# Multistep Wind Speed Forecasting Using a Novel Model Hybridizing Singular Spectrum Analysis, Modified Intelligent Optimization, and Rolling Elman Neural Network

Authors: Zhongshan Yang; Jian Wang
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Wind speed high-accuracy forecasting, an important part of the electrical system monitoring and control, is of the essence to protect the safety of wind power utilization. However, the wind speed signals are always intermittent and intrinsic complexity; therefore, it is difficult to forecast them accurately. Many traditional wind speed forecasting studies have focused on single models, which leads to poor prediction accuracy. In this paper, a new hybrid model is proposed to overcome the shortcoming of single models by combining singular spectrum analysis, modified intelligent optimization, and the rolling Elman neural network. In this model, except for the multiple seasonal patterns used to reduce interferences from the original data, the rolling model is utilized to forecast the multistep wind speed. To verify the forecasting ability of the proposed hybrid model, 10 min and 60 min wind speed data from the province of Shandong, China, were proposed in this paper as the case study. Compared to the other models, the proposed hybrid model forecasts the wind speed with higher accuracy.


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## Body

## 1. Introduction

In the past few decades, with environmental degradation and resource depletion, renewable energy [1] has received more attention. Wind energy, one of the cleanest forms of renewable energy, is developing rapidly throughout the world. With the rapid increase in the utilization of wind energy, the primary concern is the security and stability of feeding electricity into the grid [2]. Wind speed high-accuracy forecasting is an important part of the electrical system monitoring and control. However, due to the instability of wind energy and the inherent complexity, transferring electricity into the power grid is limited and costly [3, 4]. To improve the efficiency of wind power and reduce the comprehensive cost of wind energy, accurate prediction of wind speed is necessary.Many methods have been proposed to improve the forecasting accuracy of wind speed in recent decades. Based on the computational mechanism, these forecasting models can be grouped into four main categories: (i) physical models, (ii) statistical models, (iii) intelligence models, and (iv) hybrid forecasting models [5].Physical methods [6, 7], which are based on the lower atmosphere or numerical weather prediction (NWP), can accurately forecast the wind speed. However, the physical methods require long running times and are not applicable for short-term forecasting. Statistical models [8–11], which are known as time-series-based models, do not apply historical data. These models are trained with measurement data and the differences between forecasted and actual wind speed are used to adjust the model parameters. The ARMA and the ARIMA models are the most popular models used to forecast future wind speed. Many forecasting results based on statistical models show that these models are useful in the wind speed forecasting fields [12–15]. These models have numerous advantages, and the approaches need only historical wind speed data and are easy to implement. However, if the nonlinear characteristics of the wind speed series are prominent, the prediction accuracy of these methods decreases rapidly. The intelligent methods adopt artificial intelligence (AI) theories or evolutionary algorithms to forecast wind speed. Many intelligent methods are used to forecast wind speed, such as ANN (Artificial Neural Network) [16–18], FLM (Fuzzy Logic Method) [19, 20], and SVM (Support Vector Machine) [18, 21–26]. Unlike the single methods, the hybrid methods proposed by experts and scholars always combine several models to improve the accuracy of wind speed forecasting. Most of the recently proposed forecasting methods are hybrids, and the decomposition algorithms are often used to enhance their precision [27, 28].Chaotic theory has been used to handle time series in many fields [29–31]. Considering chaotic characteristics of the wind speed series, a hybrid prediction model was introduced in [32] using largest Lyapunov exponent prediction method to predict. Due to the inherent complexity of wind speed, describing the moving trend of wind speed and accurate prediction is difficult. Therefore, many studies use other methods to enhance the forecasting capacity of the original series. The specific methods include hybrid models that employ different approaches or combine different forecasting models to extract the inner traits of the original series in different aspects to perform wind speed forecasting. For the former type of hybrid models, the most common methods, such as wavelet transform (WT), singular spectrum analysis (SSA), and empirical mode decomposition (EMD), are used to preprocess the original series and forecast the wind speeds [32–35]. These data processing methods are used to eliminate the influence of outliers on the forecasting accuracy, thereby improving the forecasting accuracy.In this paper, a novel algorithm is proposed that hybridizes SSA (singular spectrum analysis), FAPSO (Firefly Algorithm and Particle Swarm Optimization), and RENN (rolling Elman neural network), to forecast wind speed. To verify the performance of the model, several hybrid models and single models are also used to forecast wind speed. In this model, besides the multiple seasonal patterns used to reduce interferences from the original data, the rolling model is utilized to forecast the multistep wind speed. To verify the forecasting ability of the proposed hybrid model, 10 min and 60 min wind speed data from the province of Shandong, China, were used as the case study.The details of the algorithm are described below, and the flow diagram is shown in Figure1.Figure 1
The flowchart of the proposed integrated forecasting model.Step 1.
The SSA is used to decompose the original wind speed datasets into several subseries. Then, the new series is reconstructed. The wind speed data used in this paper is typically a chaotic time series, and the use of SSA can eliminate the influence of outliers and improve the prediction accuracy of the wind speed forecast model.Step 2.
The hybrid optimization algorithm (FAPSO) that combines the FA with the PSO is utilized to optimize the weights and thresholds of the ENN model. The optimization algorithm can provide better initial weights and thresholds to the ENN and improve the search ability. Compared with the single optimization model, the hybrid optimization model has better optimization effects.Step 3.
Construct the ENN model for the reconstructed series. Then, use the established model to forecast the one-step wind speed. The optimized ENN model can avoid getting trapped into local optimum and the global searching ability of the algorithm is enhanced.Step 4.
The rolling ENN model is used to forecast the multistep results. Multistep wind speed forecasting with high-precision is helpful for electricity production to produce various benefits, such as avoiding a power-grid collapse, reducing production costs, and reducing the spinning reserve capacity of thermal power units.Step 5.
The Diebold-Mariano test is used to validate the accuracy and stability of the proposed model.


## 2. Methodology

In this paper, numerous methods are involved. In this section, the relative algorithms including singular spectrum analysis, the firefly algorithm, particle swarm optimization, and the hybrid model are described in detail.


### 2.1. Singular Spectrum Analysis

The singular spectrum analysis [36–38] is a signal processing technique capable of capturing the intrinsic oscillation modes of a signal. The SSA has two main stages: decomposition and reconstruction.


#### 2.1.1. Wind Speed Time Series Decomposition

The decomposition has two main steps: embedding and SVD (singular value decomposition). During the embedding stage, given a time seriesXN=(x0,…,xN-1),N is the effective length. DefineL  (1<L<N) called “window length” and K=N-L+1.To perform the embedding, the original time series is mapped into a sequence of lagged vectors of sizeLby forming K=N-L+1 lagged vectors(1)xi=xi-1,…,xi+L-2T,i=1,…,K.Then, the trajectory matrix is derived:(2)X=xi:⋯:xK=xiji,j=1L,K=x0x1⋯xK-1x1x2⋯xK⋮⋮⋱⋮xL-1xL⋯xN-1.From the trajectory matrix, both the rows and columns ofX are subseries of the original series, and X is a Hankel matrix, which means it has equal elements on antidiagonals.In the singular value decomposition step, the singular value decomposition of the matrixX can be computed through eigenvalues and eigenvectors of the matrix XXT. Suppose λi(λ1>λ2>⋯>λL>0) as the eigenvalues of XXT, and suppose Ui as the corresponding eigenvector. The singular value decomposition of the trajectory matrix X is shown below:(3)X=X1+X2+⋯+Xd,(4)Vi=XTUiλi,(5)Xi=UiViλi.The matricesXi are elementary matrices. The collection (Ui,ViT,λi) is called the ith eigentriple of the SVD. Each eigentriple consists of an eigenvector, a factor vector, and a singular value. Ui and Vi are ith left singular vectors and right singular vectors of X, respectively.


#### 2.1.2. Wind Speed Series Reconstruction

The decomposition has two main steps: grouping and diagonal averaging. During the grouping stage,I={1,2,…,d} is split into m disjoint groups I1,I2,…,Im. Let I={i1,i2,…,ik}; then, the resultant matrix XI corresponding to the group I is defined as(6)XI=Xi1+Xi2+⋯+XiK.These matrices are computed forI=I1,I2,…,Im and the expansion of (3) leads to the decomposition as follows:(7)X=XI1+XI2+⋯+XIm.In the diagonal averaging step, the grouped decomposition in (6) will be transformed into a new series of length N. Let Y be a L×K matrix with elements yij  (1≤i≤L,1≤j≤K). Diagonal averaging transfers the matrix Y to the series h1,h2,…,hm by the following formula:(8)hk=1k+1∑m=1k+1ym,K-m+2∗,0≤k≤l∗-11k∗∑m=1l∗ym,K-m+2∗,l∗-1≤k≤k∗1N-k∑m=k-k∗+2N-k∗+1ym,K-m+2∗,k∗≤k≤N,where l∗=min⁡(L,K); k∗=max⁡(L,K); N=L+K-1; if L<K, yij∗=yij; and if not yij∗=yji.The original time series can be decomposed intom series by SSA(9)xn=∑k=1mx~nk⁡,n=0,1,…,N-1,where x~n(k) is the result of diagonal averaging of the matrix XIk.


### 2.2. Elman Neural Network (ENN)

The Elman recurrent neural network, proposed by Elman, is a partial recurrent network model [39]. Compared with classic feed-forward perception and pure recurrent network, ENN has a context layer that feeds back the hidden layer outputs in the previous time-steps. The context layer can enhance the ability of processing dynamic information and improve the forecasting accuracy.The neurons contained in each layer are used to disseminate information from one layer to another. The nonlinear state space expression of Elman networks is as follows:(10)yk=gω3xk,xk=fω1xck+ω2uk-1,xck=xk-1,where y is the m-dimension output node vector; x is n-dimension hidden layer node vector; u is r-dimension input layer vector; xc is the feedback state vector; ω1, ω2, and ω3 denote their corresponding weights; g(∗) is the transfer function of the output neuron; and f(∗) is the transfer function of the hidden neuron.Then, adjust the weights of the network to minimize the squared error between the actual values and forecasting results:(11)min⁡E=∑k=1nykω-y~kω2,where y~k(ω) is a target output vector.Although the ENN has strong predictive power, the limitations are obvious. The initial weights and threshold values of ENN are randomly generated, the training speed is slow, and ENN is susceptible to falling into the local optimal value. The intelligent optimization algorithm can effectively overcome these shortcomings.


### 2.3. Firefly Algorithm and Particle Swarm Optimization (FAPSO)

The optimization algorithm is composed of the firefly algorithm and particle swarm optimization. Compared with a single optimization algorithm, the proposed optimization algorithm avoids many shortcomings and determines a better solution.


#### 2.3.1. Firefly Algorithm (FA)

The firefly algorithm, proposed by Yang, is a multimodal nature inspired metaheuristic algorithm based on the flashing behavior of fireflies [40, 41]. The algorithm has proved effective in solving linear design problems and multimodal optimization problems.The firefly algorithm has two stages, which are described as follows.Step 1.
The brightness is dependent on the intensity of light emitted by the firefly. Suppose there are a group of fireflies and the position for anith firefly is ξi, where f(ξi) indicates the fitness value of the firefly. The brightness I of a firefly is chosen to reflect the fitness value of its current position(12)Ii=fξiwhere1≤i≤n.Step 2.
All fireflies have a unique attractivenessIi, which indicates the ability to attract other fireflies. The attractiveness is related to the distance factor rij at locations xi and xj, and between the two corresponding fireflies, i and j are given by(13)rij=ξi-ξj=∑k=1dξi,k-ξj,k2.The attractiveness functionβ(r) of the firefly is computed as(14)βijrij=β0e-γrij2,where β0 is the largest attraction and γ is the coefficient of light absorption.The movement of the less bright firefly toward the most bright firefly is computed as(15)ξit+1=ξit+β0e-γrij2ξj-ξi+ηrand-12,where η is the randomization parameter and rand is a randomly selected number between the interval [0,1]. A rudimentary FA algorithm is outlined in Algorithm 1.Algorithm 1:FA.
Parameters: F i—the fitness function of firefly  i. I t e r m a x—the maximum number of iterations. N—the number of fireflies. I i—light intensity of firefly  i. γ—coefficient of light absorption. (1)/∗Set the parameters of FA./∗ (2)/∗Generate initial population of fireflies ξi randomly./∗ (3)FOR EACH  i: 1≤i≤n  DO (4) Evaluate the corresponding fitness functionFi. (5)END FOR (6)/∗Determine light intensity./∗ (7)FOR EACH  i:  1≤i≤n  DO (8) Determine light intensityIi  using  F(ξi) (10)WHILE  (g<Itermax)   DO (11)FOR EACH  i=1:n  DO (12)FOR EACH  j=1:n  DO (13)/∗Move firefly  i  towards  j  in all  d  dimensions./∗ (14)IF  (Ij>Ii)  THEN (15)rij=ξi-ξj=∑k=1dxi,k-xj,k2 (16)ξi=ξi+β0e-γr2ξj-ξi+αrand-1/2 (17)END IF (18)/∗Evaluate the new solution and update the new light intensity  Ii./∗ (19)END FOR  j (20)END FOR  i (21)END WHILE


#### 2.3.2. Particle Swarm Optimization (PSO)

Particle Swarm Optimization (developed by Kennedy and Eberhart) [42, 43] is based on the behavior of birds. The principle of PSO is assuming a location has no mass or volume, flying like a bird in multidimensional space, and not only adjusts its position but also exchanges information about its current position in search space according to its own earlier experience and that of its neighbors. In this mechanism, members of a swarm communicate their information and modify their positions and velocities using the group information according to the best position appearing in the current movement of the swarm. The particles of the swarm find the optimal point by cooperation. The particle velocity and position updating formulas are shown below:(16)vit+1=ω∗vit+c1∗rand1∗lbestit-xit+c2∗rand1∗gbestit-xit,xit+1=xit+vit+1,where ω is the inertia weight; vi is the velocity of the ith particle; c1 and c2 are acceleration coefficients (nonnegative constant); rand(1) is a random value between 0 and 1; xi(t) is the position of the ith particle; lbest(t) and gbest(t) denote local best particle of the ith particle and global best particle among local bests at time t.A rudimentary PSO algorithm is outlined in Algorithm2.Algorithm 2:PSO.
Parameters: n—the number of particles. I t e r m a x—the maximum number of iterations. w , c , v—the parameters of PSO. (1)/∗Set the parameters of PSO./∗ (2)/∗Generate initial population of particles randomly./∗ (3)FOR EACH  i: 1≤i≤n  DO (4)xi=rand(); (5)P=xiiter:1≤i≤n; (6)END FOR (7)/∗Find the best value of particles./∗ (8)WHILE  (g≤Itermax)  DO (9)/∗Find the best fitness value for each candidates./∗ (10)FOR EACH  i: xiiter∈P  DO (11) Calculate each particle fitness function. (12)IF  pbesti>fitnessxiiter  THEN (13)pbesti=fitnessxiiter (14)END IF (15)END FOR (16)/∗Choose the candidate with the best fitness value of all the candidates/∗ (17)FOR EACHi: xiiter∈P  DO (18)IF  gbest>pbesti  THEN (19)gbest=pbesti (20)xbest=xiiter (21)END IF (22)END FOR (23)/∗Update the values of all the candidates by using PSO’s evolution equations./∗ (24)FOR EACHi:  xiiter∈P  DO (25)vit+1=ω∗vit+c1∗rand()∗lbestit-xit+c2∗rand()∗gbestit-xit (26)xit+1=xit+vit+1 (27)END FOR (28)END WHILE


#### 2.3.3. Hybrid Optimization Algorithm

In this section, a modified optimization model hybridizing firefly algorithm and particle swarm optimization is proposed to improve the accuracy of wind speed forecasting. The specifics of the FAPSO are described below.Step 1.
The firefly algorithm was used to optimizen fireflies.(1.1)
Generate initial population of fireflies randomly.(1.2)
Compute the brightness of each firefly by using objective function.(1.3)
Move firefly and evaluate new fireflies.(1.4)
Rank the fireflies and find the current best as the firefly researched.(1.5)
Optimizen fireflies in total.Step 2.
The particle swarm algorithm is used to search the best particle.(2.1)
The fireflies, searched in the first step, are the initial population.(2.2)
Calculate the fitness values of particles.(2.3)
Find the global best position.(2.4)
Update the positions and velocities of the agents.(2.5)
Determine whether the termination conditions have been satisfied; if so, proceed to step (2.1) and continue to search; otherwise, continue to step (2.6).(2.6)
Find the best global position.


#### 2.3.4. Test of FAPSO

To verify the optimization performance and convergence speed of the modified algorithm, four benchmark functions are selected in this paper. These benchmark functions have different characteristics, which are used to fully investigate the optimization ability of the algorithm. The four common test functions are shown in Table1. The experimental parameters of PSO and FAPSO are shown in Table 2.Table 1 Function name Test function Variable domain Global optimum Function characteristic Sphere f 1 X =  ∑ i = 1 D x i 2 x i ∈ - 5.12,5.12 x i = 0 ,   f 1 ( X ) = 0 Unimodal Rosenbrock f 2 X =  ∑ i = 1 D - 1 100 x i 2 - x i + 1 2 + x i - 1 2 x i ∈ - 2.084,2.084 x i = 0 ,   f 2 ( X ) = 0 Unimodal Rastrigin f 3 X =  ∑ i = 1 D x i 2 - 10 cos ⁡ 2 π x i + 10 x i ∈ - 5.12,5.12 x i = 0 ,   f 3 ( X ) = 0 Multimodal Griewank f 4 X = 1 4000  ∑ i = 1 D x i 2 - ∏ i = 1 D cos ⁡ x i i + 1 x i ∈ - 5.12,5.12 x i = 0 ,   f 4 ( X ) = 0 MultimodalTable 2
The experimental parameters of PSO and FAPSO.
Experimental parameters PSO FAPSO Maximum iteration 2000 2000 Population size 40 40 w 0.729 0.729 c 1.494 1.494 v m a x 0.5 0.5 Dimension 30 30 Number of experiments 30 30Thirty experiments, searching for the minimum value point by 2000 iterations, were carried out independently. The results, including the maximum, minimum, average, and standard deviations, are displayed in Table3. The results show that the optimization effect of FAPSO is better than PSO.Table 3
Test results of PSO and FAPSO.
Test function Algorithm Min value Max value Average value Standard deviation Sphere PSO 0.4276 1.198 0.7030 0.1740 FAPSO 0.0481 0.1158 0.0837 0.0156 Rosenbrock PSO 56.2745 131.5162 90.7276 18.7055 FAPSO 35.1140 43.7005 38.3808 1.8540 Rastrigin PSO 169.0 252.1567 201.6199 21.5151 FAPSO 40.8482 71.1049 59.012 6.5202 Griewank PSO 0.0478 0.3839 0.0884 0.0607 FAPSO 0.0090 0.0200 0.0145 0.0026


### 2.4. The Proposed Hybrid Model

In this paper, a novel algorithm, SSA-FAPSO-RENN, is proposed to forecast wind speed. SSA is used to acquire the moving tendency of wind speed and enhance the forecasting abilities. The hybrid optimization algorithm (FAPSO) that combines the FA and the PSO is utilized to optimize the parameters of the ENN model. To forecast the multistep wind speed, the rolling Elman neural network (RENN) model is used.For the convenience of narrative, the proposed hybrid model is named the SSA-FAPSO-RENN model.


## 2.1. Singular Spectrum Analysis

The singular spectrum analysis [36–38] is a signal processing technique capable of capturing the intrinsic oscillation modes of a signal. The SSA has two main stages: decomposition and reconstruction.


### 2.1.1. Wind Speed Time Series Decomposition

The decomposition has two main steps: embedding and SVD (singular value decomposition). During the embedding stage, given a time seriesXN=(x0,…,xN-1),N is the effective length. DefineL  (1<L<N) called “window length” and K=N-L+1.To perform the embedding, the original time series is mapped into a sequence of lagged vectors of sizeLby forming K=N-L+1 lagged vectors(1)xi=xi-1,…,xi+L-2T,i=1,…,K.Then, the trajectory matrix is derived:(2)X=xi:⋯:xK=xiji,j=1L,K=x0x1⋯xK-1x1x2⋯xK⋮⋮⋱⋮xL-1xL⋯xN-1.From the trajectory matrix, both the rows and columns ofX are subseries of the original series, and X is a Hankel matrix, which means it has equal elements on antidiagonals.In the singular value decomposition step, the singular value decomposition of the matrixX can be computed through eigenvalues and eigenvectors of the matrix XXT. Suppose λi(λ1>λ2>⋯>λL>0) as the eigenvalues of XXT, and suppose Ui as the corresponding eigenvector. The singular value decomposition of the trajectory matrix X is shown below:(3)X=X1+X2+⋯+Xd,(4)Vi=XTUiλi,(5)Xi=UiViλi.The matricesXi are elementary matrices. The collection (Ui,ViT,λi) is called the ith eigentriple of the SVD. Each eigentriple consists of an eigenvector, a factor vector, and a singular value. Ui and Vi are ith left singular vectors and right singular vectors of X, respectively.


### 2.1.2. Wind Speed Series Reconstruction

The decomposition has two main steps: grouping and diagonal averaging. During the grouping stage,I={1,2,…,d} is split into m disjoint groups I1,I2,…,Im. Let I={i1,i2,…,ik}; then, the resultant matrix XI corresponding to the group I is defined as(6)XI=Xi1+Xi2+⋯+XiK.These matrices are computed forI=I1,I2,…,Im and the expansion of (3) leads to the decomposition as follows:(7)X=XI1+XI2+⋯+XIm.In the diagonal averaging step, the grouped decomposition in (6) will be transformed into a new series of length N. Let Y be a L×K matrix with elements yij  (1≤i≤L,1≤j≤K). Diagonal averaging transfers the matrix Y to the series h1,h2,…,hm by the following formula:(8)hk=1k+1∑m=1k+1ym,K-m+2∗,0≤k≤l∗-11k∗∑m=1l∗ym,K-m+2∗,l∗-1≤k≤k∗1N-k∑m=k-k∗+2N-k∗+1ym,K-m+2∗,k∗≤k≤N,where l∗=min⁡(L,K); k∗=max⁡(L,K); N=L+K-1; if L<K, yij∗=yij; and if not yij∗=yji.The original time series can be decomposed intom series by SSA(9)xn=∑k=1mx~nk⁡,n=0,1,…,N-1,where x~n(k) is the result of diagonal averaging of the matrix XIk.


## 2.1.1. Wind Speed Time Series Decomposition

The decomposition has two main steps: embedding and SVD (singular value decomposition). During the embedding stage, given a time seriesXN=(x0,…,xN-1),N is the effective length. DefineL  (1<L<N) called “window length” and K=N-L+1.To perform the embedding, the original time series is mapped into a sequence of lagged vectors of sizeLby forming K=N-L+1 lagged vectors(1)xi=xi-1,…,xi+L-2T,i=1,…,K.Then, the trajectory matrix is derived:(2)X=xi:⋯:xK=xiji,j=1L,K=x0x1⋯xK-1x1x2⋯xK⋮⋮⋱⋮xL-1xL⋯xN-1.From the trajectory matrix, both the rows and columns ofX are subseries of the original series, and X is a Hankel matrix, which means it has equal elements on antidiagonals.In the singular value decomposition step, the singular value decomposition of the matrixX can be computed through eigenvalues and eigenvectors of the matrix XXT. Suppose λi(λ1>λ2>⋯>λL>0) as the eigenvalues of XXT, and suppose Ui as the corresponding eigenvector. The singular value decomposition of the trajectory matrix X is shown below:(3)X=X1+X2+⋯+Xd,(4)Vi=XTUiλi,(5)Xi=UiViλi.The matricesXi are elementary matrices. The collection (Ui,ViT,λi) is called the ith eigentriple of the SVD. Each eigentriple consists of an eigenvector, a factor vector, and a singular value. Ui and Vi are ith left singular vectors and right singular vectors of X, respectively.


## 2.1.2. Wind Speed Series Reconstruction

The decomposition has two main steps: grouping and diagonal averaging. During the grouping stage,I={1,2,…,d} is split into m disjoint groups I1,I2,…,Im. Let I={i1,i2,…,ik}; then, the resultant matrix XI corresponding to the group I is defined as(6)XI=Xi1+Xi2+⋯+XiK.These matrices are computed forI=I1,I2,…,Im and the expansion of (3) leads to the decomposition as follows:(7)X=XI1+XI2+⋯+XIm.In the diagonal averaging step, the grouped decomposition in (6) will be transformed into a new series of length N. Let Y be a L×K matrix with elements yij  (1≤i≤L,1≤j≤K). Diagonal averaging transfers the matrix Y to the series h1,h2,…,hm by the following formula:(8)hk=1k+1∑m=1k+1ym,K-m+2∗,0≤k≤l∗-11k∗∑m=1l∗ym,K-m+2∗,l∗-1≤k≤k∗1N-k∑m=k-k∗+2N-k∗+1ym,K-m+2∗,k∗≤k≤N,where l∗=min⁡(L,K); k∗=max⁡(L,K); N=L+K-1; if L<K, yij∗=yij; and if not yij∗=yji.The original time series can be decomposed intom series by SSA(9)xn=∑k=1mx~nk⁡,n=0,1,…,N-1,where x~n(k) is the result of diagonal averaging of the matrix XIk.


## 2.2. Elman Neural Network (ENN)

The Elman recurrent neural network, proposed by Elman, is a partial recurrent network model [39]. Compared with classic feed-forward perception and pure recurrent network, ENN has a context layer that feeds back the hidden layer outputs in the previous time-steps. The context layer can enhance the ability of processing dynamic information and improve the forecasting accuracy.The neurons contained in each layer are used to disseminate information from one layer to another. The nonlinear state space expression of Elman networks is as follows:(10)yk=gω3xk,xk=fω1xck+ω2uk-1,xck=xk-1,where y is the m-dimension output node vector; x is n-dimension hidden layer node vector; u is r-dimension input layer vector; xc is the feedback state vector; ω1, ω2, and ω3 denote their corresponding weights; g(∗) is the transfer function of the output neuron; and f(∗) is the transfer function of the hidden neuron.Then, adjust the weights of the network to minimize the squared error between the actual values and forecasting results:(11)min⁡E=∑k=1nykω-y~kω2,where y~k(ω) is a target output vector.Although the ENN has strong predictive power, the limitations are obvious. The initial weights and threshold values of ENN are randomly generated, the training speed is slow, and ENN is susceptible to falling into the local optimal value. The intelligent optimization algorithm can effectively overcome these shortcomings.


## 2.3. Firefly Algorithm and Particle Swarm Optimization (FAPSO)

The optimization algorithm is composed of the firefly algorithm and particle swarm optimization. Compared with a single optimization algorithm, the proposed optimization algorithm avoids many shortcomings and determines a better solution.


### 2.3.1. Firefly Algorithm (FA)

The firefly algorithm, proposed by Yang, is a multimodal nature inspired metaheuristic algorithm based on the flashing behavior of fireflies [40, 41]. The algorithm has proved effective in solving linear design problems and multimodal optimization problems.The firefly algorithm has two stages, which are described as follows.Step 1.
The brightness is dependent on the intensity of light emitted by the firefly. Suppose there are a group of fireflies and the position for anith firefly is ξi, where f(ξi) indicates the fitness value of the firefly. The brightness I of a firefly is chosen to reflect the fitness value of its current position(12)Ii=fξiwhere1≤i≤n.Step 2.
All fireflies have a unique attractivenessIi, which indicates the ability to attract other fireflies. The attractiveness is related to the distance factor rij at locations xi and xj, and between the two corresponding fireflies, i and j are given by(13)rij=ξi-ξj=∑k=1dξi,k-ξj,k2.The attractiveness functionβ(r) of the firefly is computed as(14)βijrij=β0e-γrij2,where β0 is the largest attraction and γ is the coefficient of light absorption.The movement of the less bright firefly toward the most bright firefly is computed as(15)ξit+1=ξit+β0e-γrij2ξj-ξi+ηrand-12,where η is the randomization parameter and rand is a randomly selected number between the interval [0,1]. A rudimentary FA algorithm is outlined in Algorithm 1.Algorithm 1:FA.
Parameters: F i—the fitness function of firefly  i. I t e r m a x—the maximum number of iterations. N—the number of fireflies. I i—light intensity of firefly  i. γ—coefficient of light absorption. (1)/∗Set the parameters of FA./∗ (2)/∗Generate initial population of fireflies ξi randomly./∗ (3)FOR EACH  i: 1≤i≤n  DO (4) Evaluate the corresponding fitness functionFi. (5)END FOR (6)/∗Determine light intensity./∗ (7)FOR EACH  i:  1≤i≤n  DO (8) Determine light intensityIi  using  F(ξi) (10)WHILE  (g<Itermax)   DO (11)FOR EACH  i=1:n  DO (12)FOR EACH  j=1:n  DO (13)/∗Move firefly  i  towards  j  in all  d  dimensions./∗ (14)IF  (Ij>Ii)  THEN (15)rij=ξi-ξj=∑k=1dxi,k-xj,k2 (16)ξi=ξi+β0e-γr2ξj-ξi+αrand-1/2 (17)END IF (18)/∗Evaluate the new solution and update the new light intensity  Ii./∗ (19)END FOR  j (20)END FOR  i (21)END WHILE


### 2.3.2. Particle Swarm Optimization (PSO)

Particle Swarm Optimization (developed by Kennedy and Eberhart) [42, 43] is based on the behavior of birds. The principle of PSO is assuming a location has no mass or volume, flying like a bird in multidimensional space, and not only adjusts its position but also exchanges information about its current position in search space according to its own earlier experience and that of its neighbors. In this mechanism, members of a swarm communicate their information and modify their positions and velocities using the group information according to the best position appearing in the current movement of the swarm. The particles of the swarm find the optimal point by cooperation. The particle velocity and position updating formulas are shown below:(16)vit+1=ω∗vit+c1∗rand1∗lbestit-xit+c2∗rand1∗gbestit-xit,xit+1=xit+vit+1,where ω is the inertia weight; vi is the velocity of the ith particle; c1 and c2 are acceleration coefficients (nonnegative constant); rand(1) is a random value between 0 and 1; xi(t) is the position of the ith particle; lbest(t) and gbest(t) denote local best particle of the ith particle and global best particle among local bests at time t.A rudimentary PSO algorithm is outlined in Algorithm2.Algorithm 2:PSO.
Parameters: n—the number of particles. I t e r m a x—the maximum number of iterations. w , c , v—the parameters of PSO. (1)/∗Set the parameters of PSO./∗ (2)/∗Generate initial population of particles randomly./∗ (3)FOR EACH  i: 1≤i≤n  DO (4)xi=rand(); (5)P=xiiter:1≤i≤n; (6)END FOR (7)/∗Find the best value of particles./∗ (8)WHILE  (g≤Itermax)  DO (9)/∗Find the best fitness value for each candidates./∗ (10)FOR EACH  i: xiiter∈P  DO (11) Calculate each particle fitness function. (12)IF  pbesti>fitnessxiiter  THEN (13)pbesti=fitnessxiiter (14)END IF (15)END FOR (16)/∗Choose the candidate with the best fitness value of all the candidates/∗ (17)FOR EACHi: xiiter∈P  DO (18)IF  gbest>pbesti  THEN (19)gbest=pbesti (20)xbest=xiiter (21)END IF (22)END FOR (23)/∗Update the values of all the candidates by using PSO’s evolution equations./∗ (24)FOR EACHi:  xiiter∈P  DO (25)vit+1=ω∗vit+c1∗rand()∗lbestit-xit+c2∗rand()∗gbestit-xit (26)xit+1=xit+vit+1 (27)END FOR (28)END WHILE


### 2.3.3. Hybrid Optimization Algorithm

In this section, a modified optimization model hybridizing firefly algorithm and particle swarm optimization is proposed to improve the accuracy of wind speed forecasting. The specifics of the FAPSO are described below.Step 1.
The firefly algorithm was used to optimizen fireflies.(1.1)
Generate initial population of fireflies randomly.(1.2)
Compute the brightness of each firefly by using objective function.(1.3)
Move firefly and evaluate new fireflies.(1.4)
Rank the fireflies and find the current best as the firefly researched.(1.5)
Optimizen fireflies in total.Step 2.
The particle swarm algorithm is used to search the best particle.(2.1)
The fireflies, searched in the first step, are the initial population.(2.2)
Calculate the fitness values of particles.(2.3)
Find the global best position.(2.4)
Update the positions and velocities of the agents.(2.5)
Determine whether the termination conditions have been satisfied; if so, proceed to step (2.1) and continue to search; otherwise, continue to step (2.6).(2.6)
Find the best global position.


### 2.3.4. Test of FAPSO

To verify the optimization performance and convergence speed of the modified algorithm, four benchmark functions are selected in this paper. These benchmark functions have different characteristics, which are used to fully investigate the optimization ability of the algorithm. The four common test functions are shown in Table1. The experimental parameters of PSO and FAPSO are shown in Table 2.Table 1 Function name Test function Variable domain Global optimum Function characteristic Sphere f 1 X =  ∑ i = 1 D x i 2 x i ∈ - 5.12,5.12 x i = 0 ,   f 1 ( X ) = 0 Unimodal Rosenbrock f 2 X =  ∑ i = 1 D - 1 100 x i 2 - x i + 1 2 + x i - 1 2 x i ∈ - 2.084,2.084 x i = 0 ,   f 2 ( X ) = 0 Unimodal Rastrigin f 3 X =  ∑ i = 1 D x i 2 - 10 cos ⁡ 2 π x i + 10 x i ∈ - 5.12,5.12 x i = 0 ,   f 3 ( X ) = 0 Multimodal Griewank f 4 X = 1 4000  ∑ i = 1 D x i 2 - ∏ i = 1 D cos ⁡ x i i + 1 x i ∈ - 5.12,5.12 x i = 0 ,   f 4 ( X ) = 0 MultimodalTable 2
The experimental parameters of PSO and FAPSO.
Experimental parameters PSO FAPSO Maximum iteration 2000 2000 Population size 40 40 w 0.729 0.729 c 1.494 1.494 v m a x 0.5 0.5 Dimension 30 30 Number of experiments 30 30Thirty experiments, searching for the minimum value point by 2000 iterations, were carried out independently. The results, including the maximum, minimum, average, and standard deviations, are displayed in Table3. The results show that the optimization effect of FAPSO is better than PSO.Table 3
Test results of PSO and FAPSO.
Test function Algorithm Min value Max value Average value Standard deviation Sphere PSO 0.4276 1.198 0.7030 0.1740 FAPSO 0.0481 0.1158 0.0837 0.0156 Rosenbrock PSO 56.2745 131.5162 90.7276 18.7055 FAPSO 35.1140 43.7005 38.3808 1.8540 Rastrigin PSO 169.0 252.1567 201.6199 21.5151 FAPSO 40.8482 71.1049 59.012 6.5202 Griewank PSO 0.0478 0.3839 0.0884 0.0607 FAPSO 0.0090 0.0200 0.0145 0.0026


## 2.3.1. Firefly Algorithm (FA)

The firefly algorithm, proposed by Yang, is a multimodal nature inspired metaheuristic algorithm based on the flashing behavior of fireflies [40, 41]. The algorithm has proved effective in solving linear design problems and multimodal optimization problems.The firefly algorithm has two stages, which are described as follows.Step 1.
The brightness is dependent on the intensity of light emitted by the firefly. Suppose there are a group of fireflies and the position for anith firefly is ξi, where f(ξi) indicates the fitness value of the firefly. The brightness I of a firefly is chosen to reflect the fitness value of its current position(12)Ii=fξiwhere1≤i≤n.Step 2.
All fireflies have a unique attractivenessIi, which indicates the ability to attract other fireflies. The attractiveness is related to the distance factor rij at locations xi and xj, and between the two corresponding fireflies, i and j are given by(13)rij=ξi-ξj=∑k=1dξi,k-ξj,k2.The attractiveness functionβ(r) of the firefly is computed as(14)βijrij=β0e-γrij2,where β0 is the largest attraction and γ is the coefficient of light absorption.The movement of the less bright firefly toward the most bright firefly is computed as(15)ξit+1=ξit+β0e-γrij2ξj-ξi+ηrand-12,where η is the randomization parameter and rand is a randomly selected number between the interval [0,1]. A rudimentary FA algorithm is outlined in Algorithm 1.Algorithm 1:FA.
Parameters: F i—the fitness function of firefly  i. I t e r m a x—the maximum number of iterations. N—the number of fireflies. I i—light intensity of firefly  i. γ—coefficient of light absorption. (1)/∗Set the parameters of FA./∗ (2)/∗Generate initial population of fireflies ξi randomly./∗ (3)FOR EACH  i: 1≤i≤n  DO (4) Evaluate the corresponding fitness functionFi. (5)END FOR (6)/∗Determine light intensity./∗ (7)FOR EACH  i:  1≤i≤n  DO (8) Determine light intensityIi  using  F(ξi) (10)WHILE  (g<Itermax)   DO (11)FOR EACH  i=1:n  DO (12)FOR EACH  j=1:n  DO (13)/∗Move firefly  i  towards  j  in all  d  dimensions./∗ (14)IF  (Ij>Ii)  THEN (15)rij=ξi-ξj=∑k=1dxi,k-xj,k2 (16)ξi=ξi+β0e-γr2ξj-ξi+αrand-1/2 (17)END IF (18)/∗Evaluate the new solution and update the new light intensity  Ii./∗ (19)END FOR  j (20)END FOR  i (21)END WHILE


## 2.3.2. Particle Swarm Optimization (PSO)

Particle Swarm Optimization (developed by Kennedy and Eberhart) [42, 43] is based on the behavior of birds. The principle of PSO is assuming a location has no mass or volume, flying like a bird in multidimensional space, and not only adjusts its position but also exchanges information about its current position in search space according to its own earlier experience and that of its neighbors. In this mechanism, members of a swarm communicate their information and modify their positions and velocities using the group information according to the best position appearing in the current movement of the swarm. The particles of the swarm find the optimal point by cooperation. The particle velocity and position updating formulas are shown below:(16)vit+1=ω∗vit+c1∗rand1∗lbestit-xit+c2∗rand1∗gbestit-xit,xit+1=xit+vit+1,where ω is the inertia weight; vi is the velocity of the ith particle; c1 and c2 are acceleration coefficients (nonnegative constant); rand(1) is a random value between 0 and 1; xi(t) is the position of the ith particle; lbest(t) and gbest(t) denote local best particle of the ith particle and global best particle among local bests at time t.A rudimentary PSO algorithm is outlined in Algorithm2.Algorithm 2:PSO.
Parameters: n—the number of particles. I t e r m a x—the maximum number of iterations. w , c , v—the parameters of PSO. (1)/∗Set the parameters of PSO./∗ (2)/∗Generate initial population of particles randomly./∗ (3)FOR EACH  i: 1≤i≤n  DO (4)xi=rand(); (5)P=xiiter:1≤i≤n; (6)END FOR (7)/∗Find the best value of particles./∗ (8)WHILE  (g≤Itermax)  DO (9)/∗Find the best fitness value for each candidates./∗ (10)FOR EACH  i: xiiter∈P  DO (11) Calculate each particle fitness function. (12)IF  pbesti>fitnessxiiter  THEN (13)pbesti=fitnessxiiter (14)END IF (15)END FOR (16)/∗Choose the candidate with the best fitness value of all the candidates/∗ (17)FOR EACHi: xiiter∈P  DO (18)IF  gbest>pbesti  THEN (19)gbest=pbesti (20)xbest=xiiter (21)END IF (22)END FOR (23)/∗Update the values of all the candidates by using PSO’s evolution equations./∗ (24)FOR EACHi:  xiiter∈P  DO (25)vit+1=ω∗vit+c1∗rand()∗lbestit-xit+c2∗rand()∗gbestit-xit (26)xit+1=xit+vit+1 (27)END FOR (28)END WHILE


## 2.3.3. Hybrid Optimization Algorithm

In this section, a modified optimization model hybridizing firefly algorithm and particle swarm optimization is proposed to improve the accuracy of wind speed forecasting. The specifics of the FAPSO are described below.Step 1.
The firefly algorithm was used to optimizen fireflies.(1.1)
Generate initial population of fireflies randomly.(1.2)
Compute the brightness of each firefly by using objective function.(1.3)
Move firefly and evaluate new fireflies.(1.4)
Rank the fireflies and find the current best as the firefly researched.(1.5)
Optimizen fireflies in total.Step 2.
The particle swarm algorithm is used to search the best particle.(2.1)
The fireflies, searched in the first step, are the initial population.(2.2)
Calculate the fitness values of particles.(2.3)
Find the global best position.(2.4)
Update the positions and velocities of the agents.(2.5)
Determine whether the termination conditions have been satisfied; if so, proceed to step (2.1) and continue to search; otherwise, continue to step (2.6).(2.6)
Find the best global position.


## 2.3.4. Test of FAPSO

To verify the optimization performance and convergence speed of the modified algorithm, four benchmark functions are selected in this paper. These benchmark functions have different characteristics, which are used to fully investigate the optimization ability of the algorithm. The four common test functions are shown in Table1. The experimental parameters of PSO and FAPSO are shown in Table 2.Table 1 Function name Test function Variable domain Global optimum Function characteristic Sphere f 1 X =  ∑ i = 1 D x i 2 x i ∈ - 5.12,5.12 x i = 0 ,   f 1 ( X ) = 0 Unimodal Rosenbrock f 2 X =  ∑ i = 1 D - 1 100 x i 2 - x i + 1 2 + x i - 1 2 x i ∈ - 2.084,2.084 x i = 0 ,   f 2 ( X ) = 0 Unimodal Rastrigin f 3 X =  ∑ i = 1 D x i 2 - 10 cos ⁡ 2 π x i + 10 x i ∈ - 5.12,5.12 x i = 0 ,   f 3 ( X ) = 0 Multimodal Griewank f 4 X = 1 4000  ∑ i = 1 D x i 2 - ∏ i = 1 D cos ⁡ x i i + 1 x i ∈ - 5.12,5.12 x i = 0 ,   f 4 ( X ) = 0 MultimodalTable 2
The experimental parameters of PSO and FAPSO.
Experimental parameters PSO FAPSO Maximum iteration 2000 2000 Population size 40 40 w 0.729 0.729 c 1.494 1.494 v m a x 0.5 0.5 Dimension 30 30 Number of experiments 30 30Thirty experiments, searching for the minimum value point by 2000 iterations, were carried out independently. The results, including the maximum, minimum, average, and standard deviations, are displayed in Table3. The results show that the optimization effect of FAPSO is better than PSO.Table 3
Test results of PSO and FAPSO.
Test function Algorithm Min value Max value Average value Standard deviation Sphere PSO 0.4276 1.198 0.7030 0.1740 FAPSO 0.0481 0.1158 0.0837 0.0156 Rosenbrock PSO 56.2745 131.5162 90.7276 18.7055 FAPSO 35.1140 43.7005 38.3808 1.8540 Rastrigin PSO 169.0 252.1567 201.6199 21.5151 FAPSO 40.8482 71.1049 59.012 6.5202 Griewank PSO 0.0478 0.3839 0.0884 0.0607 FAPSO 0.0090 0.0200 0.0145 0.0026


## 2.4. The Proposed Hybrid Model

In this paper, a novel algorithm, SSA-FAPSO-RENN, is proposed to forecast wind speed. SSA is used to acquire the moving tendency of wind speed and enhance the forecasting abilities. The hybrid optimization algorithm (FAPSO) that combines the FA and the PSO is utilized to optimize the parameters of the ENN model. To forecast the multistep wind speed, the rolling Elman neural network (RENN) model is used.For the convenience of narrative, the proposed hybrid model is named the SSA-FAPSO-RENN model.


## 3. Experimental Simulation

In this section, the details of experimental simulation will be introduced. Wind speed series of 10 min and 60 min are used to verify the effect of the model.


### 3.1. Performance Metric

The primary concern is to determine whether the prediction model is superior to other models. The performance of the model is usually evaluated using statistical criterions.To estimate the forecasting performance, the Diebold-Mariano (DM) test and three error criterions are adopted, including MAE, MAPE, and MSE. DM test [44] is a comparison test that focuses on the predictive accuracy and can be used to evaluate the forecasting performance of the proposed hybrid model and other comparing models. The details of DM test are given as follows: (17)DM=∑n=1kLεt+n1-Lεt+n2/kS2/ks2,where L(∗) is the loss function. εt+n1 and εt+n2 are the forecast errors from two models. S2 is an estimator of the variance of dn=Lεt+n1-Lεt+n2. The hypothesis test is defined as(18)H0:Edn=0,H1:Edn≠0.The null hypothesis is that the two forecasts have the same accuracy. Under the null hypothesis, the test statistics DM are asymptotically N(0,1) distributed. If DM>zα/2, the null hypothesis will be rejected.The detailed equations of these three error criterions are given as follows.MSE (Mean Squared Error) (19) M S E = 1 T ∑ t = 1 T y t - y ^ t 2 .MAE (Mean Absolute Error) (20) M A E = 1 T ∑ t = 1 T y t - y ^ t .MAPE (Mean Absolute Percentage Error) (21) M A P E = 1 T ∑ t = 1 T y t - y ^ t y t × 100 % ,where yt and y^t denote the real and predicted values at time t, respectively.


### 3.2. Wind Speed Datasets

To verify the forecasting ability of the proposed hybrid model, 10 min and 60 min wind speed data (January 1, 2011, to November 9, 2011) from the province of Shandong, China, are proposed as the case study in this paper. In the two tests, multiple seasonal patterns are used to reduce interferences from the original data, March 1 to May 31 (spring), June 1 to August 31 (summer), September 1 to November 9 (fall), and January 1 to February 28 (winter), and the wind speed datasets are randomly selected.To further assess the forecasting accuracy, every wind speed series is divided into a training set and a validation set. In addition, an entire day of data will be used as a test set to test the forecasting ability of the models.The first case study is 10 min wind speed forecasting. The total number of available samples is 1152. The training set also includes 806 wind speed datasets and the validation set includes 140 wind speed datasets. The remaining data are used to calculate the predictive ability of these models. Figure2 shows four wind speed datasets from three wind observation sites corresponding to the four seasons.Figure 2
Four wind speed datasets (10 min speed) from three wind observation sites corresponding to the four seasons.The second case study is 60 min wind speed forecasting. The total number of available samples is 1032. The training set includes 806 wind speed datasets, and the validation set includes 202 wind speed datasets, and the remaining data are the test set. Figure3 shows four wind speed datasets (60 min wind speed) from three wind observation sites corresponding to the four seasons.Figure 3
Four wind speed datasets (60 min speed) from three wind observation sites corresponding to the four seasons.From Figures2 and 3, several features can be summarized:(a)
The data for four seasons are quite different.(b)
There are three wind observation sites. The wind speed data from the same site is similar.(c)
The intensity of the wind in winter is large but small for the wind in summer.(d)
The experimental datasets reveal the chaotic nature and intrinsic complexity of wind speed.


### 3.3. Set Parameters of the Related Models

In this paper, a novel algorithm that hybridizes SSA, FAPSO, and RENN is proposed to forecast wind speed. Parameters have a large influence on prediction accuracy.


#### 3.3.1. SSA Decomposition and Reconstruction

SSA performs well with complex irregular time series. The parameter setting of SSA is very important for the forecasting effect. The window lengthL is the only parameter in the SSA decomposition process. The window length was chosen as an integer fulfilling the conditions: 1<L<N and L≤N/2, where N is the data length. In this paper, the data length of 10 min and 60 min wind speed is 1152 and 1032, respectively; therefore, the window length L chosen is 400.ChoosingL=400 allows the trend to be extracted simultaneously. Since the trend of wind speed series is complex, many eigentriples are required to reconstruct it. In this paper, the trend is reconstructed by eigentriples 1–150 in the 10 min wind speed forecasting experiments, and the 60 min wind speed trend is reconstructed by eigentriples 1–100. The graph in Figure 4 depicts the initial series and trend of the 10 min wind speed series from the wind observation site A.Figure 4
The initial series and trend of 10 min wind speed series from the wind observation site A.


#### 3.3.2. Parameters of the Hybrid Model

Setting the parameters is very important for the prediction of wind speed. To compare the prediction effect of the model and attain a scientific conclusion, the initial parameters of these models need to be unified. The details are shown in Table4.Table 4
Parameters of the hybrid model.
Experimental parameters Default value PSO Acceleration coefficient 1.494 Inertia weight 1 Maximum number of iterations 1000 The number of particles 20 Particle velocity [−0.5, 0.5] Particle positions [−5, 5]  FA The population size 20 Maximum iteration number 50 Absorption coefficient 1 Light intensity coefficient 0.2 Step size 0.5 Firefly positions [−5, 5]  ENN Input layer 4 Middle layer 5 Output layer 1 Iteration time 1000 Training requirement accuracy 0.000001 Learning rate 0.1


### 3.4. Case Study One: 10 Min Wind Speed Forecasting

In this section, 10 min wind speed series, which are from four datasets of three wind observation sites, are used to test the forecasting capacity of the proposed hybrid model.The forecasting results of the SSA-FAPSO-RENN model are compared with the forecasting results of the BPNN, ARIMA, RENN, SSA-RENN, and SSA-PSO-RENN. The BPNN, ARIMA, and RENN models are the single models, and the others are combination models. The parameters of BPNN are the same as those of ENN. The MAE, MAPE, and MSE values are the evaluation standard.Figure5 shows the multistep predicted results, the original wind speed data from spring datasets of wind observation site A, using the different involved models. The forecasting results are given in Tables 5–7. The values in bold indicate the smallest values of MSE, MAPE, and MAE. The smallest values of MSE, MAPE, and MAE are not all the forecasting results of the proposed model, but the minimum and the predictive values of the proposed model are very close. The results may differ with different error criterions. The proposed model is shown to have better prediction accuracy for most of the sample wind speed series.Table 5
Performance evaluations of different models for the forecast of the 10 min wind speed series from the wind observation site A.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 1.5112 1.5822 1.3404 0.4362 0.4433 0.4317 2-step 2.1976 2.3299 1.8691 0.9032 0.9068 0.8717 MAE (m/s) 0.9583 0.9781 0.9151 0.5184 0.5181 0.5183 1.1455 1.182 1.057 0.7154 0.7171 0.7055 MAPE (%) 7.75 7.89 7.4 4.21 4.2 4.2 9.34 9.55 8.57 5.82 5.81 5.72 3-step MSE (m/s) 2.7135 3.0059 2.2917 1.4506 1.439 1.3738 5-step 3.1234 3.5938 2.701 2.0879 2.1161 1.9795 MAE (m/s) 1.2682 1.3242 1.1629 0.8828 0.8827 0.8655 1.3667 1.4767 1.2655 1.0698 1.0719 1.043 MAPE (%) 10.34 10.68 9.41 7.19 7.15 7.01 11.06 11.85 10.15 8.65 8.6 8.37  Summer 1-step MSE (m/s) 0.4632 0.6339 0.449 0.1755 0.1665 0.1628 2-step 0.6472 1.0769 0.6471 0.369 0.3525 0.3398 MAE (m/s) 0.5074 0.602 0.4956 0.31 0.3025 0.3023 0.6055 0.7833 0.6038 0.4504 0.4425 0.4386 MAPE (%) 6.3 7.09 6.02 3.74 3.67 3.67 7.54 9.11 7.31 5.4 5.34 5.3 3-step MSE (m/s) 0.8074 1.5554 0.8309 0.5901 0.5753 0.5581 5-step 1.0994 2.4044 1.1971 0.9631 0.9914 0.937 MAE (m/s) 0.6787 0.9364 0.6858 0.5616 0.5552 0.5497 0.8095 1.1767 0.8246 0.7215 0.725 0.7082 MAPE (%) 8.5 10.77 8.27 6.7 6.66 6.62 10.17 13.39 9.84 8.54 8.58 8.47  Fall 1-step MSE (m/s) 0.2504 0.2578 0.2398 0.1271 0.1265 0.1225 2-step 0.3173 0.3319 0.2977 0.1859 0.1886 0.1827 MAE (m/s) 0.3894 0.3852 0.3773 0.262 0.2606 0.2557 0.4337 0.4349 0.4181 0.3243 0.3258 0.3201 MAPE (%) 5.18 5.04 4.96 3.43 3.41 3.38 5.69 5.62 5.44 4.20 4.22 4.19 3-step MSE (m/s) 0.3705 0.3909 0.3449 0.2449 0.2512 0.2444 5-step 0.4391 0.492 0.4141 0.3329 0.3509 0.341 MAE (m/s) 0.4659 0.4659 0.4487 0.3714 0.3767 0.3721 0.5114 0.5322 0.4981 0.4382 0.4491 0.4491 MAPE (%) 6.07 5.97 5.8 4.79 4.85 4.85 6.6 6.75 6.39 5.63 5.74 5.83  Winter 1-step MSE (m/s) 0.5426 0.5362 0.5508 0.3184 0.2908 0.28 2-step 0.8537 0.7711 0.7896 0.5109 0.4902 0.4721 MAE (m/s) 0.5754 0.5681 0.5735 0.4421 0.4256 0.4171 0.6887 0.6718 0.6755 0.5472 0.5358 0.5273 MAPE (%) 9.38 8.23 8.27 6.36 6.19 6.03 11.08 9.68 9.64 7.78 7.61 7.52 3-step MSE (m/s) 1.1557 0.9653 0.9953 0.7207 0.7013 0.6828 5-step 1.6309 1.2887 1.3295 1.072 1.0714 1.038 MAE (m/s) 0.796 0.7404 0.7509 0.6294 0.6216 0.6127 0.9501 0.8451 0.8573 0.7537 0.752 0.7387 MAPE (%) 12.67 10.54 10.58 8.87 8.7 8.67 14.83 11.83 11.85 10.42 10.3 10.24Table 6
Performance evaluations of different models for the forecast of the 10 min wind speed series from the wind observation site B.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 1.7933 1.3933 1.3314 0.5031 0.4979 0.486 2-step 2.2254 1.7834 1.6877 0.9475 0.9395 0.9143 MAE (m/s) 0.9946 0.9531 0.934 0.5744 0.5717 0.5591 1.1299 1.0699 1.0411 0.7536 0.7498 0.7401 MAPE (%) 16.16 9.62 9.45 5.85 5.82 5.7 17.82 10.8 10.54 7.68 7.64 7.56 3-step MSE (m/s) 2.8511 2.1217 1.9957 1.3859 1.3794 1.3498 5-step 3.0363 2.5219 2.3401 1.9192 1.9249 1.8929 MAE (m/s) 1.2173 1.1613 1.1246 0.8957 0.8953 0.8853 1.2432 1.2569 1.2031 1.0468 1.0478 1.0436 MAPE (%) 19.37 11.77 11.44 9.11 9.09 9.05 19.33 12.57 12.09 10.46 10.46 10.52  Summer 1-step MSE (m/s) 0.4929 0.4557 0.4809 0.1818 0.1768 0.1642 2-step 0.5592 0.5071 0.5184 0.2389 0.2334 0.2132 MAE (m/s) 0.5475 0.518 0.5254 0.3436 0.3399 0.3255 0.5966 0.5697 0.5723 0.3872 0.3844 0.3699 MAPE (%) 8.19 7.72 7.86 5.11 5.07 4.85 8.92 8.41 8.48 5.73 5.69 5.45 3-step MSE (m/s) 0.6594 0.5928 0.5985 0.3242 0.3167 0.2976 5-step 0.8018 0.7361 0.7424 0.505 0.4903 0.512 MAE (m/s) 0.6502 0.6166 0.6169 0.4436 0.4393 0.4305 0.7195 0.6859 0.6854 0.5508 0.5428 0.5545 MAPE (%) 9.73 9.08 9.12 6.57 6.51 6.28 10.76 10.04 10.06 8.21 8.1 8.03  Fall 1-step MSE (m/s) 0.416 0.4438 0.1106 0.1936 0.19 0.1847 2-step 0.5802 0.6031 0.1103 0.3085 0.3089 0.2995 MAE (m/s) 0.4926 0.5046 0.2576 0.3432 0.3409 0.3381 0.5703 0.5767 0.257 0.4175 0.4149 0.4099 MAPE (%) 7.76 7.75 8.68 5.46 5.41 5.39 8.78 8.71 8.71 6.55 6.48 6.44 3-step MSE (m/s) 0.708 0.7263 0.1281 0.4589 0.4774 0.4601 5-step 0.8672 0.8777 0.1541 0.6675 0.7352 0.7026 MAE (m/s) 0.6275 0.626 0.279 0.506 0.5037 0.5002 0.7024 0.6854 0.3082 0.6128 0.6181 0.6193 MAPE (%) 9.56 9.37 9.46 7.85 7.75 7.75 10.61 10.15 1.05 9.35 9.3 9.39  Winter 1-step MSE (m/s) 0.5337 0.5718 0.6338 0.3426 0.3371 0.3086 2-step 0.8135 0.8456 0.9081 0.5718 0.5991 0.5244 MAE (m/s) 0.5685 0.5933 0.6283 0.4647 0.4517 0.4376 0.6836 0.7078 0.7279 0.5824 0.592 0.5574 MAPE (%) 8.23 9.71 10.14 7.47 7.24 7.05 9.85 11.59 11.68 9.26 9.45 8.85 3-step MSE (m/s) 1.0503 1.0935 1.1473 0.8221 0.8712 0.7726 5-step 1.4387 1.4821 1.5336 1.2745 1.3178 1.2256 MAE (m/s) 0.7791 0.7882 0.8049 0.6836 0.7038 0.6626 0.9043 0.9081 0.9213 0.835 0.854 0.8228 MAPE (%) 11.14 12.91 12.87 10.8 11.2 10.45 12.65 14.86 14.65 13.14 13.5 12.92Table 7
Performance evaluations of different models for the forecast of the 10 min wind speed series from the wind observation site C.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 0.9886 0.933 0.91 0.3511 0.3455 0.3409 2-step 1.3418 1.2772 1.174 0.5605 0.5547 0.554 MAE (m/s) 0.7739 0.7637 0.7531 0.4734 0.4693 0.4653 0.9002 0.888 0.8637 0.5905 0.5876 0.5863 MAPE (%) 7.79 7.48 7.39 4.67 4.63 4.59 9.1 8.8 8.53 5.86 5.83 5.82 3-step MSE (m/s) 1.6737 1.5485 1.3948 0.8204 0.8221 0.8228 5-step 2.0057 1.8882 1.6584 1.1905 1.2223 1.2007 MAE (m/s) 0.9984 0.9769 0.9369 0.7053 0.707 0.7051 1.1032 1.082 1.0265 0.8535 0.8653 0.8552 MAPE (%) 10.19 9.78 9.33 7.06 7.07 7.08 11.25 10.87 10.22 8.54 8.63 8.62  Summer 1-step MSE (m/s) 0.3861 0.3647 0.386 0.1246 0.1176 0.1108 2-step 0.6725 0.5137 0.5217 0.2596 0.2526 0.2337 MAE (m/s) 0.497 0.4851 0.4957 0.2842 0.2808 0.2711 0.6467 0.5523 0.5614 0.3861 0.3873 0.3699 MAPE (%) 7.82 7.23 7.36 4.23 4.18 4.1 10.47 8.19 8.31 5.61 5.6 5.46 3-step MSE (m/s) 0.7203 0.6296 0.6273 0.4256 0.4323 0.4018 5-step 1.0697 0.8304 0.8011 0.7011 0.7408 0.6927 MAE (m/s) 0.6478 0.6144 0.6143 0.4859 0.4911 0.4734 0.8047 0.6966 0.6876 0.6181 0.6331 0.614 MAPE (%) 10.61 9.04 9.05 6.99 7.02 6.9 13.26 10.06 9.99 8.78 8.92 8.77  Fall 1-step MSE (m/s) 0.0878 0.2985 0.4265 0.1268 0.1208 0.1185 2-step 0.1023 0.4189 0.447 0.2122 0.2017 0.2025 MAE (m/s) 0.2676 0.3903 0.5267 0.2743 0.2709 0.268 0.2534 0.4647 0.5428 0.3454 0.3427 0.3413 MAPE (%) 5.81 5.81 9.53 4.07 4.03 4 5.52 6.85 9.83 5.12 5.09 5.07 3-step MSE (m/s) 0.1209 0.5499 0.4602 0.3148 0.3045 0.31 5-step 0.1455 0.7512 0.4765 0.4895 0.4849 0.4991 MAE (m/s) 0.272 0.5365 0.5519 0.4118 0.4086 0.4119 0.2962 0.6405 0.5586 0.5204 0.512 0.5201 MAPE (%) 5.94 7.87 10 6.09 6.03 6.09 6.51 9.31 10.16 7.74 7.56 7.69  Winter 1-step MSE (m/s) 0.5355 0.7541 0.5705 0.3163 0.3245 0.3254 2-step 0.8343 1.001 0.8674 0.5561 0.568 0.5674 MAE (m/s) 0.5268 0.6375 0.5552 0.424 0.4251 0.4244 0.643 0.729 0.6783 0.532 0.5407 0.5353 MAPE (%) 9.56 11.75 10.35 7.88 7.87 7.82 11.6 13.37 12.66 9.76 9.97 9.73 3-step MSE (m/s) 1.1368 1.2756 1.1347 0.8115 0.8161 0.8175 5-step 1.7051 1.7882 1.5878 1.3017 1.2785 1.2866 MAE (m/s) 0.742 0.8183 0.7725 0.6307 0.6403 0.6351 0.9113 0.9635 0.903 0.7998 0.8049 0.7989 MAPE (%) 13.23 14.94 14.35 11.52 11.77 11.49 15.99 17.31 16.5 14.74 14.87 14.53Figure 5
The multistep predicted results of 10 min wind speed series using the different involved models.To reflect the forecasting results more directly, the results of Tables5–7 can be averaged. The average results were calculated by the results of three wind observation sites and four seasons. They are shown in Table 8.Table 8
Average results of the 10 min wind speed.
1-step 2-step 3-step 5-step ARIMA MSE (m/s) 0.6668 0.9287 1.164 1.4469 MAE (m/s) 0.5916 0.6914 0.7619 0.8602 MAPE (%) 8.33 9.64 10.61 11.92  RENN MSE (m/s) 0.6191 0.8198 0.9958 1.2446 MAE (m/s) 0.5865 0.6666 0.7291 0.8116 MAPE (%) 8.12 9.14 9.97 10.25  SSA-PSO-RENN MSE (m/s) 0.2614 0.4663 0.6989 1.0604 MAE (m/s) 0.388 0.5017 0.6021 0.7397 MAPE (%) 5.1436 6.5611 7.8166 9.5465  BPNN MSE (m/s) 0.6854 0.955 1.2046 1.5546 MAE (m/s) 0.6149 0.7192 0.8004 0.9125 MAPE (%) 7.94 9.22 10.23 11.58  SSA-RENN MSE (m/s) 0.2664 0.4687 0.6975 1.0421 MAE (m/s) 0.3929 0.5027 0.6006 0.735 MAPE (%) 5.21 6.57 7.8 9.52  SSA-FAPSO-RENN MSE (m/s) 0.253 0.4479 0.6743 1.0256 MAE (m/s) 0.3819 0.4918 0.592 0.7306 MAPE (%) 5.0657 6.4264 7.6869 9.4487Table8 indicates the following:(1)
The forecasting results of 1-step are better than 2-step, 3-step, and 5-step. For example, the MAPE values of the SSA-FAPSO-RENN model change from 5.06% to 6.43%, 7.69%, and 9.45% at 2-step, 3-step, and 5-step. This conclusion can be reached through other models.(2)
Among all involved single models, the RENN model has the best performance except for the 1-step forecasting result, and the ARIMA model has the worst performance in every forecasting step.(3)
Compared with combined models, the single model forecasting effect is relatively poor.(4)
SSA-FAPSO-RENN offered the most accurate forecast value, with MAPE values of 5.06%, 6.43%, 7.69%, and 9.45% at 1-step, 2-step, 3-step, and 5-step, respectively.(5)
The above conclusion can also be reached with MSE and MAE.(6)
The SSA can improve the forecasting performance of the RENN model.Wind speed high-accuracy forecasting, an important part of electrical system monitoring and control, is crucial to protect the safety of wind power utilization but is always a difficult and arduous task. Compared with the other forecasting models involved in this paper, the proposed hybrid model has better forecasting ability in the 10 min wind speed forecasting study.


### 3.5. Case Study Two: 60 Min Wind Speed Forecasting

In this case, one-hour wind speed series were used to test the forecasting capacity of the proposed hybrid model. Figure6 depicts the initial series and trend of the 60 min wind speed series from the wind observation site A.Figure 6
The initial series and trend of the 60 min wind speed series from wind observation site A.The forecasting results of proposed hybrid model, SSA-FAPSO-RENN, are compared with the forecasting results of BPNN, ARIMA, RENN, SSA-RENN, and SSA-PSO-RENN.Figure7 shows the multistep predicted results with the one-hour wind speed data from spring datasets of wind observation site A, using the different involved models. The estimated results of these predictions are given in Tables 9–11.Table 9
Performance evaluations of different models for the forecast of the one-hour wind speed series from the wind observation site A.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 3.5491 2.6786 2.5691 1.3928 1.3207 1.1857 2-step 4.8962 3.5233 3.3591 2.0892 2.071 2.2118 MAE (m/s) 1.488 1.2338 1.1719 0.917 0.9008 0.879 1.7826 1.4945 1.4178 1.0479 1.054 1.1309 MAPE (%) 20.5 17.55 16.54 13.59 13.2 12.54 25.23 21.59 20.23 15.19 15.12 15.95 3-step MSE (m/s) 4.9024 3.649 3.5204 2.6858 2.8108 3.7149 5-step 5.3601 3.6718 3.616 2.7484 3.1291 5.4178 MAE (m/s) 1.7797 1.5561 1.4819 1.2117 1.2498 1.4508 1.8963 1.5616 1.5003 1.2732 1.3735 1.7669 MAPE (%) 24.92 22.22 20.82 17.25 17.65 20.37 26.43 22.03 20.62 17.86 19.1 24.89  Summer 1-step MSE (m/s) 0.4484 0.3349 0.2988 0.2956 0.2991 0.2871 2-step 0.3988 0.3094 0.2506 0.3168 0.3106 0.2945 MAE (m/s) 0.5476 0.4808 0.451 0.4385 0.4395 0.4283 0.4988 0.4558 0.4016 0.4547 0.449 0.4317 MAPE (%) 15.13 14.3 13.1 12.83 12.91 12.62 13.73 13.59 11.52 13.04 12.94 12.48 3-step MSE (m/s) 0.4349 0.3361 0.245 0.3588 0.343 0.3256 5-step 0.4863 0.4093 0.2582 0.4176 0.3804 0.3691 MAE (m/s) 0.5286 0.4736 0.3973 0.4899 0.48 0.4567 0.5748 0.5177 0.4189 0.5304 0.5 0.4878 MAPE (%) 14.57 14.26 11.42 13.96 13.76 13.16 15.91 15.88 12.18 14.91 14.28 14.14  Fall 1-step MSE (m/s) 0.7452 0.5487 0.4813 0.3457 0.3399 0.3174 2-step 0.9214 0.8502 0.696 0.6654 0.6462 0.6252 MAE (m/s) 0.7107 0.6559 0.6106 0.4473 0.441 0.4305 0.7608 0.7823 0.7093 0.6265 0.6117 0.6247 MAPE (%) 12.74 11.19 10.58 7.69 7.62 7.41 13.61 12.95 12.11 10.66 10.48 10.69 3-step MSE (m/s) 1.0666 1.09 0.8224 0.9081 0.8629 0.8287 5-step 1.086 1.3202 0.8961 1.1529 1.0374 1.0159 MAE (m/s) 0.8368 0.8473 0.752 0.7382 0.7208 0.7164 0.8329 0.9354 0.7681 0.823 0.7944 0.7929 MAPE (%) 15.23 13.82 12.73 12.28 12.07 11.97 15.32 15.44 13.21 13.85 13.44 13.38  Winter 1-step MSE (m/s) 0.8732 0.5293 0.4903 0.4433 0.4465 0.4397 2-step 1.0566 0.6481 0.5735 0.5438 0.5476 0.5512 MAE (m/s) 0.7766 0.5785 0.5512 0.522 0.5217 0.511 0.8729 0.6463 0.5842 0.5677 0.5645 0.5573 MAPE (%) 13.09 10.06 9.4 8.98 8.97 8.75 15.03 11.51 10.13 9.78 9.72 9.54 3-step MSE (m/s) 1.5411 0.8365 0.7038 0.6564 0.6605 0.671 5-step 2.1199 1.0782 0.8526 0.7186 0.7207 0.7353 MAE (m/s) 1.0077 0.7348 0.6595 0.6252 0.6211 0.6212 1.1633 0.8229 0.7431 0.6621 0.6582 0.6585 MAPE (%) 17.6 13.32 11.56 10.84 10.75 10.69 20.42 15.11 13.04 11.49 11.39 11.26Table 10
Performance evaluations of different models for the forecast of the one-hour wind speed series from the wind observation site B.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 3.8189 4.002 3.9104 1.9404 1.8876 1.8199 2-step 4.7529 4.8088 4.5033 3.1605 3.4326 3.1185 MAE (m/s) 1.6305 1.5971 1.6031 1.1912 1.1649 1.143 1.8193 1.8023 1.7529 1.3794 1.4516 1.3555 MAPE (%) 13.55 13.56 13.2 9.85 9.57 9.42 15.63 15.47 14.58 11.76 12.34 11.47 3-step MSE (m/s) 6.0454 5.5072 5.0829 3.9493 4.4624 3.8992 5-step 7.7094 6.6719 6.2785 5.3395 6.2761 5.2808 MAE (m/s) 1.9603 1.9117 1.834 1.5277 1.6338 1.5078 2.1847 2.1189 2.0119 1.8245 1.9751 1.8167 MAPE (%) 16.76 16.19 15.01 12.88 13.68 12.58 18.74 17.77 16.1 15.34 16.22 14.96  Summer 1-step MSE (m/s) 0.3249 0.3033 0.289 0.1718 0.1866 0.1732 2-step 0.3683 0.3475 0.3368 0.1911 0.2304 0.1739 MAE (m/s) 0.4131 0.4046 0.3898 0.3252 0.3342 0.3171 0.4592 0.4465 0.4414 0.3442 0.3737 0.3223 MAPE (%) 12.1 12.24 11.84 9.27 9.48 8.94 13.32 13.48 13.47 10.07 10.73 9.27 3-step MSE (m/s) 0.3838 0.4068 0.4013 0.2402 0.2944 0.2029 5-step 0.4521 0.5 0.5067 0.3235 0.3757 0.2663 MAE (m/s) 0.4788 0.4694 0.4731 0.3751 0.419 0.3451 0.5333 0.5196 0.5304 0.4315 0.4705 0.3896 MAPE (%) 14.15 14.67 14.97 11.31 12.2 10.14 16.13 16.88 17.49 13.54 13.84 11.75  Fall 1-step MSE (m/s) 1.1401 0.8356 0.8062 0.5619 0.4867 0.5122 2-step 1.5353 1.0856 1.0266 0.9874 0.7936 0.7479 MAE (m/s) 0.9306 0.7736 0.7577 0.6446 0.593 0.5914 1.0578 0.86 0.8309 0.8106 0.7297 0.6882 MAPE (%) 20.25 16.2 15.68 12.62 11.71 12 22.74 17.87 17 15.81 14.37 13.91 3-step MSE (m/s) 1.5995 1.1857 1.1158 1.3834 1.0994 1.0397 5-step 1.5644 1.239 1.1143 1.5687 1.2873 1.3849 MAE (m/s) 1.062 0.8895 0.857 0.9206 0.834 0.8154 1.0399 0.9121 0.8637 0.9556 0.8883 0.9313 MAPE (%) 22.9 18.35 17.35 18.04 16.57 16.63 22.64 19.22 17.76 19.01 18.08 19.47  Winter 1-step MSE (m/s) 0.8207 0.4931 0.5235 0.4726 0.4589 0.4658 2-step 1.35 0.6226 0.6651 0.5605 0.5481 0.5595 MAE (m/s) 0.7238 0.5613 0.5871 0.4794 0.4757 0.4707 0.9529 0.6408 0.6697 0.5721 0.5701 0.571 MAPE (%) 11.63 9.17 9.62 8.03 8.01 7.95 15.67 10.43 10.97 9.43 9.47 9.54 3-step MSE (m/s) 1.9413 0.6929 0.7451 0.6288 0.605 0.6022 5-step 3.4093 0.7917 0.837 0.7601 0.6951 0.6894 MAE (m/s) 1.1112 0.6877 0.7223 0.6169 0.6143 0.6127 1.4227 0.7494 0.7776 0.6901 0.6777 0.674 MAPE (%) 18.31 11.25 11.95 10.02 10.13 10.17 23.55 12.22 12.94 10.89 11.05 11.03Table 11
Performance evaluations of different models for the forecast of the one-hour wind speed series.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 5.1762 4.5464 4.8944 1.6948 1.7761 1.5977 2-step 6.9013 5.7573 6.2777 2.6703 2.9233 2.3518 MAE (m/s) 1.8581 1.6491 1.7513 0.9221 0.9017 0.901 2.1454 1.8997 2.0602 1.1652 1.2252 1.0835 MAPE (%) 16.8 16.74 17.32 8.93 8.75 8.7 20.99 20.01 20.9 12 12.57 11.05 3-step MSE (m/s) 9.0442 6.9167 7.7342 3.7011 4.254 3.2441 5-step 12.4765 9.1089 10.1619 6.6403 7.6275 5.6975 MAE (m/s) 2.4758 2.069 2.2768 1.4621 1.5986 1.3514 2.9377 2.3763 2.613 1.9783 2.1813 1.8934 MAPE (%) 24.77 21.93 23.21 15.41 16.77 13.9 30.67 25.29 26.97 21.39 23.78 19.61  Summer 1-step MSE (m/s) 0.3986 0.2957 0.2727 0.2522 0.2437 0.2103 2-step 0.4336 0.3453 0.3165 0.3076 0.3052 0.24 MAE (m/s) 0.5323 0.453 0.4444 0.3861 0.3732 0.3495 0.5385 0.4741 0.4624 0.426 0.424 0.3665 MAPE (%) 15.19 13.55 13.48 11.67 11.34 10.63 15.31 14.32 14.24 12.81 12.89 11.11 3-step MSE (m/s) 0.4829 0.4236 0.3905 0.3509 0.3731 0.2917 5-step 0.5605 0.5792 0.5214 0.3982 0.4984 0.4136 MAE (m/s) 0.5713 0.5104 0.5026 0.4644 0.4754 0.4148 0.6208 0.6001 0.5815 0.5018 0.5576 0.5049 MAPE (%) 16.8 16 16.11 14.23 14.87 12.9 18.5 19.34 19.28 15.74 18.08 16.34  Fall 1-step MSE (m/s) 0.7161 0.54 0.5012 0.4865 0.4872 0.4901 2-step 0.8772 0.7482 0.6455 0.586 0.5816 0.5957 MAE (m/s) 0.7168 0.5844 0.5675 0.5727 0.5731 0.5733 0.7821 0.7098 0.6587 0.6175 0.6137 0.621 MAPE (%) 15.59 12.24 11.82 11.79 11.82 11.82 16.94 14.88 13.75 12.77 12.72 12.85 3-step MSE (m/s) 0.9588 0.8908 0.7079 0.6657 0.6553 0.68 5-step 0.8706 1.1014 0.7552 0.752 0.7259 0.7747 MAE (m/s) 0.8032 0.7729 0.6902 0.6707 0.6666 0.676 0.7562 0.8413 0.7071 0.703 0.6959 0.7108 MAPE (%) 17.46 16.3 14.46 13.91 13.86 14.03 16.41 18.06 15.04 14.51 14.43 14.7  Winter 1-step MSE (m/s) 1.6137 0.6872 0.6367 0.5746 0.5758 0.5683 2-step 2.079 0.7809 0.7208 0.6903 0.6766 0.678 MAE (m/s) 1.0548 0.6464 0.6414 0.641 0.6314 0.6306 1.1227 0.7051 0.7027 0.7059 0.6933 0.6993 MAPE (%) 21.17 12.57 12.5 12.31 12.03 12.02 23.1 14.09 13.95 13.77 13.45 13.55 3-step MSE (m/s) 2.4407 0.8516 0.781 0.7784 0.7552 0.7616 5-step 3.1868 0.9485 0.869 0.821 0.8172 0.816 MAE (m/s) 1.204 0.74 0.7345 0.7416 0.718 0.7279 1.3773 0.774 0.7643 0.7534 0.7495 0.7516 MAPE (%) 24.88 15.01 14.71 14.52 13.99 14.15 27.5 15.91 15.38 14.73 14.49 14.5Figure 7
The multistep predicted results of one-hour wind speed series using the different involved models.From Tables9–11, the values in bold indicate the smallest values of MSE, MAPE, and MAE, and the minimum values and the predictive value of the proposed model are approximately the same; compared to 10 min forecasting results, one-hour wind speed forecasting has more error; with the increase of prediction steps, precision will rise; some wind speed series are more adaptable to other models, but overall, the proposed model has better prediction ability.To reflect the forecasting results more directly, the results of Tables9–11 can be averaged. The average results are shown in Table 12.Table 12
Average results of one-hour wind speed.
1-step 2-step 3-step 5-step ARIMA MSE (m/s) 1.6354 2.1309 2.5701 3.2735 MAE (m/s) 0.9486 1.0661 1.1516 1.2783 MAPE (%) 15.65 17.61 19.03 21.02  RENN MSE (m/s) 1.2826 1.644 1.9423 2.359 MAE (m/s) 0.7939 0.891 0.9484 1.0233 MAPE (%) 12.92 14.4 15.36 16.67  SSA-PSO-RENN MSE (m/s) 0.71 1.0889 1.4313 1.9642 MAE (m/s) 0.6125 0.73 0.8369 0.9602 MAPE (%) 10.45 12.23 13.86 15.68  BPNN MSE (m/s) 1.3162 1.6523 1.8989 2.285 MAE (m/s) 0.8015 0.9098 0.9719 1.0608 MAPE (%) 13.28 15.02 16.11 17.76  SSA-RENN MSE (m/s) 0.72 1.064 1.3589 1.8034 MAE (m/s) 0.6239 0.7265 0.8203 0.9272 MAPE (%) 10.63 12.26 13.72 15.27  SSA-FAPSO-RENN MSE (m/s) 0.6723 1.0123 1.3551 1.9051 MAE (m/s) 0.6021 0.7043 0.808 0.9482 MAPE (%) 10.23 11.78 13.39 15.5Table12 indicates the following:(7)
The forecasting results of 1-step are better than 2-step, 3-step, and 5-step. The MAPE values of the proposed model change from 10.23% to 11.78%, 13.39%, and 15.50% at 2-step, 3-step, and 5-step. Other models can also reach this conclusion.(8)
Among all involved single models, the RENN model has the best performance in every forecasting step, and the ARIMA model has the worst performance. The MAPE values of the ARIMA model are 15.65%, 17.61%, 19.03%, and 21.02% at 1-step, 2-step, 3-step, and 5-step, respectively. However, the values of RENN reduce to 12.92%, 14.40%, 15.36%, and 16.67%.(9)
SSA-FAPSO-RENN offered the most accurate forecast value. In 1-step forecasting experiment, the MAPE values of the combined model were reduced by 20.8%, 3.8%, and 2.1% compared with the Elman model, SSA Elman, and SSAFA Elman, respectively. Similar conclusions can be obtained from multistep experiments.(10)
The above conclusion can also be achieved with MSE and MAE.The forecasting results are generally good. The proposed hybrid model can be used to forecast 60 min wind speed. Compared with traditional single models and other models involved in this paper, the proposed model has the best forecasting ability. The forecasting results show that the model has a better performance in the 10 min wind speed forecasting study than the 60 min study.The average results of study one and study two are shown in Table13. The proposed model has the best performance as evaluated by MAPE, MAE, and MSE. The Diebold-Mariano values of the SSA-PSO-RENN and SSA-RENN models are larger than the upper limits at the 10% significance level, the DM values of the RENN and the BP models are larger than the upper limits at the 5% significance level, and the DM value of the ARIMA models is larger than the upper limits at the 1% significance level. The result indicates that the proposed hybrid model is significantly better than other models.Table 13
The average results of study one and study two.
ARIMA BP RENN SSA-RENN SSA-PSO-RENN Proposed model MSE (m/s) 1.7270 1.4440 1.3634 0.9276 0.9602 0.9182 MAE (m/s) 0.9187 0.8489 0.8063 0.6661 0.6714 0.6574 MAPE (%) 14.23 12.64 12.10 10.12 10.16 9.94 DM 2.57 ∗ ∗ 2.67 ∗ 2.45 ∗ ∗ 1.89 ∗ ∗ ∗ 1.74 ∗ ∗ ∗ — ∗ is 1% significance level; ∗∗ is 5% significance level; ∗∗∗ is 10% significance level.


## 3.1. Performance Metric

The primary concern is to determine whether the prediction model is superior to other models. The performance of the model is usually evaluated using statistical criterions.To estimate the forecasting performance, the Diebold-Mariano (DM) test and three error criterions are adopted, including MAE, MAPE, and MSE. DM test [44] is a comparison test that focuses on the predictive accuracy and can be used to evaluate the forecasting performance of the proposed hybrid model and other comparing models. The details of DM test are given as follows: (17)DM=∑n=1kLεt+n1-Lεt+n2/kS2/ks2,where L(∗) is the loss function. εt+n1 and εt+n2 are the forecast errors from two models. S2 is an estimator of the variance of dn=Lεt+n1-Lεt+n2. The hypothesis test is defined as(18)H0:Edn=0,H1:Edn≠0.The null hypothesis is that the two forecasts have the same accuracy. Under the null hypothesis, the test statistics DM are asymptotically N(0,1) distributed. If DM>zα/2, the null hypothesis will be rejected.The detailed equations of these three error criterions are given as follows.MSE (Mean Squared Error) (19) M S E = 1 T ∑ t = 1 T y t - y ^ t 2 .MAE (Mean Absolute Error) (20) M A E = 1 T ∑ t = 1 T y t - y ^ t .MAPE (Mean Absolute Percentage Error) (21) M A P E = 1 T ∑ t = 1 T y t - y ^ t y t × 100 % ,where yt and y^t denote the real and predicted values at time t, respectively.


## 3.2. Wind Speed Datasets

To verify the forecasting ability of the proposed hybrid model, 10 min and 60 min wind speed data (January 1, 2011, to November 9, 2011) from the province of Shandong, China, are proposed as the case study in this paper. In the two tests, multiple seasonal patterns are used to reduce interferences from the original data, March 1 to May 31 (spring), June 1 to August 31 (summer), September 1 to November 9 (fall), and January 1 to February 28 (winter), and the wind speed datasets are randomly selected.To further assess the forecasting accuracy, every wind speed series is divided into a training set and a validation set. In addition, an entire day of data will be used as a test set to test the forecasting ability of the models.The first case study is 10 min wind speed forecasting. The total number of available samples is 1152. The training set also includes 806 wind speed datasets and the validation set includes 140 wind speed datasets. The remaining data are used to calculate the predictive ability of these models. Figure2 shows four wind speed datasets from three wind observation sites corresponding to the four seasons.Figure 2
Four wind speed datasets (10 min speed) from three wind observation sites corresponding to the four seasons.The second case study is 60 min wind speed forecasting. The total number of available samples is 1032. The training set includes 806 wind speed datasets, and the validation set includes 202 wind speed datasets, and the remaining data are the test set. Figure3 shows four wind speed datasets (60 min wind speed) from three wind observation sites corresponding to the four seasons.Figure 3
Four wind speed datasets (60 min speed) from three wind observation sites corresponding to the four seasons.From Figures2 and 3, several features can be summarized:(a)
The data for four seasons are quite different.(b)
There are three wind observation sites. The wind speed data from the same site is similar.(c)
The intensity of the wind in winter is large but small for the wind in summer.(d)
The experimental datasets reveal the chaotic nature and intrinsic complexity of wind speed.


## 3.3. Set Parameters of the Related Models

In this paper, a novel algorithm that hybridizes SSA, FAPSO, and RENN is proposed to forecast wind speed. Parameters have a large influence on prediction accuracy.


### 3.3.1. SSA Decomposition and Reconstruction

SSA performs well with complex irregular time series. The parameter setting of SSA is very important for the forecasting effect. The window lengthL is the only parameter in the SSA decomposition process. The window length was chosen as an integer fulfilling the conditions: 1<L<N and L≤N/2, where N is the data length. In this paper, the data length of 10 min and 60 min wind speed is 1152 and 1032, respectively; therefore, the window length L chosen is 400.ChoosingL=400 allows the trend to be extracted simultaneously. Since the trend of wind speed series is complex, many eigentriples are required to reconstruct it. In this paper, the trend is reconstructed by eigentriples 1–150 in the 10 min wind speed forecasting experiments, and the 60 min wind speed trend is reconstructed by eigentriples 1–100. The graph in Figure 4 depicts the initial series and trend of the 10 min wind speed series from the wind observation site A.Figure 4
The initial series and trend of 10 min wind speed series from the wind observation site A.


### 3.3.2. Parameters of the Hybrid Model

Setting the parameters is very important for the prediction of wind speed. To compare the prediction effect of the model and attain a scientific conclusion, the initial parameters of these models need to be unified. The details are shown in Table4.Table 4
Parameters of the hybrid model.
Experimental parameters Default value PSO Acceleration coefficient 1.494 Inertia weight 1 Maximum number of iterations 1000 The number of particles 20 Particle velocity [−0.5, 0.5] Particle positions [−5, 5]  FA The population size 20 Maximum iteration number 50 Absorption coefficient 1 Light intensity coefficient 0.2 Step size 0.5 Firefly positions [−5, 5]  ENN Input layer 4 Middle layer 5 Output layer 1 Iteration time 1000 Training requirement accuracy 0.000001 Learning rate 0.1


## 3.3.1. SSA Decomposition and Reconstruction

SSA performs well with complex irregular time series. The parameter setting of SSA is very important for the forecasting effect. The window lengthL is the only parameter in the SSA decomposition process. The window length was chosen as an integer fulfilling the conditions: 1<L<N and L≤N/2, where N is the data length. In this paper, the data length of 10 min and 60 min wind speed is 1152 and 1032, respectively; therefore, the window length L chosen is 400.ChoosingL=400 allows the trend to be extracted simultaneously. Since the trend of wind speed series is complex, many eigentriples are required to reconstruct it. In this paper, the trend is reconstructed by eigentriples 1–150 in the 10 min wind speed forecasting experiments, and the 60 min wind speed trend is reconstructed by eigentriples 1–100. The graph in Figure 4 depicts the initial series and trend of the 10 min wind speed series from the wind observation site A.Figure 4
The initial series and trend of 10 min wind speed series from the wind observation site A.


## 3.3.2. Parameters of the Hybrid Model

Setting the parameters is very important for the prediction of wind speed. To compare the prediction effect of the model and attain a scientific conclusion, the initial parameters of these models need to be unified. The details are shown in Table4.Table 4
Parameters of the hybrid model.
Experimental parameters Default value PSO Acceleration coefficient 1.494 Inertia weight 1 Maximum number of iterations 1000 The number of particles 20 Particle velocity [−0.5, 0.5] Particle positions [−5, 5]  FA The population size 20 Maximum iteration number 50 Absorption coefficient 1 Light intensity coefficient 0.2 Step size 0.5 Firefly positions [−5, 5]  ENN Input layer 4 Middle layer 5 Output layer 1 Iteration time 1000 Training requirement accuracy 0.000001 Learning rate 0.1


## 3.4. Case Study One: 10 Min Wind Speed Forecasting

In this section, 10 min wind speed series, which are from four datasets of three wind observation sites, are used to test the forecasting capacity of the proposed hybrid model.The forecasting results of the SSA-FAPSO-RENN model are compared with the forecasting results of the BPNN, ARIMA, RENN, SSA-RENN, and SSA-PSO-RENN. The BPNN, ARIMA, and RENN models are the single models, and the others are combination models. The parameters of BPNN are the same as those of ENN. The MAE, MAPE, and MSE values are the evaluation standard.Figure5 shows the multistep predicted results, the original wind speed data from spring datasets of wind observation site A, using the different involved models. The forecasting results are given in Tables 5–7. The values in bold indicate the smallest values of MSE, MAPE, and MAE. The smallest values of MSE, MAPE, and MAE are not all the forecasting results of the proposed model, but the minimum and the predictive values of the proposed model are very close. The results may differ with different error criterions. The proposed model is shown to have better prediction accuracy for most of the sample wind speed series.Table 5
Performance evaluations of different models for the forecast of the 10 min wind speed series from the wind observation site A.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 1.5112 1.5822 1.3404 0.4362 0.4433 0.4317 2-step 2.1976 2.3299 1.8691 0.9032 0.9068 0.8717 MAE (m/s) 0.9583 0.9781 0.9151 0.5184 0.5181 0.5183 1.1455 1.182 1.057 0.7154 0.7171 0.7055 MAPE (%) 7.75 7.89 7.4 4.21 4.2 4.2 9.34 9.55 8.57 5.82 5.81 5.72 3-step MSE (m/s) 2.7135 3.0059 2.2917 1.4506 1.439 1.3738 5-step 3.1234 3.5938 2.701 2.0879 2.1161 1.9795 MAE (m/s) 1.2682 1.3242 1.1629 0.8828 0.8827 0.8655 1.3667 1.4767 1.2655 1.0698 1.0719 1.043 MAPE (%) 10.34 10.68 9.41 7.19 7.15 7.01 11.06 11.85 10.15 8.65 8.6 8.37  Summer 1-step MSE (m/s) 0.4632 0.6339 0.449 0.1755 0.1665 0.1628 2-step 0.6472 1.0769 0.6471 0.369 0.3525 0.3398 MAE (m/s) 0.5074 0.602 0.4956 0.31 0.3025 0.3023 0.6055 0.7833 0.6038 0.4504 0.4425 0.4386 MAPE (%) 6.3 7.09 6.02 3.74 3.67 3.67 7.54 9.11 7.31 5.4 5.34 5.3 3-step MSE (m/s) 0.8074 1.5554 0.8309 0.5901 0.5753 0.5581 5-step 1.0994 2.4044 1.1971 0.9631 0.9914 0.937 MAE (m/s) 0.6787 0.9364 0.6858 0.5616 0.5552 0.5497 0.8095 1.1767 0.8246 0.7215 0.725 0.7082 MAPE (%) 8.5 10.77 8.27 6.7 6.66 6.62 10.17 13.39 9.84 8.54 8.58 8.47  Fall 1-step MSE (m/s) 0.2504 0.2578 0.2398 0.1271 0.1265 0.1225 2-step 0.3173 0.3319 0.2977 0.1859 0.1886 0.1827 MAE (m/s) 0.3894 0.3852 0.3773 0.262 0.2606 0.2557 0.4337 0.4349 0.4181 0.3243 0.3258 0.3201 MAPE (%) 5.18 5.04 4.96 3.43 3.41 3.38 5.69 5.62 5.44 4.20 4.22 4.19 3-step MSE (m/s) 0.3705 0.3909 0.3449 0.2449 0.2512 0.2444 5-step 0.4391 0.492 0.4141 0.3329 0.3509 0.341 MAE (m/s) 0.4659 0.4659 0.4487 0.3714 0.3767 0.3721 0.5114 0.5322 0.4981 0.4382 0.4491 0.4491 MAPE (%) 6.07 5.97 5.8 4.79 4.85 4.85 6.6 6.75 6.39 5.63 5.74 5.83  Winter 1-step MSE (m/s) 0.5426 0.5362 0.5508 0.3184 0.2908 0.28 2-step 0.8537 0.7711 0.7896 0.5109 0.4902 0.4721 MAE (m/s) 0.5754 0.5681 0.5735 0.4421 0.4256 0.4171 0.6887 0.6718 0.6755 0.5472 0.5358 0.5273 MAPE (%) 9.38 8.23 8.27 6.36 6.19 6.03 11.08 9.68 9.64 7.78 7.61 7.52 3-step MSE (m/s) 1.1557 0.9653 0.9953 0.7207 0.7013 0.6828 5-step 1.6309 1.2887 1.3295 1.072 1.0714 1.038 MAE (m/s) 0.796 0.7404 0.7509 0.6294 0.6216 0.6127 0.9501 0.8451 0.8573 0.7537 0.752 0.7387 MAPE (%) 12.67 10.54 10.58 8.87 8.7 8.67 14.83 11.83 11.85 10.42 10.3 10.24Table 6
Performance evaluations of different models for the forecast of the 10 min wind speed series from the wind observation site B.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 1.7933 1.3933 1.3314 0.5031 0.4979 0.486 2-step 2.2254 1.7834 1.6877 0.9475 0.9395 0.9143 MAE (m/s) 0.9946 0.9531 0.934 0.5744 0.5717 0.5591 1.1299 1.0699 1.0411 0.7536 0.7498 0.7401 MAPE (%) 16.16 9.62 9.45 5.85 5.82 5.7 17.82 10.8 10.54 7.68 7.64 7.56 3-step MSE (m/s) 2.8511 2.1217 1.9957 1.3859 1.3794 1.3498 5-step 3.0363 2.5219 2.3401 1.9192 1.9249 1.8929 MAE (m/s) 1.2173 1.1613 1.1246 0.8957 0.8953 0.8853 1.2432 1.2569 1.2031 1.0468 1.0478 1.0436 MAPE (%) 19.37 11.77 11.44 9.11 9.09 9.05 19.33 12.57 12.09 10.46 10.46 10.52  Summer 1-step MSE (m/s) 0.4929 0.4557 0.4809 0.1818 0.1768 0.1642 2-step 0.5592 0.5071 0.5184 0.2389 0.2334 0.2132 MAE (m/s) 0.5475 0.518 0.5254 0.3436 0.3399 0.3255 0.5966 0.5697 0.5723 0.3872 0.3844 0.3699 MAPE (%) 8.19 7.72 7.86 5.11 5.07 4.85 8.92 8.41 8.48 5.73 5.69 5.45 3-step MSE (m/s) 0.6594 0.5928 0.5985 0.3242 0.3167 0.2976 5-step 0.8018 0.7361 0.7424 0.505 0.4903 0.512 MAE (m/s) 0.6502 0.6166 0.6169 0.4436 0.4393 0.4305 0.7195 0.6859 0.6854 0.5508 0.5428 0.5545 MAPE (%) 9.73 9.08 9.12 6.57 6.51 6.28 10.76 10.04 10.06 8.21 8.1 8.03  Fall 1-step MSE (m/s) 0.416 0.4438 0.1106 0.1936 0.19 0.1847 2-step 0.5802 0.6031 0.1103 0.3085 0.3089 0.2995 MAE (m/s) 0.4926 0.5046 0.2576 0.3432 0.3409 0.3381 0.5703 0.5767 0.257 0.4175 0.4149 0.4099 MAPE (%) 7.76 7.75 8.68 5.46 5.41 5.39 8.78 8.71 8.71 6.55 6.48 6.44 3-step MSE (m/s) 0.708 0.7263 0.1281 0.4589 0.4774 0.4601 5-step 0.8672 0.8777 0.1541 0.6675 0.7352 0.7026 MAE (m/s) 0.6275 0.626 0.279 0.506 0.5037 0.5002 0.7024 0.6854 0.3082 0.6128 0.6181 0.6193 MAPE (%) 9.56 9.37 9.46 7.85 7.75 7.75 10.61 10.15 1.05 9.35 9.3 9.39  Winter 1-step MSE (m/s) 0.5337 0.5718 0.6338 0.3426 0.3371 0.3086 2-step 0.8135 0.8456 0.9081 0.5718 0.5991 0.5244 MAE (m/s) 0.5685 0.5933 0.6283 0.4647 0.4517 0.4376 0.6836 0.7078 0.7279 0.5824 0.592 0.5574 MAPE (%) 8.23 9.71 10.14 7.47 7.24 7.05 9.85 11.59 11.68 9.26 9.45 8.85 3-step MSE (m/s) 1.0503 1.0935 1.1473 0.8221 0.8712 0.7726 5-step 1.4387 1.4821 1.5336 1.2745 1.3178 1.2256 MAE (m/s) 0.7791 0.7882 0.8049 0.6836 0.7038 0.6626 0.9043 0.9081 0.9213 0.835 0.854 0.8228 MAPE (%) 11.14 12.91 12.87 10.8 11.2 10.45 12.65 14.86 14.65 13.14 13.5 12.92Table 7
Performance evaluations of different models for the forecast of the 10 min wind speed series from the wind observation site C.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 0.9886 0.933 0.91 0.3511 0.3455 0.3409 2-step 1.3418 1.2772 1.174 0.5605 0.5547 0.554 MAE (m/s) 0.7739 0.7637 0.7531 0.4734 0.4693 0.4653 0.9002 0.888 0.8637 0.5905 0.5876 0.5863 MAPE (%) 7.79 7.48 7.39 4.67 4.63 4.59 9.1 8.8 8.53 5.86 5.83 5.82 3-step MSE (m/s) 1.6737 1.5485 1.3948 0.8204 0.8221 0.8228 5-step 2.0057 1.8882 1.6584 1.1905 1.2223 1.2007 MAE (m/s) 0.9984 0.9769 0.9369 0.7053 0.707 0.7051 1.1032 1.082 1.0265 0.8535 0.8653 0.8552 MAPE (%) 10.19 9.78 9.33 7.06 7.07 7.08 11.25 10.87 10.22 8.54 8.63 8.62  Summer 1-step MSE (m/s) 0.3861 0.3647 0.386 0.1246 0.1176 0.1108 2-step 0.6725 0.5137 0.5217 0.2596 0.2526 0.2337 MAE (m/s) 0.497 0.4851 0.4957 0.2842 0.2808 0.2711 0.6467 0.5523 0.5614 0.3861 0.3873 0.3699 MAPE (%) 7.82 7.23 7.36 4.23 4.18 4.1 10.47 8.19 8.31 5.61 5.6 5.46 3-step MSE (m/s) 0.7203 0.6296 0.6273 0.4256 0.4323 0.4018 5-step 1.0697 0.8304 0.8011 0.7011 0.7408 0.6927 MAE (m/s) 0.6478 0.6144 0.6143 0.4859 0.4911 0.4734 0.8047 0.6966 0.6876 0.6181 0.6331 0.614 MAPE (%) 10.61 9.04 9.05 6.99 7.02 6.9 13.26 10.06 9.99 8.78 8.92 8.77  Fall 1-step MSE (m/s) 0.0878 0.2985 0.4265 0.1268 0.1208 0.1185 2-step 0.1023 0.4189 0.447 0.2122 0.2017 0.2025 MAE (m/s) 0.2676 0.3903 0.5267 0.2743 0.2709 0.268 0.2534 0.4647 0.5428 0.3454 0.3427 0.3413 MAPE (%) 5.81 5.81 9.53 4.07 4.03 4 5.52 6.85 9.83 5.12 5.09 5.07 3-step MSE (m/s) 0.1209 0.5499 0.4602 0.3148 0.3045 0.31 5-step 0.1455 0.7512 0.4765 0.4895 0.4849 0.4991 MAE (m/s) 0.272 0.5365 0.5519 0.4118 0.4086 0.4119 0.2962 0.6405 0.5586 0.5204 0.512 0.5201 MAPE (%) 5.94 7.87 10 6.09 6.03 6.09 6.51 9.31 10.16 7.74 7.56 7.69  Winter 1-step MSE (m/s) 0.5355 0.7541 0.5705 0.3163 0.3245 0.3254 2-step 0.8343 1.001 0.8674 0.5561 0.568 0.5674 MAE (m/s) 0.5268 0.6375 0.5552 0.424 0.4251 0.4244 0.643 0.729 0.6783 0.532 0.5407 0.5353 MAPE (%) 9.56 11.75 10.35 7.88 7.87 7.82 11.6 13.37 12.66 9.76 9.97 9.73 3-step MSE (m/s) 1.1368 1.2756 1.1347 0.8115 0.8161 0.8175 5-step 1.7051 1.7882 1.5878 1.3017 1.2785 1.2866 MAE (m/s) 0.742 0.8183 0.7725 0.6307 0.6403 0.6351 0.9113 0.9635 0.903 0.7998 0.8049 0.7989 MAPE (%) 13.23 14.94 14.35 11.52 11.77 11.49 15.99 17.31 16.5 14.74 14.87 14.53Figure 5
The multistep predicted results of 10 min wind speed series using the different involved models.To reflect the forecasting results more directly, the results of Tables5–7 can be averaged. The average results were calculated by the results of three wind observation sites and four seasons. They are shown in Table 8.Table 8
Average results of the 10 min wind speed.
1-step 2-step 3-step 5-step ARIMA MSE (m/s) 0.6668 0.9287 1.164 1.4469 MAE (m/s) 0.5916 0.6914 0.7619 0.8602 MAPE (%) 8.33 9.64 10.61 11.92  RENN MSE (m/s) 0.6191 0.8198 0.9958 1.2446 MAE (m/s) 0.5865 0.6666 0.7291 0.8116 MAPE (%) 8.12 9.14 9.97 10.25  SSA-PSO-RENN MSE (m/s) 0.2614 0.4663 0.6989 1.0604 MAE (m/s) 0.388 0.5017 0.6021 0.7397 MAPE (%) 5.1436 6.5611 7.8166 9.5465  BPNN MSE (m/s) 0.6854 0.955 1.2046 1.5546 MAE (m/s) 0.6149 0.7192 0.8004 0.9125 MAPE (%) 7.94 9.22 10.23 11.58  SSA-RENN MSE (m/s) 0.2664 0.4687 0.6975 1.0421 MAE (m/s) 0.3929 0.5027 0.6006 0.735 MAPE (%) 5.21 6.57 7.8 9.52  SSA-FAPSO-RENN MSE (m/s) 0.253 0.4479 0.6743 1.0256 MAE (m/s) 0.3819 0.4918 0.592 0.7306 MAPE (%) 5.0657 6.4264 7.6869 9.4487Table8 indicates the following:(1)
The forecasting results of 1-step are better than 2-step, 3-step, and 5-step. For example, the MAPE values of the SSA-FAPSO-RENN model change from 5.06% to 6.43%, 7.69%, and 9.45% at 2-step, 3-step, and 5-step. This conclusion can be reached through other models.(2)
Among all involved single models, the RENN model has the best performance except for the 1-step forecasting result, and the ARIMA model has the worst performance in every forecasting step.(3)
Compared with combined models, the single model forecasting effect is relatively poor.(4)
SSA-FAPSO-RENN offered the most accurate forecast value, with MAPE values of 5.06%, 6.43%, 7.69%, and 9.45% at 1-step, 2-step, 3-step, and 5-step, respectively.(5)
The above conclusion can also be reached with MSE and MAE.(6)
The SSA can improve the forecasting performance of the RENN model.Wind speed high-accuracy forecasting, an important part of electrical system monitoring and control, is crucial to protect the safety of wind power utilization but is always a difficult and arduous task. Compared with the other forecasting models involved in this paper, the proposed hybrid model has better forecasting ability in the 10 min wind speed forecasting study.


## 3.5. Case Study Two: 60 Min Wind Speed Forecasting

In this case, one-hour wind speed series were used to test the forecasting capacity of the proposed hybrid model. Figure6 depicts the initial series and trend of the 60 min wind speed series from the wind observation site A.Figure 6
The initial series and trend of the 60 min wind speed series from wind observation site A.The forecasting results of proposed hybrid model, SSA-FAPSO-RENN, are compared with the forecasting results of BPNN, ARIMA, RENN, SSA-RENN, and SSA-PSO-RENN.Figure7 shows the multistep predicted results with the one-hour wind speed data from spring datasets of wind observation site A, using the different involved models. The estimated results of these predictions are given in Tables 9–11.Table 9
Performance evaluations of different models for the forecast of the one-hour wind speed series from the wind observation site A.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 3.5491 2.6786 2.5691 1.3928 1.3207 1.1857 2-step 4.8962 3.5233 3.3591 2.0892 2.071 2.2118 MAE (m/s) 1.488 1.2338 1.1719 0.917 0.9008 0.879 1.7826 1.4945 1.4178 1.0479 1.054 1.1309 MAPE (%) 20.5 17.55 16.54 13.59 13.2 12.54 25.23 21.59 20.23 15.19 15.12 15.95 3-step MSE (m/s) 4.9024 3.649 3.5204 2.6858 2.8108 3.7149 5-step 5.3601 3.6718 3.616 2.7484 3.1291 5.4178 MAE (m/s) 1.7797 1.5561 1.4819 1.2117 1.2498 1.4508 1.8963 1.5616 1.5003 1.2732 1.3735 1.7669 MAPE (%) 24.92 22.22 20.82 17.25 17.65 20.37 26.43 22.03 20.62 17.86 19.1 24.89  Summer 1-step MSE (m/s) 0.4484 0.3349 0.2988 0.2956 0.2991 0.2871 2-step 0.3988 0.3094 0.2506 0.3168 0.3106 0.2945 MAE (m/s) 0.5476 0.4808 0.451 0.4385 0.4395 0.4283 0.4988 0.4558 0.4016 0.4547 0.449 0.4317 MAPE (%) 15.13 14.3 13.1 12.83 12.91 12.62 13.73 13.59 11.52 13.04 12.94 12.48 3-step MSE (m/s) 0.4349 0.3361 0.245 0.3588 0.343 0.3256 5-step 0.4863 0.4093 0.2582 0.4176 0.3804 0.3691 MAE (m/s) 0.5286 0.4736 0.3973 0.4899 0.48 0.4567 0.5748 0.5177 0.4189 0.5304 0.5 0.4878 MAPE (%) 14.57 14.26 11.42 13.96 13.76 13.16 15.91 15.88 12.18 14.91 14.28 14.14  Fall 1-step MSE (m/s) 0.7452 0.5487 0.4813 0.3457 0.3399 0.3174 2-step 0.9214 0.8502 0.696 0.6654 0.6462 0.6252 MAE (m/s) 0.7107 0.6559 0.6106 0.4473 0.441 0.4305 0.7608 0.7823 0.7093 0.6265 0.6117 0.6247 MAPE (%) 12.74 11.19 10.58 7.69 7.62 7.41 13.61 12.95 12.11 10.66 10.48 10.69 3-step MSE (m/s) 1.0666 1.09 0.8224 0.9081 0.8629 0.8287 5-step 1.086 1.3202 0.8961 1.1529 1.0374 1.0159 MAE (m/s) 0.8368 0.8473 0.752 0.7382 0.7208 0.7164 0.8329 0.9354 0.7681 0.823 0.7944 0.7929 MAPE (%) 15.23 13.82 12.73 12.28 12.07 11.97 15.32 15.44 13.21 13.85 13.44 13.38  Winter 1-step MSE (m/s) 0.8732 0.5293 0.4903 0.4433 0.4465 0.4397 2-step 1.0566 0.6481 0.5735 0.5438 0.5476 0.5512 MAE (m/s) 0.7766 0.5785 0.5512 0.522 0.5217 0.511 0.8729 0.6463 0.5842 0.5677 0.5645 0.5573 MAPE (%) 13.09 10.06 9.4 8.98 8.97 8.75 15.03 11.51 10.13 9.78 9.72 9.54 3-step MSE (m/s) 1.5411 0.8365 0.7038 0.6564 0.6605 0.671 5-step 2.1199 1.0782 0.8526 0.7186 0.7207 0.7353 MAE (m/s) 1.0077 0.7348 0.6595 0.6252 0.6211 0.6212 1.1633 0.8229 0.7431 0.6621 0.6582 0.6585 MAPE (%) 17.6 13.32 11.56 10.84 10.75 10.69 20.42 15.11 13.04 11.49 11.39 11.26Table 10
Performance evaluations of different models for the forecast of the one-hour wind speed series from the wind observation site B.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 3.8189 4.002 3.9104 1.9404 1.8876 1.8199 2-step 4.7529 4.8088 4.5033 3.1605 3.4326 3.1185 MAE (m/s) 1.6305 1.5971 1.6031 1.1912 1.1649 1.143 1.8193 1.8023 1.7529 1.3794 1.4516 1.3555 MAPE (%) 13.55 13.56 13.2 9.85 9.57 9.42 15.63 15.47 14.58 11.76 12.34 11.47 3-step MSE (m/s) 6.0454 5.5072 5.0829 3.9493 4.4624 3.8992 5-step 7.7094 6.6719 6.2785 5.3395 6.2761 5.2808 MAE (m/s) 1.9603 1.9117 1.834 1.5277 1.6338 1.5078 2.1847 2.1189 2.0119 1.8245 1.9751 1.8167 MAPE (%) 16.76 16.19 15.01 12.88 13.68 12.58 18.74 17.77 16.1 15.34 16.22 14.96  Summer 1-step MSE (m/s) 0.3249 0.3033 0.289 0.1718 0.1866 0.1732 2-step 0.3683 0.3475 0.3368 0.1911 0.2304 0.1739 MAE (m/s) 0.4131 0.4046 0.3898 0.3252 0.3342 0.3171 0.4592 0.4465 0.4414 0.3442 0.3737 0.3223 MAPE (%) 12.1 12.24 11.84 9.27 9.48 8.94 13.32 13.48 13.47 10.07 10.73 9.27 3-step MSE (m/s) 0.3838 0.4068 0.4013 0.2402 0.2944 0.2029 5-step 0.4521 0.5 0.5067 0.3235 0.3757 0.2663 MAE (m/s) 0.4788 0.4694 0.4731 0.3751 0.419 0.3451 0.5333 0.5196 0.5304 0.4315 0.4705 0.3896 MAPE (%) 14.15 14.67 14.97 11.31 12.2 10.14 16.13 16.88 17.49 13.54 13.84 11.75  Fall 1-step MSE (m/s) 1.1401 0.8356 0.8062 0.5619 0.4867 0.5122 2-step 1.5353 1.0856 1.0266 0.9874 0.7936 0.7479 MAE (m/s) 0.9306 0.7736 0.7577 0.6446 0.593 0.5914 1.0578 0.86 0.8309 0.8106 0.7297 0.6882 MAPE (%) 20.25 16.2 15.68 12.62 11.71 12 22.74 17.87 17 15.81 14.37 13.91 3-step MSE (m/s) 1.5995 1.1857 1.1158 1.3834 1.0994 1.0397 5-step 1.5644 1.239 1.1143 1.5687 1.2873 1.3849 MAE (m/s) 1.062 0.8895 0.857 0.9206 0.834 0.8154 1.0399 0.9121 0.8637 0.9556 0.8883 0.9313 MAPE (%) 22.9 18.35 17.35 18.04 16.57 16.63 22.64 19.22 17.76 19.01 18.08 19.47  Winter 1-step MSE (m/s) 0.8207 0.4931 0.5235 0.4726 0.4589 0.4658 2-step 1.35 0.6226 0.6651 0.5605 0.5481 0.5595 MAE (m/s) 0.7238 0.5613 0.5871 0.4794 0.4757 0.4707 0.9529 0.6408 0.6697 0.5721 0.5701 0.571 MAPE (%) 11.63 9.17 9.62 8.03 8.01 7.95 15.67 10.43 10.97 9.43 9.47 9.54 3-step MSE (m/s) 1.9413 0.6929 0.7451 0.6288 0.605 0.6022 5-step 3.4093 0.7917 0.837 0.7601 0.6951 0.6894 MAE (m/s) 1.1112 0.6877 0.7223 0.6169 0.6143 0.6127 1.4227 0.7494 0.7776 0.6901 0.6777 0.674 MAPE (%) 18.31 11.25 11.95 10.02 10.13 10.17 23.55 12.22 12.94 10.89 11.05 11.03Table 11
Performance evaluations of different models for the forecast of the one-hour wind speed series.
Error ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model  ARIMA BPNN RENN SSA-RENN SSA-PSO-RENN Proposed model Spring 1-step MSE (m/s) 5.1762 4.5464 4.8944 1.6948 1.7761 1.5977 2-step 6.9013 5.7573 6.2777 2.6703 2.9233 2.3518 MAE (m/s) 1.8581 1.6491 1.7513 0.9221 0.9017 0.901 2.1454 1.8997 2.0602 1.1652 1.2252 1.0835 MAPE (%) 16.8 16.74 17.32 8.93 8.75 8.7 20.99 20.01 20.9 12 12.57 11.05 3-step MSE (m/s) 9.0442 6.9167 7.7342 3.7011 4.254 3.2441 5-step 12.4765 9.1089 10.1619 6.6403 7.6275 5.6975 MAE (m/s) 2.4758 2.069 2.2768 1.4621 1.5986 1.3514 2.9377 2.3763 2.613 1.9783 2.1813 1.8934 MAPE (%) 24.77 21.93 23.21 15.41 16.77 13.9 30.67 25.29 26.97 21.39 23.78 19.61  Summer 1-step MSE (m/s) 0.3986 0.2957 0.2727 0.2522 0.2437 0.2103 2-step 0.4336 0.3453 0.3165 0.3076 0.3052 0.24 MAE (m/s) 0.5323 0.453 0.4444 0.3861 0.3732 0.3495 0.5385 0.4741 0.4624 0.426 0.424 0.3665 MAPE (%) 15.19 13.55 13.48 11.67 11.34 10.63 15.31 14.32 14.24 12.81 12.89 11.11 3-step MSE (m/s) 0.4829 0.4236 0.3905 0.3509 0.3731 0.2917 5-step 0.5605 0.5792 0.5214 0.3982 0.4984 0.4136 MAE (m/s) 0.5713 0.5104 0.5026 0.4644 0.4754 0.4148 0.6208 0.6001 0.5815 0.5018 0.5576 0.5049 MAPE (%) 16.8 16 16.11 14.23 14.87 12.9 18.5 19.34 19.28 15.74 18.08 16.34  Fall 1-step MSE (m/s) 0.7161 0.54 0.5012 0.4865 0.4872 0.4901 2-step 0.8772 0.7482 0.6455 0.586 0.5816 0.5957 MAE (m/s) 0.7168 0.5844 0.5675 0.5727 0.5731 0.5733 0.7821 0.7098 0.6587 0.6175 0.6137 0.621 MAPE (%) 15.59 12.24 11.82 11.79 11.82 11.82 16.94 14.88 13.75 12.77 12.72 12.85 3-step MSE (m/s) 0.9588 0.8908 0.7079 0.6657 0.6553 0.68 5-step 0.8706 1.1014 0.7552 0.752 0.7259 0.7747 MAE (m/s) 0.8032 0.7729 0.6902 0.6707 0.6666 0.676 0.7562 0.8413 0.7071 0.703 0.6959 0.7108 MAPE (%) 17.46 16.3 14.46 13.91 13.86 14.03 16.41 18.06 15.04 14.51 14.43 14.7  Winter 1-step MSE (m/s) 1.6137 0.6872 0.6367 0.5746 0.5758 0.5683 2-step 2.079 0.7809 0.7208 0.6903 0.6766 0.678 MAE (m/s) 1.0548 0.6464 0.6414 0.641 0.6314 0.6306 1.1227 0.7051 0.7027 0.7059 0.6933 0.6993 MAPE (%) 21.17 12.57 12.5 12.31 12.03 12.02 23.1 14.09 13.95 13.77 13.45 13.55 3-step MSE (m/s) 2.4407 0.8516 0.781 0.7784 0.7552 0.7616 5-step 3.1868 0.9485 0.869 0.821 0.8172 0.816 MAE (m/s) 1.204 0.74 0.7345 0.7416 0.718 0.7279 1.3773 0.774 0.7643 0.7534 0.7495 0.7516 MAPE (%) 24.88 15.01 14.71 14.52 13.99 14.15 27.5 15.91 15.38 14.73 14.49 14.5Figure 7
The multistep predicted results of one-hour wind speed series using the different involved models.From Tables9–11, the values in bold indicate the smallest values of MSE, MAPE, and MAE, and the minimum values and the predictive value of the proposed model are approximately the same; compared to 10 min forecasting results, one-hour wind speed forecasting has more error; with the increase of prediction steps, precision will rise; some wind speed series are more adaptable to other models, but overall, the proposed model has better prediction ability.To reflect the forecasting results more directly, the results of Tables9–11 can be averaged. The average results are shown in Table 12.Table 12
Average results of one-hour wind speed.
1-step 2-step 3-step 5-step ARIMA MSE (m/s) 1.6354 2.1309 2.5701 3.2735 MAE (m/s) 0.9486 1.0661 1.1516 1.2783 MAPE (%) 15.65 17.61 19.03 21.02  RENN MSE (m/s) 1.2826 1.644 1.9423 2.359 MAE (m/s) 0.7939 0.891 0.9484 1.0233 MAPE (%) 12.92 14.4 15.36 16.67  SSA-PSO-RENN MSE (m/s) 0.71 1.0889 1.4313 1.9642 MAE (m/s) 0.6125 0.73 0.8369 0.9602 MAPE (%) 10.45 12.23 13.86 15.68  BPNN MSE (m/s) 1.3162 1.6523 1.8989 2.285 MAE (m/s) 0.8015 0.9098 0.9719 1.0608 MAPE (%) 13.28 15.02 16.11 17.76  SSA-RENN MSE (m/s) 0.72 1.064 1.3589 1.8034 MAE (m/s) 0.6239 0.7265 0.8203 0.9272 MAPE (%) 10.63 12.26 13.72 15.27  SSA-FAPSO-RENN MSE (m/s) 0.6723 1.0123 1.3551 1.9051 MAE (m/s) 0.6021 0.7043 0.808 0.9482 MAPE (%) 10.23 11.78 13.39 15.5Table12 indicates the following:(7)
The forecasting results of 1-step are better than 2-step, 3-step, and 5-step. The MAPE values of the proposed model change from 10.23% to 11.78%, 13.39%, and 15.50% at 2-step, 3-step, and 5-step. Other models can also reach this conclusion.(8)
Among all involved single models, the RENN model has the best performance in every forecasting step, and the ARIMA model has the worst performance. The MAPE values of the ARIMA model are 15.65%, 17.61%, 19.03%, and 21.02% at 1-step, 2-step, 3-step, and 5-step, respectively. However, the values of RENN reduce to 12.92%, 14.40%, 15.36%, and 16.67%.(9)
SSA-FAPSO-RENN offered the most accurate forecast value. In 1-step forecasting experiment, the MAPE values of the combined model were reduced by 20.8%, 3.8%, and 2.1% compared with the Elman model, SSA Elman, and SSAFA Elman, respectively. Similar conclusions can be obtained from multistep experiments.(10)
The above conclusion can also be achieved with MSE and MAE.The forecasting results are generally good. The proposed hybrid model can be used to forecast 60 min wind speed. Compared with traditional single models and other models involved in this paper, the proposed model has the best forecasting ability. The forecasting results show that the model has a better performance in the 10 min wind speed forecasting study than the 60 min study.The average results of study one and study two are shown in Table13. The proposed model has the best performance as evaluated by MAPE, MAE, and MSE. The Diebold-Mariano values of the SSA-PSO-RENN and SSA-RENN models are larger than the upper limits at the 10% significance level, the DM values of the RENN and the BP models are larger than the upper limits at the 5% significance level, and the DM value of the ARIMA models is larger than the upper limits at the 1% significance level. The result indicates that the proposed hybrid model is significantly better than other models.Table 13
The average results of study one and study two.
ARIMA BP RENN SSA-RENN SSA-PSO-RENN Proposed model MSE (m/s) 1.7270 1.4440 1.3634 0.9276 0.9602 0.9182 MAE (m/s) 0.9187 0.8489 0.8063 0.6661 0.6714 0.6574 MAPE (%) 14.23 12.64 12.10 10.12 10.16 9.94 DM 2.57 ∗ ∗ 2.67 ∗ 2.45 ∗ ∗ 1.89 ∗ ∗ ∗ 1.74 ∗ ∗ ∗ — ∗ is 1% significance level; ∗∗ is 5% significance level; ∗∗∗ is 10% significance level.


## 4. Conclusions

Wind power systems need to further develop accurate and reliable technology for short-term wind speed forecasting. Due to the influence of various meteorological factors, wind speed series are intermittent and randomly characterized, making it difficult to forecast wind speed using a single model. The focus of recent research has been the development of new methods and combinations of methods. However, individual models do not always achieve desirable performance. Hybrid models can decrease negative influences that are intrinsic in each of the individual models. These models can use the advantages of each individual model and are less sensitive, in certain cases, to the factors that cause the individual models to perform undesirably. Therefore, the hybrid model is more effective than individual models for wind speed forecasting.To forecast the 10 min and one-hour wind speed more accurately, a new hybrid model, SSA-FAPSO-RENN, is proposed, which can overcome many limitations of single models, such as poor prediction accuracy and artificial parameters. The forecasting results show that the proposed model can improve the accuracy of 10 min and 60 min wind speed forecasting. Compared with other models involved in this paper, the prediction precision of the proposed model is the largest.



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*Source: 3623412-2016-12-04.xml*