# A Hybrid Approach for Multi-Step Wind Speed Forecasting Based on Multi-Scale Dominant Ingredient Chaotic Analysis, KELM and Synchronous Optimization Strategy

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Variational Mode Decomposition

- Step 1: Initialize ${\widehat{m}}_{k}^{1}$, ${\omega}_{k}^{1}$, ${\beta}^{1}$ and n = 1;
- Step 2: Update ${\widehat{m}}_{k}$ and ${\omega}_{k}$ by Formulas (3) and (4);
- Step 3: Update $\widehat{\beta}$ based on Formula (5);
- Step 4: If ${\sum}_{k}\Vert {\widehat{m}}_{k}^{n+1}-{\widehat{m}}_{k}^{n}{\Vert}_{2}^{2}/\Vert {\widehat{m}}_{k}^{n}{\Vert}_{2}^{2}<\epsilon $ stop updating; else n = n + 1, and turn to Step 2.

#### 2.2. Singular Spectrum Analysis

- (1)
- Embedding. The original time sequence x = {x
_{i}| i =1, 2, ⋯, N} is reconstructed into a Hankel matrix [32] to begin with SSA, which is defined as:$$H=\left[\begin{array}{cccc}{x}_{1}& {x}_{2}& \cdots & {x}_{t}\\ {x}_{2}& {x}_{3}& \cdots & {x}_{t+1}\\ \vdots & \vdots & \ddots & \vdots \\ {x}_{l}& {x}_{l+1}& \cdots & {x}_{N}\end{array}\right]$$ - (2)
- SVD. On the basis of the time series embedded, the i-th eigentriple (σ
_{i}, U_{i}, V_{i}) can be obtained by decomposing the matrix H with SVD, thus deducing the Hankel matrix H as follows:$$H={H}_{1}+{H}_{2}+\dots +{H}_{l},{H}_{i}={\sigma}_{i}{U}_{i}{V}_{i}^{T}$$_{i}is the singular value, and U_{i}and V_{i}denote the singular vectors of matrixes HH^{T}and H^{T}H, respectively. - (3)
- Grouping. Several discrete subsets of matrices H
_{Z}can be partitioned into the grouping procedure. For Z = {Z_{1}, Z_{2}, …, Z_{r}}, the matrix H_{Z}, corresponding to group Z, can be defined as follows:$${H}_{Z}={H}_{Z1}+{H}_{Z2}+\dots +{H}_{Zr}$$ - (4)
- Diagonal averaging. A new series with length N, corresponding to each matrix grouped in Equation (8) can be transformed in this procedure. Let matrix X to be a W×Q matrix with elements x
_{ij}, where i ≤ 1 ≤ W and 1 ≤ j ≤ Q. Let x_{ij}* = x_{ij}when W < Q, otherwise, let x_{ij}* = x_{ji}. Then, the restructured sequence V_{m}(m = 1, 2, …, N) can be obtained as:$${V}_{m}=\{\begin{array}{l}\frac{1}{m}{\displaystyle \sum _{i=1}^{m}{x}_{i,m-i+1}^{*}}\hspace{1em}for\hspace{1em}1\le m<{W}^{*}\\ \frac{1}{{W}^{*}}{\displaystyle \sum _{i=1}^{{W}^{*}}{x}_{i,m-i+1}^{*}}\hspace{1em}for\hspace{1em}{W}^{*}\le m\le {Q}^{*}\\ \frac{1}{N-m+1}{\displaystyle \sum _{i=m-{Q}^{*}+1}^{T-{Q}^{*}}{x}_{i,m-i+1}^{*}}\hspace{1em}for\hspace{1em}{Q}^{*}<m\le N\end{array}$$

#### 2.3. Phase Space Reconstruction

_{i}|i = 1, 2, ⋯, N} can be denoted as follows:

_{i}(i = 1, 2, …, L) denotes the i-th space vector in the phase space. The corresponding output matrix of the forecasting engine could be deduced by the following formula:

_{i}represents the forecasting value corresponding to the i-th vector of the phase space matrix.

#### 2.4. Kernel Extreme Learning Machine

**H**β =

**T**, which is shown as below:

**H**

^{†}represents the Moore–Penrose generalized inverse of matrix

**H**. Due to the fact that both of the smallest training errors and smallest norms of the output weights are considered in ELM, a better generalization performance for the networks could be obtained. For this purpose, the regularization coefficient C was adopted in the optimization phase, with which the output weights β could be described as [15]:

**I**denotes an identity matrix of dimension N. For the cases where the hidden layer feature mapping h(∙) would be unknown, the kernel matrix for kernel extreme learning machine (KELM) can be defined as [15]:

^{2}denotes the kernel parameter. To achieve better generalization for the performance of the networks, the regularization coefficient C and the kernel parameter σ

^{2}need to be set appropriately [15].

## 3. The Proposed Approach

#### 3.1. Multi-Scale Dominant Ingredient Chaotic Analysis

_{1}= {1, 2, …, s} and Z

_{2}= {s + 1, s + 2, …, l}, with which the matrix H

_{Z}could be represented as H

_{Z}= H

_{Z}

_{1}+ H

_{Z}

_{2}. It is worth noting that the parameter s that determines the dominant ingredients could affect the prediction accuracy to some extent [36]. Additionally, the residual of VMD, i.e., ${m}_{r}=f-{{\displaystyle \sum}}_{k=1}^{K}{m}_{k}$ is integrated with all the residuary ingredients of all the sub-series for the ulterior improvement of the forecasting model’s capabilities. Subsequently, PSR, which has been widely utilized for chaotic time series analysis [9,37], is implemented to construct the inputs and outputs of the forecasting models, corresponding to each predictive component. Nevertheless, the time delay τ and the embedded dimension d could affect the recovery of the PSR dynamic system, with which the prediction performance would be significant restricted. It can be seen that the key to constructing the proposed hybrid forecasting model is to assign appropriate parameters to each module. To this end, an improved hybrid grey wolf optimizer-sine cosine algorithm (IHGWOSCA)-based synchronous optimization strategy is proposed, to achieve better parameter optimization and forecasting performance, which will be detailed later.

#### 3.2. An Improved Hybrid Grey Wolf Optimizer-Sine Cosine Algorithm

_{1}and r

_{2}are random vectors in the scopes of [0, 1]; the components of $\stackrel{\rightharpoonup}{a}$ linearly decrease from 2 to 0 over the course of the iteration in the normal GWO and HGWO-SCA. Besides, the transition between the exploration and exploitation stage depends on the components of $\stackrel{\rightharpoonup}{a}$ and $\stackrel{\rightharpoonup}{A}$, of which half of the iterations are divided toward exploration, when $\stackrel{\rightharpoonup}{\left|\mathrm{A}\right|}>1$ and the remaining iterations are assigned for exploitation when $\stackrel{\rightharpoonup}{\left|\mathrm{A}\right|}<1$ [39]. Hence, to better improve the corresponding abilities of these two phases, a cosine function-based decreasing formula for updating the components of $\stackrel{\rightharpoonup}{a}$ is proposed in this study:

Algorithm 1. The pseudo code of the proposed IHGWOSCA algorithm | |

1: | Initialization the population ${\stackrel{\rightharpoonup}{X}}_{i}$ (i = 1, 2, …, N) |

2: | Initialize a, $\stackrel{\rightharpoonup}{A}$ and $\stackrel{\rightharpoonup}{C}$ |

3: | Calculate the fitness of each search member |

4: | ${\stackrel{\rightharpoonup}{X}}_{\alpha}$: the best search agents, ${\stackrel{\rightharpoonup}{X}}_{\beta}$: the second-best search agent, ${\stackrel{\rightharpoonup}{X}}_{\delta}$: the third-best search agent |

5: | While (t < maximum number of iterations) |

6: | For each search agent: |

7: | Update the position of the current search agent on the basis of Equations (25) and (26) |

8: | End for: |

9: | Update a, $\stackrel{\rightharpoonup}{A}$, and $\stackrel{\rightharpoonup}{C}$ by Equations (21), (19) and (20), respectively. |

10: | Calculate the fitness of all grey wolves |

11: | Save the position information owned by the β and δ wolves with Equations (23) and (24), while the position information for α wolf are updated as below: |

12: | If rand () < 0.5 |

13: | Then: |

14 | ${\stackrel{\rightharpoonup}{D}}_{\alpha}$ = rand () × sin (0.5⋅π⋅rand ()) × |${\stackrel{\rightharpoonup}{C}}_{1}$ × ${\stackrel{\rightharpoonup}{X}}_{\alpha}$ − $\stackrel{\rightharpoonup}{D}$| |

15: | Else: |

16: | ${\stackrel{\rightharpoonup}{D}}_{\alpha}$ = rand () × cos (0.5⋅π⋅rand ()) × |${\stackrel{\rightharpoonup}{C}}_{1}$ × ${\stackrel{\rightharpoonup}{X}}_{\alpha}$ − $\stackrel{\rightharpoonup}{D}$| |

17: | ${\stackrel{\rightharpoonup}{X}}_{1}$ = ${\stackrel{\rightharpoonup}{X}}_{\alpha}$ − ${\stackrel{\rightharpoonup}{A}}_{1}$ · ${\stackrel{\rightharpoonup}{D}}_{\alpha}$ |

18: | End if |

19: | End else |

20: | t = t+1 |

21: | End while |

22: | Return ${\stackrel{\rightharpoonup}{X}}_{\alpha}$ |

#### 3.3. Optimization Strategy

#### 3.4. Specific Procedures

^{2}for KELM;

_{r}of VMD;

_{r}.

^{2}. Repeat Steps 2.4 to 2.5 until k = K + 1, then accumulate the predicted results of all the components to obtain the ultimate forecasting value, and then calculate the fitness value by Equation (27).

## 4. Experimental Design

#### 4.1. Data Collection

#### 4.2. Experimental Description

_{RMSE}, P

_{MAE}, and P

_{MAPE}are as follows:

^{2}) in KELM are searched for the scopes of [2, 10], [0, 1], [1, 2000], [1, 167], [1, 15], [1, 40], [1, 1000], and [1, 1000], orderly. For the relevant comparative models, the regularization coefficient C and the kernel parameter σ

^{2}of all the SVR- and KELM-based models are optimized by GS, where the searching scopes are in intervals [2

^{−8}, 2

^{8}] and [2

^{−5}, 2

^{5}], respectively. For the VMD-based comparative models, the parameter α is set as the default value of 2000, and the parameters K and γ are optimized by GS [13], of which K is searched in [2, 10] with increasing step 1, and γ is searched in [0, 1] with increasing step 0.1. Meanwhile, for the SSA- and PSR-based models, including EMD-SSA-PSR-KELM and VMD-SSA-PSR-KELM, the window length l of the Hankel matrix is given as 500, and the corresponding parameter s is set as 105, as suggested in [36]. Besides, the parameters τ and d of PSR are set as 1 and 10, respectively. Furthermore, the optimal parameters within the proposed models are obtained by the proposed IHGWOSCA algorithm in different horizons for all experimental cases, as illustrated in Table 2.

#### 4.3. Contrasting Analyses

- (1)
- Comparing the metrics RMSE, MAE, and MAPE, obtained by SVR and KELM in all experimental cases, it can be observed that KELM generally possesses lower metrics than SVR, which means that a better forecasting performance could be obtained by KELM. For instance, in the cases of SG Mar. and SG Jun., the one-step prediction results in terms of MAPE for these two models are 11.37%, 10.35%, and 13.69% 12.76%, of which the reducing ratios of MAPE for KELM are 8.92% and 6.75%, respectively. Furthermore, this trend would be pronounced in the multi-step predictions. In the three-step and five-step predictions in the case of SG Sep., the three employed indicators obtained by SVR and KELM are 0.97 m/s, 0.71 m/s, 23.04% (SVR, three-step), 0.96 m/s, 0.71 m/s, 18.91% (KELM, three-step) and 1.27 m/s, 0.97 m/s, 29.97 (SVR, five-step), 1.19 m/s, 0.91 m/s, 23.27% (KELM, five-step) orderly. It can be seen that the decreasing percentages of the index MAPE for KELM in three-step and five-step forecasting are 17.91% and 22.37%, respectively, with which the superiority of KELM could be demonstrated effectively. It is worth noting that satisfactory results could not be directly achieved by the single models such as SVR and KELM, which could be attributed to the strong non-stationarity and non-linearity of the original wind speed time series. To this end, signal preprocessing technologies are necessary to enhance prediction performance.
- (2)
- Following the comparison of KLEM, EMD-KELM and VMD-KELM, it can be indicated that time-frequency signal processing approaches could greatly improve the prediction accuracy for wind speed. In the case of SG Dec., the evaluation metrics obtained by EMD-KELM in one-step predictions are 0.63 m/s, 0.50 m/s, 6.76%, which are deceased by 48.37%, 43.45%, and 43.74%, compared to the single-model KELM. Meanwhile, the metrics reducing ratio in the three-step and five-step predictions obtained by comparing KELM and EMD-KELM are 52.37%, 50.16%, 49.15% and 47.87%, 46.07%, and 42.20%, respectively. From further comparison of the results of EMD-KELM and VMD-KELM, it can be indicated that the three evaluation indicators obtained by VMD-KELM are decreased by averaging 75.05%, 64.90%, and 56.17% in three experimental prediction horizons, respectively. Hence, it can be concluded that VMD could improve the forecasting accuracy better than EMD. Similar conclusions could be drawn by the same analysis for the remaining experimental cases.
- (3)
- The proposed dominant ingredient chaotic analysis combining SSA and PSR could improve the prediction model performance in an ulterior manner. In the case of SG Mar., compared with EMD-KELM, the metrics obtained by EMD-SSA-PSR-KELM are 0.52 m/s, 0.38 m/s, 4.71%, 0.80 m/s, 0.60 m/s, 7.69%, 0.93 m/s, 0.70 m/s, 9.12% in three variously predicted horizons, of which the corresponding decreasing percentages are averaged by 25.67%, 6.50% and 12.04%. Meanwhile, compared with VMD-KELM, the metrics of VMD-SSA-PSR-KELM have been averaged, decreasing by 31.11%, 8.02% and 9.30% in different predicted horizons, respectively. The comparisons of EMD-KELM, EMD-SSA-PSR-KELM and VMD-KELM, and VMD-SSA-PSR-KELM indicate that the proposed dominant ingredient chaotic analysis could further enhance the forecasting performance on the basis of the signal decomposition approaches implemented. Nevertheless, the performance of the proposed dominant ingredient chaotic analysis would be restricted by the parameters that are settled in SSA and PSR, which makes parameter optimization necessary.
- (4)
- Comparing VMD-SSA-PSR-KELM and the proposed model, both of these two models possess the same frameworks, while the parameters in the proposed one are optimized by the proposed IHGWOSCA algorithm synchronously. In the case of SG Mar. as the example, the metrics decreasing the percentage between VMD-SSA-PSR-KELM and the proposed model in terms of MAPE are 23.05%, 8.71%, and 15.94% in three predicted horizons, respectively. It can be concluded that the forecasting performance obtained by the synchronous optimization strategy-based model is much better; in other words, the appropriate parameters in each module could be optimized by the proposed IHGWOSCA effectively. Additionally, for one-step prediction in all cases, the average decline ratios between these two models in terms of RMSE, MAE, and MAPE are 32.86%, 32.94%, 31.89%, respectively. Furthermore, compared with SVR models in all experimental cases, the maximum decreasing ratio of MAPE in the one-, three- and five-step predictions are 95.57% (in the case of SG Jun.), 91.62% (in the case of SG Sep.), and 90.79% (in the case of SG Sep.), respectively, with which it can be indicated that a large promotion in performance could be achieved by the proposed model.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ADMM | Alternating direction method of multipliers |

ANN | Artificial neural network |

AI | Artificial intelligence |

AR | Autoregressive |

ARIMA | Autoregressive integrated moving average |

ARMA | Autoregressive moving average |

ELM | Extreme learning machine |

EMD | Empirical mode decomposition |

GS | Grid search |

GWO | Grey wolf optimizer |

HGWO-SCA | Hybrid grey wolf optimizer-sine cosine algorithm |

IHGWOSCA | Improved hybrid grey wolf optimizer-sine cosine algorithm |

IMF | Intrinsic mode function |

KELM | Kernel extreme learning machine |

Kurt. | Kurtosis |

MAE | Mean absolute error |

MAPE | Mean absolute percentage error |

Max. | Maximum |

Min. | Minimum |

NWP | Numerical weather prediction |

PSR | Phase space reconstruction |

RMSE | Root mean square error |

SCA | Sine cosine algorithm |

SG | Sotavento Galicia |

Skew. | Skewness |

SLFN | Single hidden layer feed-forward network |

SSA | Singular spectrum analysis |

Std. | Standard deviation |

SVD | Singular value decomposition |

SVR | Support vector regression |

VMD | Variational mode decomposition |

WT | Wavelet transform |

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**Figure 1.**Comparison of the proposed function and the original one for updating the components of over the course of iterations.

**Figure 3.**The procedures of the proposed IHGWOSCA-based synchronous optimization hybrid forecasting approach.

**Figure 5.**The multi-step forecasting results of various hybrid models in different cases: (

**a**) the case of SG Mar.; (

**b**) the case of SG Jun.

**Figure 6.**The multi-step forecasting results of various hybrid models in different cases: (

**a**) the case of SG Sep.; (

**b**) the case of SG Dec.

**Figure 7.**Comparison between all of the multi-step experimental results in terms of RMSE, MAE, and MAPE in different cases: (

**a**) the case of SG Mar.; (

**b**) the case of SG Jun.

**Figure 8.**Comparison of all of the multi-step experimental results in terms of RMSE, MAE, and MAPE in different cases: (

**a**) the case of SG Sep.; (

**b**) the case of SG Dec.

Cases | Statistic Indices | |||||
---|---|---|---|---|---|---|

Max. (m/s) | Min. (m/s) | Mean (m/s) | Stew. | Kurt. | Std. | |

SG March | 17.38 | 2.56 | 8.53 | 0.56 | 2.96 | 2.68 |

SG June | 11.00 | 0.35 | 4.5 | 0.27 | 2.94 | 2.13 |

SG September | 13.39 | 0.35 | 5.77 | 0.6 | 2.6 | 3.19 |

SG December | 16.84 | 0.35 | 6.7 | 0.44 | 3.21 | 3.12 |

**Table 2.**Optimal parameters of the proposed models in different prediction horizons for all of the experimental cases.

Cases | Horizons | Parameters | |||||||
---|---|---|---|---|---|---|---|---|---|

K | α | γ | s | τ | d | C | σ^{2} | ||

SG March | One-step | 10 | 877 | 0.98 | 167 | 1 | 13 | 358.13 | 118.06 |

Three-step | 10 | 770 | 0.19 | 118 | 1 | 26 | 514.82 | 242.26 | |

Five-step | 10 | 968 | 0.29 | 117 | 1 | 15 | 419.53 | 129.29 | |

SG June | One-step | 10 | 645 | 0.43 | 167 | 1 | 15 | 1000 | 129.48 |

Three-step | 10 | 1213 | 0.91 | 93 | 1 | 6 | 1000 | 52.61 | |

Five-step | 10 | 62 | 0.48 | 64 | 1 | 31 | 829.72 | 166.47 | |

SG September | One-step | 10 | 213 | 0.79 | 154 | 1 | 12 | 1000 | 43.67 |

Three-step | 10 | 127 | 0.4 | 86 | 1 | 18 | 988.64 | 330.17 | |

Five-step | 10 | 122 | 0.09 | 84 | 1 | 14 | 875.89 | 129.23 | |

SG December | One-step | 10 | 728 | 0.99 | 167 | 1 | 9 | 1000 | 201.19 |

Three-step | 10 | 223 | 0.94 | 83 | 1 | 10 | 686.65 | 116.63 | |

Five-step | 10 | 531 | 0.61 | 53 | 1 | 13 | 688.51 | 263.35 |

Cases | Models | One-Step | Three-Step | Five-Step | ||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | MAPE | RMSE | MAE | MAPE | RMSE | MAE | MAPE | ||

(m/s) | (m/s) | (%) | (m/s) | (m/s) | (%) | (m/s) | (m/s) | (%) | ||

SG March | SVR | 1.06 | 0.81 | 11.37 | 1.53 | 1.18 | 17.33 | 1.83 | 1.42 | 21.29 |

KELM | 1.05 | 0.80 | 10.35 | 1.50 | 1.16 | 15.20 | 1.76 | 1.36 | 17.67 | |

EMD-LSSVM | 0.68 | 0.50 | 6.54 | 0.83 | 0.64 | 8.43 | 1.05 | 0.80 | 10.39 | |

VMD-KELM | 0.12 | 0.09 | 1.21 | 0.19 | 0.15 | 1.98 | 0.33 | 0.26 | 3.44 | |

EMD-SSA-PSR-KELM | 0.52 | 0.38 | 4.71 | 0.80 | 0.60 | 7.69 | 0.93 | 0.70 | 9.12 | |

VMD-SSA-PSR-KELM | 0.08 | 0.06 | 0.84 | 0.17 | 0.14 | 1.87 | 0.30 | 0.24 | 3.11 | |

Proposed | 0.06 | 0.05 | 0.65 | 0.16 | 0.13 | 1.71 | 0.27 | 0.20 | 2.61 | |

SG June | SVR | 0.64 | 0.52 | 13.69 | 0.93 | 0.74 | 19.79 | 0.97 | 0.77 | 22.05 |

KELM | 0.63 | 0.51 | 12.76 | 0.88 | 0.71 | 17.41 | 0.96 | 0.76 | 18.37 | |

EMD-LSSVM | 0.36 | 0.28 | 7.27 | 0.48 | 0.39 | 10.07 | 0.54 | 0.44 | 11.34 | |

VMD-KELM | 0.05 | 0.04 | 1.00 | 0.11 | 0.09 | 2.29 | 0.20 | 0.16 | 4.24 | |

EMD-SSA-PSR-KELM | 0.22 | 0.17 | 4.40 | 0.43 | 0.33 | 8.96 | 0.49 | 0.39 | 10.22 | |

VMD-SSA-PSR-KELM | 0.04 | 0.03 | 0.75 | 0.10 | 0.08 | 2.08 | 0.17 | 0.13 | 3.60 | |

Proposed | 0.03 | 0.02 | 0.61 | 0.09 | 0.07 | 1.82 | 0.11 | 0.09 | 2.33 | |

SG September | SVR | 0.63 | 0.45 | 13.44 | 0.97 | 0.71 | 23.04 | 1.27 | 0.97 | 29.97 |

KELM | 0.61 | 0.44 | 11.96 | 0.96 | 0.71 | 18.91 | 1.19 | 0.91 | 23.27 | |

EMD-LSSVM | 0.42 | 0.29 | 9.29 | 0.58 | 0.41 | 12.20 | 0.71 | 0.53 | 14.98 | |

VMD-KELM | 0.08 | 0.06 | 1.70 | 0.12 | 0.09 | 2.76 | 0.20 | 0.16 | 4.69 | |

EMD-SSA-PSR-KELM | 0.33 | 0.22 | 7.21 | 0.49 | 0.34 | 10.58 | 0.65 | 0.47 | 13.67 | |

VMD-SSA-PSR-KELM | 0.07 | 0.05 | 1.40 | 0.10 | 0.08 | 2.25 | 0.18 | 0.14 | 4.37 | |

Proposed | 0.03 | 0.03 | 0.70 | 0.08 | 0.06 | 1.93 | 0.12 | 0.09 | 2.76 | |

SG December | SVR | 1.23 | 0.89 | 12.29 | 1.89 | 1.38 | 18.95 | 2.22 | 1.63 | 21.96 |

KELM | 1.22 | 0.89 | 12.01 | 1.89 | 1.37 | 18.28 | 2.15 | 1.57 | 20.53 | |

EMD-LSSVM | 0.63 | 0.50 | 6.76 | 0.90 | 0.68 | 9.30 | 1.12 | 0.85 | 11.87 | |

VMD-KELM | 0.16 | 0.12 | 1.71 | 0.32 | 0.24 | 3.24 | 0.50 | 0.37 | 5.14 | |

EMD-SSA-PSR-KELM | 0.41 | 0.31 | 4.24 | 0.74 | 0.55 | 7.35 | 0.99 | 0.73 | 10.18 | |

VMD-SSA-PSR-KELM | 0.14 | 0.10 | 1.33 | 0.29 | 0.21 | 2.81 | 0.47 | 0.35 | 4.75 | |

Proposed | 0.09 | 0.07 | 0.87 | 0.22 | 0.16 | 2.28 | 0.36 | 0.28 | 3.72 |

**Table 4.**Percentage of the promotion between the comparative models and the proposed model in all experimental cases for multi-step prediction.

Cases | Extant Models vs Proposed | One-Step | Three-Step | Five-Step | ||||||
---|---|---|---|---|---|---|---|---|---|---|

P_{RMSE} (%) | P_{MAE} (%) | P_{MAPE} (%) | P_{RMSE} (%) | P_{MAE} (%) | P_{MAPE} (%) | P_{RMSE} (%) | P_{MAE} (%) | P_{MAPE} (%) | ||

SG March | SVR | 94.03 | 94.11 | 94.31 | 89.38 | 89.23 | 90.14 | 84.97 | 85.56 | 87.74 |

KELM | 93.97 | 94.00 | 93.75 | 89.19 | 88.97 | 88.76 | 84.42 | 84.98 | 85.23 | |

EMD-KELM | 90.70 | 90.50 | 90.11 | 80.46 | 80.2 | 79.74 | 73.95 | 74.61 | 74.86 | |

VMD-KELM | 47.03 | 47.76 | 46.56 | 15.78 | 15.39 | 13.89 | 17.91 | 21.75 | 24.18 | |

EMD-SSA-PSR-KELM | 87.78 | 87.32 | 86.26 | 79.71 | 78.72 | 77.78 | 70.60 | 70.97 | 71.37 | |

VMD-SSA-PSR-KELM | 22.14 | 24.49 | 23.05 | 5.97 | 8.07 | 8.71 | 9.28 | 14.41 | 15.94 | |

SG June | SVR | 95.45 | 95.58 | 95.57 | 90.58 | 90.84 | 90.81 | 88.59 | 88.81 | 89.41 |

KELM | 95.40 | 95.55 | 95.25 | 90.05 | 90.49 | 89.56 | 88.44 | 88.73 | 87.29 | |

EMD-KELM | 91.98 | 91.87 | 91.66 | 81.96 | 82.87 | 81.94 | 79.48 | 80.30 | 79.41 | |

VMD-KELM | 41.97 | 42.32 | 39.64 | 20.05 | 23.03 | 20.50 | 44.31 | 46.62 | 44.89 | |

EMD-SSA-PSR-KELM | 86.89 | 86.49 | 86.22 | 79.55 | 79.83 | 79.72 | 77.25 | 78.07 | 77.16 | |

VMD-SSA-PSR-KELM | 21.68 | 22.90 | 19.68 | 12.17 | 16.25 | 12.80 | 33.13 | 35.72 | 35.12 | |

SG September | SVR | 94.51 | 94.29 | 94.81 | 91.63 | 90.92 | 91.62 | 90.60 | 90.52 | 90.79 |

KELM | 94.32 | 94.16 | 94.17 | 91.55 | 90.88 | 89.79 | 89.95 | 89.82 | 88.14 | |

EMD-KELM | 91.85 | 91.24 | 92.50 | 85.97 | 84.28 | 84.17 | 83.14 | 82.53 | 81.57 | |

VMD-KELM | 56.66 | 56.78 | 58.96 | 34.81 | 30.39 | 30.04 | 40.35 | 41.00 | 41.18 | |

EMD-SSA-PSR-KELM | 89.39 | 88.60 | 90.33 | 83.41 | 81.26 | 81.75 | 81.52 | 80.39 | 79.81 | |

VMD-SSA-PSR-KELM | 48.58 | 49.90 | 50.28 | 19.70 | 17.39 | 14.26 | 35.44 | 35.70 | 36.76 | |

SG December | SVR | 92.97 | 92.63 | 92.90 | 88.56 | 88.14 | 87.96 | 83.73 | 82.99 | 83.08 |

KELM | 92.90 | 92.6 | 92.73 | 88.51 | 88.05 | 87.52 | 83.23 | 82.39 | 81.91 | |

EMD-KELM | 86.26 | 86.91 | 87.09 | 75.87 | 76.03 | 75.46 | 67.82 | 67.33 | 68.70 | |

VMD-KELM | 45.80 | 45.89 | 48.98 | 32.69 | 30.69 | 29.62 | 27.79 | 25.07 | 27.76 | |

EMD-SSA-PSR-KELM | 79.08 | 78.98 | 79.41 | 70.56 | 70.06 | 68.95 | 63.47 | 62.22 | 63.50 | |

VMD-SSA-PSR-KELM | 39.04 | 34.46 | 34.56 | 25.47 | 22.39 | 18.69 | 24.07 | 19.98 | 21.72 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fu, W.; Wang, K.; Zhou, J.; Xu, Y.; Tan, J.; Chen, T.
A Hybrid Approach for Multi-Step Wind Speed Forecasting Based on Multi-Scale Dominant Ingredient Chaotic Analysis, KELM and Synchronous Optimization Strategy. *Sustainability* **2019**, *11*, 1804.
https://doi.org/10.3390/su11061804

**AMA Style**

Fu W, Wang K, Zhou J, Xu Y, Tan J, Chen T.
A Hybrid Approach for Multi-Step Wind Speed Forecasting Based on Multi-Scale Dominant Ingredient Chaotic Analysis, KELM and Synchronous Optimization Strategy. *Sustainability*. 2019; 11(6):1804.
https://doi.org/10.3390/su11061804

**Chicago/Turabian Style**

Fu, Wenlong, Kai Wang, Jianzhong Zhou, Yanhe Xu, Jiawen Tan, and Tie Chen.
2019. "A Hybrid Approach for Multi-Step Wind Speed Forecasting Based on Multi-Scale Dominant Ingredient Chaotic Analysis, KELM and Synchronous Optimization Strategy" *Sustainability* 11, no. 6: 1804.
https://doi.org/10.3390/su11061804