Research on Multi-Step Prediction of Short-Term Wind Power Based on Combination Model and Error Correction

Abstract

The instability of wind power poses a great threat to the security of the power system, and accurate wind power prediction is beneficial to the large-scale entry of wind power into the grid. To improve the accuracy of wind power prediction, a short-term multi-step wind power prediction model with error correction is proposed, which includes complete ensemble empirical mode decomposition adaptive noise (CEEMDAN), sample entropy (SE), improved beetle antennae search (IBAS) and kernel extreme learning machine (KELM). First, CEEMDAN decomposes the original wind power sequences into a set of stationary sequence components. Then, a set of new sequence components is reconstructed according to the SE value of each sequence component to reduce the workload of subsequent prediction. The new sequence components are respectively sent to the IBAS-KELM model for prediction, and the wind power prediction value and error prediction value of each component are obtained, and the predicted values of each component are obtained by adding the two. Finally, the predicted values of each component are added to obtain the final predicted value. The prediction results of the actual wind farm data show that the model has outstanding advantages in high-precision wind power prediction, and the error evaluation indexes of the combined model constructed in this paper are at least 34.29% lower in MAE, 34.53% lower in RMSE, and 36.36% lower in MAPE compared with other models. prediction decreased by 30.43%, RMSE decreased by 29.67%, and MAPE decreased by 28.57%, and the error-corrected three-step prediction decreased by 55.60%, RMSE decreased by 50.00%, and MAPE decreased by 54.17% compared with the uncorrected three-step prediction, and the method significantly improved the prediction accuracy.

1. Introduction

Energy is the power source of economic development and the material basis for the survival and development of society. Since the industrial revolution, traditional fossil energy has been overused, the energy and environmental crisis has become increasingly serious, and the energy structure urgently needs to accelerate the transformation, in recent years, renewable energy power generation technology has had strong development momentum [1], and the power system is gradually transforming to a high proportion of new energy structure. Wind energy is one of the main renewable clean energy sources, wind power generation has become the main measure to promote the energy transition and combat climate change in most countries in the world [2]. However, using wind power is extremely challenging for current power systems. One reason is that the output power of wind farms has strong intermittency and fluctuation due to the characteristics of wind energy [3], and the large amount of wind power connected to the grid will lead to voltage fluctuations and frequency changes [4], which will significantly increase the instability of power system operation [5] and bring great safety risks to the power system. The other reason is that wind power cannot be dispatched [6] and the phenomenon of wind curtailment is serious, which causes huge energy waste and greatly hinders the further development of the wind power generation industry [7].

A stable wind power is beneficial to the accumulation and conversion of wind energy, which is the premise of the large-scale development of wind power [8]. Wind power prediction is the main means to reduce the instability of wind power [9]. After years of development, accurate wind power prediction has become the basis for wind power economic dispatch and participation in energy trading decisions in the electricity market [10], and plays an irreplaceable role in supporting the effective consumption of wind power and safe operation of the power grid. Therefore, it is crucial to improve the accuracy of wind power prediction.

In the early stage of wind power development, most wind power prediction systems around the world used relatively simple and easy-to-implement single prediction models, mainly including physical models, and traditional statistical models. The physical method has the advantage in long-term prediction, while the statistical method has an outstanding effect in short-term prediction [11]. The physical model uses weather prediction data and geographic data for prediction, and its prediction cycle is too long, the short-term wind speed prediction ability is poor, and the prediction error is large. The traditional statistical model mainly includes time series models [12], feedforward neural network models [13], back propagation neural network models [14], and support vector machine models [15]. the advantage of statistical prediction is that when the data is complete, the prediction error can reach the minimum value and the prediction accuracy is high. Through the simulation training adjustment, the input in the non-training set can also be given the appropriate output. Literature [16] constructs a sequence-to-sequence model based on LSTM. the proposed sequence-to-sequence recurrent models are trained only by using the data itself in a univariate sense. Thus, it minimizes the computation cost while increasing the forecasting power. The results show that this model is more efficient than artificial neural networks and supports vector machine models. However, with the rapid development of wind power, due to the inherent defects of poor generalization ability and unstable performance, a single prediction model has become increasingly difficult to meet the requirements of accurate prediction. For example, by comparing the prediction results of the AMRMA model shown in the literature [17], it can be seen that its prediction accuracy is lower than that of the model in this paper. Therefore, in order to overcome these limitations of a single model, the hybrid prediction model has attracted more and more attention in the field of wind power prediction. Literature [18] proposed that CEEMDAN was used to decompose the original data into different components, then each component of the combination method was determined by weight for prediction, and CLSFPA was used for optimization. The results proved that the combination model indeed improved the prediction accuracy.

The original wind power data is unstable and has high noise, so using the original data to establish the prediction model will produce large errors. In order to solve this problem and improve the prediction accuracy, the combination of data preprocessing combined with the prediction model of “decomposition-prediction-reconstruction” has become a hot research topic. In the literature [19], wavelet transform was used to decompose wind power data to effectively reduce the irregularity of data, and in the literature [20,21], wavelet analysis was applied to wind power data processing to establish a power prediction model based on wavelet hybrid neural network, which significantly improved the prediction accuracy. However, the selection of wavelet basis is difficult and lacks adaptivity, and the prediction results vary widely with different basis function selections. In contrast, empirical modal decomposition is based on the time scale of the data itself to decompose the signal without setting any basis functions, and the decomposition is more stable and has better generalization performance. Based on this, the literature [22] and [23] respectively used empirical mode decomposition and Complementary Ensemble Empirical Mode Decomposition (CEEMD) to preprocess wind power data. However, there are many sequential components after the empirical mode decomposition, which makes the subsequent prediction models more computationally intensive. To remedy this deficiency, the literature [24,25] and [26] reconstruct the decomposed sequences by calculating the sample entropy of the sequence components after the decomposition of the complete ensemble empirical mode decomposition adaptive noise and the variational mode decomposition algorithm, respectively, which greatly reduces the amount of operations and reduces the prediction time.

Although performing data preprocessing can significantly improve the prediction accuracy, it is still difficult to select the initial values of nonlinear parameters of the prediction model when feeding the data into the model for prediction [27]. In recent years, novel intelligent algorithms have emerged in machine learning parameter optimization. The literature [28] used the cuckoo search algorithm to optimize the least squares support vector machine, the literature [29] used the Jaya algorithm to optimize the support vector machine, the literature [30] used the Particle Swarm Optimization algorithm to optimize the extreme learning machine (ELM), the literature [31] applied the enhanced crow search algorithm to optimize the extreme learning machine, and The literature [32] used an improved dragonfly algorithm to optimize the parameters of the support vector machine, and the results all proved that the intelligent algorithm optimizer is effective in improving the prediction accuracy. Literature [33] shows that compared with PSO and other algorithms, the BAS algorithm has a simple single structure, which is conducive to reducing the computational complexity of the optimization process. Compared with existing prediction models, ELM, and KELM models have become the main methods for wind power prediction due to their better results. ELM model can effectively deal with nonlinear regression problems. Literature [34] proposes the ELM algorithm for wind power prediction, and test results show that ELM has better prediction performance compared with traditional models such as SVM and ANN. KELM is an improved ELM. Literature [35] points out that KELM has better robustness and stability than ELM.

In view of the advantages of the above methods, this paper proposes to use CEEMDAN and sample entropy to decompose and reconstruct the original wind power data. The reconstructed sequence components are fed into the KELM model optimized by improved beetle antennae search for prediction, and error correction is used to further improve the prediction accuracy. Finally, a CEEMDAN-SE-IBAS-KELM combined wind power prediction model with error correction is constructed. Compared with many existing models, the results of the proposed model show that it is effective in improving prediction accuracy.

The main contributions of this study are as follows:

(1)

Construct the CEEMDAN-SE model to preprocess data and provide data basis for subsequent models.

(2)

The BAS optimizer has been improved to propose the IBAS optimizer.

(3)

IBAS-KELM model is proposed to predict wind power.

(4)

Accurate wind power prediction is conducive to improving the stability of wind energy, which is of great significance for large-scale wind power grid connection and effective power dispatching.

2. Data Pre-Processing Methods

This section introduces the CEEMDAN model and SE model, which are proposed to reduce the volatility and intermittency of wind power data.

2.1. Complete Ensemble Empirical Mode Decomposition Adaptive Noise

The CEEMDAN algorithm decomposes the data into several sequence components according to the characteristics of the data and does not need to manually set the number of decompositions. It not only eliminates the modal aliasing problem by adding adaptive white noise to each decomposition stage but also makes the signal reconstruction complete and noiseless [36]. CEEMDAN adds adaptive white noise to the original wind power sequence v(t) as shown in Equation (1).

$$v_e(t)=v(t)+{\alpha }_k{n}_e(t)$$

(1)

where v_e(t) is the wind power sequence with white noise added for the eth time; n_e(t) is the Gaussian white noise obeying normal distribution added for the eth time; α_k is the kth signal-to-noise ratio.

Finally, the wind power signal after the addition of white noise is decomposed into a set of intrinsic mode function (IMF) and a residual component r(t) arranged by frequency and amplitude.

$$v(t)={\displaystyle \sum _{k=1}^{K}im{f}_{\theta }}(t)+r(t)$$

(2)

where K is the sum of the IMF components; imf_θ is the θth IMF component; and r(t) is the residual component.

2.2. Sample Entropy Principle

The entropy value is a characterization parameter used to measure the complexity of a time series. The larger the entropy value, the higher the complexity of the series. Two time series with similar entropy values have similar complexity. The SE algorithm is very resistant to interference and has a high measurement accuracy for both deterministic and random signals [37]. The specific steps of the SE algorithm are expressed as follows.

(1)

For a given original sequence X = {x(n), n = 1, 2,..., N}, containing N data points.

(2)

Given the embedding dimension m, compute the reconstruction vector xm i of the original sequence X.

(3)

For any value of i, calculate the Chebyshev distance dij between xm i and other vectors xm j. Find the number of dij that is below the similarity tolerance k and define it as B_i.

(4)

Calculate the mean value of Bm i(k).

(5)

Similarly, calculate B^m+1(k) when the embedding dimension m + 1.

(6)

When the number of original sequences is finite, then the SE value can be calculated by the following equation:

$$SE(m,\mathit{k},N)=-\ln \frac{B^{m+1}(k)}{B^m(k)}$$

(3)

There are three parameters to be set in the sample entropy algorithm: m, k and N. m are generally taken as 1 or 2; k is taken as (0.15~0.25) SD, where SD is the standard deviation of time series sample points, and N is the number of sample points.

3. Improved Beetle Antennae Search

In this section, the IBAS optimizer is proposed to solve the difficult problem of KELM parameter selection. Next, the optimization performance of the proposed optimizer is proved to be more efficient by comparing it with other algorithms.

3.1. Beetle Antennae Search

The beetle antennae search (BAS) algorithm is an intelligent optimization algorithm that mimics the foraging of a beetle [38]. When a beetle forages, it decides the next move by comparing the scent intensity of the two long antennae on the left and right, and if the scent intensity of the right antennae exceeds that of the left, the beetle flies to the right next, otherwise, it flies to the left. Since the BAS algorithm requires only one beetle, the amount of operations is very small, the running time is short [39], and the convergence is very fast. The specific steps of BAS implementation are as follows.

(1)

Determine the dimension of the search space as n, the center of mass of the beetle denoted as x, the left and right antennae of the beetle as x_l and x_r, respectively, and the distance between the two antennae as d, where x_l, x_r are both n-dimensional vectors.

(2)

Initialize the position of the beetle, and since the beetle head is oriented randomly, generate a random n-dimensional unit vector β to represent the beetle head orientation:

$$\beta =\frac{rands(n,1)}{\left|\right|rands(n,1)\left|\right|}$$

(4)

where rands(n,1) is the generated n-dimensional vector.

(3)

Determine the spatial location of the left and right antennae of the beetle.

$$\{\begin{array}{l}{\mathit{x}}_{li}={\mathit{x}}_i+d^i\beta \\ {\mathit{x}}_{ri}={\mathit{x}}_i-d^i\beta \end{array}$$

(5)

where i is the number of iterations; x_li and x_ri are the spatial positions of the left and right antennae of the beetle in the ith iteration, respectively; dⁱ is the distance between the two antennae of the beetle in the ith iteration.

(4)

Calculate the left and right antennae adaptation values as well as determine the step size factor.

$$\{\begin{array}{l}{f}_l=f({\mathit{x}}_l)\\ f_r=f({\mathit{x}}_r)\\ {\delta }^i=d^i/2\end{array}$$

(6)

where f_l and f_r are the adaptation values of the left and right antennae, respectively; δⁱ is the movement step of the beetle at the i-th iteration.

(5)

Beetle update position.

$${\mathit{x}}^i={\mathit{x}}^{i-1}+{\delta }^i\beta sign(f_l-f_r)$$

(7)

where sign() is the sign function.

(6)

Update the distance between the two antennae.

$$d^i=0.95d^{i-1}$$

(8)

(7)

Determine whether the end condition is satisfied, and end if it is satisfied, and repeat steps (2) to (6) if it is not until the end condition is met.

3.2. Improvement Strategies

(1)

Dynamic inertia weights

BAS has a good global search ability, but the local search ability needs to be improved. Dynamic inertia weight is a mechanism that can effectively coordinate the search ability, which can dynamically balance the global search ability and local search ability of the optimization algorithm [40], which enables the beetle to successfully find the optimal solution in the search process. The weight w is taken as follows:

$$w=w_{\max }-{(i/i_{\max })}^3\times (w_{\max }-w_{\min })$$

(9)

where, w_max = 0.9; w_min = 0.4; and i_max is the maximum number of iterations.

(2)

Lévy flight trajectory optimization strategy

In order to improve the global convergence ability of the BAS algorithm and jump out of local convergence thus improving the accuracy of the algorithm, the Lévy flight trajectory mechanism is introduced in the later stages of the algorithm for local development, which helps to jump out of the local optimal solution [41] and better balance the search capability with the development capability [42]. The motion of the Lévy flight L is defined as follows:

$$L=\frac{\mu \cdot \mathsf{\Phi }}{{\left|v\right|}^{1/\lambda }}$$

(10)

where μ and v are standard normal distributions; λ = 1.5; and Φ is a normal distribution obeying N(0,σ2 Φ).

The dynamic inertia weight w and Lévy flight L are introduced into the position update formula of the beetle to optimize the position update mode of the beetle, so that the beetle adjusts according to the improved movement strategy when self-learning, and the position update mode of the improved beetle antennae search algorithm is expressed as follows.

$${\mathit{x}}^i=w\cdot {\mathit{x}}^{i-1}+L\cdot {\delta }^i\beta sign(f_l-f_r)$$

(11)

3.3. Improved Algorithm Testing

In order to analyze the performance of the IBAS algorithm, six standard test functions are selected in this summary to test and analyze its performance in finding the best performance. Li et al. [43] used the unimodal standard function to test the convergence ability of the algorithm and the ability to test the algorithm to jump out of the local optimal using the multimodal function. The IBAS algorithm was tested together with the BAS algorithm, the flower pollination algorithm (FPA) and the particle swarm algorithm (PSO), and the final test results were compared to observe the performance of the IBAS algorithm. f₁–f₃ of the selected test functions are unimodal functions and f₄–f₆ are multimodal functions. The specific information of the six standard test functions selected is shown in

Table 1. Standard test functions.

Function	Range	Optima	Attribute
$f_1={\displaystyle \sum _{i=1}^n{\mathit{x}}_i^2}$	[−100,100]	0	Unimodal
$f_2={\displaystyle \sum _{i=1}^{n-1}\left[100{\left({\mathit{x}}_{i+1}-{\mathit{x}}_i^2\right)}^2+{\left({\mathit{x}}_i-1\right)}^2\right]}$	[−30,30]	0	Unimodal
$f_3={\displaystyle \sum _{i=1}^n{([{\mathit{x}}_i+0.5])}^2}$	[−100,100]	0	Unimodal
$f_4={\displaystyle \sum _{i=1}^n\left[{\mathit{x}}_i^2-10\cos (2\pi {\mathit{x}}_i)+10\right]}$	[−5.12,5.12]	0	Multimodal
$\begin{array}{l}{f}_5=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\mathit{x}}_i^2}}\right)\\ -\exp \left(\frac{1}{n}{\displaystyle \sum _{i=1}^n\cos 2\pi \mathit{x}}\right)+20+e\end{array}$	[−32,32]	0	Multimodal
$f_6=\frac{1}{4000}{\displaystyle \sum _{i=1}^n{\mathit{x}}_i^2}-{\displaystyle \prod _{i=1}^n\cos \left(\frac{1}{\sqrt{i}}\right)+1}$	[−8,8]	0	Multimodal

.

To ensure the fairness of the test, the number of iterations of the four algorithms, IBAS, BAS, FPA, and PSO, is uniformly set to 1000 and the dimension is set to 30, and the other parameters of the algorithms are set, as shown in

Table 2. Algorithm parameter setting.

Algorithm	Parameters Setting
IBAS&BAS	N = 1, d = 1, ωmax = 0.9, ωmin = 0.4, i = 1000, eta = 0.95, n = 30
FPA	N = 30,p = 0.8, d = 30
PSO	N = 30, c1 = c2 = 1.49445, ω = 0.6, Vmax = 1, Vmin = −1, popmax = 5.12, popmin = −5.12

.

In Table 2, N is the population size, and FPA and PSO are set to 30 except for IBAS and BAS, which require only one beetle; d is the initial spacing between the left and right antennae of the beetle; p is the flower transition probability, and c1 and c2 are the learning factors. For performance testing, in order to reduce the effect of algorithm randomness on the results, each test function was run 30 times with IBAS, BAS, FPA, and PSO, and the minimum and average values of the simulation results of each algorithm for each test function were found. All algorithms are tested in Windows 11 operating system and Matlab R2022a operating environment. The results of the algorithm tests are shown in

Table 3. Algorithm test results.

ID	Statistical Indicators	IBAS	BAS	FPA	PSO
f1	AVG	7.38 × 10−35	3.03 × 10−24	4.50 × 10−9	1.32 × 10−1
	MIN	3.32 × 10−36	7.88 × 10−25	2.44 × 10−11	6.06 × 10−2
f2	AVG	1.22 × 10−1	3.10 × 101	3.27 × 101	4.77 × 101
	MIN	9.80 × 10−2	2.90 × 101	3.01 × 101	4.56 × 101
f3	AVG	2.46 × 10−8	2.37 × 10−7	1.76 × 10−1	2.31 × 10−2
	MIN	5.16 × 10−9	1.12 × 10−7	3.22 × 10−2	1.43 × 10−2
f4	AVG	0	0	4.40 × 10−4	7.44 × 101
	MIN	0	0	3.60 × 10−4	1.19 × 100
f5	AVG	8.88 × 10−16	4.64 × 10−13	3.24 × 10−3	2.42 × 100
	MIN	7.84 × 10−16	9.86 × 10−14	2.37 × 10−3	1.56 × 100
f6	AVG	0	0	5.87 × 10−6	7.95 × 10−2
	MIN	0	0	4.23 × 10−6	4.09 × 10−2

.

For the unimodal test functions f₁–f₃, except for the test function f₂ in which the difference between BAS and PSO and FPA algorithms converge to the minimum value is not large, the IBAS and BAS algorithms in the other two test functions occupy a significant advantage in the minimum value and the average value, which indicates that IBAS and BAS algorithms have better search capability for unimodal functions. The IBAS algorithm performs better than the BAS algorithm in these three tests, which indicates that the improved algorithm has better search and optimization performance. For the multimodal test functions, f₄–f₆, the minimum and average values of IBAS and BAS algorithms are 0 in f₄ and f₆ test results, which indicates that the BAS algorithm has not only stronger global search capability but also better stability of the search optimization compared with PSO and FPA algorithms. For function f₅, although the test results of the IBAS algorithm did not converge to 0, the search accuracy was much better than that of PSO and FPA algorithms, and it was also significantly better than that of the BAS algorithm, indicating that IBAS can better jump out of the local optimum of the multimodal test function and has better overall search ability than the other three algorithms.

4. Model Combination

In this section, KELM, IBAS optimizing KELM principle, multi-step prediction method, and error correction are introduced. Next, the implementation steps of the proposed combination model are described.

4.1. KELM Parameter Optimization

4.1.1. Extreme Learning Machine

ELM is a fast and efficient single-layer feedforward neural network [44]. ELM can generate random weights and thresholds before training, and only the number of hidden layers and the activation function need to be set to find the optimal solution. The mathematical model of ELM is as follows.

$$H\alpha =Q$$

(12)

where H is the output matrix of the hidden layer, α is the weight matrix, and Q is the target expectation output matrix.

The training objective of ELM is to find the best combination of weights W = (w, b, β) such that the following equation holds.

$$\min E(W)=\min \left|\right|H\alpha -Q\left|\right|$$

(13)

where w is the input weight of the hidden layer; b is the bias of the hidden layer.

In the ELM algorithm, the input w and b, and the output matrix H are uniquely determined. At this point, the optimal solution of ELM is as follows:

$$\stackrel{\wedge }{\alpha }=H^{+}Q$$

(14)

where H⁺ is the generalized inverse of the Moore-Penrose of the output matrix H.

4.1.2. Kernel Extreme Learning Machine

KELM is an improved algorithm based on ELM combined with kernel functions, which makes KELM improve the generalization performance and stability performance of the model and reduce the computational cost while retaining the advantages of ELM [45]. The KELM kernel matrix Ω_ELM is as follows.

$$\{\begin{array}{l}{\Omega }_{\mathrm{ELM}}=H{H}^T\\ {\Omega }_{\mathrm{ELM}}\left(i,j\right)=h({\mathit{x}}_i)h(h_j)=K({\mathit{x}}_i,{\mathit{x}}_j)\end{array}$$

(15)

where, h(x) is the output function of the hidden layer; K(x_i,x_j) is the kernel function, and the RBF kernel function with good generalization ability is chosen in this paper.

$$K({\mathit{x}}_i,{\mathit{x}}_j)=\exp (-\gamma {\Vert {\mathit{x}}_i,{\mathit{x}}_j\Vert }^2)$$

(16)

where γ is a kernel parameter greater than 0.

Then the output weights and output results of KELM are shown as follows:

$$\{\begin{array}{l}\alpha ={\mathit{H}}^T{\left(\frac{\mathit{I}}{C}+\mathit{H}{\mathit{H}}^T\right)}^{-1}\mathit{Q}\\ \hat{\mathit{Q}}={\left[\begin{array}{c}K\left(\mathit{x},{\mathit{x}}_1\right)\\ ⋮\\ K\left(\mathit{x},{\mathit{x}}_N\right)\end{array}\right]}^T{\left(\frac{\mathit{I}}{C}+\mathit{H}{\mathit{H}}^T\right)}^{-1}\mathit{Q}\end{array}$$

(17)

where I is the diagonal matrix; C is the penalty coefficient; N is the number of training samples.

4.1.3. IBAS Optimized KELM Principle

Since the kernel parameter g and penalty coefficient C of KELM are randomly generated, both the training accuracy and time are affected by the randomness. The IBAS algorithm is used to optimize the two parameters of KELM, and the optimized parameters are used for the training of the KELM model to obtain the final KELM prediction model. The flow chart of the specific steps of KELM optimization by the IBAS algorithm is shown in Figure 1.

4.2. Multi-Step Prediction Method

Multi-step wind power prediction is commonly used in multiple output prediction methods and multi-step iterative prediction methods. The multi-output prediction method outputs multiple predicted values at once, while the multi-step iterative prediction method predicts the next wind power data each time and iterates the predicted wind power as the known value of the power data after the prediction, which can be obtained by multiple predicted values. Since wind power short-term multi-step prediction uses iterative prediction methods with higher accuracy [46], this paper uses multi-step iterative prediction methods.

4.3. Multi-Step Prediction Error Correction Principle

The errors generated by the KELM model when predicting wind power includes not only include the errors caused by randomness, but also include the regular errors of the KELM prediction model due to its own model characteristics [47]. Since the errors include this part of deterministic and regular error, this part of errors can be considered to correct the predicted wind power data to improve the prediction accuracy. The error sequences between the trained predicted data and the trained real data after the training of the KELM model are retained, and the error sequences between the predicted wind power data and the real wind power are calculated, and the two sequences are formed into a new error sequence. The errors are predicted using the IBAS-KELM prediction model, and the error prediction values are superimposed with the predicted wind power values to obtain the final wind power data.

4.4. Overall Prediction Modeling

The CEEMDAN-SE-IBAS-KELM combined wind power prediction model combined with error correction constructed in this paper is implemented in the following steps.

(1)

Decompose the original wind power sequences into several components using the CEEMDAN algorithm.

(2)

Calculate the entropy values of each component, and reconstruct the sequences with similar entropy values into a new sequence.

(3)

The training data of each new sequence component after reconstruction are used as the input quantity of the prediction model respectively, and the parameters of the KELM model are optimized by the IBAS algorithm to obtain the optimized prediction model and training error sequences of each component.

(4)

The new sequence prediction data after reconstruction are fed into the respective trained IBAS-KELM prediction models, and the power prediction values and prediction error sequences of each new sequence component are obtained.

(5)

Combine the training error sequences of each new sequence component with the prediction error sequences to form an error sequence, and use the IBAS-KELM model to predict the errors to obtain the error prediction values of each sequence component.

(6)

Add the error prediction value and the power prediction value to get the prediction value of each sequence component, superimpose the prediction values of each component to get the final prediction value, and use the final prediction value and the true value to calculate the error evaluation index and analyze the results.

5. Example Simulation and Result Analysis

In this section, an example analysis will be introduced to prove the effectiveness of the combined model through comparison with other models. The example analysis is divided into four sections: experimental data and evaluation indicators, experimental data preprocessing, wind power prediction experiment, and prediction error analysis.

5.1. Experimental Data and Evaluation Index

The experimental wind power data studied in this paper were obtained from the Sotavento wind farm in Galicia, Spain [48]. The wind farm has a sampling interval of 10 min and a rated power of 17 MW. 2448 sample points are used to validate the actual wind power measurement data from March 1 to 17, 2021, with the first 2304 data as the training set and 144 sample points on the last day as the test set.

In order to evaluate the prediction model performance more objectively, three error evaluation indexes are selected in this paper, as follows.

(1)

Mean absolute error (MAE), which is defined as follows.

$$E_{MAE}=\frac{1}{M}{\displaystyle \sum _{t=1}^M|y_t-f_t|}$$

(18)

where f_t is the actual value of the tth sample in the test sample; y_t is the model predicted value of the tth sample in the test sample; M is the corresponding sample number.

(2)

Root mean square error (RMSE), defined as follows.

$$E_{RMSE}=\sqrt{\frac{1}{M}{\displaystyle \sum _{t=1}^M{(y_t-f_t)}^2}}$$

(19)

(3)

Mean absolute error percentage (MAPE), defined as follows.

$$E_{MAPE}=\frac{1}{M}{\displaystyle \sum _{t=1}^M\left|\frac{y_t-f_t}{f_t}\right|}\times 100\%$$

(20)

5.2. Experimental Data Pre-Processing

Considering the correlation of wind power data, in practical application, short-term wind power prediction mostly adopts iterative prediction, that is, the first several wind power values are used to predict the next wind power value. In this paper, the autocorrelation coefficient p of the original wind power sequences is calculated to determine the number of input variables. The results of the autocorrelation coefficient calculation for the wind power sequences are shown in

Table 4. Autocorrelation coefficient of power sequences.

Delay	1	2	3	4	5	6
p	0.995	0.986	0.974	0.960	0.945	0.931
Delay	7	8	9	10	11	…
p	0.919	0.909	0.900	0.893	0.886	…

. The autocorrelation function diagram is shown in Figure 2.

From Table 4, it can be seen that the autocorrelation coefficient p of the first 9 data is 0.9 and above, which has a strong correlation. Therefore, in this paper, the first 8 data are selected as the input quantity, the 9th data is the predicted value, and so on to iteratively predict the power data of all test sets.

First, the original wind power data sequences are preprocessed using the CEEMDAN algorithm and CEEMD algorithm, and the results show that both CEEMDAN algorithm and CEEMD algorithm decompose the data sequences into nine IMF components and one RES component, and since the difference between the two decomposition plots is not large, this paper only places each sequence component after CEEMDAN decomposition, as shown in Figure 3 below.

Second, the sequences of empirical modal decomposition algorithms decompose more sequence components, and both CEEMD and CEEMDAN in this paper decompose the original wind power sequence into 10 sequence components, which not only ignores the connectivity between the sequences, but also increases the workload of the subsequent prediction model. Calculating the entropy values of these 10 components and combining the components with similar entropy values into a new component can significantly reduce the workload and working time of subsequent prediction with almost no reduction in prediction accuracy. The results of calculating the entropy values of these 10 components are shown in Figure 4.

From Figure 4, we can see that the entropy values of the wind power sequences are basically arranged from large to small after CEEMD and CEEMDAN decomposition. The entropy values of component 1 and component 2 after CEEMDAN decomposition are much larger than the other components, indicating that their randomness is also much larger than the other components, so these two components are kept unchanged. The entropy values of component 3 to component 6 are not very different, so they are combined into a new sequence; the entropy values of components 7 to 10 are also very different, so they are also combined. After entropy calculation and reconstruction, the original sequence of 10 components is reconstructed into a new sequence of 4 components.

5.3. Wind Power Prediction Experiment

Since the accuracy of multi-step iterative prediction decreases with increasing step length [49,50], it is particularly important to obtain accurate first-step prediction values. To verify the prediction accuracy of the combined prediction model proposed in this paper, KELM, IBAS-KELM, FPA-KELM (KELM improved by the flower pollination algorithm), PSO-KELM (KELM improved by the particle swarm algorithm), CEEMDAN-SE-IBAS-KELM, and the CEEMDAN-SE-KELM combined with error correction proposed in this paper were constructed -IBAS-KELM combined with error correction. The combined prediction models proposed in this paper are also used to achieve wind power prediction 1 step ahead (10 min), 2 steps ahead (20 min), and 3 steps ahead (30 min).

The maximum number of iterations of the IBAS algorithm is set to 100, and the population is 1 while the maximum number of iterations of other optimization algorithms is set to 100 and the population is set to 30 and the parameters of both CEEMD and CEEMDAN are chosen to be 0.2 times standard deviation with 100 times white noise added.

The results of the CEEMDAN-SE-IBAS-KELM prediction model for the wind power of 144 sample points of Sotavento wind farms on March 17, 2021 with 1-step ahead, 2-step ahead and 3-step ahead are shown in Figure 5. The errors of 1-step, 2-step and 3-step ahead with the true values and the results of the error prediction values are shown in Figure 6. The results of the 1, 2, and 3-step prediction values of the CEEMDAN-SE-IBAS-KELM prediction model with error correction, are shown in Figure 7. The error evaluation index of each prediction model, as shown in

Table 5. Error evaluation indexes of each prediction model.

Model	MAE	RMSE	MAPE
ELM	0.415	0.506	0.041
KELM	0.345	0.441	0.036
PSO-KELM	0.289	0.369	0.030
FPA-KELM	0.232	0.296	0.024
BAS-KELM	0.217	0.277	0.023
IBAS-KELM	0.175	0.224	0.018
CEEMD-IBAS-KELM	0.105	0.139	0.011
CEEMDAN-IBAS-KELM	0.066	0.087	0.007
One-step prediction	0.069	0.091	0.007
Two-step prediction	0.133	0.169	0.014
Three-step prediction	0.232	0.288	0.024
Modified One-step prediction	0.048	0.064	0.005
Modified Two-step prediction	0.078	0.103	0.007
Modified Three-step prediction	0.103	0.144	0.011

.

As can be seen from Table 5, the KELM model with the introduction of the kernel function decreased MAE by 16.87%, RMSE by 12.85%, and MAPE by 12.19% compared with the ELM model, which improved the model accuracy and also made the model more stable, indicating that the introduction of the kernel function significantly improved the prediction performance of the model; by comparing the three error evaluation indexes of the IBAS-KELM, CEEMD-IBAS-KELM and CEEMDAN-IBAS-KELM models, it can be seen that the prediction model without decomposition has the largest error, and the error of the decomposed prediction model is significantly reduced. It can be concluded that the prediction accuracy of prediction models based on modal decomposition has been greatly improved, indicating that the signal decomposition technology can effectively reduce the volatility of the original data. Compared with the prediction model without modal decomposition, the prediction models based on modal Compared with BAS-KELM, FPA-KELM and PSO-KELM models, the MAE decreased by 19.35%, 24.57% and 39.45%, respectively; the RMSE decreased by 19.13%, 24.32% and 39.29%; MAPE decreased by 21.74%, 25.00%, 40.00%, respectively, showing not only the effectiveness of the algorithm improvement but also the more effective merit-seeking ability of the IBAS algorithm. In the CEEMDAN-IBAS-KELM model compared with the CEEMD-IBAS-KELM model, MAE decreased by 37.14%, RMSE decreased by 37.41% and MAPE decreased by 36.36%, indicating the better decomposition ability of CEEMDAN. The one-step prediction is the result of the CEEMDAN-SE-IBAS-KELM model prediction, and it can be seen that MAE, RMSE, and MAPE are almost unchanged after SE reconstruction, but make the subsequent prediction model operation much lower. The one-step prediction after error correction decreases by 30.43%, RMSE decreases by 29.67%, and MAPE decreases by 28.57% compared with the uncorrected MAE, which proves that the error compensation correction can significantly improve the prediction accuracy.

From Figure 6, it can be seen that the prediction accuracy decreases with the increase of the prediction step, and the trend of the error between the predicted and true values in one, two, and three steps is approximately the same, which indicates that the prediction model causes regular errors due to its own characteristics, and it is proved that the prediction error correction proposed in this paper is scientific. And the predicted error of IBAS-KELM is approximately the same as the real error trend, which proves the practicality of using the prediction error value to correct the power prediction value.

From Table 5, it can be seen that the two-step prediction is 0.064 larger than the one-step prediction in terms of MAE, 0.078 larger in terms of RMSE, and 0.007 larger in terms of MAPE; the three-step prediction is 0.099 larger than the two-step prediction in terms of MAE, 0.119 larger in terms of RMSE, and 0.011 larger in terms of MAPE. It is proved that the prediction accuracy decreases faster and faster as the prediction step increases, which is caused by the fact that the later prediction values are affected by the prediction errors in the previous steps, and the errors will keep accumulating as the prediction step increases. The MAE of the two-step prediction with error correction decreases by 41.35%, RMSE decreases by 39.05%, and MAPE decreases by 50.00% compared with the uncorrected two-step prediction; the MAE of the three-step prediction with error correction decreases by 55.60%, RMSE decreases 50.00%, and MAPE decreases 54.17% compared with the uncorrected three-step prediction. It not only further proves the effectiveness of error correction, but also shows that as the step length increases, the error correction produces more and more influence, and the correction effect increases with the step length, which proves the superiority of error correction.

5.4. Prediction Error Analysis

Due to the randomness and volatility of wind power as well as the constraints of technology development level, the prediction error of wind power is unavoidable. With the advancement of big data analysis technology, it becomes an opportunity and challenge for wind power prediction research to effectively mine the hidden uncertainty information and laws in model prediction errors. By probabilistically fitting the wind power prediction error distribution and quantifying the possible fluctuation range of the prediction error in the probabilistic form, it is beneficial to make the expected optimal decision in the decision problem considering wind power uncertainty, thus making up for the lack of uncertainty information in deterministic prediction and providing more comprehensive information for wind farms.

In order to more intuitively demonstrate the fluctuation range of different prediction model errors, this paper will analyze the relative error of model predictions as a percentage of rated power (i.e., the prediction error minimum value), the prediction error minimum value (hereafter referred to as error) is defined as follows.

$$P_E=\frac{P_{act}(t)-P_{pre}(t)}{P_{cap}}$$

(21)

where P_E is the standardized value of prediction error; P_act(t) is the real value of wind power; P_pre(t) is the predicted value of wind power; P_cap is the rated power of wind farm, i.e., 17 MW.

Three representative prediction models, KELM, IBAS-KELM, and CEEMDAN-SE-IBAS-KELM, were selected, and the frequency density histograms of the errors of these three prediction models were plotted, while the frequency density histograms were curve-fitted with normal distribution curves, and the mean and variance of the fitted curves were calculated, and in order to observe the distribution of the errors of these three models, the fitted curves of these three prediction model errors are put into one graph for comparative analysis. In this paper, the errors are divided into 12 intervals for frequency density calculation, where the horizontal coordinate is the error of the model and the vertical coordinate is the frequency density value, and the higher the frequency density value is, the more the number of errors in the interval. The frequency density histogram and fitted curve of each prediction model error are shown in Figure 8.

From Figure 8a,b, we can see that the mean and error of the normal distribution curve fitted by the IBAS-KELM model are reduced compared with the KELM model, where the mean error is reduced by 83.53% and the variance is reduced by 32.99%, and the KELM model error is distributed in the interval [−5%, 5%] and the maximum frequency density value is about 25, while the IBAS-KELM model error is distributed in the interval [−3%, 3%], and the maximum frequency density value near the zero value is about 42, indicating that the KELM model can significantly reduce the overall error after the optimization of IBAS algorithm. From (b) and (c), it can be seen that the variance of the fitted curve of the combined CEEMDAN-SE-IBAS-KELM prediction model is reduced by 61.36% compared with that of the IBAS-KELM model, and most of the errors are concentrated in the interval [−1%, 1%], and the maximum frequency density value near the zero value is about 95, which proves that the wind power data are processed by the CEEMDAN-SE algorithm and the overall prediction results are significantly reduced. SE algorithm, the overall error of the prediction results becomes smaller and more stable. From (d), it can be seen that compared with the KELM model, the mean value of the IBAS-KELM model is closer to zero, but the overall error distribution is still relatively scattered. The prediction results of the CEEMDAN-SE-IBAS-KELM combined model constructed in this paper are not only small in mean value, but the overall error is also concentrated around the zero value with almost no large deviation.

In order to show the prediction ability of different combination models more intuitively, the percentage of the number of prediction sample points in the error intervals of [−0.1%, 0.1%], [−0.3%, 0.3%], [−0.5%, 0.5%] and [−1%, 1%] of the total number of 144 prediction sample points for the combination models constructed in this paper and other comparison combination models are counted. The results of the error interval statistics for each prediction model are shown in Figure 9 below.

In Figure 9, the statistical results of the percentage of the number of predicted values between the CEEMDAN-SE-IBAS-KELM model and other comparative models in different error intervals are shown, and it can be seen that in all error intervals, the percentage of predicted values of the model constructed in this paper is more than that of other models. In the error interval [−0.1%, 0.1%], the proportion of the model constructed in this paper is close to 20%, which is 8.33% higher than that of the second place, indicating that the prediction accuracy of the model constructed in this paper is high, and the number of prediction values distributed in the low error interval is the largest. In the error interval [−0.3%, 0.3%], the model constructed in this paper is the only model with more than 45% of all prediction models. In the error interval [−0.5%, 0.5%], the percentages of different prediction models were 67.36%, 52.08%, 38.18%, 31.25%, 31.25%, 23.61%, 20.83%, and 11.81%, respectively, and the constructed model was 15.28%, 29.18%, 36.11%, 36.11%, 43.75%, 46.53%, and 55.55% higher than the other prediction models, respectively. In the interval [−1%, 1%], the number of predicted values of the BAS-KLL model and IBAS-KEL model accounted for more than 50%, and the prediction model using the data decomposition algorithm accounted for more than 70%, and the model constructed in this paper accounted for 93.75%, which was 13.89% higher than that of CEEMD-IBAS-KELM model. In summary, in the short-term wind power prediction, most of the prediction points of the model constructed in this paper are distributed in a low error interval, which shows that the CEEMDAN-SE-IBAS-KELM combination model constructed in this paper has excellent prediction ability.

In order to more intuitively demonstrate the prediction accuracy of the model after error correction, the one-step iterative prediction results and the one-step iterative prediction results with error correction are selected, and the frequency density histograms and fitted curves of the errors of these two models are plotted, as shown in Figure 10.

From Figure 10a,b, it can be seen that the mean difference between the single-step iterative error with error correction and the normal distribution curve fitted by the single-step iterative error without error correction is small but the variance is reduced by 35.29%, and the error of the model without error correction is basically distributed in the interval [−1%, 1%] and the maximum frequency density value near the zero value is about 95, while the error of the model with error correction The errors of the model with error correction are basically distributed in the interval [−0.5%, 0.5%], and the maximum frequency density value near the zero value is about 130, indicating that the combined CEEMDAN-SE-IBAS-KELM model with error correction has higher prediction accuracy. From Figure 10c, it can be seen that the overall distribution of the fitted curves of the combined error correction model is more concentrated, indicating that the error of the model becomes smaller after the error correction, and the overall aggregation is near the zero value.

In order to more intuitively represent the predictive ability of the over-the-top one-two-three step model with error correction and the over-the-top one-two-three step model without error correction, the number of predictions with different model error intervals in [−0.1%, 0.1%], [−0.3%, 0.3%], [−0.5%, 0.5%], and [−1%, 1%] as a percentage of the total number of 144 predictions are counted in this paper, and the specific results of the error interval evaluation for each prediction model are shown in

Table 6. Error interval evaluation of prediction results.

Model	[−0.1%,0.1%]	[−0.3%,0.3%]	[−0.5%,0.5%]	[−1%,1%]
One-step prediction	19.44%	45.14%	67.36%	93.75%
Two-step prediction	6.25%	25.69%	40.28%	68.75%
Three-step prediction	2.78%	13.19%	25.00%	43.75%
Modified One-step prediction	20.83%	61.81%	84.72%	98.61%
Modified Two-step prediction	16.67%	49.31%	65.28%	92.36%
Modified Three-step prediction	11.81%	34.03%	54.17%	84.72%

below.

It can be analyzed from Table 6 that compared with the combined CEEMDAN-SE-IBAS-KELM model without the error correction, the percentage of the number of predicted values in different error intervals for the combined CEEMDAN-SE-IBAS-KELM model with the error correction for the one, two, and three steps ahead is significantly improved. In the error interval [−0.1%, 0.1%], the percentage of the prediction results of the combined error-corrected model with one, two, and three steps ahead of the combined model increased by 1.39%, 10.42%, and 9.03%, respectively; in the error interval [−0.3%, 0.3%], the percentage of the prediction results of the combined model with one, two, three steps ahead of the combined model increased by 16%, respectively, compared with that of the uncombined model with one, two, three steps ahead of the combined model. In the error interval [−0.5%, 0.5%], the prediction results of the model with error correction increase by 17.36%, 25%, 29.17%, and 35.42%, respectively, compared with the prediction results of the model without the combination of one, two, and three steps ahead; in the error interval [−1%, 1%], the prediction results of the model with error correction increase by 17.36%, 25%, 29.17%, and 35.42%, respectively, compared with the prediction results of the model without the combination of one, two, and three steps ahead. In the error interval [−1%, 1%], the prediction results of the model with error correction increase by 4.86%, 23.61%, and 40.97%, respectively, compared with the prediction results of the model without the combination of the one, two, and three steps ahead. In summary, the combined CEEMDAN-SE-IBAS-KELM model with error correction has higher prediction accuracy and is more suitable for multi-step iterative wind power prediction.

6. Conclusions

Aiming at the problem that wind power is highly volatile and intermittent. In this paper, a multi-step short-term wind power prediction method of CEEMDAN-SE-IBAS-KELM combined with error correction is proposed, and after comparing the experimental results of several prediction models, it is concluded that the prediction effect of this model is better than that of PSO-KELM, FPA-KELM, BAS-KELM, and CEEMD-IBAS-KELM models. The main contributions of this research are as follows:

(1)

Using the combination of decomposition algorithm and entropy algorithm to preprocess the data not only can greatly improve the prediction accuracy, but also can significantly reduce the operation of the subsequent prediction model.

(2)

The ability of the algorithm to jump out of local convergence is enhanced according to the characteristics of the BAS algorithm itself, which not only preserves the advantage of the algorithm itself of extremely fast computing speed, but also improves the algorithm’s ability to find the best.

(3)

The model will produce regular errors due to its own characteristics, predict this part of the error to correct the predicted wind power data, and the results prove the scientific superiority of this method.

(4)

MAE, RMSE, and MAPE are used to analyze the prediction results of each model. Compared with other models, the three evaluation indexes of the combined CEEMDAN-SE-IBAS-KELM prediction model after error correction are the smallest, which shows that the prediction model proposed in this paper has high accuracy and a certain reference value.

(5)

Accurate prediction of wind power can improve the safety and stability of power system operation, promote the effective absorption of the power grid, make full use of wind energy resources, reduce wind abandonment, and promote the large-scale development of clean energy technologies. Accurate wind prediction has unlocked the potential of offshore wind power.

The proposed model has a high prediction accuracy for short-term wind power prediction, but this method also has some limitations. For example, the autocorrelation between wind power is used for prediction, but the impact of wind speed, wind direction, temperature, and other factors on the model prediction accuracy is not considered. Future research will focus on solving these limitations.