Multistep Wind Power Prediction Using Time-Varying Filtered Empirical Modal Decomposition and Improved Adaptive Sparrow Search Algorithm-Optimized Phase Space Reconstruction–Echo State Network

Abstract

Accurate wind power prediction is vital for improving grid stability. In order to improve the accuracy of wind power prediction, in this study, a hybrid prediction model combining time-varying filtered empirical modal decomposition (TVFEMD), improved adaptive sparrow search algorithm (IASSA)-optimized phase space reconstruction (PSR) and echo state network (ESN) methods was proposed. First, the wind power data were decomposed into a set of subsequences by using TVFEMD. Next, PSR was used to construct the corresponding phase space matrix for sequences, which were then divided into training sets, validation sets, and testing sets. Then, ESN was used for subsequence prediction. Finally, the predicted values of all the subseries were used to determine the final predicted power. To enhance the model performance, the sparrow search algorithm was improved in terms of the discoverer position update strategy, the follower position update strategy, and the population structure. IASSA was employed to synchronously optimize multiple parameters of PSR-ESN. The results revealed that the proposed model has higher applicability and prediction accuracy than existing models.

1. Introduction

Due to the exacerbation of the global energy crisis, renewable energy sources, such as wind, solar, and hydropower, have garnered increased attention. While wind energy represents an environmentally friendly source of energy, its unpredictable nature renders the integration of wind power into the power grid a cause for concern, frequently resulting in a decline in the grid’s power quality [1].

Therefore, accurate prediction of wind power can effectively improve the stability of the power system. With the development of new power grids, the prediction of wind pow-er forecasting plays an increasingly important role [2,3]. In recent years, power forecasting has become increasingly important in electric system planning studies. The progress in wind power research is of mutual scientific value to the development of other renewable energy sources, such as photovoltaic power generation, hydroelectric power, and nuclear energy. With the increasing penetration of smart grids, it is obvious that power forecasts play an essential role in the intelligence of the power grid [4,5,6].

Existing methods for improving wind power prediction mainly include physical models, statistical models, and machine learning (ML) techniques [7,8,9]. Physical models combine physical factors such as weather, temperature, and wind direction to estimate future wind power [10,11,12]. However, the development of such models is limited by data resources. Statistical models predict future wind power by mining information from his-torical data, mainly by using autoregressive, moving autoregressive, and multi-class au-toregressive moving average (ARMA) [13,14]. However, solving the nonlinearity problem in time series by using these models is difficult, thereby hindering high-accuracy predic-tion. Compared with physical models and statistical models, prediction models based on ML methods, such as extreme learning machine (ELM), backpropagation neural network (BPNN), recurrent neural network (RNN), convolutional Neural Networks (CNN), and long short-term memory (LSTM), can better analyze nonlinear time series and have thus been favored by numerous researchers [15,16,17,18,19]. Ding et al. [20] employed the numerical weather prediction wind speed, trend, and detail terms as the inputs of the weighted time series and used the two-way gated recursive unit neural network to correct the wind speed error of the weather forecast and used the modified data to predict the final wind speed. Due to the volatile nature of wind power, a single model cannot provide accurate predic-tions for wind power generation. Ruiz-Aguilar et al. [21] proposed a hybrid prediction model to decompose the clustering preprocessing method by using empirical modal de-composition (EMD) and PE, that inputting the clustered components into the artificial neural network for prediction. However, due to the modal aliasing phenomenon encoun-tered in EMD, the final prediction performance has limited room for improvement. Zhang et al. [22] used complete ensemble empirical mode decomposition adaptive noise (CEEMDAN) to process data, which effectively addresses issues related to modal decom-position following signal partitioning, and effectively resolves the transference of white noise from high to low frequency, thereby elevating the quality of subsequent analyses. Liu and Zhao [23] proposed a prediction model for convolutional networks and long short-term memory (LSTM) networks based on clustering EMD and experimentally demonstrated that this method yields high-accuracy prediction results [24]. Wind power, being a time series signal, is essentially associated with data processing. Consequently, employing intelligent data processing techniques can significantly enhance its efficiency [25,26].

The echo state network (ESN) is a novel form of RNN and can effectively predict nonlinear, chaotic time series by using the echo characteristics of its internal memory pool [27]. In the ESN, the input layer weight, savings pool weight, and connection of ESN are randomly set; thus, it offers the advantages of fast calculation speed and strong generalization ability and is widely used in the field of wind power prediction [28]. However, in ML-based prediction models, setting hyperparameters is challenging. Setting appropriate hyperparameter can greatly affect the prediction results; thus, various optimization algorithms have been proposed to obtain high-accuracy prediction results [29]. Li et al. [30] used the crow search algorithm to improve the ELM, improved the input weight of the ELM, solved the problem of unstable bias of the hidden layer, and improved the accuracy of the model. Samadianfard et al. [31] combined multilayer perceptrons with the whale optimization algorithm (WOA) to enhance the prediction capability. In the present study, an improved adaptive Sparrow Search Algorithm (IASSA) was developed to enhance the search effectiveness of the original SSA and improve the prediction performance of Echo State Network (ESN) by synchronously optimizing Phase Space Reconstruction (PSR) parameters. The application of IASSA has demonstrated significant effectiveness in enhancing the accuracy of wind power prediction.

For wind power forecasting, Du et al. [32] utilized long short-term memory (LSTM) networks. LSTM, as a specialized type of recurrent neural network, has the capability to learn and retain time-series patterns from historical data, thus enabling accurate forecasts. However, when confronted with the non-linearity and non-stationarity inherent in wind power data, LSTM may encounter challenges in achieving high predictive accuracy. Tian et al. [16] employed the Backpropagation Neural Network (BPNN) for wind power forecasting. BPNN is trained using a backpropagation algorithm, which can address the non-linearity of wind power forecasting to a certain extent. However, BPNN has inherent limitations. Firstly, when dealing with the non-stationarity of wind power data, its predictive performance is often restricted. Additionally, BPNN may get stuck in local optima, thereby compromising predictive precision. In contrast, we proposed a short-term wind power prediction model by combining time-varying filtered EMD (TVFEMD), PSR, ESN, and an IASSA-based synchronous optimization strategy. First, TVFEMD was used to preprocess the original wind power time series to reduce the non-stationarity of the wind power time series. Next, the decomposed subsequences were converted into a phase space matrix using PSR. The training, validation, and test sets were divided, and each set of subsequences was predicted using the ESN. At this stage, IASSA was used to synchronize the optimization of parameters in PSR and ESN. Finally, the prediction results of all subcomponents were accumulated to obtain the final predicted value of wind power.

In conclusion, our method exhibits superior performance in tackling the complexity and challenges of wind power forecasting compared to LSTM, BPNN, and other methods such as SSA-ESN and EMD-SSA-PSR-ESN. Our experimental results further confirm that our method significantly outperforms the aforementioned techniques, thereby affirming the efficacy and superiority of our approach. The novel contributions of this paper primarily encompass the following aspects:

1. The proposed model decomposes the wind power of the original time series through TVFEMD, which effectively solves the problem of it being difficult to predict non-stationary signals. A synchronous optimization strategy based on phase space reconstruction (PSR) and echo state network (ESN) is proposed.

2. In order to enhance the performance of the prediction model, the sparrow search algorithm (SSA) is improved in three aspects, namely, finder position update strategy, follower location update strategy and population structure. Additionally, the proposed IASSA is verified in the test function.

3. It is proposed that IASSA synchronously optimize PSR-ESN to achieve optimal prediction.

2. Fundamental Principle

2.1. Time-Varying Filtering-Based Empirical Mode Decomposition

TVFEMD is an improved EMD method based on time-varying filters proposed by Li et al. [33]. TVFEMD provides an alternative scheme for single-component signals, namely local narrowband signals, which greatly improves the problem of modal aliasing encountered in EMD. TVFEMD performs time-varying filtering of the signal in order to determine the local cutoff frequency. The main steps involved are as follows:

(1) The original time series x(t) is transformed using the Hilbert technique to obtain the instantaneous amplitude A(t) and instantaneous frequency φ′(t).

(2) The local maximum value A({t_max}) and minimum value A({t_min}) of A(t) are interpolated to obtain β₁(t) and β₂(t), respectively, and the instantaneous mean a₁(t) and instantaneous envelope a₂(t) are calculated:

$$\begin{array}{l}{a}_1(t)=\frac{{\beta }_1(t)+{\beta }_2(t)}{2}\\ a_2(t)=\frac{{\beta }_2(t)-{\beta }_1(t)}{2}\end{array}$$

(1)

(3) φ′({t_min})A2({t_min}) and φ′({t_max})A2({t_max}) are interpolated to obtain η₁(t) and η₂(t), respectively, to calculate the instantaneous frequency:

$$\begin{array}{l}{\phi }_1^{\prime }(t)=\frac{{\eta }_1(t)}{2a_1^2(t)-2a_1(t)a_2(t)}+\frac{{\eta }_2(t)}{2a_1^2(t)+2a_1(t)a_2(t)}\\ {\phi }_2^{\prime }(t)=\frac{{\eta }_1(t)}{2a_2^2(t)-2a_1(t)a_2(t)}+\frac{{\eta }_2(t)}{2a_2^2(t)+2a_1(t)a_2(t)}\end{array}$$

(2)

(4) The cutoff frequency φ′_bis(t) is calculated:

$${\phi }_{bis}^{\prime }(t)=\frac{\left[{\phi }_1^{\prime }(t)+{\phi }_2^{\prime }(t)\right]}{2}=\frac{{\eta }_2(t)-{\eta }_1(t)}{4a_1(t)a_2(t)}$$

(3)

(5) To solve the intermittent problem, the cutoff frequency φ′_bis(t) is rearranged.

(6) The extreme value of the signal $h(t)=\cos \left[{\displaystyle \int {\phi }_{\mathrm{bis}}^{\prime }}(t)\mathrm{d}t\right]$ is taken as the node, x(t) is approximated using B-spline interpolation, and the approximation result m(t) is obtained.

(7) It is determined whether the cutoff criterion is met, and the bandwidth threshold ξ is set. If θ(t) ≤ ξ, x(t) is determined as a subcomponent; otherwise, x₁(t) = x(t) − m(t), and steps 1–7 are repeated:

$$\theta (t)=\frac{B_{Loughlin }(t)}{{\phi }_{avg }(t)}$$

(4)

where φ_avg(t) is the weighted mean instantaneous frequency, and B_Loughlin(t) is the instantaneous Loughlin bandwidth.

2.2. Phase Space Reconstruction

PSR is a mathematical method for analyzing chaotic time series [34] and can be used to convert discrete one-dimensional time series x(t) into a d-dimensional vector with delay time τ. In this study, the historical load data were reconstructed into the input and output vectors of the prediction model by using PSR, as follows:

$$X_{input }={\left[\begin{array}{llll}{X}_1\hfill & X_2\hfill & \cdots \hfill & X_L\hfill \end{array}\right]}^T=\left[\begin{array}{cccc}{x}_1& x_{1+\tau }& \cdots & x_{1+(d-1)\tau }\\ ⋮& ⋮& \ddots & ⋮\\ x_i& x_{i+\tau }& \cdots & x_{i+(d-1)\tau }\\ ⋮& ⋮& \ddots & ⋮\\ x_L& x_{L+\tau }& \cdots & x_{L+(d-1)\tau }\end{array}\right]$$

(5)

$$X_{\mathrm{output} }={\left[\begin{array}{llll}{x}_{1+h+(d-1)\tau }\hfill & x_{2+h+(d-1)\tau }\hfill & \dots \hfill & x_n\hfill \end{array}\right]}^T$$

(6)

where $\left\{x_i\mid i=1,2,\dots ,n\right\}$ represents the original load sequence, n is the length of the sequence, $L=n-(d-1)·\tau -h$, and h is the prediction step.

In this experiment, we adopted a combined PSR-ESN model; PSR was used to construct the corresponding phase space matrix for sequences, which were then divided into training sets, validation sets, and testing sets. Then, ESN was used for the subsequence prediction.

2.3. Echo State Network

The ESN is a variant of the RNN. The network structure of the ESN includes three parts, namely an input layer, a reserve pool, and an output layer, as shown in Figure 1 [35]. Because the input layer weight matrix and the reserve pool internal connection matrix are randomly generated and fixed, the computational amount of model training is reduced.

The ESN solves the fitting regression problem of time series by replacing the fully connected hidden layer with a sparsely connected reserve pool; the updated state of the network along with the output equation is as follows:

$$x(t)=(1-a)x+a·\tanh (Rx(t-1)+W_{in}u(t))$$

(7)

$$y(t)=W_{out} x(t)$$

(8)

where $\tanh (·)$ represents the activation function and is used to obtain the network echo characteristics, a is the leakage rate used to control the update weight of the ESN network, W_in represents a matrix of input weights randomly generated in the range [−1, 1], R is the connection matrix with a sparse structure inside the reserve pool, u(t) represents the input at time t, x(t) represents the t-moment state of the reserve pool, and y(t) is the output at time t. The output matrix W_out of the ESN can be solved via ridge regression, with the following optimization objectives:

$$\min {‖W_{out}X-Y‖}_2^2+\lambda {‖W_{out}‖}_2^2$$

(9)

$$W_{out}=Y{X}^T{\left(X{X}^T+\lambda I\right)}^{-1}$$

(10)

where $\lambda$ is the regularization coefficient used to prevent the phenomenon of overfitting in the ESN training set, and $I$ is the identity matrix. The prediction data are substituted into Equations (7) and (8) to obtain the final prediction result.

The ESN structure is simple and practical; however, its prediction performance is affected by the parameter settings, such as the reserve pool connection matrix scaling parameter R_h, number of reserve pool network nodes N, input data scaling coefficient I_S, reserve pool sparsity degree S, and leakage value a. Using the appropriate parameter settings can effectively enhance the prediction ability of the ESN.

3. IASSA Algorithm and its Optimization Strategy

3.1. Sparrow Search Algorithm

SSA is a meta-heuristic optimization algorithm [36] inspired by the foraging and anti-predation behavior of sparrows [36,37]. In SSA, individuals are classified as discoverers, followers, and watchers. The natural behavior and mathematical expression of the sparrow can be described as follows:

Discoverers: Each generation of discoverers represents a point in the population that is closer to food, and its main function is to provide directions for the entire population to explore food. The mathematical expression of the location update of the discoverer is as follows:

$$x_{i,j}^{t+1}=\left\{\begin{array}{c}{x}_{i,j}^t·\exp \left(\frac{-i}{\alpha \times ite{r}_{\max }}\right),R_2<ST\\ x_{i,j}^t+Q·L,\hspace{1em}{R}_2\ge ST\end{array}\right.$$

(11)

where $x_{i,j}^t$ represents the position of the i-th sparrow in the j-dimensional when the number of iterations is t, iter_max is the maximum number of iterations, R₂ and ɑ are randomly generated values between 0 and 1, ST is the safety threshold, Q is a random number that follows a normal distribution, and L is a vector with all 1s of the elements. When $R_2<ST$, the discoverer can search extensively for food; in contrast, $R_2\ge ST$ indicates that there is danger and that the discoverer needs to withdraw from the danger zone.

Followers: The role of the follower is to follow the discoverer for food; their update strategy is as follows:

$$x_{i,j}^{t+1}=\left\{\begin{array}{l}Q·\exp \left(\frac{x_{{}_{worst}}^t-x_{i,j}^t}{i^2}\right), \mathrm{if} i>n/2\hfill \\ x_{best}^{t+1}+\left|x_{i,j}-x_{best}^{t+1}\right|·A^{+}·L, \mathrm{otherwise} \hfill \end{array}\right.$$

(12)

where $x_{worst}^t$ and $x_{best}^{t+1}$, respectively, represent individuals with the worst current fitness and the best individuals, and A is a vector with a value of 1 or −1 randomly generated for each element. $i>n/2$ indicates that the current follower is in a poor position in the population and that the food is widely searched for by diverging from the worst individual; otherwise, the search for food is performed by competing with the best individuals.

Vigilantes: Vigilantes are a random proportion of individuals in the population, whose main role is to alert the foraging area. Their mathematical expression is as follows:

$$x_{i,j}^{t+1}=\left\{\begin{array}{ll}{x}_{\mathrm{best} }^t+\beta ·\left|x_{i,j}^t-x_{\mathrm{best} }^t\right|,\hfill & f_i\ne f_g\hfill \\ x_{\mathrm{best} }^t+k·\left(\frac{x_{i,j}^t-x_{\mathrm{best} }^t}{\left|f_i-f_w\right|+\epsilon }\right),\hfill & f_i=f_g\hfill \end{array}\right.$$

(13)

where $\beta$ is the update step with a mean of 0 and a variance of 1; k is a random number between −1 and 1; $f_i$, $f_g$, and $f_w$ are fitness functions of the current alert, optimal individual, and worst individual, respectively; and $\epsilon$ is used to avoid constants with a denominator of 0. $f_i\ne f_g$ indicates that the sparrow is at the edge of the population and thus approaches the optimal individual; and $f_i=f_g$ indicates that the sparrow is at the center of the population, feels threatened and moves closer to its own species, reducing the risk of predation.

3.2. Improved Adaptive Sparrow Search Algorithm

Although SSA offers the advantages of a comprehensive search mechanism and fast convergence speed, it easily falls into local optimum tendencies. In addition, the population ratio of discoverers to followers in SSA must be set empirically; this affects the optimization performance of SSA. In order to improve the comprehensive optimization capability of SSA, we proposed IASSA in this paper.

Improvement 1: In the discoverer position update strategy of the original SSA, the use of the mathematical model as a method to control the flight distance by the normally distributed random number Q is not obvious because the random number generated by the standard normal distribution reduces the probability of the discoverer withdrawing from the danger zone at a distance; this affects the global search ability of the algorithm. Therefore, generating Q via the Cauchy distribution allows the discoverer to quickly fly to a distant location. Moreover, varying the safety threshold RT with the iteration time allows the discoverer to search extensively for food in the early stage of the iteration, and as such the discoverer has a greater probability of escaping the current position in the late iteration to ensure population diversity in the later stage of the iteration. The formulas for generating the random number Q and the safety threshold ST are as follows:

$$x_{i,j}^{t+1}=\left\{\begin{array}{l}{x}_{i,j}^t·\exp \left(\frac{-i}{\alpha \times ite{r}_{\max }}\right),\hspace{1em}{R}_2<ST\\ x_{i,j}^t+Cauchy(0,1)·L,\hspace{1em}{R}_2\ge ST\end{array}\right.$$

(14)

$$ST=1-t/ite{r}_{\max }$$

(15)

Improvement 2: In the follower’s renewal strategy, the follower searches for food by competing with the best individual. The process of the follower approaching the optimal individual is single, and food can be searched on only one side of the optimal individual close to the direction of the follower. Inspired by the whale optimization algorithm (WOA) [38], in this study we realized the purpose of searching for food around the optimal individual by updating the position by encircling the optimal individual with a spiral of the better followers. The specific process is as follows:

$$D^{\prime }=\left|x_{best }^t-x(t)\right|$$

(16)

$$x_{i,j}^{t+1}=\left\{\begin{array}{l}Q·\exp \left(\frac{x_{{}_{worst}}^t-x_{i,j}^t}{i^2}\right), \mathrm{if} i>n/2\hfill \\ D^{\prime }·e^{bl}·\cos (2\pi l)+x_{best }^t, \mathrm{otherwise} \hfill \end{array}\right.$$

(17)

where b is a constant that defines the shape of the logarithmic spiral, and l is a random number in [−1, 1].

Improvement 3: In the original SSA, the proportional population selection of discoverers and followers was a complex process. This is because the discoverer mainly provides the ability to search globally, while the follower represents the local search capability in the algorithm. The selection of discoverer and follower ratios also affects the search efficiency of the algorithm; thus, in this paper, we proposed an adaptive population structure strategy and determined the proportion of discoverers in the population by using the following formula:

$$P=P_{\max }-(P_{\max }-P_{\min })·\cos (\frac{\pi }{2}·\frac{t}{ite{r}_{\max }})$$

(18)

where P is the proportion of t-generation discoverers in the population, and P_max and P_min are, respectively, the upper and lower limits of the proportion of discoverers. P decays from P_max to P_max iteratively.

In conclusion, these three improvements are interconnected and collectively optimize the sparrow search algorithm. The first improvement enhances global search capability, the second improvement increases local search and exploration capabilities, and the third improvement further enhances search efficiency and adaptability through the adaptive population structure strategy. Together, these improvements enable the improved sparrow search algorithm to achieve better performance and effectiveness in problem-solving scenarios.

3.3. Algorithm Evaluation

To verify the performance of the proposed algorithm, the algorithm was evaluated using multiple benchmark functions, as shown in

Table 1. Single-peak and multipeak test functions.

Function	Dimension	Optimal Value
$F_1(x)={\displaystyle \sum _{i=1}^n{x}_i^2}$	30	0
$F_2(x)={\displaystyle \sum _{i=1}^n\left|x_i\right|}+{\displaystyle \prod _{i=1}^n\left|x_i\right|}$	30	0
$F_3(x)={\displaystyle \sum _{i=1}^n{\left({\displaystyle \sum _{j=1}^i{x}_j}\right)}^2}$	30	0
$F_4(x)={\displaystyle \sum _{i=1}^n-}{x}_i\sin \left(\sqrt{\left|x_i\right|}\right)$	30	0
$\begin{array}{l}{F}_5(x)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i^2}}\right)-\exp \left(\frac{1}{n}{\displaystyle \sum _{i=1}^n\cos }\left(2\pi x_i\right)\right)\\ +20+e\end{array}$	30	0
$\begin{array}{l}{F}_6(x)=0.1\left\{{\sin }^2\left(3\pi x_1\right)+{\displaystyle \sum _{i=1}^n{\left(x_i-1\right)}^2}\left[1+{\sin }^2\left(3\pi x_i+1\right)\right]\right.\\ \left.+{\left(x_n-1\right)}^2\left[1+{\sin }^2\left(2\pi x_n\right)\right]\right\}+{\displaystyle \sum _{i=1}^{n}u}\left(x_i,5,100,4\right)\end{array}$	30	0

. In addition, the proposed algorithm was compared with other commonly used optimization algorithms to better demonstrate the improvement effect of the IASSA.

The optimization results of different algorithms for the benchmark function are listed in

Table 2. Test function optimization results.

	GWO		PSO		SSA		IASSA
	AVE	STD	AVE	STD	AVE	STD	AVE	STD
F1	6.440 × 10−9	4.361 × 10−9	0.2602	0.131	2.607 × 10−6	7.343 × 10−6	1.459 × 10−58	5.459 × 10−58
F2	6.493 × 10−6	3.116 × 10−6	1.243	0.398	0.0046	0.0047	1.967 × 10−30	5.021 × 10−30
F3	7.002	10.113	4.370	1.473	3.354 × 10−4	5.799 × 10−4	6.249 × 10−42	3.410 × 10−41
F4	−5.985 × 103	9.992	−3.351 × 103	4.544	−7.712 × 103	8.516	−7.722 × 102	8.395
F5	0.0087	0.016	5.789	2.073	8.297 × 10−8	2.712 × 10−7	0	0
F6	32.3543	6.092	2.508 × 105	1.099 × 105	1.654 × 10−5	2.941 × 10−5	1.544 × 10−5	5.064 × 10−6

, the convergence effect of different algorithms is illustrated in Figure 2 and the convergence time of different algorithms for various functions are outlined in

Table 3. The convergence time of different algorithms.

	F1 (s)	F2 (s)	F3 (s)	F4 (s)	F5 (s)	F6 (s)
IASSA	0.016875	0.017637	0.043607	0.019618	0.01944	0.33775
SSA	0.017871	0.018149	0.044131	0.022935	0.020401	0.40221
GWO	0.017566	0.018148	0.038441	0.021897	0.019067	0.32584
PSO	0.0076562	0.007721	0.028181	0.013613	0.010778	0.39815

. The comparative analysis revealed that the IASSA has better convergence efficiency in unimodal functions F1–F3 and can jump out of local optimal solutions in multimodal functions F4–F6, and it also has a certain advantage in time. To ensure impartiality, the population size and iteration count were set to 30 and 200 across all algorithms. Each aforementioned algorithm was independently executed 30 times, with their respective mean values (AVE) and standard deviations (STD) recorded each time. Herein, the mean value represents the optimal performance of the algorithm, whereas the standard deviation signifies its stability.

The maximum number of iterations was set to 200 based on previous research and preliminary experimentation. It was observed that, when the iteration count exceeded 200, the convergence of various algorithm components significantly improved or approached stability. Considering the constraints of computational resources and time, the iteration count was set to 200.

4. Wind Power Prediction Model Design

We developed a wind power hybrid prediction model by using TVFEMD, PSR, and ESN. To improve the prediction accuracy, the IASSA optimization algorithm was used to determine the optimal parameters of PSR and ESN simultaneously. The parameters τ and d in PSR and Rh, N, IS, and S in the ESN were encoded together in the IASSA. In addition, the root mean-square error (RMSE) was used as the fitness function to solve the parameters. The steps involved in the hybrid model are shown in Figure 3 and are described as follows:

(1)

Raw wind power data were collected.

(2)

The original wind power time series was divided into a series of subsequences by using TVFEMD.

(3)

All subsequences were converted into phase space matrices by using PSR and divided into the training set, validation set, and test set.

(4)

The ESN model was trained using the training set to verify that the RMSE of the set was the minimum, and all parameters in PSR and ESN were optimized synchronously using IASSA.

(5)

The optimized optimal parameters were introduced into PSR-ESN, and the test set was ran in order to obtain the prediction results of the subseries.

(6)

The prediction results of all subcomponents were accumulated to obtain the final wind power prediction results.

5. Case Analysis

The wind power generation data of the Irish energy system were used for experimental analysis, the wind power generation time was collected from 0:00 on 1 January 2020 to 0:00 on 16 January 2020, and the sampling interval of experimental data was 15 min, yielding a total of 1152 pieces of data.

5.1. Description of the Experiment

In order to verify the prediction performance of the proposed TVFEMD-ISSA-PSR-ESN model, the single models BPNN and LSTM, and the combined models SSA-ESN, EMD-SSA-PSR-ESN, and TVFEMD-SSA-PSR-ESN were compared. For the performance evaluation of the prediction models, three indicators were used: root mean-square error (RMSE), mean absolute error (MAE), and mean absolute proportional error (MAPE):

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(Y_i-Y_i^{*}\right)}^2}}$$

(19)

$$MAE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|Y_i-Y_i^{*}\right|}$$

(20)

$$MAPE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|\frac{100\times \left(Y_i-Y_i^{*}\right)}{Y_i}\right|}$$

(21)

For the model involved in the experiment, the B-spline interpolation n and bandwidth threshold ξ for TVFEMD decomposition were set as 26 and 0.3, respectively. The parameters, namely delay time, embedding dimension d, network update weight a, reserve pool connection matrix scaling parameter Rh, reserve pool node number N, input data scaling coefficient W_in, and reserve pool sparsity S, were determined using the proposed IASSA and SSA optimization algorithms. Other parameters are presented in

Table 4. Parameter settings of all experimental methods.

Method	Search Method	Parameter	Parameter Methods
SSA/ IASSA	Preset	Population size	30
	Preset	Number of iterations	100
	Preset	Proportion of discoverers	0.2/[0.2, 0.8]
ESN	SSA/IASSA	The number of reserve pool nodes is N	[20, 500]
	SSA/IASSA	Sparseness degree S	[0.01, 0.1]
	SSA/IASSA	Reserve pool connection matrix scaling parameter Rh	[0.01, 1]
	SSA/IASSA	Enter the data scaling factor IS	[0.01, 1]
	SSA/IASSA	Update weight a	[0.01, 1]
LSTM/ BPNN	Preset	Number of hidden layers	1
	Preset	The number of hidden layer nodes	50
TVFEMD	Preset	B-spline interpolation n	26
	Preset	Bandwidth threshold ξ	0.3
PSR	SSA/IASSA	Embedding dimension D	[1, 10]
	SSA/IASSA	Delay time τ	[1, 50]

. In the experiment, the training set, validation set, and test set accounted for 60%, 15%, and 25% of all subsequences, respectively.

5.2. Data Decomposition and Normalization

After the TVFEMD decomposition of the original wind power data, seven subsequences were obtained and were recorded as IMF1–IMF7, as shown in Figure 4. In order to better train the law of subsequences, all subsequences were normalized separately. The normalization formula is:

$$y=\left(y_{\max }-y_{\min }\right)·\frac{x-x_{\min }}{x_{\max }-x_{\min }}+y_{\min }$$

(22)

where y is the normalized data; x is the original wind power data; x_max and x_min are, respectively, the maximum and minimum values of wind power; y_max and y_min are, respectively, the maximum and minimum values of the mapping space; and the mapped space is [−1, 1].

5.3. Prediction Results and Analysis

In this subsection, six experimental models are discussed in detail, and the prediction results are evaluated to demonstrate the validity of the proposed combined model. All experimental model prediction performance indicators, that is, RMSE, MAE, and MASE, including 1-, 3-, and 5-step-ahead predictions, are presented in Table 4, and the error radar chart is shown in Figure 5.

As can be seen from

Table 5. Multistep prediction errors of different models.

Model	1 Step Ahead			3 Step Ahead			5 Step Ahead
	RMSE	MAE	MAPE	RMSE	MAE	MAPE	RMSE	MAE	MAPE
	(MW)	(MW)	(%)	(MW)	(MW)	(%)	(MW)	(MW)	(%)
SSA-ESN	29.5899	20.1982	3.4523	66.6408	46.6874	7.8592	98.3090	70.9405	11.7082
BPNN	32.8771	22.2273	3.7655	74.5271	51.0994	8.4672	128.4560	86.8865	13.9469
LSTM	39.4565	31.8201	5.3473	68.2564	47.9236	8.0453	98.4259	71.4935	11.9273
EMD-SSA-PSR-ESN	22.9729	16.0040	2.5961	32.1147	23.6209	3.6597	56.0199	41.9148	6.5519
TVFEMD-SSA-PSR-ESN	5.0939	3.9508	0.6301	12.5648	9.4510	1.5488	23.2008	17.8810	2.9592
This article method	3.8821	2.9876	0.5043	9.1991	6.9535	1.1556	13.8121	10.4647	1.6578

and Figure 5, in the case of 1, 3, and 5 steps ahead, the proposed method predicted the wind power better, and the prediction error gradually increased with the increase in the number of prediction steps. A comparison of the evaluation indicators of different prediction models revealed the following points:

• A comparison of the evaluation indicators RMSE, MAE, and MAPE of the SSA-ESN, BPNN, and LSTM models revealed that SSA-ESN had better prediction performance. In the 1-step, 3-step, and 5-step predictions, compared with BPNN and LSTM, the RMSE decreased by 3.2872 and 9.8666 (1 step), 7.8863 and 1.6156 (3 steps), and 30.1470 and 0.1169 (5 steps), respectively; the MAE decreased by 2.0291 and 11.6219 (1 step), 4.4120 and 1.2362 (3 steps), and 15.9460 and 0.5530 (5 steps); and the MAPE decreased by 0.3132 and 1.8950 (1 step), 0.6080 and 0.1861 (3 steps), and 2.2387 and 0.2191 (5 steps), respectively.
• A comparative analysis of the models SSA-ESN, EMD-SSA-PSR–ESN, and TVFEMD-SSA-PSR–ESN revealed that the prediction performance of the model were greatly improved upon the addition of the signal decomposition method and PSR-ESN. The RMSE decreased by 17.8790 (1 step), 19.5499 (3 steps), and 32.8191 (5 steps); the MAE decreased by 12.0532 (1 step), 14.1699 (3 steps), and 24.0338 (5 steps); and the MAPE decreased by 1.9660 (1 step), 2.1109 (3 steps), and 3.5927 (5 steps).
• Due to the drawbacks of the traditional SSA optimization methods, as well as PSR and ESN, requiring many parameters, SSA often falls into local optimal solutions. Therefore, in this study, the proposed IASSA was combined with PSR and ESN on the basis of TVFEMD to strengthen the prediction ability of the model through better parameter optimization performance. Comparing the TVFEMD-SSA-PSR-ESN model with the proposed model, the RMSE decreased by 1.2118 (1 step), 3.3657 (3 steps), and 9.3887 (5 steps); the MAE decreased by 0.9632, 2.4975, and 7.4163; and the MAPE decreased by 0.1258 (1 step), 0.3932 (3 steps), and 1.3014 (5 steps), respectively. Thus, the optimization performance of the IASSA not only has a certain effect on the classical test function but also exhibits good capacity to solve the actual parameter optimization problem.

In conclusion, the data analyses presented here highlight the effectiveness of the SSA-ESN model in terms of predictive performance when compared to BPNN and LSTM models. Furthermore, the inclusion of signal decomposition methods, PSR, and the proposed IASSA algorithm demonstrates significant improvements in the models’ predictive capabilities. These findings underscore the importance of selecting appropriate techniques and algorithms to optimize data analysis models for the achievement of enhanced performances and better results in real-world applications.

In order to verify the performance of the proposed method more intuitively, the experimental prediction effect was further visualized and analyzed. The error curves, fitting curves, and error distributions of the multistep prediction of the six models are shown in Figure 6, Figure 7, and Figure 8, respectively. To demonstrate the advantages of the proposed model more intuitively, the distribution value of the error was studied; the abscissa of the proposed method was found to be smaller, and its MAE and standard deviation were the smallest—3.2511 and 4.1478 (1 step), 7.5864 and 9.8941 (3 steps), and 11.2086 and 14.6149 (5 steps), respectively. This demonstrated the superiority of the proposed method.

6. Conclusions

In this paper, we proposed a hybrid model based on TVFEMD, PSR, ESN, and IASSA to predict short-term multistep wind power generation. In the proposed model, first, the wind power of the original time series is decomposed into multiple subcomponents by using TVFEMD, which effectively solves the problem of the difficult prediction of non-stationary signals. Next, the subcomponents are constructed by PSR to construct the output vectors of the prediction model, and the ESN is then used as the final prediction model to predict the data. Because PSR and ESN involve the adjustment of multiple parameters, the IASSA algorithm is used to optimize PSR and ESN parameters synchronously. Through a comprehensive experimental analysis of real wind power, the effectiveness of the proposed model is verified.

Based on the experimental results, the following conclusions can be drawn:

(1)

A comparison of the performance of EMD and TVFEMD decomposition of the original wind power time series revealed that TVFEMD, combined with the prediction model, yielded an improved prediction effect: the difference was expressed via number of steps, namely, 1 step, 3 steps, and 5 steps, and the effect was more obvious.

(2)

Compared with the original SSA, the proposed IASSA exhibited a better optimization performance in the classical test function and the actual prediction application. The accuracy was effectively improved in the three performance evaluations of root mean-square error (RMSE), mean absolute error (MAE) and mean absolute proportional error (MAPE).

(3)

Compared with six existing models, the experimental analysis of multistep prediction revealed that the proposed prediction model combining TVFEMD, PSR, ESN, and IASSA synchronous optimization strategies were able to effectively improve the prediction accuracy.

Based on the above three points, the combined prediction model proposed in this paper improved the accuracy of wind power research and provided a new feasible strategy for the multistep prediction of wind power.