A novel hybrid system based on a new proposed algorithm—Multi-Objective Whale Optimization Algorithm for wind speed forecasting

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A novel hybrid system based on a new proposed algorithm—Multi-Objective Whale Optimization Algorithm for wind speed forecasting ⁎

Jianzhou Wang, Pei Du , Tong Niu, Wendong Yang School of Statistics, Dongbei University of Finance and Economics, Dalian 116025, China

H I G H L I G H T S propose a new algorithm—Multi-Objective Whale Optimization Algorithm (MOWOA). • We hybrid system based on MOWOA is proposed for wind speed forecasting. • AThenovel proposed model is compared to sixteen models for wind speed prediction. • The new • proposed hybrid system demonstrates higher prediction accuracy and reliability.

A R T I C L E I N F O

A B S T R A C T

Keywords: Wind speed forecasting Multi-Objective Whale Optimization Algorithm Hybrid forecasting system Forecasting accuracy and stability

In recent years, managers and researchers have paid increasing attention to accurate and stable wind speed prediction due to its significant effect on power dispatching and power grid security. However, most previous research has focused only on enhancing either accuracy or stability, with few studies addressing the two issues, simultaneously. This task is challenging due to the intermittency and complex fluctuations of wind speed. Therefore, we proposed a novel hybrid system based on a newly proposed called the MOWOA, which includes four modules: a data preprocessing module, optimization module, forecasting module, and evaluation module. An effective decomposing technique is also applied to eliminate redundant noise and extract the primary characteristics of wind speed data. In order to obtain high accuracy, and stability for wind speed prediction simultaneously, and overcome the weaknesses of single objective optimization algorithms, the optimization module of the proposed MOWOA is utilized to optimize the weights and thresholds of the Elman neutral network used in the forecasting module. Finally, the evaluation module, which includes hypothesis testing, evaluation criteria, and three experiments, is introduced perform comprehensive evaluation on the system. The results indicate that the proposed MOWOA performs better than the two recently developed MOALO and MODA algorithms, and that the proposed hybrid model outperforms all sixteen models used for comparison, which demonstrates its superior ability to generate forecasts in terms of forecasting accuracy and stability.

1. Introduction Wind energy is one type of the most promising renewable energy resources and an excellent alternative to fossil energy for addressing environmental problems and the current energy crisis [1], because it is clean, widely available, inexhaustible, and economical. More importantly, it has become the fastest growing renewable energy resource for electricity generation and is receiving increasing attention globally [2]. At the end of 2016, the global cumulative installed wind capacity reached approximately 486,749 MW [3]. Fig. 1 shows the global top 10 wind energy installation capacities from January to December 2016. However, the random and unstable characteristics of the wind speed

⁎

tend to increase operating costs and reduce the reliability and stability of electricity grids [4]. Therefore, with the goal of addressing these problems and improving the utilization efficiency of wind power conversion, the accurate and stable forecasting of wind speed has become a popular research topic [5,6]. During the past few decades, an increasing number of technologies have been proposed for predicting wind speed. These approaches can largely be divided into five categories [7]: physical models, conventional statistical models, spatial correlation models, artificial intelligence models, and hybrid forecasting models. Physical models, which are suitable for long-term wind speed forecasting, consider not only historical data, but also make use of physical parameters,

Corresponding author. E-mail address: [email protected] (P. Du).

http://dx.doi.org/10.1016/j.apenergy.2017.10.031 Received 4 July 2017; Received in revised form 2 September 2017; Accepted 5 October 2017 0306-2619/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Wang, J., Applied Energy (2017), http://dx.doi.org/10.1016/j.apenergy.2017.10.031

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Fig. 1. Distribution of wind power all over the world in 2016.

short-term wind speed and ultimately achieved higher precision than previous methods. Similarly, Xiao et al. [24] developed a novel hybrid forecasting architecture, which integrated improved BA, SSA, and a general regression neural network (GRNN) for multi-step wind speed forecasting. Numerical experiments demonstrated that their proposed hybrid model can obtain the most accurate forecasting results for onestep to three-step wind speed forecasting. Sun et al. [25] proposed a new dynamic integrated approach by utilizing phase space reconstruction, data preprocessing approaches, and a core vector regression model, optimized by the competition over resource heuristic algorithm, for wind speed forecasting. The results indicated that their proposed integrated method can significantly improve forecasting effectiveness and statistically outperform the benchmark models used for comparison. However, to the best of our knowledge, most previous studies are based on single-objective optimization algorithms for improving prediction accuracy, which neglect the importance of forecasting stability improvement, despite it being vital to the effectiveness of forecasting models. More importantly, both accuracy and stability are very significant, especially when evaluating the prediction performance of different models. It is noteworthy that the problem of achieving high accuracy and strong stability simultaneously for wind speed forecasting belongs to the set of multi-objective optimization problems (MOPs) rather than the set of single-objective optimization problems (SOPs). Therefore, multi-objective optimization algorithms should be introduced into the field of time series forecasting to achieve more accurate and stable prediction results, simultaneously. Fortunately, recent soft-computing technique development has yielded many new multi-objective algorithms, such as the multi-objective BA (MOBA) [26], binary coded elitist non-dominated sorting GA (NSGA-II) [27], multi-objective ant lion optimizer (MOALO) [28], multi-objective dragonfly algorithm (MODA) [29], etc. Currently, the application of multi-objective optimization techniques can be observed in many fields of research, such as mechanical engineering [30], the design of water distribution networks [31], and other fields [32].

including temperature, density, speed, and topography information [8,9]. In contrast, statistical models, such as the autoregressive (AR) model, autoregressive moving average (ARMA) model, and the widely used autoregressive integrated moving average (ARIMA) model, are more appropriate for short-term wind speed prediction, which is simple to implement by various historical data. However, these models cannot handle the non-linear features of wind speed due to their linear correlation structure [10]. Typically, spatial correlation models primarily utilize the spatial relationships between wind speeds at different sites. In some cases, they can achieve satisfactory prediction accuracy [11]. However, their information requirements, including wind speed and delay times, add complexity and cost to the implementation of spatial correlation forecasting [12]. Fortunately, with the rapid development of soft-computing techniques, many different intelligent algorithms, including artificial neural networks (ANNs), support vector machines (SVMs) [13], and other mixed models have been successfully developed and widely applied for wind speed forecasting [14–16]. Due to the inherent weakness of each model, as well as the intermittency and complex fluctuations of wind speed, individual forecasting model cannot always capture the characteristics of time series, especially when comes to the non-linear traits of wind speed. Thus, in order to obtain an advanced forecasting method for higher accuracy levels and wider forecasting horizons, approaches, called hybrid models [17] have emerged. These models incorporate the individually superior features of multifarious algorithms, including forecasting models (i.e., SVM, ANNs, etc.), intelligent optimization algorithms (i.e., the bat algorithm (BA) [18], Whale Optimization Algorithm (WOA) [19], etc.), and data-processing algorithms (i.e., singular spectrum analysis (SSA) [20], variational mode decomposition (VMD) [21], complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) [22], etc.). Recently, with the aim of achieving superior forecasting results, many hybrid forecasting models have been successfully presented and have improved the precision of wind speed predictions to some extent. For example, Wang et al. [23] employed ensemble empirical mode decomposition (EEMD), genetic algorithm (GA) and ANNs to predict 2

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2. Multi-Objective Whale Optimization Algorithm

However, there is no single optimization technique for solving all optimization problems, which has been logically proven by the “No Free Lunch (NFL)” [33], although related studies indicate that multi-objective algorithms can effectively approximate the true Pareto optimal solutions of most MOPs. Therefore, based on this theorem and the basic WOA, we have developed a novel multi-objective optimization algorithm called the Multi-Objective Whale Optimization Algorithm (MOWOA). The proposed algorithm can not only overcome the drawbacks of single-objective optimization algorithms (such as their focus on single objective functions) and but also simultaneously achieve high accuracy and excellent stability for wind speed forecasting. Therefore, in order to obtain higher accuracy and better stability for wind speed prediction simultaneously, a novel hybrid system involving CEEMD, the widely used Elman neural network (ENN), and our proposed MOWOA algorithm is successfully presented in this paper. This system consists of four modules: a data preprocessing module, optimization module, forecasting module and evaluation module. The proposed MOWOA, which is used in the optimization module, is first tested on four multi-objective benchmark problems and compared to two recently developed meta-heuristics: MOALO and MODA. After verifying the effectiveness of the proposed MOWOA as well as achieving nonstationary wind speed decomposition by using CEEMD in data preprocessing module, the hybrid forecasting MOWOA-ENN (i.e. ENN optimized by MOWOA) model is established and applied for wind speed forecasting in the forecasting module. Finally, the evaluation module, which includes hypothesis testing, evaluation criteria, and three experiments using eight wind speed data sets collected from wind farms of China, is adopted to provide a comprehensive evaluation of the proposed hybrid system. The main originalities of this paper can be summarized as follows:

In this part, the concepts of multi-objective optimization and the related description of the MOWOA are all described as follows. 2.1. Multi-objective optimization Single-objective optimization algorithms have only one objective function and can easily compare solutions to find a global optimum by applying the conventional relational operators: > , ≥, < , ≤ and =. However, these operators cannot be adopted in the process of multiobjective optimization due to the existence of more than one objective function in multi-objective optimization problems. With the hope of addressing this problem, scientists present a new concept: dominates, which can be utilized to find best solutions. Without loss of generality, the relative definitions of the maximization problems, Pareto Dominance, Pareto optimality, Pareto optimal set, and Pareto optimal front are given as follows [34]: Definition 1. Minimization problem: In multi-objective optimization, a minimization problem can be written as:

Minimize : F (x ) = {f1 (x ),f2 (x ),…,fo (x )} Subject to: gj (x ) ⩾ 0, j = 1,2,…,m hj (x ) = 0, j = 1,2,…,p Lj ⩽ x j ⩽ Uj, j = 1,2,…,n

(1)

where n, o, m and p represent the number of the variables, objective functions, inequality constraints and equality constraints, respectively. While gj and hj are the j-th inequality and equality constraints. [Lj ,Uj] stand for the boundaries of j-th variable.

(a) Development of a new robust multi-objective algorithm. A new algorithm called the Multi-Objective Whale Optimization Algorithm is successfully developed in this paper and not only outperforms two recently devolved algorithms (MOALO and MODA) in terms of computation time, but also provides a novel viable option for solving multi-objective optimization problems. (b) Focusing on accuracy and stability simultaneously. Most previous studies focus only on improving accuracy or stability, with few focusing on the accuracy and stability simultaneously. Therefore, in this paper, we present a novel hybrid system based on the MOWOA, which can not only obtain high accuracy and stability for wind speed prediction simultaneously, but also overcome the weaknesses of single objective optimization algorithms. (c) Focusing not only on single-step forecasting, but also on multistep forecasting. Widely utilized one-step wind speed forecasting results are sometimes insufficient to guarantee the reliability and controllability of wind power systems. Thus, in addition to singlestep forecasting, multi-step wind speed predictions with various time intervals are also adopted and discussed in this paper to provide additional future wind speed information. (d) Scientific and reasonable model evaluation system. Hypothesis testing, eight evaluation criteria, and experiments using ten wind speed datasets are all introduced to provide a comprehensive evaluation of the proposed system. The results indicate that the developed hybrid model can outperform all sixteen models used for comparison, which demonstrates its superior performance for generating forecasts in terms of forecasting accuracy and stability.

Definition 2. Pareto Dominance: Two vectors x = (x1,x2,…,x n ) and y = (y1,y2 ,…,yn ) . x dominates y (i. e . x ≻ y ) iff:

∀ i ∈ [1,n], [f (x i ) ⩾ f (yi )] ∧ [ ∃ i ∈ [1,n]: f (x i )]

(2)

Definition 3. Pareto optimality: A Pareto-optimal x belongs to X iff:

∄ y ∈ X s. t . F (y ) ≻ F (x )

(3)

Definition 4. Pareto optimal set:

Ps : ={x ,y ∈ X |∃ F (y ) ≻ F (x )}

(4)

Definition 5. Pareto optimal front: A set including the value of objective functions for Pareto solutions set:

Pf : ={F (x )|x ∈ Ps}

(5)

2.2. MOWOA The developed MOWOA is a multi-objective version of the WOA proposed by Mirjalili and Lewis in 2016 [19]. The special hunting behavior (i.e., spiral bubble-net feeding technique) of humpback whales was the primary inspiration for this algorithm. Based on related articles [19,35–37], the core mathematical model of this algorithm is introduced in two phases.

The rest of this paper is organized as follows. Section 2 develops a novel multi-criterion optimization algorithm. Section 3 describes the wind speed decomposition involved in the hybrid forecasting system. And three experiments and their corresponding forecasting results as well as analysis are given in Section 4. The discussions are introduced in Section 5. Finally, the last Section 6 presents the conclusions of this paper.

(a) Searching and encircling prey The mathematical model about searching prey can be shown as follows: 3

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D = |C ⊗ Xrand −X |

(6)

X (t + 1) = Xrand −A ⊗ D

(7)

empirical mode decomposition (EMD) [39], but also completely neutralizes noise by adding paired white noise, eventually obtaining components with less noise and greater significance. Based on related articles [38,40], the mainly steps of this decomposition method is given as follows.

where the coefficient vectors A and D can be expressed by:

A = 2 ⊗ a ⊗ r −a

(8)

C=2⊗r

(9)

Step 1: Add white noise in pairs (i.e. the positive white noise Wni (t ) and the negative white noise −Wni (t ) ) to the original data x (t ) to generate one pair of polluted signals with positive and negative noise respectively.

where r is a random number between [0, 1], a is linearly decreasing from 2 to 0.

D = |C ⊗ X ∗ (t )−X (t )|

(10)

X (t + 1) = X ∗ (t )−A ⊗ D

(11)

⎧ Pni = x (t ) + Wni (t ) ⎨ ⎩ Nni = x (t )−Wni (t )

where t is the current iteration, X represents the position vector, and X ∗ is the position vector of the best solution obtained so far. Here, if |A| ⩾ 1, searching prey used by Eqs. (6) and (7), otherwise, perform shrinking mechanism for encircling prey by applying Eqs. (10) and (11).

(14)

Step 2: According to the EMD technique, decompose the polluted signals (Pni, Nni ) into a finite set of Intrinsic Mode Functions (IMFs) components. M

⎧ + ⎪ Pni (t ) = ∑ cij (t ) ⎪ j=1 M ⎨ ⎪ Nni (t ) = ∑ cij− (t ) ⎪ j=1 ⎩

(b) Spiral updating position It is assumed that the humpback whales either take the shrinking encircling mechanism or update their positions by applying the spiral model, the probabilities of these two cases are all 50%. This process can be mathematically simulated by the following formulas:

X ∗ (t )−A

⊗D if X (t + 1) = ⎧ bl ⊗ cos(2πl ) + X ∗ (t ) if ′ ⎨ ⊗ D e ⎩

cij+

where and are the j-th IMF of the i-th trial with positive and negative noise, respectively. Moreover, M indicates the number of IMFs. Step 3: Repeat the above two Steps 1 and 2 K times using different white noises to obtain two collections of IMF components i.e. the collection of IMFs of the i-th time with positive noises {cij+ (t )}iK=,M 1,j = 1 and positive noises {cij− (t )}iK=,M 1,j = 1. Step 4: Obtain the ensemble means of the whole IMFs, and the j-th IMF component cj (t ) of this algorithm can be calculated by:

p < 0.5 p ⩾ 0.5

(12)

where p is a random number in [0, 1], l is in [−1, 1] and b is constant for describing the spiral shape. In order to perform multi-objective optimization using the WOA, several significant mechanisms, such as an archive, roulette wheel selection, etc., are adopted in the MOWOA. The archive is a simple storage unit, which is responsible for storing the non-dominated Pareto optimal answers achieved up to the current iteration. It is worth noting that a maximum number of members for the archive exists in this algorithm. Thus, non-dominated solutions achieved during the process of iteration should be compared with the archived members to update the archive. If a new solution dominates one or more solutions in the archive, or if neither the new solution nor the archive members dominate each other, the new solutions will be allowed to enter the archive. Otherwise, the archive should not be updated. Additionally, if the archive is full, the solutions with the most populated neighborhoods should be removed from the archive to accommodate new solutions. The corresponding probability for this accommodation can be expressed as:

Pi =

Ni , c>1 c

(15)

cij−

cj (t ) =

1 2K

K

∑

(cij+ (t ) + cij− (t ))

(16)

i=1

3.2. Elman neural network (ENN) As a member of the recurrent neural networks architecture, ENN is a two-layer network, with additional feedback connections from the output of the hidden layer to its input layer. It was proposed by Elman in 1990 [41]. The special connections are called context nodes. These nodes make ENN more sensitive to historical data and improve its ability to handle the dynamic information. Moreover, the main input layer of the ENN is U (k ) and the output layer is Y (k ) . The architecture of ENN can be expressed by the following formulas:

(13)

where c is a constant and Ni represents the number of solutions in the vicinity of the i-th solution. More importantly, with the goal of improving the distribution of solutions in the archive, a roulette wheel method with probability Pi = c / Ni is adopted in this algorithm. The pseudo code for the MOWOA algorithm is presented in Algorithm 1 in Section 3.3.

No

⎡ yk = f0 ⎢ ∑ bo + ⎣ o=1

Nh

∑

Nh

Ni

who ⊗ fh (bh +

h=1

∑ i=0

wih μi +

∑ j=0

⎤ wjh ah (k−1)) ⎥ ⎦ (17)

In this section, the related methods of the proposed hybrid forecasting system are introduced in detail, including CEEMD, Elman neural network, hybrid MOWOA-ENN model.

where wih , wjh and who are the weights that connects the nodes between the input layer and the hidden layer, between the recurrent and the hidden layer, and between the hidden layer and the output layer, respectively. Moreover, bh and bo represent the biases of hidden and output layer, fh (·) and f0 (·) are hidden and output functions, respectively [42]. Moreover, Fig. 2 presents the architecture of Elman neural network, which can aid in understanding its operation mechanism.

3.1. Data preprocessing module: Complementary CEEMD (CEEMD)

3.3. The MOWOA-ENN model

As an improved version of EEMD, CEEMD was first proposed by Yeh et al. in 2010 [38]. It not only addresses the mode mixing problem of

Considering one criterion, such as accuracy or stability, of the forecasting results is not sufficient. In this subsection, we describe two

3. Proposed hybrid forecasting model

4

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Fig. 2. The architecture of Elman neural network.

14 15 16 17 18 19 20

objective functions focusing on obtaining high precision and stability simultaneously. First, the widely used function O1 (·) from the conventional single-objective algorithm is employed to obtain high prediction accuracy. It’s well known that errors in forecasting are one of the most important indicators for evaluating the performance of forecasting models. Thus, in this study, the stability of forecasting errors is utilized as the other objective function O2 (·) , which reflects the stability of forecasting models. The pseudo-code of the hybrid MOWOA-ENN model as well as the two functions is presented in Algorithm 1.

21 22 23 24 25 26 27 28 29 30 31

Algorithm 1. MOWOA-ENN.

Objective functions: 1

Minimize

N

̂ yt )2 ⎧O1 (y ) = MSE = N ∑t = 1 (yt − ⎨O2 (y ) = std (y − t ̂ yt ), t = 1,2,…,N ⎩

Input:

x t(0) = (x (0) (1),x (0) (2),…,x (0) (q)) —the training samples x f(0) = (x (0) (q + 1),x (0) (q + 2),…,x (0) (q + l)) —the test samples

32 33 34

Output:

x f(0) ̂ = (x (0) ̂ (q + 1),x (0) ̂ (q + 2),…,x (0) ̂ (q + l)) —the forecasting data Parameters: IterMax—the maximum iterations n—the number of wolves Fi—the fitness function of i-th whale xi—the position of i-th whale t—current iteration number d—the number of dimension. 1 /∗Set the parameters of MOWOA. ∗/ 2 /∗Initialize population of n whales xi (i = 1, 2, … n) randomly. ∗/ 3 /∗Define the archive size. ∗/ 4 FOR EACH i: 1 ≤ i ≤ n DO 5 Compute the corresponding fitness function using ranking process 6 END FOR 7 /∗Determine the best search agent X∗. ∗/ 8 WHILE (t < IterMax) DO 9 FOR EACH i = 1: n DO 10 FOR EACH j = 1: m DO 11 /∗Update a, A and C, l and p ∗/ 12 /∗Select a random antlion from the archive. ∗/ 13 /∗Select the elite using Roulette wheel from the archive. ∗/

35 36 37 38 39 40 41 42 43 44 45 46 47 48

5

IF (p < 0.5) THEN IF (|A| < 1) THEN /∗Update the position of the current search agent. ∗/ D = |C ⊗ X ∗ (t )−X (t )| IF (|A| ≥ 1) THEN Select a random search agent (Xrand) /∗Update the position of the current search agent. ∗/ X (t + 1) = X ∗ (t )−A ⊗ D END IF END IF END IF ELSE IF (p ≥ 0.5) THEN /∗Update the position of the current search agent. ∗/ X ( t+ 1) = D′ ⊗ e bl ⊗ cos(2πl) + X ∗ (t ) END IF END FOR END FOR /∗Check if any search agent goes beyond the search space and amend it. ∗/ /∗Calculate the objective values. ∗/ /∗Find the non-dominated solutions. ∗/ /∗Update the archive in regard to the obtained non-dominated solutions. ∗/ IF the archive is full DO /∗ Delete some solutions from the archive to hold the new solutions. ∗/ Using Roulette wheel and Pi = Ni / c, c > 1 END IF IF the archive is full DO /∗ Update the boundaries to cover the new solution(s). ∗/ END IF t = it + 1 END WHILE RETURN archive Obtain X∗ according to archive Set parameters of ENN according to X∗. Use xt to train and update the parameters of the ENN Import wind speed series into ENN to achieve the predicted value x f̂ .

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4. Experiments and analysis

studies to evaluate forecasting performance [35]. However, there are no set rules for selecting accuracy metrics. Therefore, in this paper, we use the eight common evaluation metrics presented in Table 2 to evaluate the performance of the forecasting models. The traits of these accuracy metrics are described below. Mean absolute error (MAE) reflects the overall level of errors. Normalized mean squared error (NMSE) can be used as an estimator, to reflect the deviations between the forecasted and observed values. Root of mean squared error (RMSE) is used because it can easily reflect the degree of differences between the observed and forecasted values. Mean absolute percent error (MAPE) is a measure of the prediction accuracy of a forecasting method in statistics. Median absolute percentage error (MdAPE) is used in evaluation metrics when forecasting methods must be compared across different series. Directional Change (DC) reflects prediction movement directions or turning points. Pearson's correlation coefficient measures the correlation between the predicted sequence and the observed sequence. Finally, the index of agreement (IA) is also a useful measure of model performance allows for sensitivity to differences in observed and predicted sequences, as well as proportionality changes [45].

In this section, three experiments called Experiment I, Experiment II, and Experiment III are established to verify the performance of the proposed hybrid forecasting system. Experiment I consists of two parts and its purpose is to compare the proposed MOWOA with the MOALO and MODA to verify the performance of the MOWOA. Experiment II is designed to compare the proposed hybrid forecasting model with several benchmark models, including ENN, WOA-ENN and CEEMD-WOAENN, etc, to test the forecasting performance of the proposed CEEMDMOWOA-ENN model. Furthermore, in an effort to comprehensively test the proposed hybrid forecasting model, four other forecasting models are adopted in Experiment III as additional comparison points. The models used for comparison are LSSVM, persistence model, SSA-CSENN, and VMD-BBO-BPNN. Because there are no clear rules to aid researchers in setting the number of neurons in an ANN, the number of nodes is chosen by trial and error [43,44] in this study. The experiments and analysis are described in detail in the following subsections. 4.1. Data description

4.3. Experiment I: Tests of MOWOA and CEEMD-MOWOA-ENN In this paper, nine 10-min wind speed data sets from six sites, as well as an additional 30-min wind speed dataset collected from wind farms of China, are utilized as illustrative examples to determine the effectiveness and efficiency of the proposed method. Wherein the total number of each data set is 2880, and the first 2304 observations are taken as the training set, and the remaining 576 observations are regarded as the testing set. Moreover, the statistical information i.e. maximum, minimum and average values etc. of the whole datasets involved in this paper are shown in Table 1.

In this experiment, four test functions and two well-known metaheuristics: MOALO and MODA are used to test and compare the developed MOWOA algorithm. Moreover, four different seasons’ historical wind speed data sets of Site 1 are utilized to verify the performance of the proposed CEEMD-MOWOA-ENN forecasting model by comparing with CEEMD-MOALO-ENN and CEEMD-MODA-ENN. 4.3.1. Test of MOWOA Four test functions (shown in Appendix A), the performance metric of inverted generational distance (IGD), written below [34], and the two well-known meta-heuristics MOALO and MODA are applied to verify the quality of the proposed algorithm. For every test function, 50

4.2. Evaluation metrics Many accuracy metrics have been developed and used in various Table 1 Descriptive statistics of ten wind speed data sets. Seasons and Sites Four seasons of Site 1 (Experiment I/III)

Spring

Summer

Fall

Winter

Four different sites (Experiment II/III)

Site 2

Site 3

Site 4

Site 5

Multi-step ahead forecasting (Section 5.4)

10-min

30-min

Data

Numbers

Mean (m/s)

Max. (m/s)

Median (m/s)

Min. (m/s)

Std. (m/s)

All samples Training Testing All samples Training Testing All samples Training Testing All samples Training Testing

2880 2304 576 2880 2304 576 2880 2304 576 2880 2304 576

7.1694 7.2633 6.7941 6.0194 6.0929 5.7253 5.6902 5.6518 5.8436 6.9409 7.1134 6.2509

17.5000 17.5000 15.2000 14.1000 14.1000 13.5000 12.8000 12.8000 12.4000 17.2000 17.2000 10.9000

7.1000 7.2000 6.1000 5.9000 6.1000 5.6000 5.2000 5.1000 5.4000 6.5000 6.6000 6.2000

0.8000 0.8000 1.2000 0.8000 0.8000 1.7000 0.8000 0.8000 1.6000 1.4000 1.8000 1.4000

3.0145 2.8265 3.6502 2.7451 2.9208 1.8587 2.5510 2.5140 2.6909 2.6479 2.7858 1.8508

All samples Training Testing All samples Training Testing All samples Training Testing All samples Training Testing

2880 2304 576 2880 2304 576 2880 2304 576 2880 2304 576

5.3180 5.3131 5.7394 6.1434 6.1325 5.1408 7.1212 7.1155 8.9634 4.9832 4.9597 6.5997

13.7000 13.7000 9.7000 20.0000 20.0000 10.5000 19.8000 19.8000 18.6000 16.6000 16.6000 19.5000

4.8000 4.8000 5.9000 5.3000 5.3000 5.1000 6.8000 6.8000 8.4000 4.3000 4.3000 4.7000

0.7000 0.7000 1.0000 0.8000 0.8000 1.2000 0.5000 0.5000 2.1000 0.2000 0.2000 0.8000

2.8260 2.8151 1.7827 3.6329 3.6169 2.1345 3.1980 3.1850 2.9809 2.9034 2.8987 4.7976

All samples Training Testing All samples Training Testing

2880 2304 576 1680 1440 240

8.1241 8.2375 7.6703 6.0443 6.2285 4.9388

18.7000 17.4000 18.7000 15.3000 15.3000 10.2000

8.0000 8.1000 6.7500 6.0000 6.1000 4.7500

0.8000 0.8000 1.0000 0.6000 0.6000 1.1000

3.3331 3.0360 4.2955 2.6983 2.7665 1.9068

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Table 2 Error metrics. Metric

Definition

Equation

MAE

The mean absolute forecasting error of t times forecasting results

MAE =

NMSE

The normalized average of the squares of the errors

RMSE

Average of prediction error squares

MAPE

The average of absolute error

MdAPE

The median of N absolute percentage error

r

Pearson's correlation coefficient

DC

The directional change of the forecasting results

NMSE =

N

∑t = 1 |yt −yt |̂ 1 N

MAPE =

N

∑t = 1 1 N

RMSE = 1 N

(yt̂ − yt ) yt̂ ·yt

r=

The index of agreement of forecasting results

2

N

∑t = 1 (yt −yt )̂ 2 N

∑t = 1

yt − yt̂ yt

MdAPE = median (

DC = IA

1 N

× 100%

yt − yt̂ yt

× 100%)

∑tN= 1 (yt − y )(yt̂ − y ̂) ∑tN= 1 (yt − y )2 ∑tN= 1 (yt̂ − y ̂)2

0, otherwise N −1 ∑t = 1 at ,at = ⎧ ⎨ ⎩1, if (yt +̂ 1−yt )(yt + 1−yt ) > 0

100 N−1

N

N

̂ yt )2/ ∑t = 1 (|yt − ̂ y | + |yt −y |)2 IA = 1− ∑t = 1 (yt −

Note: yt ̂ is the t-th forecasting value, N is the total number, y ̂ and y are the average of the forecasting and observed values, respectively. Table 3 Results of the multi-objective algorithms (using IGD) on the four test functions adopted in this paper. Algorithm

ZDT1

Algorithm

Ave

Std.

Median

Best

Worst

MOALO MODA MOWOA

0.006571 0.003886 0.001199

0.005144 0.001770 0.000181

0.004238 0.003209 0.001188

0.002430 0.002045 0.000891

0.024140 0.009824 0.001704

Algorithm

ZDT3

MOALO MODA MOWOA

Ave

Std.

Median

Best

Worst

0.024807 0.024757 0.024472

0.000595 0.000379 0.000237

0.024742 0.024695 0.024420

0.023550 0.024278 0.024178

0.026117 0.025794 0.025219

ZDT2 Ave

Std.

Median

Best

Worst

MOALO MODA MOWOA

0.010416 0.003909 0.001109

0.007407 0.003985 0.000120

0.007396 0.003073 0.001092

0.002349 0.002028 0.000859

0.027703 0.030180 0.001354

Algorithm

ZDT1 with linear front

MOALO MODA MOWOA

Ave

Std.

Median

Best

Worst

0.006662 0.003413 0.001110

0.004815 0.001142 0.000144

0.004713 0.003148 0.001074

0.002210 0.002006 0.000898

0.025447 0.008042 0.001553

Note: The values in bold indicate the best values of IGD.

seasons. The following findings can be derived from this data:

experiments were implemented to search for the Pareto optimal solutions. Note that we applied 100 iterations, 40 search whales, and an archive size of 100 in every experiment. Furthermore, the qualitative results for the best obtained Pareto optimal values of the algorithms are presented in Table 3 and in Figs. 2 and 3. It can be clearly observed that the developed MOWOA performs well and is superior to MOALO and MODA.

IGD =

1 N

(a) For all of wind speed datasets in this subsection, compared to the other two hybrid models, (i.e., CEEMD-MOALO-ENN and CEEMDMODA-ENN), the CEEMD-MOWOA-ENN model achieves the highest perdition accuracy (measured using the error metrics in Table 2) with the lowest computer run time. In other words, the proposed MOWOA provides extremely competitive results and outperforms the MOALO and MODA methods. (b) For the Spring, Summer, Fall, and Winter wind speed data sets from Site 1, the proposed hybrid model can satisfactorily approximate the actual values when compared to the two other models. This means that the proposed model is not affected by seasonal changes due to its superiority forecasting ability. For example, the MAPE values of the developed hybrid model for all four seasons are within in 4.6% and their fluctuation ranges are all within approximately 0.2%. (c) One can see that the error metrics adopted in this paper can adequately reflect the forecasting performance of the models. In other words, the error metrics are all effective, meaning be used to accurate evaluate the abilities of the forecasting models. Remark. For the datasets for all four seasons at Site 1, the best values of MAE, NMSE, RMSE, and MAPE, etc. of the developed hybrid forecasting model indicate that the proposed CEEMD-MOWOA-ENN model can achieve excellent forecasting accuracy due to the strong performance of the MOWOA, which supports the conclusions in Section 4.3.1.

N

∑i =1 di2

(18)

where di represents the Euclidean distance between the i-th true Pareto optimal solution and the nearest ones obtained by algorithms, N is the number of true Pareto optimal solutions. Remark. Compared to MODA and MOALO in Table 3 and Figs. 3 and 4, we can see that the developed MOWOA not only achieves the best values for IGD, but also finds more Pareto optimal solutions with higher accuracy and better robustness. Additionally, our algorithm provides a novel viable option for solving multi-objective optimization problems.

4.3.2. Test of CEEMD-MOWOA-ENN model This subsection is aimed at comparing the CEEMD-MOWOA-ENN model with the CEEMD-MOALO-ENN and CEEMD-MODA-ENN models to further prove the validity of the MOWOA. Table 4 presents the forecasting results and corresponding computer run times of these three hybrid forecasting models using four seasonal wind speed datasets. Fig. 5 shows the forecasting results of the three models in all four

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Fig. 3. Obtained Pareto optimal solutions by MOWOA, MODA and MOALO for ZDT1 and ZDT2. (Note: PF represents Pareto Font).

Site 5, where the MAPE and MdAPE values of CEEMD-WOA-ENN and CEEMD-ENN are 5.7238%, 6.3756% and 4.1930%, 5.2515%, respectively. The reason for this phenomenon is likely that the single-objective optimization WOA adopted in the CEEMD-WOAENN model cannot always handle non-linear wind speed time series with periodicity, trends, and randomness due to its own limitations. (b) In comparison II, by comparing the ENN model with the EEMD-ENN model, and the EEMD-ENN model with the CEEMD-ENN model (i.e., ENN vs. EEMD-ENN and ENN vs. CEEMD-ENN), we can see that EEMD-ENN and CEEMD-ENN significantly outperform the basic ENN model due to their higher prediction accuracy and hit rates for directional prediction. For example, for Site 2 in Table 5, the RMSE and MAPE values of the ENN, EEMD-ENN, and CEEMDENN methods are 0.5971, 0.3271, and 0.2866, and 8.8348%, 4.7769%, and 4.1544%, respectively. Furthermore, by comparing the CEEMD-ENN model to the EEMD-ENN model, it can be seen that the CEEMD decomposition method performs better than EEMD. (c) Comparison III is established to illustrate the high quality of the multi-objective optimization MOWOA by comparing the proposed CEEMD-MOWOA-ENN model with the CEEMD-WOA-ENN model. Based on the above analysis, as well as the simulation results in Table 5, we can draw the conclusion that the hybrid model using the MOWOA can significantly outperform the hybrid model using the WOA. This conclusion is support by the MAPE improvement values for the proposed model at the four sites, which are 0.3171%, 0.3872%, 0.3043%, and 0.2364%, respectively. (d) In comparison IV (comparing the entire models with each other), it can be observed that the hybrid model developed in this study outperforms the other models at generating forecasts in terms of forecasting accuracy and stability. For example, for Site 3, the MAPE (MdAPE) values of the comparison models (i.e., WNN, BPNN,

4.4. Experiment II Four types of model comparisons are designed in this subsection. Comparisons I and II include the WNN, BPNN, ENN, EEMD-ENN, CEEMD-ENN, WOA-ENN, CEEMD-ENN, and CEEMD-WOA-ENN models, which are built to emphasize important usages of the data decomposition technique and the optimization algorithm. Comparison III (i.e., CEEMD-WOA-ENN vs. CEEMD-MOWOA-ENN) is performed to compare the single-objective optimization algorithm (i.e., WOA) with the proposed multi-objective optimization algorithm (i.e., MOWOA) and is aimed at further demonstrating the superiority of the proposed hybrid forecasting method. Finally, comparison IV is implemented to compare the developed forecasting model with the aforementioned other methods, which further demonstrates the superiority of the forecasting model presented in this paper. The forecasting error values of the comparison models and the developed hybrid model are all displayed in Table 5 and Fig. 6. More detailed comparisons are outlined below: (a) For Comparison I, the WNN, BPNN, and ENN models were first built to compare with each other in order to determine the best one, which was found to be the ENN model. The subsequent comparisons (i.e., WOA-ENN vs. ENN and CEEMD-WOA-ENN vs. WOAENN) are designed to show the contribution of the WOA algorithm to the ENN model. Based on the results listed in Table 5, it can be observed that, without using decomposition methods, WOA can enhance the prediction accuracy of ENN, but not greatly. For example, for Site 4 in Table 5, the MAPE values of ENN and WOAENN are 8.0417% and 8.0022%, respectively. However, when adding the CEEMD technique, the prediction accuracy of CEEMDWOA-ENN is slightly lower than that of CEEMD-ENN, except for at

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Fig. 4. Obtained Pareto optimal solutions by MOWOA, MODA and MOALO for ZDT3 and ZDT1 with linear front.

model using the MOWOA enhances the forecasting accuracy to a large degree.

ENN, WOA-ENN, EEMD-ENN, CEEMD-ENN, and CEEMD-WOAENN) and the presented model are 10.8237% (7.8406%), 9.8773% (7.2458%), 9.8000% (7.3398%), 9.7572% (7.3905%), 5.5023% (3.9478%), 4.8658% (3.5903%), 4.9671% (3.7231%), and 4.5799% (3.4179%), respectively. Remark. For all observation sites, the proposed CEEMD-MOWOA-ENN model achieves the best values for MAE, NMSE, RMSE, MAPE, MdAPE, r, DC, and IA, meaning that the hybrid model can achieve excellent forecasting performance. Additionally, by comparing the other models in Table 5 to the proposed hybrid model, one can see that the proposed

4.5. Experiment III In order to deeply investigate and analyze the applicability and superiority of the proposed model as well as the proposed MOWOA, four other forecasting models (LSSVM, persistence model, SSA-CS-ENN, and VMD-BBO-BPNN) are used for additional experiments to verify the effectiveness and applicability of the CEEMD-MOWOA-ENN model. All

Table 4 Forecasting results of three hybrid models in four seasons at Site 1. Seasons

Models

MAE

NMSE

RMSE

MAPE

MdAPE

r

DC

IA

Spring

CEEMD-MOALO-ENN CEEMD-MODA-ENN CEEMD-MOWOA-ENN

0.2556 0.2630 0.2336

0.0043 0.0051 0.0039

0.3626 0.3676 0.3330

4.7055 5.0530 4.3752

3.2671 3.3716 2.8618

0.9951 0.9950 0.9959

78.6087 77.3913 82.6087

0.9975 0.9974 0.9979

Summer

CEEMD-MOALO-ENN CEEMD-MODA-ENN CEEMD-MOWOA-ENN

0.2748 0.3022 0.2459

0.0049 0.0060 0.0040

0.3882 0.4240 0.3465

5.1013 5.6160 4.5576

3.8614 4.2369 3.4469

0.9780 0.9736 0.9825

80.1739 77.3913 82.6087

0.9888 0.9866 0.9911

Fall

CEEMD-MOALO-ENN CEEMD-MODA-ENN CEEMD-MOWOA-ENN

0.2722 0.2579 0.2397

0.0047 0.0041 0.0036

0.3861 0.3608 0.3383

4.9976 4.7338 4.3975

3.6744 3.3711 3.1980

0.9897 0.9910 0.9921

81.0435 81.5652 83.8261

0.9948 0.9954 0.9960

Winter

CEEMD-MOALO-ENN CEEMD-MODA-ENN CEEMD-MOWOA-ENN

0.2944 0.2841 0.2693

0.0041 0.0040 0.0036

0.3908 0.3805 0.3610

4.9187 4.7634 4.5148

3.7712 3.5653 3.3445

0.9774 0.9786 0.9808

79.6522 81.7391 81.5652

0.9885 0.9890 0.9902

Average

CEEMD-MOALO-ENN CEEMD-MOALO-ENN CEEMD-MOALO-ENN

0.2743 0.2768 0.2471

0.0045 0.0048 0.0038

0.3819 0.3832 0.3447

4.9308 5.0416 4.4613

3.6435 3.6362 3.2128

0.9851 0.9846 0.9878

79.8696 79.5217 82.6522

0.9924 0.9921 0.9938

Note: The values in bold indicate the best values of error metrics.

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Fig. 5. Forecasting results of the three hybrid models in four seasons.

5.1. Diebold-Mariano test

of the experimental forecast errors are listed in Tables 6 and 7. Conclusions very similar to those from experiments I and II can be drawn from this experiment. Thus, the conclusions from this experiment are discussed in relatively lower detail. According to the prediction errors listed in Tables 6 and 7, we can easily draw similar conclusions to those drawn from experiments I and II. One can see that the proposed CEEMD-MOWOA-ENN method has the best forecasting results when compared to the other four forecasting models considered here (i.e., LSSVM model, Persistence model, SSA-CSENN model, and VMD-BBO-BPNN model), which further demonstrates that the proposed hybrid model is more suitable for wind speed time series forecasting and that it has significant practical application ability. It can be also seen that the forecasting abilities of the proposed model, based on the powerful CEEMD decomposition technique and the proposed MOWOA, perform better than those of the comparison models. The discussion and analysis above also clearly demonstrate that the proposed hybrid model, CEEMD-MOWOA-ENN, is a highly robust, highly accurate, and practical forecasting model for wind speed.

This subsection further compares the forecasting abilities of the developed hybrid model with the comparison models by using an effective hypothesis testing method, called the DM test. Considering a significance level α , the null hypothesis H0 indicates that there is no significant difference in the prediction performances of the proposed model and the comparison model.H1 represents a disagreement with the H0 . The related formulas can be expressed by:

H0: E [L (errori1)] = E [L (errori2)] H1:

E [L (errori1)]

≠

(19)

E [L (errori2)]

(20) p

wherein L is the loss function of the forecasting errors, errori ,p = 1,2 are the forecasting errors of the two comparison models. Moreover, the DM test statistics can be defined by: n

DM =

Remark. The persistence model performs better than LSSVM model, but its prediction accuracy is lower than that of the SSA-CS-ENN, VMDBBO-BPNN and proposed CEEMD-MOWOA-ENN models. However, among the five models, CEEM-MOWOA-ENN has the best performance in terms of total error metrics, indicating that CEEMD and the MOWOA have a significant impact on improving prediction accuracy and forecasting stability.

∑i = 1 (L (errori1)−L (errori2))/n S 2/ n s2

s2

(21)

L (εi1)−L (εi2) .

wherein is an estimation for the variance of di = Assuming the given significance level α , the calculated values of DM are compared with Zα/2 . If the DM statistic falls outside the interval [−Zα /2,Zα /2], then H0 will be rejected. This would indicate that there is a significant difference between the forecasting performances of the proposed model and the comparison models, meaning H1 will be accepted. Table 8 shows that the upper limit at the 1% significance level is smaller than the DM test values between our proposed model and the comparison models for the three experiments above. It can also be seen from Table 8 that, except for the proposed model vs. CEEMD-WOA-ENN at Site 5 in Experiment II, the smallest value for |DM| in Table 8 is 2.9707 (the proposed model vs. CEEMD-MODA-ENN in Fall of Experiment I), which is larger than Z0.01/2 = 2.58. As a result, we can draw the conclusion that H0 should be rejected and that the null hypothesis can be accepted under a 1% significance level, which also means that there is 99% probability of accepting the alternative hypothesis. This

5. Discussions In order to provide a more detailed discussion of the experimental results and achieve enhanced prediction accuracy for wind speed, various issues, such as hypothesis testing (performance of the forecasting model), stability of the forecasting model, running time and multi-step ahead forecasting of the proposed model, etc. are considered in this section.

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Table 5 Forecasting results of the six forecasting models at Sites 2, 3, 4 and 5. Sites

Forecasting Models

MAE

NMSE

RMSE

MAPE

MdAPE

r

DC

IA

Site 2

WNN BPNN ENN WOA-ENN EEMD -ENN CEEMD -ENN CEEMD-WOA-ENN CEEMD-MOWOA-ENN

0.4828 0.4540 0.4563 0.4552 0.2492 0.2178 0.2206 0.2032

0.0161 0.0149 0.0145 0.0146 0.0042 0.0031 0.0033 0.0028

0.6333 0.6001 0.5971 0.5973 0.3271 0.2866 0.2928 0.2678

9.1192 8.8240 8.8348 8.8186 4.7769 4.1544 4.2489 3.9318

7.0159 6.5371 6.5562 6.4774 3.5235 3.1004 3.1660 3.0310

0.9422 0.9425 0.9434 0.9432 0.9830 0.9870 0.9864 0.9886

41.7391 51.1304 47.8261 47.6522 81.0435 84.3478 84.5217 85.0435

0.9695 0.9705 0.9710 0.9709 0.9914 0.9934 0.9931 0.9942

Site 3

WNN BPNN ENN WOA-ENN EEMD-ENN CEEMD-ENN CEEMD-WOA-ENN CEEMD-MOWOA-ENN

0.4828 0.4464 0.4429 0.4422 0.2446 0.2181 0.2220 0.2067

0.0213 0.0182 0.0179 0.0183 0.0054 0.0043 0.0045 0.0038

0.6514 0.6044 0.5988 0.5999 0.3146 0.2835 0.2904 0.2697

10.8237 9.8773 9.8000 9.7572 5.5023 4.8658 4.9671 4.5799

7.8406 7.2458 7.3398 7.3905 3.9478 3.5903 3.7231 3.4179

0.9527 0.9591 0.9599 0.9597 0.9891 0.9912 0.9907 0.9920

47.3043 48.6957 49.2174 48.5217 77.9130 79.8261 81.0435 82.9565

0.9750 0.9790 0.9795 0.9794 0.9945 0.9955 0.9953 0.9959

Site 4

WNN BPNN ENN WOA-ENN EEMD-ENN CEEMD-ENN CEEMD-WOA-ENN CEEMD-MOWOA-ENN

0.7630 0.6750 0.6515 0.6506 0.3802 0.3361 0.3340 0.3108

0.0168 0.0136 0.0129 0.0128 0.0039 0.0030 0.0032 0.0028

1.0087 0.9147 0.8876 0.8841 0.4977 0.4431 0.4434 0.4143

9.3406 8.2812 8.0417 8.0022 4.6583 4.0854 4.1082 3.8039

6.7120 5.6866 5.3114 5.2252 3.4635 2.9690 2.9724 2.6414

0.9434 0.9523 0.9550 0.9555 0.9860 0.9890 0.9889 0.9904

48.8696 53.0435 53.7391 52.5217 80.5217 82.0870 82.0870 83.6522

0.9710 0.9748 0.9762 0.9764 0.9928 0.9943 0.9943 0.9950

Site 5

WNN BPNN ENN WOA-ENN EEMD -ENN CEEMD -ENN CEEMD-WOA-ENN CEEMD-MOWOA-ENN

0.7820 0.6687 0.5804 0.5549 0.3590 0.3969 0.2805 0.2734

0.0337 0.0282 0.0251 0.0251 0.0079 0.0070 0.0062 0.0057

1.0895 0.9543 0.7950 0.7623 0.4899 0.6253 0.3831 0.3762

13.3290 12.1901 11.3186 11.2397 6.7384 6.3756 5.7238 5.4874

10.5563 8.9816 8.2159 8.1059 5.3475 5.2515 4.1930 4.0153

0.9750 0.9835 0.9875 0.9877 0.9957 0.9944 0.9968 0.9969

45.3913 49.2174 52.1739 51.6522 76.0000 78.7826 84.5217 83.8261

0.9871 0.9891 0.9927 0.9934 0.9973 0.9954 0.9984 0.9985

Average

WNN BPNN ENN WOA-ENN EEMD -ENN CEEMD -ENN CEEMD-WOA-ENN CEEMD-MOWOA-ENN

0.6276 0.5610 0.5328 0.5257 0.3083 0.2922 0.2643 0.2485

0.0220 0.0187 0.0176 0.0177 0.0054 0.0044 0.0043 0.0038

0.8457 0.7684 0.7196 0.7109 0.4073 0.4096 0.3524 0.3320

10.6531 9.7931 9.4988 9.4544 5.4190 4.8703 4.7620 4.4508

8.0312 7.1128 6.8558 6.7998 4.0706 3.7278 3.5136 3.2764

0.9533 0.9593 0.9615 0.9615 0.9885 0.9904 0.9907 0.9920

45.8261 50.5217 50.7391 50.0870 78.8696 81.2609 83.0435 83.8696

0.9756 0.9783 0.9799 0.9800 0.9940 0.9947 0.9953 0.9959

Note: The values in bold indicate the best values of error metrics.

displays the values of computation time for the experiments for the comparison algorithms and proposed model on all of the datasets from different sites. It can be observed that the run time of the proposed model is 343.6163 s in Experiment I and 362.6783 s in Experiment II. Additionally, it can be seen that the computation time of the developed model is shorter than the two hybrid comparison models (i.e., CEEMDMOALO-ENN and CEEMD-MODA-ENN), which demonstrates that, when compared to MODA and MOALO, the proposed MOWOA can find Pareto optimal solutions in less time. Furthermore, we can see that the Persistence model requires the shortest time at 0.5 s. However, the computation time of the proposed technique is slightly longer than some of the comparison models, but this seems warranted due to its superior modeling abilities. Additionally, a high-performance computer can be used to reduce computation times.

confirms that the developed hybrid model performs significantly better than the eleven comparison hybrid and single models used for comparison, with a significance level of 99%. In other words, the proposed model achieves a significant improvement in accuracy compared to other models. 5.2. Stability of the forecasting models It is well known that performance variance can be utilized to demonstrate the forecasting stability of a model. The smaller the variance is, the stronger the stability will be. Additionally, the error in forecasts, as one of the most important indicators, can be used to evaluate the performance of forecasting models. Thus, in this subsection, the variance in the forecasting errors of the models is utilized to evaluate the stabilities of the hybrid model and the comparison models. It can be seen in Table 9 that the variance values of proposed hybrid model are smaller than those of the comparison models, meaning that the proposed model is more stable than the comparison models.

5.4. Supplementary work: Multi-step ahead forecasting We now consider the fact that widely utilized one-step wind speed forecasting system are sometimes insufficient to guarantee the reliability and controllability of wind power systems. In order to address this issue, it is necessary to perform multi-step wind speed forecasting with different time intervals to provide additional future information regarding wind speed. Therefore, in order to test the multi-step ahead forecasting performance of the proposed model and better forecasting

5.3. Run time Moreover, all the experimental wind speed data are performed in the MATLAB R2014b environment running on Windows 7 with a 64-bit 3.30 GHz Intel Core i5 4590 CPU and 8.00 GB of RAM. Table 10 11

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Fig. 6. Forecasting results of the comparison models and proposed model.

Table 6 Forecasting results of the forecasting models in four seasons at Site 1. Seasons

Forecasting Models

MAE

NMSE

RMSE

MAPE

MdAPE

r

DC

IA

Spring

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.4840 0.4816 0.3633 0.2619 0.2336

0.0145 0.0151 0.0100 0.0049 0.0039

0.6763 0.6875 0.4745 0.3630 0.3330

8.7637 8.6442 7.0986 4.8950 4.3752

6.4786 6.4665 4.7969 3.4345 2.8618

0.9827 0.9822 0.9915 0.9951 0.9959

47.6522 — 68.6957 76.6957 82.6087

0.9911 0.9911 0.9957 0.9975 0.9979

Summer

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.5153 0.5134 0.3829 0.3159 0.2459

0.0182 0.0181 0.0095 0.0067 0.0040

0.7188 0.7255 0.5231 0.4279 0.3465

9.7805 9.6767 7.2645 6.1343 4.5576

7.0504 6.8729 5.5743 4.5646 3.4469

0.9235 0.9236 0.9596 0.9734 0.9825

47.4783 — 69.9130 74.9565 82.6087

0.9604 0.9606 0.9792 0.9859 0.9911

Fall

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.5261 0.5274 0.3488 0.2841 0.2397

0.0167 0.0160 0.0077 0.0048 0.0036

0.7513 0.7342 0.4714 0.3969 0.3383

9.6902 9.5479 6.6236 5.3017 4.3975

6.9528 7.1946 5.2532 4.1356 3.1980

0.9602 0.9627 0.9845 0.9894 0.9921

49.2174 — 70.2609 77.0435 83.8261

0.9796 0.9810 0.9922 0.9943 0.9960

Winter

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.4903 0.5019 0.3328 0.2850 0.2693

0.0121 0.0125 0.0056 0.0040 0.0036

0.6583 0.6821 0.4384 0.3817 0.3610

8.2933 8.3774 5.6853 4.7816 4.5148

5.9861 6.2500 4.1688 3.4991 3.3445

0.9351 0.9320 0.9715 0.9789 0.9808

50.7826 — 71.8261 78.2609 81.5652

0.9662 0.9650 0.9854 0.9887 0.9902

Average

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.5039 0.5061 0.3570 0.2867 0.2471

0.0154 0.0154 0.0082 0.0051 0.0038

0.7012 0.7073 0.4769 0.3924 0.3447

9.1319 9.0616 6.6680 5.2782 4.4613

6.6170 6.6960 4.9483 3.9085 3.2128

0.9504 0.9501 0.9768 0.9842 0.9878

48.7826 — 70.1739 76.7392 82.6522

0.9743 0.9744 0.9881 0.9916 0.9938

Note: The values in bold indicate the best values of error metrics.

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Table 7 Forecasting results of the five forecasting models at Sites 2, 3, 4, and 5. Seasons

Forecasting Models

MAE

NMSE

RMSE

MAPE

MdAPE

r

DC

IA

Site 2

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.4567 0.4580 0.3834 0.2839 0.2032

0.0146 0.0152 0.0101 0.0062 0.0028

0.5978 0.6049 0.4743 0.3766 0.2678

8.8473 8.7559 7.6138 5.6025 3.9318

6.5352 6.5574 5.9988 3.8410 3.0310

0.9429 0.9425 0.9640 0.9779 0.9886

47.1304 — 65.0435 74.0870 85.0435

0.9707 0.9706 0.9816 0.9881 0.9942

Site 3

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.4414 0.4415 0.3412 0.2402 0.2067

0.0180 0.0181 0.0118 0.0055 0.0038

0.5982 0.6024 0.4445 0.3170 0.2697

9.7454 9.6652 7.9681 5.3929 4.5799

7.2154 7.1946 5.6273 3.7563 3.4179

0.9599 0.9600 0.9780 0.9892 0.9920

49.3913 — 65.2174 77.3913 82.9565

0.9795 0.9797 0.9888 0.9943 0.9959

Site 4

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.6525 0.6582 0.5100 0.3647 0.3108

0.0131 0.0133 0.0074 0.0040 0.0028

0.8887 0.8922 0.6561 0.4707 0.4143

8.0658 8.1602 6.4405 4.6087 3.8039

5.2859 5.5301 4.8152 3.3435 2.6414

0.9549 0.9552 0.9755 0.9879 0.9904

53.5652 — 68.5217 78.9565 83.6522

0.9767 0.9772 0.9874 0.9934 0.9950

Site 5

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.9350 0.5339 0.3809 0.2957 0.2734

0.0412 0.0259 0.0126 0.0072 0.0057

1.8051 0.7419 0.5026 0.4001 0.3762

13.5053 10.9156 8.2526 6.1083 5.4874

10.0413 7.4537 5.8836 4.4544 4.0153

0.9401 0.9880 0.9945 0.9969 0.9969

50.7826 — 71.4783 77.9130 83.8261

0.9559 0.9940 0.9972 0.9982 0.9985

Average

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

0.6214 0.5229 0.4039 0.2961 0.2485

0.0217 0.0181 0.0105 0.0057 0.0038

0.9725 0.7104 0.5194 0.3911 0.3320

10.041 9.3742 7.5688 5.4281 4.4508

7.2695 6.6840 5.5812 3.8488 3.2764

0.9495 0.9614 0.9780 0.9880 0.9920

50.2174 — 67.5652 77.0870 83.8696

0.9707 0.9804 0.9888 0.9935 0.9959

Note: The values in bold indicate the best values of error metrics.

min and 30-min wind speed intervals with high prediction performance (measured by the error criteria) and the highest accuracy for directional measurement (measured by the IA and DC criteria).

results for the future, this supplementary work on multi-step ahead forecasting uses two datasets (i.e., 10-min and 30-min interval wind speeds) listed in Table 1. The forecasting results of the comparison models (i.e., ARIMA, Persistence model, and CEEMD-MOWOA-WNN) and the proposed model are shown in Table 11. It can be observed that for one-step, twostep, or three-step forecasting using 10-min or 30-min wind speed data intervals, the proposed model always achieves the lowest MAPE values and the highest IA and DA values among the tested models. In other words, the proposed model can effectively multi-step ahead forecast 10-

5.5. Summary Based on the analysis of experiments I-III, as well as the DM test and the discussions above, the following conclusions can be drawn from this study:

Table 8 Diebold–Mariano test of different models. DM test

Models

Spring

Summer a

Fall

Winter

CEEMD-MOALO-ENN CEEMD-MODA-ENN

5.8980 7.4815a

6.6364 6.7220a

5.6813 2.9707a

9.6442 6.5171a

6.9650a 5.9228a

DM test

Models

Site 2

Site 3

Site 4

Site 5

Average

Experiment II

WNN BPNN ENN WOA-ENN EEMD-ENN CEEMD-ENN CEEMD-WOA-ENN

13.2987a 12.9226a 13.2946a 13.2564a 9.2178a 7.6360a 8.9512a

11.3819a 11.7738a 11.6207a 11.4881a 7.8552a 5.4515a 8.1952a

13.3548a 12.0346a 12.3726a 12.3262a 9.8261a 6.6313a 8.4927a

12.0071a 9.6254a 9.3012a 8.6658a 6.9508a 5.5512a 1.2448

12.5106a 11.5891a 11.6473a 11.4341a 8.4625a 6.3175a 6.7210a

DM test

Models

Spring

Summer

Fall

Winter

Average

a

Experiment III

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN

8.6166 7.8385a 8.4042a 8.4042a

DM test

Models

Site 2

a

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN

a

Average

Experiment I

Experiment III

a

a

a

9.0155 8.5265a 6.1564a 6.1564a Site 3 a

a

11.6046 11.5138a 8.7378a 8.7378a

13.3027 13.1648a 13.7462a 13.7462a

1% significance level.

13

a

8.6299 10.9785a 8.3186a 8.3186a

12.2996 12.0988a 6.2090a 6.2090a

9.6404a 9.8606a 7.2720a 7.2720a

Site 4

Site 5

Average

a

11.8082 12.1888a 10.0229a 10.0229a

a

a

5.9100 8.1574a 6.7193a 6.7193a

10.6564a 11.2562a 9.8066a 9.8066a

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series forecasting due to its high accuracy and stable forecasting abilities. Remark. Compared to the benchmark models, CEEMD-MOWOA-ENN achieves better performance based on eight model evaluation criteria, demonstrating that significant resources can be saved if MAPE is decreased. Based on the description above and the conclusions presented in this subsection, we can state that the MOWOA is a robust multi-objective optimization method, that, constitutes a new option for solving multi-objective optimization problems. Additionally, the proposed CEEMD-MOWOA-ENN hybrid model, has been shown to be a robust, highly accurate, and practical forecasting model for wind speed.

Table 9 Test results of the variance about the forecasting errors. Variance value

Models

Spring

Summer

Fall

Winter

Average

Experiment I

CEEMDMOALO-ENN CEEMDMODA-ENN CEEMDMOWOA-ENN

0.1311

0.1507

0.1484

0.1527

0.1457

0.1348

0.1797

0.1301

0.1447

0.1473

0.1109

0.1200

0.1143

0.1302

0.1189

Variance value

Models

Site 2

Site 3

Site 4

Site 5

Average

Experiment II

WNN BPNN ENN WOA-ENN EEMD-ENN CEEMD-ENN CEEMDWOA-ENN CEEMDMOWOA-ENN

0.4008 0.3598 0.3566 0.3568 0.1069 0.0822 0.0857

0.4204 0.3653 0.3585 0.3599 0.0989 0.0803 0.0843

1.0175 0.8266 0.7799 0.7726 0.2475 0.1958 0.1961

1.1645 0.8730 0.6145 0.5740 0.2313 0.3608 0.1464

0.7508 0.6062 0.5274 0.5158 0.1711 0.1798 0.1281

0.0717

0.0727

0.1714

0.1415

0.1143

Variance value

Models

Spring

Summer

Fall

Winter

Average

Experiment III

LSSVM Persistence model SSA-CS-ENN VMD-BBOBPNN CEEMDMOWOA-ENN

0.4574 0.4726

0.5166 0.5264

0.5645 0.5390

0.4320 0.4653

0.4926 0.5008

0.2251 0.1318

0.2736 0.1829

0.2188 0.1575

0.1920 0.1457

0.2274 0.1545

0.1109

0.1200

0.1143

0.1302

0.1189

Variance value

Models

Site 2

Site 3

Site 4

Site 5

Average

Experiment III

LSSVM Persistence model SSA-CS-ENN VMD-BBOBPNN CEEMDMOWOA-ENN

0.3574 0.3659

0.3579 0.3628

0.7840 0.7961

3.0215 0.5502

1.1302 0.5187

0.1069 0.1418

0.0989 0.1005

0.2475 0.2216

0.2313 0.1585

0.1711 0.1556

0.0717

0.0727

0.1714

0.1415

0.1143

5.6. How can the proposed model be integrated into the power system? Wind speed forecasting is a crucial for wind power systems and is generally regarded as a challenging task due to the uncertainty and fluctuations in wind speed. Therefore, accurate forecasts are needed for a variety of utility tasks, such as generation scheduling, maintenance scheduling, security analysis, and energy transactions. The detailed roles of the powerful forecasting model into the study and design of a power system including wind parks are described as follows: (a) Contributing to wind turbine power generation Accurate and stable wind speed forecasting models can provide sufficient information for decision-makers who are making plans for wind turbine power generation. For example, based on the forecasted wind speed values, decision-makers can make a detailed schedule for adjusting wind turbines to ensure the maximum yield of wind energy. If wind speed values are larger (or smaller) than the capacity of the wind turbine, the turbines should be shut down to avoid damage (thereby reducing the operating costs of wind farms). Moreover, the formula for converting wind energy into wind power can be shown below:

exp[−(vc / c ) k ]−exp[−(vr / c ) k ] Pa = ⎧ −exp[−(vf / c ) k ] ⎫ × Pr ⎨ ⎬ (vr / c ) k −(vc / c ) k ⎩ ⎭

Note: The values in bold indicate the best values of error metrics.

where vc , vf , and vr are the cut-in, cut-off, and nominal wind speed values (m/s), respectively. Additionally, c is the Weibull scale parameter (m/s), while Pa and Pr are the average power output of the wind turbine (kW) and rated electrical power of the wind turbine (kW), respectively. It can be seen from the equation above that accurate and stable wind speed forecasting plays a vital role in wind turbine power generation.

(a) The proposed hybrid model based on the MOWOA not only has stronger forecasting ability and more stable forecasting results than the benchmarks, but also shows remarkably improved forecasting accuracy. (b) The MOWOA can perform better than two recently developed algorithms (i.e., MOALO and MODA) with shorter computation time, and can be used to improve the forecasting performance of ENN, particularly when utilized in CEEMD-MOWOA-ENN. With the goal of achieving outstanding forecasting results, researchers should pay more attention to data processing methods, optimization algorithms, etc., which have a significant impact on improving prediction accuracy and forecasting stability. (c) Whether for different seasonal wind speed datasets or wind speed datasets from different sites, the proposed hybrid model can always satisfactorily approximate actual values. The MAPE values for the proposed method are all within in 4.6% and the fluctuation ranges are all within approximately 0.2%. This indicates that the proposed model is not affected by seasonal changes and site-specific effects due to its superior forecasting ability and the effectiveness of the CEEMD method. (d) We presented a novel robust multi-objective optimization algorithm called the MOWOA, which not only outperforms two recently developed algorithms (i.e., MOALO and MODA), but also adds a new optimization method for solving multi-objective optimization problems. Similarly, the proposed hybrid forecasting model is a new viable option for one-step and multi-step ahead wind speed time

(b) Contributing to power system operation and scheduling It is well known that the balance of power supply and demand is very crucial because it plays a fundamental role in sustainable energy management and economically efficient operation. On one hand, overload will result in an increase in start-up and long-term costs due to the inherent difficulties in storing electricity. On the other hand, under-load will negatively affect the quality of power supply, rendering it incapable of satisfying regular power demands and potentially compromising the safety and stability of the power system. This was the cause of the serious power outages that occurred in India in 2012. With the use of a powerful forecasting model that can provide excellent prediction results and help decision-makers to implement suitable solutions in a timely manner, the problems mentioned above can be avoided. 6. Conclusions Accurate and reliable wind speed prediction not only plays a crucial role in power dispatching and the stable operation of power grids, but also has environmental benefits, as well as economic and social 14

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Table 10 Computation time of the different forecasting models. Computation time (s)

Models

Spring

Summer

Fall

Winter

Average

Experiment I

CEEMD-MOALO-ENN CEEMD-MODA-ENN CEEMD-MOWOA-ENN

388.1379 416.9264 347.2688

388.0680 417.3805 341.6970

381.0662 412.1830 340.3655

368.3381 395.8093 345.1339

381.4026 410.5748 343.6163

Computation time (s)

Models

Site 2

Site 3

Site 4

Site 5

Average

Experiment II

WNN BPNN ENN WOA-ENN EEMD-ENN CEEMD-ENN CEEMD-WOA-ENN CEEMD-MOWOA-ENN

7.5036 7.3446 10.8933 386.2553 10.3119 12.2655 392.3849 367.1780

7.2091 7.0193 10.9436 395.1292 10.3310 12.0897 387.9307 362.8119

7.3486 7.2723 11.0417 391.5671 10.3095 11.5541 367.7602 358.6443

7.2365 7.0963 10.9030 398.1982 10.2088 12.0318 385.1764 362.0791

7.3244 7.1831 10.9454 392.7875 10.2903 11.9853 383.3131 362.6783

Computation time (s)

Models

Spring

Summer

Fall

Winter

Average

Experiment III

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

2688.8488 0.5007 449.4872 60.6039 347.2688

2659.8639 0.4953 446.7153 56.8201 341.6970

2733.1375 0.4845 416.4164 60.3547 340.3655

3040.1007 0.4935 451.9967 59.8799 345.1339

2780.4877 0.4935 441.1539 59.4147 343.6163

Computation time (s)

Models

Site 2

Site 3

Site 4

Site 5

Average

Experiment III

LSSVM Persistence model SSA-CS-ENN VMD-BBO-BPNN CEEMD-MOWOA-ENN

2302.1848 0.5370 403.4429 66.5006 367.1780

2685.5416 0.5188 461.8247 59.7018 362.8119

2714.3862 0.5280 452.2416 60.8298 358.6443

2397.6418 0.5206 447.4905 60.3222 362.0791

2524.9386 0.5261 441.2499 61.8386 362.6783

Table 11 Multi-step ahead forecasting results of 10-min and 30min intervals wind speed datasets. Time

Multi-Step ahead

Forecasting Models

MAE

NMSE

RMSE

MAPE

MdAPE

r

DC

IA

10-min

One-step ahead

ARIMA Persistence model CEEMD-MOWOA-WNN CEEMD-MOWOA-ENN

0.5064 0.4962 0.3388 0.2375

0.0144 0.0141 0.0076 0.0039

0.6785 0.6701 0.5001 0.3187

8.4971 8.2642 5.3932 4.2247

6.2523 5.8998 3.4799 2.7234

0.9875 0.9878 0.9953 0.9973

47.6522 — 76.8696 84.6957

0.9936 0.9939 0.9964 0.9986

Two-step ahead

ARIMA Persistence model CEEMD-MOWOA-WNN CEEMD-MOWOA-ENN

0.6553 0.7375 0.4442 0.3528

0.0243 0.0314 0.0154 0.0099

0.9001 0.9877 0.6436 0.4974

10.9240 12.3216 7.4655 6.4554

7.3887 8.6785 4.5702 3.9328

0.9778 0.9736 0.9905 0.9933

44.0000 39.6522 62.2609 70.2609

0.9887 0.9866 0.9940 0.9966

Three-step ahead

ARIMA Persistence model CEEMD-MOWOA-WNN CEEMD-MOWOA-ENN

0.7576 0.8811 0.5367 0.4756

0.0282 0.0455 0.0216 0.0175

1.0464 1.1811 0.7597 0.6809

12.0770 14.7551 8.9039 8.7614

8.3064 9.9296 5.7148 4.8889

0.9701 0.9622 0.9861 0.9875

44.6181 44.9653 59.5486 62.6736

0.9843 0.9808 0.9916 0.9935

One-step ahead

ARIMA Persistence model CEEMD-MOWOA-WNN CEEMD-MOWOA-ENN

0.6770 0.6575 0.3616 0.3184

0.0508 0.0492 0.0178 0.0117

0.9577 0.9348 0.4816 0.4601

16.1500 15.4881 9.5974 7.5652

10.4136 10.5662 5.4297 4.8878

0.8693 0.8797 0.9734 0.9704

47.6987 —0.0000 80.3347 82.8452

0.9318 0.9376 0.9814 0.9846

Two-step ahead

ARIMA Persistence mode CEEMD-MOWOA-WNN CEEMD-MOWOA-ENN

0.7339 0.8642 0.4699 0.4386

0.0610 0.0739 0.0274 0.0235

1.0007 1.1458 0.6638 0.6401

17.6272 20.1952 11.8435 10.5000

12.7402 16.0627 6.6018 6.2166

0.8557 0.8190 0.9399 0.9418

51.4644 46.4435 70.7113 73.2218

0.9234 0.9033 0.9643 0.9695

Three-step ahead

ARIMA Persistence model CEEMD-MOWOA-WNN CEEMD-MOWOA-ENN

0.9096 1.0483 0.6104 0.5716

0.0791 0.1017 0.0396 0.0364

1.1902 1.3326 0.8453 0.8153

21.6503 24.4244 14.8520 13.8944

16.6104 18.3673 9.8180 8.3895

0.7923 0.7549 0.8965 0.9039

49.5833 43.7500 63.7500 62.9167

0.8869 0.8655 0.9410 0.9483

30-min

Note: The values in bold indicate the best values of error metrics.

benefits. However, the intermittency and complex fluctuations in wind speeds make it difficult to accurately predict. Thus, in this paper, we presented a novel hybrid CEEMD-MOWOA-ENN model for short-term wind speed forecasting. Experiments using ten wind speed datasets collected from wind farms of China were used as illustrative examples to evaluate the effectiveness and efficiency of the proposed hybrid model. The experimental results clearly indicate that both the accuracy

and stability of the proposed model are superior to those of the single and hybrid models used for comparison. Additionally, based on the simulations and analysis presented in this paper, several advantages of the proposed method can be summarized follows. First, a novel multi-objective optimization algorithm called the MOWOA is successfully proposed in this paper. It not only outperforms two recently developed algorithms (i.e., MOALO and MODA), but also 15

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speed time series. Its excellent performance and reasonable prediction accuracy suggest that it can be also considered in other forecasting fields, such as stock index forecasting, wave time series forecasting, traffic flow forecasting, power load forecasting, and wind power forecasting.

constitutes a new option for solving multi-objective optimization problems. Second, most previous studies on wind speed focused only on improving prediction accuracy, which ignores the significance of forecasting stability. The proposed hybrid model using the MOWOA not only demonstrates excellent fitting ability, but is also able to approximate actual wind speed values with high accuracy and stability. Third, in addition to single-step forecasting, multi-step wind speed predictions with different time intervals are also adopted and discussed in this paper. The results indicate that the proposed hybrid model outperforms all sixteen models used for comparison. Finally, the proposed hybrid forecasting model serves as a novel viable option for forecasting wind

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 71671029).

Appendix A Test functions

ZDT1: Minimize : f1 (x ) = x1

ZDT2: Minimize : f1 (x ) = x1

Minimize : f2 (x ) = g (x ) × h (f1 (x ),g (x ))

where : G (x ) = 1 +

9 N−1

N ∑i = 2

h (f1 (x ),g (x )) = 1−

Minimize : f2 (x ) = g (x ) × h (f1 (x ),g (x ))

where : G (x ) = 1 +

xi

9 N−1

N

∑i = 2 x i f1 (x ) 2

( )

f1 (x )

h (f1 (x ),g (x )) = 1−

g (x )

0 ⩽ x i ⩽ 1, 1 ⩽ i ⩽ n

g (x )

0 ⩽ x i ⩽ 1, 1 ⩽ i ⩽ n

ZDT3:

ZDT1 with linear PF:

Minimize : f1 (x ) = x1 Minimize : f2 (x ) = g (x ) × h (f1 (x ),g (x ))

Minimize : f1 (x ) = x1 Minimize : f2 (x ) = g (x ) × h (f1 (x ),g (x ))

where : G (x ) = 1 +

9 29

N

where : G (x ) = 1 +

∑i = 2 x i

h (f1 (x ),g (x )) = 1−

h (f1 (x ),g (x )) = 1− g1(x )

g (x )

0 ⩽ x i ⩽ 1, 1 ⩽ i ⩽ n

f1 (x ) g (x )

N

∑i = 2 x i f (x )

f1 (x )

( ) sin(10πf (x ))

−

9 N−1

1

0 ⩽ x i ⩽ 1, 1 ⩽ i ⩽ n

Appendix B List of abbreviations

AR ANNs ARMA ARIMA BPNN BA BBO CEEMD CEEMDAN CS DC ENN EMD EEMD GRNN GA IA IMFs IGD

auto regressive artificial neural networks autoregressive moving average autoregressive integrated moving average back propagation neural network bat algorithm biogeography-based optimization complementary EEMD CEEMD with noise cuckoo search directional change Elman neural network empirical mode decomposition ensemble EMD generalized regression neural network genetic algorithm index of agreement intrinsic Mode Functions inverted generational distance

LSSVM MAE MOPs MOWOA MODA MOALO MAPE MdAPE NMSE NSGA-II NFL SSA SVMs SOPs RMSE VMD WOA WNN

16

least squares support vector machine mean absolute error multi-objective optimization problems multi-objective WOA multi-objective dragonfly algorithm multi-objective ALO mean absolute percentage error Median absolute percentage error normalized mean square error non-dominated sorting genetic algorithm No Free Lunch singular spectrum analysis support vector machines single-objective optimization problems root mean square error variational mode decomposition whale optimization algorithm wavelet neural network

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