Short-Term Wind Power Interval Forecasting Based on Hybrid Modal Decomposition and Improved Optimization

Abstract

Accurate wind power prediction can effectively alleviate the pressure of the power system peak frequency regulation, and is more conducive to the economic dispatch of the power system. To enhance wind power forecasting accuracy, a hybrid approach for wind power interval prediction is proposes in this study. Firstly, an Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN) is applied to decompose the initial wind power sequence into multiple modes, and Variational Mode Decomposition is used to further decompose the high-frequency non-stationary components. Next, Fuzzy Entropy (FE) is utilized to assess the complexity of the post-decomposed Intrinsic Mode Functions (IMFs), and different forecasting methods are employed accordingly, the point predictions were obtained by linearly summing the component predictions.Additionally, an improved sparrow search algorithm (ISSA) is used to seek the optimal hyperparameters of the prediction algorithm. Finally, the prediction intervals are constructed using the point prediction results based on kernel density estimation (KDE). The root mean square errors (RMSE) of deterministic predictions are 2.8458 MW and 1.8605 MW, with uncertainty coverage rates of 95.83% and 97.92% at a 95% confidence level.

Key words
uncertainty forecasts; wind power; combinatorial decomposition; ISSA; KDE

Introduction

Due to the continuous development of economy and society, the environmental pollution caused by fossil energy is becoming more and more serious, and non-fossil energy is gradually becoming the main force of energy (Yan et al. 2022). Wind energy as one of the non-fossil energy sources, due to its clean and non-polluting characteristics of the wind power industry to promote the rapid development of wind power, but the wind power generation of its own intermittency and randomness, on the reliability of the power system operation has brought serious challenges. Therefore, the establishment of a stable and reliable wind power prediction model can effectively improve the power quality and enhance the reliability of the power system (Suo et al. 2023, Zhang et al. 2022). The types of wind power prediction are categorized as physical, statistical and artificial intelligence models. Physical models are based on numerical weather predictions (NWP) considering factors such as topography, climate, and season to establish wind power prediction models. However, these models are computationally intensive and complex (Castorrini et al. 2022). Statistical modeling combines statistical methods to create a prediction of the wind power value at the next moment in time, without taking into account physical factors such as climate and topography, and is therefore widely used for short-term forecasts (Chen 2022). Statistical models include grey prediction models, time series forecasting models, etc. Nevertheless, relying on local patterns, statistical models can lead to long-term forecasting errors.

Limited by the high computational complexity and intricate processes of physical models and the potential for long-term errors in statistical models, artificial intelligence models have been widely applied in wind power forecasting. Artificial intelligence models include machine learning models such as Support Vector Machine (SVM) (Bi & Qiu 2019), Least-Squares SVM (LSSVM) (Li & Jin 2018), as well as neural network models like Backpropagation (BP) (Feng & Wencheng 2023), Elman Neural Network (Shi et al. 2015), Long Short-Term Memory (LSTM) (Bao et al. 2022, Gao et al. 2023), Bidirectional LSTM (Bilstm) (Li et al. 2022), and Gated Recurrent Unit (GRU) (Xiao et al. 2023, Zhao et al. 2023). Combining signal processing with artificial neural networks has been shown to achieve better forecasting results for wind power. Widely used signal processing methods include Wavelet Decomposition (WD) (Motlagh et al. 2021), Empirical Mode Decomposition (EMD), Ensemble Empirical Mode Decomposition (EEMD) (Wu et al. 2023), Complementary Empirical Mode Decomposition (CEEMD) (Zhao & Wang 2020), Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) (Zhang et al. 2020), Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN) (Xiao & Zhang 2023), and Variational Mode Decomposition (VMD) (Liu 2020). However, WD requires the prior determination of the mother wavelet (Li et al. 2020), EMD is prone to endpoint effects and mode mixing issues (Nguyen & Phan 2023), EEMD and CEEMD may have some residual noise (Wu et al. 2023, Zhang et al. 2020, Zhao & Wang 2020). CEEMDAN suffers from residual noise and pseudo-mode problems, while ICEEMDAN effectively addresses these issues (Bommidi et al. 2023). Wang et al. (Wang et al. 2022) decomposed the wind power sequence into multiple sub-sequences using VMD and then input them into an LSTM network for prediction, but the forecasting results depended on hyperparameter selection. He et al. (He & Tsang 2021) applied ICCEMDAN to decompose university electricity consumption data into multiple sub-sequences, which were individually predicted using a LSTM optimized by Bayesian optimization, and finally reconstructed to obtain the final prediction. Yang et al. (Yang et al. 2023) used EWT to decompose wind speed sequences and optimized the hyperparameters of LSSVM and Gaussian Process Regression (GPR) models using the Differential Evolution Grey Wolf Optimization (DE-GWO) algorithm. They sequentially predicted the decomposed components from low to high frequencies using ARIMA, LSSVM, and GPR models. However, the high-frequency components generated after decomposition suffered from spectral aliasing, leading to significant prediction errors. Liu et al. (Liu et al. 2018) used VMD for a first-round decomposition of wind speed data and applied Singular Spectrum Analysis (SSA) to further extract the trend information of the data, resulting in improved prediction accuracy. Xiang et al. (Xiang et al. 2020) used CEEMDAN for a second-round decomposition and fed all the sub-sequences into an LSTM network to obtain wind speed predictions with higher accuracy. However, the large number of sub-components generated by the second-round decomposition made the computation relatively cumbersome. To address this, aggregation processing was combined to significantly reduce the computational burden and enhance the model’s prediction accuracy and stability (Suo et al. 2023). Xiong et al. (Xiong et al. 2023) employed CEEMD for a first-round decomposition, used Sample Entropy (SE) to reduce sequence complexity, treated the sequences with Random Forest in the second round, and then applied BiLSTM and ELM models to separately predict high-frequency versus low-frequency sequences. They optimized the model parameters using the Improved Spider Monkey Algorithm (IRSA) and finally aggregated the results to obtain the final wind power prediction. However, CEEMD suffered from noise residue issues and RF’s instability, rendering the overall model unstable. Xiao et al (Xiao et al. 2023) who used quadratic decomposition for wind power series, calculated complexity using SE, and aggregated the components before inputting them into a Bayesian optimized BiLSTM network to obtain point predictions with good forecasting performance. However, the model does not treat the modal components of the different features obtained from the decomposition, and the Bayesian optimized BiLSTM network had limited accuracy in predicting low-frequency components, which in turn affected the model’s overall predictive performance.

Wind speed forecasting involves both deterministic and non-deterministic predictions. While deterministic forecasting can closely approximate the true wind speed, it may fail to represent the uncertainty in the wind power sequence. Non-deterministic or interval forecasting, on the other hand, can better quantify the prediction bias caused by the uncertainty in wind power, thus providing a range of possible outcomes (Xiao & Zhang 2023). Previous literature has applied Kernel Density Estimation (KDE) to interval forecasting (Niu et al. 2022, Zhang et al. 2023) and achieved promising results by combining point predictions to obtain interval prediction results.

In view of this, this paper comes up with a hybrid wind power interval prediction model. First, considering the stochastic and intermittent nature of wind power sequences, a two-stage decomposition process using ICEEMDAN, FE, and VMD (IVF) is established to address the issue of excessive sub-component decomposition caused by second-round decomposition. FE is used to aggregate the decomposed components, reducing the non-stationarity of the wind power sequence. Next, GRU and ELMAN models are employed to predict different components, leveraging the time characteristics of the wind power sequence. To select the optimal hyperparameters for the GRU network, an Improved Sparrow Search Algorithm (ISSA) is proposed for optimization. Finally, to overcome the limitations of point predictions, a kernel density estimation model is constructed based on the error sequence of the point predictions to perform wind power interval forecasting. The innovative aspects and contributions of this paper are as follows:

1. Combining the advantages of composite models in wind power forecasting, this study constructs a (IVF-ISSA-GRU/ELMAN) model for wind power prediction, incorporating second-order signal decomposition, deep learning models, evolutionary algorithms, and aggregation framework.

2. ICEEMDAN is used to decompose wind power data. To minimize the complexity of the high-frequency components after decomposition, VMD is applied to process the high-frequency components obtained from ICEEMDAN, enhancing their trend and stability.

3. FE is used to analyze the complexity of the decomposed components, aggregating and classifying components with correlation. Different components are predicted using GRU and ELMAN models.

4. Two improvement strategies are employed to enhance the optimization capability of SSA for optimizing GRU model parameters. Optimizing the position update formulation of the SSA algorithm using the golden sine strategy, and the golden section coefficients c1 and c2 are improved, with the introduction of the Tent chaotic map for population initialization.

5. To fully exploit wind power forecasting information, this study performs interval forecasting by combining wind power point prediction results with their error sequences, using KDE to construct wind power interval predictions.

Basic Methods

Signal processing methods

ICEEMDAN

ICEEMDAN addresses the issues of residual noise and pseudo-modes by selecting white noise from the EMD-decomposed modal components (Xiao et al. 2023). The basic steps of ICEEMDAN decomposition are as follows:

1. Use the EMD method to decompose the original wind power signal x once, obtaining the first-order residual r1, as shown in Equation (1).

   $$z^{\left(i\right)}=z+\beta E_k\left({\omega }^{\left(i\right)}\right)$$(1)
   $$r_1=⟨\tau \left(x^i\left(t\right)\right)⟩$$(2)
   Where $E_k\left(\right)$ represents the kth-order modal component from EMD decomposition; ${\omega }^{\left(i\right)}$represents Gaussian white noise; $z^{\left(i\right)}$ is the time series with white noise added in the i-th iteration; $\beta ={\phi }_{0}std\left(x\left(t\right)\right)/std\left(E_k\left({\eta }^i\left(t\right)\right)\right)$, ${\phi }_0$ is the reciprocal of the expected signal-to-noise ratio between the added noise signal and the analysis signal, where i denotes the number of noise additions, std represents the standard deviation, $⟨.⟩$ denotes the overall mean.

2. When $k=1$, calculate the modal component obtained from the first-order residual:

   $$IM{F}_1\left(t\right)=x\left(t\right)-r_1\left(t\right)$$(3)

3. Obtain the second-order residual and the second-order modal component:

   $$r_2\left(t\right)=⟨\tau (r_1\left(t\right)+{\alpha }_1{E}_2\left({\eta }^i\left(t\right)\right))⟩$$(4)
   $$IM{F}_2\left(t\right)=r_1\left(t\right)-r_2\left(t\right)$$(5)

4. Similarly in Step 3, calculate the kth-order residual and modal component:

   $$r_k\left(t\right)=⟨\tau (r_{k-1}\left(t\right)+{\alpha }_{k-1}{E}_k\left({\eta }^i\left(t\right)\right))⟩$$(6)
   $$IM{F}_k\left(t\right)=r_{k-1}\left(t\right)-r_k\left(t\right)$$(7)

5. Repeat Step 4 to obtain all the residuals and IMF components.

Second decomposition stage based on VMD

VMD decomposes time series by iteratively searching for the optimal solution to a variational problem with constraints, making it a fully non-recursive time-frequency domain signal processing method. It performs well in the decomposition and denoising of high-frequency non-stationary sequences, ensuring data reliability while minimizing noise. The basic steps of VMD decomposition are as follows:

1. Construct the variational problem with the expression shown in Equation (2).

   $$\{\begin{array}{c}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{min}\left\{\sum _{k=1}^k{\parallel {\partial }_t[\delta \left(t\right)+\frac{j}{\pi t}*u_k\left(t\right)]e^{-j{\omega }_{k}t}\parallel }_2^2\right\}\hfill \\ s.t\sum _{k=1}^k{u}_k=s\hfill \end{array}$$(8)
   Where $\left\{{\omega }_k\right\}$ denotes the set of central frequencies for each component after decomposition; $\left\{u_k\right\}$ represents the set of modes after decomposition; * denotes the convolution process; $\delta \left(t\right)$ denotes the unit pulse signal; $\partial \left(t\right)$ is the Dirac function; $k$ is the number of modal components; $s$ is the original load sequence at moment t; ${\parallel \parallel }_2^2$and denotes the number of L2 paradigms.

2. Introduce Lagrange multipliers $\lambda \left(t\right)$ and the second-order penalty factor $\alpha$ to transform the constrained variational problem into an unconstrained one, resulting in the augmented Lagrange expression shown below (Equation (3)):

   $$\begin{array}{cc}\hfill L(\left\{u_k\right\},\left\{{\omega }_k\right\},\lambda )& =\alpha \sum _{k=1}^k{\parallel {\partial }_t\left[(\delta \left(t\right)+\frac{j}{\pi t})\right]e^{-j{\omega }_{k}t}\parallel }_2^2+\hfill \\ \hfill & {\parallel f\left(t\right)-\sum _{k=1}^k{u}_k\left(t\right)\parallel }_2^2+⟨\lambda \left(t\right),f\left(t\right)-\sum _{k=1}^k{u}_k\left(t\right)⟩\hfill \end{array}$$(9)
   Where $\lambda$ is the Lagrange multiplication operator; $\alpha$is the quadratic penalty factor.

3. Use the alternating direction method of multipliers to update the "saddle points" of the Equation (3), namely, $u_k^{n+1}$, ${\omega }_k^{n+1}$, and ${\lambda }_k^{n+1}$. Obtain the optimal solution and assess the accuracy of each iteration until the desired accuracy is achieved, terminating the iterations. The update formulas are as follows:

   $${\hat{u}}_k^{n+1}\left(\omega \right)=\frac{\hat{f}\left(\omega \right)-\sum _{i\ne k}{\hat{u}}_i\left(\omega \right)+\frac{\hat{\lambda }\left(\omega \right)}{2}}{1+2\alpha {(\omega -{\omega }_k)}^2}$$(10)
   $${\omega }_k^{n+1}=\frac{{\int }_0^{\infty }\omega \left|u_k^{n+1}\left(\omega \right)\right|d\omega }{{\int }_0^{\infty }\left|u_k^{n+1}\left(\omega \right)\right|d\omega }$$(11)
   $${\lambda }^{n+1}\left(\omega \right)={\lambda }^n\left(\omega \right)+\lambda [f\left(\omega \right)-\sum _k\left|u_k^{n+1}\left(\omega \right)\right|]$$(12)

4. Repeat updating $u_k^{n+1}$, ${\omega }_k^{n+1}$, ${\lambda }_k^{n+1}$, and until the termination condition $\sum _k\left(\parallel u_k^{n+1}-u_k^n\parallel /{\parallel u_k^n\parallel }^2\right)<\epsilon$ is satisfied, then terminate the iteration. Where ${\hat{u}}_k^{n+1}\left(\omega \right)$ is the Vienna filter; $\hat{f}\left(\omega \right)$, ${\hat{u}}_i\left(\omega \right)$, $\hat{\lambda }\left(\omega \right)$ are the Fourier transforms of $f\left(t\right)$, $u\left(t\right)$, and $\lambda \left(t\right)$, respectively.

Wind power modal decomposition reconstruction based on fuzzy entropy (FE)

Fuzzy entropy is an improved version of sample entropy that combines fuzzy set theory and adopts fuzzy membership degree as the threshold criterion in entropy calculation. Compared to sample entropy, it can better measure the complexity of time series. The calculation steps of fuzzy entropy are as follows:

1. Perform phase space reconstruction on the time series $x\left(i\right)$:

   $$X\left(i\right)=\{x\left(i\right),x(i+1),\dots ,x(i+m-1)\}-u\left(i\right)$$(13)
   Where $X\left(i\right)$ is the reconstructed sequence, and $u\left(i\right)$ is the mean of m consecutive $x\left(i\right)$ values.

2. Define the distance between two vectors $X\left(i\right)$ and $X\left(j\right)$, expressed as:

   $$d_{ij}^m=max\left\{|(x(i+k)-u\left(i\right))-(x(j+k)-u\left(i\right))|\right\}$$(14)

3. Introduce the fuzzy membership degree function to define the similarity degree $A_{ij}^m$, as follows:

   $$A_{ij}^m=\{\begin{array}{ccc}\hfill & 1,\hfill & \hfill {\mathrm{\text{d}}}_{ij}^m=0\\ \hfill & exp[-ln2{\left(\frac{{\mathrm{\text{d}}}_{ij}^m}{r}\right)}^2],\hfill & \hfill {\mathrm{\text{d}}}_{ij}^m>0\end{array}$$(15)
   Where r is the similarity tolerance parameter, and $r=R\times \delta$ , where $\delta$ is the standard deviation of the original one-dimensional time series.

4. Define the function $C_i^m\left(r\right)$, as follows:

   $$C_i^m\left(r\right)=\frac{1}{N-m}{\sum }_{j=1,i\ne 1}^{N-m+1}{A}_{ij}^m$$(16)

   From the above equation, the relationship dimensionality in m-dimension can be obtained as:

   $${\varphi }^{m+1}\left(r\right)=\frac{1}{N-m}{\sum }_{j=1,i\ne 1}^{N-m+1}{C}_i^m\left(r\right)$$(17)

5. Set m=m+1, repeat steps (1)-(4) to obtain ${\varphi }^{m+1}\left(r\right)$.

6. Finally, obtain the expression for FE as follows:

   $$FE(m,r,N)=ln{\varphi }^m\left(r\right)-ln{\varphi }^{m+1}\left(r\right)$$(18)

Prediction algorithm

GRU (Gated Recurrent Unit) proposed by Cho et al, and it is an improvement of LSTM (Xiao et al. 2023). GRU combines the input and forget gates of LSTM into an update gate, which controls the information retained from the previous time step to the current state. It also replaces the output gate with a reset gate, which controls previous moment’s state information message written to current candidate set. The structure of GRU is given as Figure 1, and the update formula is as follows:

$$\{\begin{array}{c}{z}_t=\sigma (w_z\cdot [h_{t-1},x_t,1])\hfill \\ r_t=\sigma (w_r\cdot [h_{t-1},x_t,1])\hfill \\ {\tilde{h}}_t=tanh(w_h\cdot [r_t\odot h_{t-1},x_t,1])\hfill \\ h_t=(1-z_t)\odot h_t+z_t\odot h_{t-1}\hfill \end{array}$$(19)

Where $w_z$, $w_r$, $w_h$, and wo represent the weight values; $z_t$, $r_t$ is the outputs of the update gate and reset gate, respectively; $x_t$ is the input at the current moment; $h_{t-1}$ is the hidden state vector of the previous cell; $\left[\right]$denotes vector concatenation; $\odot$represents Hadamard product.

Figure 1
GRU Network Structure.

Improved sparrow search algorithm (ISSA)

When SSA initializes the population, the location information of the population is randomly generated. There exists a limited range of search, relatively poor biodiversity, and easy to fall into the local optimum, so the following strategy is used to optimize the SSA (Huang 2021). Its update steps are as follows:

1. Tent Chaotic Map for Population Initialization

   In this study, the Tent chaotic map accepted as initialize the population. It exhibits randomness, ergodicity, and sensitivity to initial values, which can enhance the convergence speed and global search capability of the original algorithm (Gao et al. 2022). The Tent mapping has a simple structure and presents a relatively uniform distribution of density in its resulting distribution. Its mathematical expression is as follows:

   $$x_{n+1}=\{\begin{array}{cc}2x_n\hfill & 0<x_n<0.5\hfill \\ 2(1-x_n)\hfill & 0.5<x_n<1\hfill \end{array}$$(20)

2. Position Update Improvement Strategy

   The Golden Sine Strategy (Tanyildizi & Demir 2017, Zhang & Liu 2022)(Golden-SA) is based on the relationship between the sine function unit and the circle. It scans the entire unit circle to enhance its global search capability and introduces the golden section factor in the strategy to narrow the search range and enhance its local search solution space ability. This strategy enhances both the global search ability and the local exploitation capability of the sparrow population . The improved position update formula for the discoverer is:

   $$X_{i,j}^t=\{\begin{array}{cc}\hfill & X_{i,j}^{t-1}\cdot |sin\left(R_1\right)|-R_2\cdot sin\left(R_1\right)\cdot \hfill \\ \hfill & |c_1\times X_{best}^{t-1}-c_2\times X_{best}^{t-1}|,\hfill & \hfill R_2<ST\\ \hfill & X_{i,j}^{t-1}+Q\cdot L,\hfill & \hfill R_2\ge ST\end{array}$$(21)
   Where $R_1\in rand[0,2\pi ]$, which is a random value in the range [0,2], $R_2\in rand[0,\pi ]$. The sparrow individual’s position information $x_i^t=[x_{i,1}^t,x_{i,2}^t,x_{i,3}^t,\cdots ,x_{i,d}^t]$, ($i=1,2,3,\dots ,n$), $P_i^t$ represents the optimal value of the sparrow’s global position at the tth iteration; $c_1=\pi {(1-2\tau )}^2$, $c_2=\pi {(2\tau -1)}^2$ , and $\tau$ is the golden section coefficient $\tau =(\sqrt{5}-1)/2$. When $R_2<ST$, the finder continues to perform the search work; when, the scout in the population discovers the predator and sends an alert signal to the other sparrows in the population, and the finder flies quickly to other areas by performing the position update according to Eq. (4). The joiner update equation is:
   $$X_{i,j}^{t+1}=\{\begin{array}{ccc}\hfill & Q\cdot exp\left(\frac{X_{worst}^t-X_{i,j}^t}{i^2}\right)\hfill & \hfill i>\frac{n}{2}\\ \hfill & X_{best}^{t+1}+|X_{i,j}-X_{best}^{t+1}|A^{+}L\hfill & \hfill i\le \frac{n}{2}\end{array}$$(22)
   Where $A=A^T{\left(A{A}^T\right)}^{-1}$, $A$ is a column vector of individual sparrows of the same latitude, with each element inside randomly assigned a value of 1 or -1; the scout’s position update formula is:
   $$X_{i,j}^{t+1}=\{\begin{array}{ccc}\hfill & X_{best}^t+\beta \cdot |X_{i,j}^t-X_{best}^t|\hfill & \hfill if{f}_i<f_g\\ \hfill & X_{i,j}^t+k\cdot \left(\frac{|X_{i,j}^t-X_{worst}^t|}{(f_i-f_w)+\epsilon }\right)\hfill & \hfill if{f}_i=f_g\end{array}$$(23)
   where $k\in [-1,1]$; $f_i$, $f_g$, $f_w$ are the individual fitness value of the ith sparrow, the current global best fitness value, and the worst fitness value, respectively; is the step control coefficient and ; is an infinitesimal constant set to prevent the denominator from being zero.

Error confidence interval based on KDE

KDE is a non-parametric estimation method. The theory is to use a moving window to estimate the sample density and get the probability density value of the point by weighted average, and finally get the probability density function of the wind power sequence. The construction method of the prediction interval is as follows:

$$\{\begin{array}{cc}\hfill & U_{re}=y_{re}+\left|F_{\frac{1-\alpha }{2}}\right|\hfill \\ \hfill & L_{re}=y_{re}+\left|F_{\frac{\alpha }{2}}\right|\hfill \end{array}$$(24)

Where $y_{re}$ represents the point prediction result of the wind power series; $F_{\frac{1-\alpha }{2}}$ is the upper $\frac{\alpha }{2}$ percentile of the point prediction result error sequence, $F_{\frac{\alpha }{2}}$ is the lower $\frac{\alpha }{2}$ percentile, and $\alpha$ is the interval confidence level; $U_{re}$ is the upper limit of the interval prediction, and $L_{re}$ is the lower limit of the interval prediction.

Model Establishment

Modeling process

1. Data Collection: Wind power data is collected every 15 minutes and outliers are removed from the dataset for preprocessing.

2. Data Decomposition: Apply the ICEEMDAN method to perform one decomposition on the dataset obtained in Step 1, the initial eigenmode components are obtained. Calculate their FE values, and for components with higher FE values, apply VMD for secondary decomposition to reduce the signal-to-noise ratio of the decomposition data.

3. Modal Aggregation: For all components obtained in Step 2, use FE to analyze the sub-components and aggregate them based on their correlation. Use GRU and ELMAN for prediction based on the characteristics of the restructured IMF component.

4. Parameter Optimization: To select the optimal hyperparameters for the GRU network, ISSA is used to optimize the four hyperparameters of the GRU network.

5. Uncertainty Prediction: To accurately model the future data trend and obtain detailed interval information, use the error sequence obtained from the deterministic projection results and KDE for interval prediction.

Performance metrics

This study uses the Coefficient Of Determination (R), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE) as predictive indexes to test the predictive performance of the points. The prediction interval coverage probability (PICP) is used as the evaluation metric for interval prediction to test the performance of each prediction model.

$$RMSE=\sqrt{\frac{1}{M}\sum _{i=1}^M{(y_i-{\hat{y}}_i)}^2}$$(25)

$$MAE=\frac{\sum _{i=1}^M|y_i-{\hat{y}}_i|}{M}$$(26)

$$R^2=1-\frac{\sum _{i=1}^M{(y_i-{\hat{y}}_i)}^2}{\sum _{i=1}^M{(y_i-{\bar{y}}_i)}^2}$$(27)

$$PICP=\frac{1}{n}{\sum }_{i=1}^n{p}_i\times 100\%$$(28)

Where $y_i$ is the true value, $y_i^{\wedge }$ is the predicted value, $y_i^{-}$ is the mean value of the samples, and M is the total number of sample points. If the true value falls within the prediction interval, $y_i\in (L_i,U_i)$, $p_i=1$; otherwise, $p_i=0$.

The flowchart of the IVF-ISSA-GRU-ELMAN model is shown in Figure 2.

Figure 2
The structure of IVF-ISSA-GRU/ELMAN model.

Calculus analysis

Data introduction

In this experiment, data was observed from a wind farm in Xinjiang, China, for two months, March and July, of a certain year. Wind power data was collected every 15 minutes.The first 30 days of data (a total of 2880 sampling points) were used as the training set, and the last day’s data (a total of 96 sampling points) were used as the test set for analysis. Table 1 presents basic information about the wind power for the two months, and Figure 3 shows the wind power data for these two months.

Table I
Basic statistical information of two months of wind power data.

Month	Mean(MW)	Max(MW)	Min(MW)	Std(MW)
March	61.9953	201.2530	0.4343	63.4630
July	55.2594	177.9613	0.3405	49.7350

Figure 3
Wind speed data of the two months.

Data processing

Due to the stochastic nature and strong fluctuations in the wind power series, ICEEMDAN was used to perform one decomposition on the wind power series, resulting in IMF components and a residue, as shown in Figures 4a and 5a. Fig. 6 shows the response data plots of each modal component, and Tables 2 and 3 show the specific values of FE for each modal obtained from decomposition.

Figure 4
Results of two-stage decomposition. (a) Decomposition of March wind power series using ICEEMDAN; (b) Decomposition of IMF3 using VMD.

Figure 5
Results of two-stage decomposition. (a) Decomposition of July wind power series using ICEEMDAN; (b) Decomposition of IMF3 using VMD.

Figure 6
FE value for each component. (a) March FE values for each component; (b) July FE values for each component.

Table II
Modal FE values of wind power series for March.

March
ICEEMDAN Mode	FE	Recombination Mode	FE	VMD Mode	FE
IMF1	0.3612			IMF11	0.5392
IMF2	0.4281	R${}_1$(IMF2,3,4,5,11)	0.6053
IMF3	0.5407			IMF12	0.3115
IMF4	0.4869	R${}_2$(IMF6,12)	0.4765
IMF5	0.4087			IMF13	0.188
IMF6	0.302	R${}_3$(IMF7,13,14,15)	0.3873
IMF7	0.1497			IMF14	0.1654
IMF8	0.0588	R${}_4$(IMF8,9,RES)	0.1495
IMF9	0.0086			IMF15	0.207
RES(IMF10)	7.67×10-5

Table III
Modal FE values of wind power series for July.

July
ICEEMDAN Mode	FE	Recombination Mode	FE	VMD Mode	FE
IMF1	0.4145			IMF12	0.3569
IMF2	0.5057	R${}_1$(IMF2,3,4)	0.6846
IMF3	0.6088			IMF13	0.3055
IMF4	0.5223	R${}_2$(IMF5,12,13,16)	0.6968
IMF5	0.3874			IMF14	0.223
IMF6	0.2545
IMF7	0.1428	R${}_3$(IMF6,7,14,15)	0.5454	IMF15	0.2087
IMF8	0.0443
IMF9	0.0052
IMF10	0.0026	R4(IMF8,9,10,RES)	0.0368	IMF16	0.33199
RES(IMF11)	5.79E-06

The high-frequency IMF component concentrates the wind power time series details are more complex and more difficult to predict compared to the low-frequency component, and according to Table 2 and Table 3, it is known that the fuzzy entropy value of IMF3 is the highest, so the VMD decomposition is performed for IMF3.Therefore, VMD was applied to perform a second decomposition on IMF3 to reduce its complexity and eliminate noise, as shown in Figures 4b and 5b. To facilitate subsequent predictions, FE was used to quantify the complexity of the sub-components obtained from the two decompositions, and the components were categorized. As shown in Table 2, the components obtained from the ICEEMDAN decomposition for March were labeled as 1-10, and the components obtained from VMD decomposition (IMF3) were labeled as 11-15. The IMF fuzzy entropy obtained from the two decompositions was recombined based on the principle of similar entropy values. R1 was reconstructed from IMF2-5 from the first decomposition and IMF11 from the second decomposition. R2 was reconstructed from IMF6 and IMF12. R3 was composed of IMF7, 13, and 15. R4 was composed of IMF8-10. For the data from July, after ICEEMDAN decomposition, the components were labeled as 1-11, and the components obtained from VMD decomposition (IMF3) were labeled as 12-16. The IMF fuzzy entropy obtained from the two decompositions is shown in Figure 6, and the recombination was based on the principle of similar entropy values. R1 was reconstructed from IMF2-4 from the first decomposition. R2 was composed of IMF5, 12, 13, and 16. R3 was composed of IMF6, 7, 14, and 15. R4 was composed of IMF8-11. The threshold for determining IMF complexity in this study was set at 0.25. A sequence was considered complex when it exceeded 0.25, and simple when it was ≤0.25. Based on the FE results, for March, IMF1, IMF2, and IMF4 were considered complex sequences and were predicted using ISSA-GRU; IMF3 was considered a simple sequence and was predicted using ELMAN. For July, IMF1-3 were considered complex sequences and were predicted using ISSA-GRU; IMF4 was considered a simple sequence and was predicted using ELMAN.

Comparison results

To validate the effectiveness of our proposed model, a total of 8 models were established, including single models, hybrid models, and distribution comparison experimental models. LSTM and GRU served as the single model control group, while two hybrid models mentioned in the literature (VMD-ICEEMDAN-LSTM and CEEMDAN-SE-VMD-LSTM[35]) were used as the hybrid model control group. The models included VMD-GRU, where the original data was decomposed using VMD, and GRU was used to predict each component; VMD-ISSA-GRU, where the components were predicted using ISSA-optimized GRU after VMD decomposition; and VMD-ISSA-GRU/ELMAN, where components were predicted using either ISSA-GRU or ELMAN based on their complexity levels, for comparison testing.

The warning value of sparrow population was set ST=0.8 and the number of discoverers SD=0.4; the four hyperparameters (L1, L2, Lr, and iteration number K) of the LSTM network were in the range of [10,200], [10,200], [0.001,0.02], and [10,500] for the optimization search, respectively; The LSTM and the GRU solver were both Adam, and the loss function was the sigmoid function, and the gradient threshold is set to 1.Three evaluation metrics, RMSE, MAE and R, were used to evaluate the model deterministic results, and PICP was used to evaluate the non-deterministic prediction results. Models 1-8 represented LSTM, GRU, ICEEMDAN-LSTM, CEEMDAN-SE-VMD-LSTM, VMD-GRU, VMD-ISSA-GRU, VMD-ISSA-GRU/ELMAN, and IVF-ISSA-GRU/ELMAN, respectively. Table 4 presented a comparison of wind power prediction indicators for March and July, while Figures 7 and 8 illustrated error analysis graphs for March and July, respectively.

Table IV
Prediction results of each model.

Month	March			July
Measurement Model	RMSE (MW)	MAE (MW)	R	RMSE (MW)	MAE (MW)	R
LSTM	53.6756	43.8006	0.5667	33.6682	26.0803	0.33292
GRU	42.4948	30.9671	0.6578	27.8759	21.3486	0.54271
ICEEMDAN-LSTM	23.4777	17.9175	0.8775	23.3278	15.1432	0.67975
CEEMDAN-SE-VMD-LSTM	20.8041	16.2833	0.9038	17.1339	13.0226	0.82724
VMD-GRU	17.9028	13.0719	0.9288	23.6342	17.624	0.67129
VMD-ISSA-GRU	3.9577	3.2696	0.9964	6.813	5.1272	0.97323
VMD-ISSA-GRU/ELMAN	3.863	3.1793	0.9967	6.7719	4.9615	0.9735
Textual model	2.8458	2.159	0.9983	1.8605	1.4568	0.99814

Figure 7
Error analysis of March dataset.

Figure 8
Error analysis of July dataset.

By analyzing Table 4, Figures 7 and 8, it can be obtained that:

1. The predictions of LSTM and GRU models for the two-month data sets are similar, with significant differences from the actual values in peak and valley sections, having RMSE values between [42.4948, 53.6756] and [27.8759, 33.6682], and MAE values between [30.9671, 443.8006] and [21.3486, 26.0803], respectively. The graphs visually demonstrate that the predictive accuracy of single models is lower than that of hybrid models.

2. The RMSE and MAE of the VMD-GRU model for the March data set decreased from 42.4948 and 30.9671 to 17.9028 and 13.0719, respectively, compared with the single GRU model. The RMSE and MAE of the VMD-GRU model for the July data decreased from 27.8759 MW and 21.3486 MW to 23.6342 MW and 17.624 MW respectively. Additionally, both data sets had an increase in R by 0.332 and 0.129, respectively. Furthermore, Model 7 outperformed Model 8, with RMSE and MAE for the March data set decreasing from 3.863MW and 3.1793MW to 2.8458MW and 2.159MW, and R increasing from 0.9967 to 0.9983. For the July data set, RMSE and MAE decreased from 6.7719MW and 4.9615MW to 1.8605MW and 1.4568MW, respectively, and R increased from 0.9735 to 0.99811. This indicates that the second decomposition improved the error of the first decomposition and enhanced the model’s predictive performance.

3. The results of the distribution comparison models showed that incorporating the ISSA algorithm to optimize the VMD-ISSA-GRU model for both data sets resulted in a 71%-78% improvement in RMSE and a 70%-75% improvement in MAE. Both comparisons verified that ISSA could optimize model parameters and improve predictive accuracy. Additionally, compared to Model 6, Model 7, which predicted different features using GRU and ELMAN networks, had an increase in RMSE by 0.095MW and 0.041MW, and an increase in MAE by 0.09MW and 0.17MW, for the March and July data sets, respectively. This demonstrates that predicting different features using GRU and ELMAN networks can effectively enhance prediction accuracy.

4. Our proposed ICEEMDAN-VMD-FE-ISSA-GRU/ELMAN model exhibited better prediction results with significantly lower errors compared to the other 7 models. The RMSE values for our model were 2.8458MW and 1.8605MW for March and July, respectively. The MAE values were 2.159MW and 1.4568MW, and R reached 0.99825 and 0.99814, respectively. Compared to the hybrid model control groups (Models 3 and 4) mentioned in the literature, our proposed model showed an improvement in RMSE of [15.2734, 21.4673] and an improvement in MAE of [11.5658, 13.6904] for the July data set, indicating the best fit.

Nondeterministic prediction and effect evaluation

Although our proposed model achieved more accurate wind power deterministic projection results compared to other models, single wind power values obtained from point predictions may not provide sufficient reference for wind farm decision-making. Therefore, based on the point prediction errors mentioned above, we performed interval prediction on the test set. After obtaining the distribution of the error sequence, we used KDE to obtain the 95% and 90% confidence intervals. To verify the method of KDE, we selected the normal distribution estimation method and performed KDE calculations using the Epanechnikow and Triangle kernel functions, and evaluated the prediction results using PICP and PINAW.

Figures 9 and 10 show the interval prediction results based on the Epanechnikow kernel function for March and July, respectively.

Figure 9
March interval prediction results.

Figure 10
July interval prediction results.

Figures 9 and 10 depict the 90% and 95% confidence interval prediction results for our proposed model for the two months. The 95% confidence interval contains 92 and 94 points in the actual wind power measurements for March and July, respectively. Furthermore, for the fluctuation range of wind power, our prediction model includes most of the peak values of wind power, indicating a better ability to predict wind power fluctuations.

Table V
Interval prediction evaluation.

Dataset		March	July
Confidence	Model	PICP	PICP
95%	NDE	93.75%	92.71%
	KDE-Triangle	93.75%	96.88%
	KDE-Epanechnikow	95.83%	97.92%
90%	NDE	90.62%	87.50%
	KDE-Triangle	91.67%	91.67%
	KDE-Epanechnikow	91.67%	93.75%

Table 5 presents the interval evaluation result indicators. According to Table 5, the method based on the Epanechnikow kernel function has higher coverage probability (PICP) compared to the Triangle kernel function and NDE estimation method at the same confidence level. For the July data set at 95% confidence level, the PICP of our proposed model is 97.92%, compared to 91.97% and 87.50% for the Triangle kernel function model and NDE estimation method, respectively. This demonstrates that our proposed model has a higher coverage probability compared to the other two models, making it more valuable for real-world decision-making, and proves the effectiveness and superiority of our proposed interval prediction model.

Conclusions

This study proposed a hybrid wind power interval prediction model based on TVF, GRU, ISSA, and ELMAN. Through verification on two datasets, the following conclusions were drawn:

1. The RMSE of deterministic prediction results of the proposed model in this paper is only 2.8458 MW and 1.8605 MW, which has the smallest value and the smallest error. The largest R-value and the highest goodness-of-fit. The non-deterministic predictions reach 95.83% and 97.92% coverage at 95% confidence level, our model demonstrated superior predictive performance and provided more accurate wind power predictions.

2. The application of ICEEMDAN and VMD for wind power data processing through secondary decomposition effectively reduced the complexity of high-frequency components from the first decomposition, mitigating adverse effects caused by strong stationarity. The use of FE fuzzy entropy technique for classifying and aggregating components from the secondary decomposition simplified the model while enhancing its predictive performance.

3. The application of the ISSA evolutionary algorithm improved the parameters of the GRU network, thereby enhancing the wind power point prediction. By utilizing GRU and ELMAN networks for prediction based on the characteristics of components after the secondary decomposition, the time features of wind power sequences were fully utilized, leading to more effective prediction of wind power sequences.

4. Using the KDE-Epanechnikow method to estimate the error sequence obtained from point predictions enabled us to fully explore its uncertainty information, reducing the limitations of point prediction results and obtaining reliable interval prediction results.

ACKNOWLEDGMENTS

This research was funded by the Key Project of Education Department of Hebei Province, grant number [ZD2020182, ZD2021021]. This research was funded by the Doctoral Fund of Hebei University of Engineering, grant number [20120134]. We sincerely thank the Key Laboratory of Smart Water Conservancy in Hebei Province and the Doctoral Special Fund Project of Hebei University of Engineering for providing simulation data. The authors declare no conflict of interest.

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Publication Dates

• Publication in this collection
  21 Oct 2024
• Date of issue
  2024

History

• Received
  9 Aug 2023
• Accepted
  5 Jan 2024