original wind speed time series $X\left(n\right)$ can be decomposed as $X\left(n\right)={\sum }_{j=1}^J\overline{IM{F}_j}\left(n\right)+r_j\left(n\right)$, where $\overline{IM{F}_j}\left(n\right)$ or $r_j\left(n\right)$ can represent different features of the wind speed.

2.3. Generator OBLS for HGANN

BLS [31] can provide incremental structural learning. It achieves better forecasting results in time-series forecasting. Furthermore, because of its shallow network structure, BLS has higher error convergence performance than CNN. Compared with BLS, OBLS can provide both higher convergence performance and predictive accuracy. This is because OBLS uses IPSO to improve the network hyper-parameter of optimization. Therefore, OBLS has faster convergence and higher error convergence than CNN. Therefore, instead of using CNN as the generator of GAN, we use OBLS as the generator of HGANN to solve the problem of target confusion during HGANN training, which can improve the generalization ability and error convergence of HGANN, and thus make HGANN more suitable for wind speed prediction. The following is the detailed process of the OBLS algorithm.

Randomly generate n particles so that the dimensions of the particles are a three-dimensional vector $\left\{NF,NW,NE\right\}$ corresponding to the three parameters of BLS, respectively. Initialize the particle position $x_{id}\in \left(1,100\right),$ and speed $v_{id}\in \left(-1, 1\right)$. Determine the learning factors $c_1=1.5$ and $c_2=1.5$, inertia weights $w_{max}=1.0$ and $w_{min}=0.4$, and the maximum number of iterations $ite{r}_{max}=100$.

Assume the input wind speed series data $X\left(n\right)$ and project the data using ${\varnothing }_i\left(X\left(n\right)W_{ei}+{\beta }_{ei}\right)$ to represent $i$th mapped feature $Z_i$, where $W_{ei}$ represents random weight with the proper dimensions. The $j$th group of enhancement nodes ${\varnothing }_j\left(Z_i{W}_{hj}+{\beta }_{hj}\right)$ is denoted as $H_i$.${\varnothing }_i$ and ${\varnothing }_j$ can be different functions. The $i$th mappings can be denoted as:

$$Z_i={\varnothing }_i\left(X\left(n\right)W_{ei}+{\beta }_{ei}\right),i=1,2,\dots n$$

(5)

The feature nodes are denoted as $Z^n\triangleq \left[Z_1,Z_2\dots Z_n\right]$, where $W_{hj}$ and ${\beta }_{hj}$ are random weights. The enhanced nodes are denoted as:

$$H_j={\varnothing }_j\left(Z_i{W}_{hj}+{\beta }_{hj}\right),j=1,2,\dots m$$

(6)

Let $H^m\triangleq \left[H_1,H_2\dots H_m\right]$ where the symbol $\triangleq$ means “noted as”; then the output of the BLS can be denoted as:

$$Y=\left\{Z^n|H^m\right\}{W}^n$$

(7)

where the $W^n$ is the final target weight needed by OBLS and is obtained through the ridge regression algorithm, that is, $W^n\triangleq {\left\{Z^n|H^m\right\}}^{+}Y$.

Let $\left\{M\right\}=\left\{Z^n|H^m\right\}$; then ${\left\{Z^n|H^m\right\}}^{+}$ can be expressed as follows:

$${\left\{Z^n|H^m\right\}}^{+}=\underset{\mathsf{\lambda }\to 0}{\lim }{\left\{\mathsf{\lambda }I+\left\{M\right\}{\left\{M\right\}}^T\right\}}^{-1}{\left\{M\right\}}^T$$

(8)

where λ is $l_2$ regularization.

The IPSO [32] is introduced to iterate to optimize the parameters of BLS: $\left\{NF,NW,NE\right\}$. When the iteration of IPSO is consistently performed, the position and speed of the particles are continually updated through the following equation:

$$v_{id}=w{v}_{id}+c_1{r}_1\left(p_{id}-x_{id}\right)+c_2{r}_2\left(p_{gd}-x_{gd}\right)$$

(9)

$$x_{id}=x_{id}+\gamma v_{id}$$

(10)

Here, $\gamma$ is the velocity coefficient; the value of the inertia weight $w$ is $w=w_{max}-\left(w_{max}-w_{min}\right)\ast 1/iter$. When reaching the maximum iterative number $ite{r}_{max}$, the iteration is stopped and the optimal value of $\left\{NF,NW,NE\right\}$ can be obtained.

The generator takes the wind speed subsequence $\left\{x_1,x_2,\dots x_n\right\}$ as input, which is generated by CEEMDAN. Then the generator generates a new wind speed sequence $\left\{y_1^{\prime },y_2^{\prime },\dots y_n^{\prime }\right\}$, which is statistically similar to the wind speed sequence $\left\{y_1,y_2,\dots y_n\right\}$.

From Equations (8)–(10), OBLS does not require layer-to-layer coupling. Since there are no multi-layer connections, OBLS does not need to use gradient descent to update the weights, so the computational cost of OBLS is significantly lower than that of deep learning. When the accuracy of OBLS does not meet the requirements, its accuracy can be improved by increasing the “width” of the network nodes. Compared with the increase in the amount of calculation by increasing the number of layers in the deep network, the increase in that by increasing the “width” of the network nodes in OBLS is negligible.

2.4. Discriminators for HGANN

To maintain the stability of the generated samples, we used the discriminator of WGAN [33] as the discriminator of HGANN. In HGANN, the discriminator takes $\left\{x_i,y_i^{\prime }\right\}$ or $\left\{x_i,y_i\right\}$ as input. The training goal of discriminator is to discriminate $\left\{x_i,y_i^{\prime }\right\}$ as false and $\left\{x_i,y_i\right\}$ as true. The discriminator is trained by minimizing the distance function (${\mathcal{L}}_D$) (loss function), which is defined as follows:

$${\mathcal{L}}_D=\mathcal{L}\left(D\left(\left\{x_i,y_i\right\}\right),1\right)+\mathcal{L}\left(D\left(\left\{x_i,y_i^{\prime }\right\}\right),0\right)+GP$$

(11)

$$P=\frac{\mathsf{\lambda }}{m}{\displaystyle \sum }_i^m{\left[{\nabla }_{x_i,y_i^{\prime }}D{\left(x_i,y_i^{\prime }\right)}^i\right]}^2$$

(12)

here, $D\left(\ast \right)$ represents the output of the D; $\mathcal{L}$ is the binary cross entropy, defined as:

$$\mathcal{L}=-\left[klog\left(s\right)+\left(1-k\right)log\left(1-s\right)\right]$$

(13)

Based on this loss function, the discriminator can achieve an output of 1 when the input is $\left\{x_i,y_i\right\}$ and an output of 0 when the input is $\left\{x_i,y_i^{\prime }\right\}$, and then discriminates the wind speed sequence $\left\{y_1,y_2,\dots y_n\right\}$.

The discriminator outputs a scalar of “0” or “1.” The scalar of “0” or “1” has two purposes: (1) It can influence and then adjust the weights of the neural network in the discriminator and maximize Equation (14) through a backpropagation algorithm. (2) It can be passed to the generator to assist the PSO algorithm to find the optimal hyper-parameters of the OBLS and then calculate the value of the fitness function $F_c$, which is defined as follows:

$$F_c=\frac{1}{n}{\displaystyle \sum }_{i=1}^{n}D\left(G\left(x\left(n\right)\right)\right)$$

(14)

where $G\left(\ast \right)$ represents the output of the generator.

2.5. Prediction Steps of the Proposed HGANN Model

We propose the HGANN model for ultra-short-term wind speed prediction. The flow chart of the prediction process of the proposed model is shown in Figure 3. CEEMDAN is used to decompose the raw wind speed data $\left\{x_1,x_2,\dots x_n\right\}$ into multiple modes $\overline{IM{F}_j}\left(n\right)$. These $\overline{IM{F}_j}\left(n\right)$ are separately sent into the generator (OBLS) of HGANN to obtain virtual samples $\left\{y_1^{\prime },y_2^{\prime },\dots y_n^{\prime }\right\}$. The discriminator (WGAN) takes $\left\{x_i,y_i^{\prime }\right\}$ or $\left\{x_i,y_i\right\}$ as input and then outputs scalars “1” or “0.” The scalars “1” or “0” are passed to the generator (OBLS) and the discriminator (WGAN) to participate in iterative model updates. Through the continuous iterative update, OBLS obtains the optimal value of $\left\{NF,NW,NE\right\}$. The final wind speed forecasting values $\left\{y_1,y_2,\dots y_n\right\}$ can be obtained by stacking all forecast values.

3. Case Analysis

3.1. Data Description

To demonstrate the applicability of the proposed model in different locations, we used datasets from the 50Hertz wind farm in Germany and the Mahuangshan wind farm in China [26]: HER and MHS, respectively. The HER datasets are freely available at http://www.netztransparenz.de/ (accessed on 5 October 2021). Both data sets are recorded for one year and wind speeds are measured in 15 min intervals. We selected wind speed series from both HER and MHS datasets for March, June, September, and December, representing spring, summer, autumn, and winter, respectively. Experiments using the four wind speed series of spring, summer, autumn, and winter can verify the applicability of our model at different periods. Each series contains 2880 samples.

Table 1. Seasonal statistics of the wind speed data.

Season	Mean (m/s)	Median (m/s)	Max (m/s)	Min (m/s)	Standard Deviation (m/s)
HER data
Spring	1.89	1.53	7.25	0.03	1.47
Summer	0.82	0.63	4.27	0	0.75
Autumn	1.23	0.83	6.65	0	1.19
Winter	1.90	1.60	5.68	0.07	1.35
MHS data
Spring	5.78	5.70	16.50	0.40	2.25
Summer	5.46	5.45	12.50	0.40	2.21
Autumn	5.18	5.20	16.50	0.40	2.33
Winter	4.50	4.27	14.07	0	2.28

shows information on the selected wind speed data for spring, summer, autumn, and winter.

In our experiments, the first 80% of the wind speed sequence was used as the training set, and the rest was used as the test set for ultra-short-term wind prediction. Table 1 displays the information of the four datasets. The experiments were implemented in MATLAB R2021b on a 64-bit personal computer with Intel(R) core i5-9300 CPU/16.00 GB RAM.

3.2. Evaluation Index

To comprehensively evaluate the prediction performance of HGANN, four evaluate indicators were given. MAE can accurately reflect the average value of the absolute error. MAPE divides the absolute error by the corresponding actual value. RMSE represents the sample standard deviation between the predicted value and the actual observation value, which has a very sensitive reflection and can reflect the accuracy of the prediction well. SSE represents the total error of the model. Their definitions are as follows:

$$MAE=\frac{1}{N}{\displaystyle \sum }_{i=1}^N\left|y_i-{\widehat{y}}_i\right|$$

(15)

$$MAPE=\frac{1}{N}{\displaystyle \sum }_{i=1}^N\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|$$

(16)

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(17)

$$SSE={\displaystyle \sum }_{j=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2$$

(18)

where ${\widehat{y}}_i$ is the predicted value and $y_i$ is the actual value.

3.3. Comparable Methods

To verify the prediction performance of the proposed HGANN, it was compared with 10 advanced predictive models, involving PSO-ANFIS [34], VMD-GA-BP [35], EEMD-GPR-LSTM [36], EMD-ISSA-LSTM [37], MWS-CE-ENN [20], CNN [38], WGAN [39], BLS [31], OBLS, and WGAN-OBLS.

Table 2. Parameter settings of the models.

Model	Parameter Setting
PSO-ANFIS	$ite{r}_{max}$$=300, n_p$$=40, c_1$$=1.0, c_2$$=2.0, n_{r }$$=4, n_v$= 52
VMD-GA-BP	$k$$=11, ite{r}_{max}$$=150, e_p$$=100, n_p$$=40, l_r$$=0.1, n_{b1}$= 9
EEMD-GPR-LSTM	$k$$=11, e_p$$=200, n_{b1 }$$=100, n_{b2}$$=100, s_1$$=50, \sigma$= 20
MWS-CE-ENN	$e_p$$=1000, l_r$$=0.1, p_r$$=0.000001, n_p$$=40, n_i$$=5, n_{b1 }$$=6, n_o$$=1, ite{r}_{max}$$=100, n_{std}$= 0.2
EMD-ISSA-LSTM	$e_p$$=100, l_r$$=0.005, n_p$$=10, ite{r}_{max}$$=20, v_e$$=0.6, p_d$$=0.7, p_e$= 0.2
Proposed Model	$n_{std}$$=0.01,n_p$$=40, ite{r}_{max}$$=100, c_1$$=1.5, c_2$$=1.5, e_p$$=50, l_r$$=0.002, \mathsf{\lambda }$$={10}^{-30}, n_{b1}$$=48, n_{b2}$$=96, n_{b3}$= 384

lists the parameter settings of six comparison methods. BLS, WGAN, PSO-BLS, and PSO-WGAN-OBLS are the same as those of HAGNN to perform the ablation experiment for HAGNN.

In Table 2, $ite{r}_{max}$ is the iterative number; $e_p$ is the number of network iterations; $n_p$ is population size; $c_1$ and $c_2$ are personal and global learning coefficients, respectively; $n_r$, $n_v$, $n_i$, and $n_o$ are the number of rules, variables, input nodes, and output nodes, respectively; $k$ is the decomposition number of VMD/EEMD; $n_{bi}$ is the number of the ith hidden nodes;$l_r$ is the learning rate of the network; $p_r$ is the training requirement accuracy; $n_{std}$ is the noise standard deviation in ICEEMDAN/CEEMDAN; $v_e$ is the early warning value; $p_d$ and $p_s$ are the proportion of discoverers and sparrows aware of danger, respectively; and $\mathsf{\lambda }$ is the regularization parameter for ridge regression.

3.4. Experimental Results

(1)

Experiment I: Comparison Between Different Forecasting Methods

We experimentally verified the effectiveness and advancement of the proposed HGANN by comparing it with PSO-ANFIS, VMD-GA-BP, EEMD-GPR-LSTM, MWS-CE-ENN, and EMD-ISSA-LSTM. Considering that wind data characteristics show strong seasonality, experiments were conducted using wind series from multiple seasons to further validate the predictive performance of the model. We chose the HER and MHS datasets for March, June, September, and December for this experiment. The training and testing processes of each of the compared models were repeated 10 times. The experimental results of the different datasets are presented in

Table 3. Forecast results of different models for the HER data.

Season	Metrics	Proposed Model	PSO-ANFIS	VMD-GA-BP	EEMD-GPR-LSTM	MWS-CE-ENN	EMD-ISSA-LSTM
Spring	RMSE	0.0065	0.0092	0.0128	0.0105	0.0093	0.0131
	SSE	0.0180	0.0365	0.0817	0.0471	0.0370	0.0737
	MAPE	0.0300	0.0402	0.0732	0.0493	0.0632	0.0536
	MAE	0.0048	0.0068	0.0108	0.0083	0.0071	0.0096
Summer	RMSE	0.0112	0.0172	0.0134	0.0126	0.0152	0.0117
	SSE	0.0537	0.1279	0.0768	0.0679	0.0989	0.0586
	MAPE	0.0420	0.0641	0.0632	0.0596	0.0730	0.0500
	MAE	0.0081	0.0123	0.0099	0.0114	0.0119	0.0093
Autumn	RMSE	0.0088	0.0120	0.0137	0.0126	0.0132	0.0204
	SSE	0.0331	0.0615	0.0806	0.0676	0.0746	0.1779
	MAPE	0.0422	0.0514	0.0524	0.1177	0.0619	0.0851
	MAE	0.0059	0.0077	0.0092	0.0089	0.0082	0.0129
Winter	RMSE	0.0063	0.0090	0.0067	0.0093	0.0104	0.0086
	SSE	0.0172	0.0354	0.0190	0.0370	0.0463	0.0317
	MAPE	0.0575	0.0880	0.0683	0.0722	0.0923	0.0620
	MAE	0.0043	0.0064	0.0047	0.0069	0.0073	0.0056

and

Table 4. Forecast results of different models for the MHS data.

Season	Metrics	Proposed Model	PSO-ANFIS	VMD-GA-BP	EEMD-GPR-LSTM	MWS-CE-ENN	EMD-ISSA-LSTM
Spring	RMSE	0.0237	0.0344	0.0264	0.0276	0.0339	0.0291
	SSE	0.2404	0.5054	0.2977	0.3250	0.4918	0.3624
	MAPE	0.0724	0.0830	0.1007	0.1055	0.0990	0.0866
	MAE	0.0146	0.0244	0.0187	0.0198	0.0229	0.0203
Summer	RMSE	0.0156	0.0267	0.0179	0.0205	0.0189	0.0236
	SSE	0.0104	0.3045	0.1375	0.1799	0.1529	0.2384
	MAPE	0.0433	0.0857	0.0588	0.0697	0.0680	0.0765
	MAE	0.0110	0.0200	0.0136	0.0171	0.0149	0.0192
Autumn	RMSE	0.0166	0.0245	0.0190	0.0183	0.0214	0.0262
	SSE	0.1179	0.2570	0.1543	0.1433	0.1960	0.2937
	MAPE	0.0410	0.0641	0.0505	0.0439	0.0762	0.0658
	MAE	0.0131	0.0185	0.0149	0.0137	0.0170	0.0211
Winter	RMSE	0.0266	0.0394	0.0309	0.0325	0.0286	0.0312
	SSE	0.3034	0.6645	0.4075	0.4521	0.3501	0.4166
	MAPE	0.0828	0.1698	0.1540	0.1215	0.0937	0.1200
	MAE	0.0190	0.0284	0.0228	0.0276	0.0193	0.0257

, where the first-best predictions are highlighted. Figure 4 depicts the wind speed prediction results of the proposed model for the HER dataset.

Interestingly, it can be seen from Table 3 and Table 4 that the proposed HGANN had the best prediction performance for each of the RMSE, SSE, MAPE, and MAE indicators on four datasets among all models. Wind speed forecasts at different times or at different locations may yield different results. Notably, our model achieved promising predictive results on both the geographically distinct German dataset HER and the Chinese dataset MHS. Furthermore, our model showed competitive prediction performance for wind series in different seasons. This indicates that our model can be extended to more general environments for wind speed prediction.

The abscissa and ordinate in Figure 4 represent the actual wind speed and the predicted wind speed, respectively; the blue line indicates that the predicted value is equal to the actual value. The ordinate of the green point is the predicted value, so the fit of the green point to the straight line reflects the accuracy of the prediction. As can be seen from Figure 4, the green points are very close to the blue line, which indicates that our model can predict wind speed effectively.

(2)

Experiment II: Multi-Step Prediction Experiment

Multi-step forecasting can be built based on single-step forecasting. Compared to single-step forecasting, multi-step forecasting is more practical for power systems. Therefore, in wind speed prediction, multi-step prediction is of high practical value. The experiment aimed to demonstrate the predictive performance of the HGANN model in multi-step forecasting. We selected 2880 samples from 23 August to 22 September from the HER dataset for the one-step, two-step, and three-step experiments. Performance metrics involved RMSE, SSE, MAPE, and MAE. Benchmark models covered PSO-ANFIS, VMD-GA-BP, EEMD-GPR-LSTM, MWS-CE-ENN, and EMD-ISSA-LSTM. The training and testing processes of each model were repeated 10 times. The experimental results of the proposed model and the benchmark models are shown in

Table 5. Multi-step prediction results for 15 min wind speed.

Model	RMSE			SSE			MAPE			MAE
	1-Step	2-Step	3-Step	1-Step	2-Step	3-Step	1-Step	2-Step	3-Step	1-Step	2-Step	3-Step
Proposed	0.0091	0.0136	0.0178	0.0356	0.0679	0.1361	0.0463	0.0712	0.1087	0.0061	0.0098	0.0134
PSO-ANFIS	0.0120	0.0174	0.0238	0.0615	0.1299	0.2428	0.0514	0.0885	0.1270	0.0077	0.0119	0.0161
VMD-GA-BP	0.0137	0.0151	0.0203	0.0806	0.0976	0.1764	0.0524	0.0721	0.1123	0.0092	0.0101	0.0213
EEMD-GPR-LSTM	0.0126	0.0164	0.0192	0.0676	0.1151	0.1578	0.1177	0.1353	0.1536	0.0089	0.0129	0.0157
MWS-CE-ENN	0.0132	0.0176	0.0213	0.0746	0.1326	0.1942	0.0619	0.0826	0.1121	0.0082	0.0172	0.0224
EMD-ISSA-LSTM	0.0109	0.0139	0.0183	0.0508	0.0827	0.1433	0.0503	0.0919	0.1325	0.0086	0.0134	0.0192

.

From Table 5, it can be seen that the one-step, two-step, and three-step prediction results of the proposed HGANN model provided lower RMSE, SSE, MAPE, and MAE values than those of the benchmark models. For instance, the proposed HGANN provided 0.0091 (one-step), 0.0136 (two-step), and 0.0178 (three-step) on RMSE, compared with EMD-ISSA-LSTM, which provided the predictive results of 0.0109 (one-step), 0.0139 (two-step), and 0.0183 (three-step). Furthermore, we also provide clear visual results of multi-step predictions for the six models in Figure 5. The results from Table 5 and Figure 5 indicate that the proposed HGANN model had the best robustness and the highest wind speed prediction accuracy among all compared models.

(3)

Experiment III: Ablation Experiment Between Single Models and Hybrid Models.

To verify the rationality of the proposed HGANN model, it was compared with WGAN-OBLS, OBLS, WGAN, BLS, and CNN on the HER dataset. The generator of HGANN is OBLS and its discriminator is the discriminator of WGAN. To emphasize the effectiveness of OBLS, it was compared with the generator of WGAN, namely, CNN. Similarly, all compared models were repeatedly trained and tested 10 times. In the HER dataset, 2880 data from 23 August to 22 September were selected for this experiment. The experimental results are shown in

Table 6. Forecasting performances of the proposed model and reference models.

Indicators	Proposed Model	WGAN-OBLS	OBLS	WGAN	CNN	BLS
RMSE	0.0088	0.0119	0.0122	0.0138	0.0221	0.0308
SSE	0.0331	0.0603	0.0642	0.0819	0.2090	0.4058
MAPE	0.0422	0.0506	0.0546	0.0571	0.0601	0.0648
MAE	0.0059	0.0076	0.0080	0.0086	0.0161	0.0164

, where the first-best predictions are highlighted with dark gray backgrounds. The forecast results for 22 September 2019 are plotted in Figure 6, which also shows the forecast errors in superimposed shades.

Figure 6 shows that among all the compared models, our proposed model had the best curve fitting and the smallest predicted error. The suggested model consistently outperformed WGAN-OBLS, PSO-BLS, WGAN, OBLS, CNN, and BLS, as shown in Table 6. This further demonstrates the advantages of our proposed model, as it combines CEEMDAN, OBLS, and WGAN.

Furthermore, first, compared with WGAN-OBLS without CEEMDAN, the proposed model had better predictive performance due to its covering CEEMADN and WGAN-OBLS, thus showing the effectiveness of CEEMDAN. Second, compared with WGAN or OBLS, WGAN-OBLS provided better predictive performance due to combing OBLS and WGAN, thus showing the effectiveness of OBLS and WGAN in WGAN-OBLS. Third, compared with BLS, OBLS provided better predictive performance due to using the improved PSO, thus showing the effectiveness of PSO in OBLS. Fourth, OBLS had better predictive results compared to the generator of WGAN, namely, CNN. This demonstrates the advantage of OBLS over CNN as a generator. This may be due to the flexible structure and better error convergence of OBLS.

4. Discussion

Our model was compared with five advanced models to evaluate its performance and advantages in various wind sequence experiments. Experimental results show that the proposed model had better predictive performance. The reasons behind this fact are given as follows.

First, the wind speed data were one-year data from wind farms in Germany and China, which cover complex fluctuation characteristics. Therefore, our HGANN model uses CEEMDAN to smoothen the volatility of the data and improve the predictive performance.

Second, HGANN uses OBLS as the generator to provide a special shallow broad incremental learning network structure, which can not only be beneficial for improving prediction accuracy for one-dimensional wind speed prediction compared to CNN but also greatly decrease computational cost using pseudo-inverse operations to determine the network weights instead of using convolution operations.

Third, in our HGANN model, the proposed OBLS uses an improved PSO to optimize the hyper-parameters of its network, which can search in a wider range and obtain the optimal parameters over BLS. Therefore, OBLS has better generalization ability than BLS.

Finally, HGANN can better extract the deeper features of wind speed data by playing a minimum–maximum game between the generator and discriminator for wind speed prediction.

5. Conclusions

Although existing various hybrid predictive models have provided competitive performance in ultra-short-term wind speed prediction, they still need to be further improved—for instance, how to effectively reduce the computational cost of hybrid predictive models, and how to effectively deal with the multicollinearity problem of the hybrid forecasting model based on weighted strategy, which leads to the problem of reduced forecasting accuracy. To enhance the predictive power and decrease the computational cost, this paper proposes the HGANN model for ultra-short-term wind speed forecasting. HGANN is a generative adversarial network in which the generator and discriminator play against each other to obtain wind speed predictions with high accuracy. In HGANN, we developed OBLS and the convolutional structures as the generator and the discriminator, respectively, which enables them to obtain effective synergies to improve predictive performance. Particularly, OBLS involves a special shallow broad incremental learning network structure, which can effectively deal with one-dimensional wind speed data. Furthermore, the shallow network structure of OBLS can also significantly decrease computational cost via using pseudo-inverse operations rather than convolution operations. In addition, the proposed OBLS applies an improved PSO to obtain the optimal network hyper-parameters. CEEMDAN performs noise reduction and decomposition of the wind data. Via the above rational combination, the proposed HGANN provides high predictive accuracy and generalization ability with low computational cost in ultra-short-term wind speed prediction. The experimental results indicate the above fact. For instance, the RMSE predictive errors of the proposed model were 29.35%, 49.22%, 38.09%, and 30.10% compared to the four state-of-art predictive models PSO-ANFIS, VMD-GA-BP, EEMD-GPR-LSTM, and MWS-CE-ENN on the spring wind data of the HER dataset, respectively. In the future, we plan to use parallel computing to speed up the process of PSO optimization of BLS during training. Furthermore, the proposed HGANN will be extended to a wider range of applications, such as financial time-series forecasting, electricity-load forecasting, traffic forecasting, etc.