1. Introduction
In recent decades, wind energy has become more and more important as an emerging renewable energy source. Nevertheless, due to uncertainties such as sunshine, topography, and pressure, as well as the intermittent and random effects of wind speed, wind power possesses a large amount of uncertainties, with which wind power operation would be challenging. However, accurate short-term wind speed forecasting plays a vital role in meeting such a challenge, with which the dispatch plan could be adjusted in time, as well as the optimal unit combination plan being formulated effectively [
1]. Additionally, due to the fact that the dynamic behavior of future wind speed trends could be excavated by multi-step forecasting, satisfactory multi-step prediction results are important indicators of wind speed forecasting. Therefore, it is necessary to develop an accuracy multi-step short-term wind speed prediction model imminently, to thus improve economic benefits for wind farms.
Over the past few decades, a great variety of methods have been developed to achieve wind speed prediction, which can be roughly divided into four categories [
2]: physical models, conventional statistical models, spatial correlation models, and artificial intelligence (AI) models. As a well-known physical model, numerical weather prediction (NWP) [
3], achieves wind speed prediction, considering various relevant meteorological factors such as humidity, pressure, wind speed, and direction, etc. However, the drawbacks of the long operation time and the large amount of computing resources make such models difficult to construct. Another popular forecasting approach, namely statistical models could extract potential information contained in the historical wind speed series, among which autoregressive (AR) [
4], autoregressive moving average (ARMA) [
5], and autoregressive integrated moving average (ARIMA) [
6] have been widely investigated. Nevertheless, due to the strong nonlinearity and non-stationarity within the wind speed time series, the capabilities of such models would be restricted significantly. Spatial correlation models that consider the spatial relationship of the wind speed information collected from various wind farms are investigated to be an effective tool for wind speed forecasting [
7,
8]. Whereas the formulation of such models is more difficult to implement than conventional statistical models, due to the difficulty in wind speed data collection and the timely transmission from various space-related sites [
2]. The other category, namely AI models, have been developed rapidly in the field of wind speed prediction over the past few decades. The superior performance in dealing with nonlinear and non-stationary time series has been approved by a number of scholars. Among the varied methods, artificial neural networks (ANNs) [
9,
10] possess strong robustness, as well as the ability to fully approximate complex nonlinear relationships, whereas the network structures are difficult to determine, as well as being time consuming for examination. In contrast, the appropriate parameters in support vector regression (SVR) [
11,
12] are easier to determine, of which the nonlinear forecasting problems could be solved by proper kernel transformations. Compared with the AI approaches mentioned above, extreme learning machine (ELM) [
1,
13,
14] is widely utilized in the field of wind speed forecasting due to the fast computing speed and strong generalization capability. Additionally, to enhance the generalization performance of ELM, the regularization coefficient is employed to solve optimization problems, as well as replacing hidden nodes by kernel functions, thus weakening the randomness of the predicted results [
15].
Generally, the forecasting performance obtained by directly applying a prediction model would be yield terrible results, which can be attributed to the non-linear and non-stationary traits within the wind speed time series. To this end, a large number of data preprocessing methods has been developed for reducing the non-stationarity of the raw wind speed data, which has been proven to be beneficial for improving prediction accuracy [
16]. Time-frequency signal decomposition methods such as wavelet transform (WT) [
17], empirical mode decomposition (EMD) [
18], and variational mode decomposition (VMD) [
19] have been widely utilized in wind speed prediction. Among the approaches, VMD possesses a more adaptive ability than WT, as well as owning a more solid mathematical theoretical basis than EMD [
17,
18], with which VMD has been widely applied into various fields [
20,
21]. Nevertheless, the decomposition efficiency of VMD is affected by the mode number and the quadratic penalty term as well as updating steps to some extent [
21,
22], which makes parameter optimizations for VMD necessary. On the basis of the comparisons analyzed above, VMD will be employed in this study to preliminary preprocessing of the raw wind speed time series.
To further enhance the forecasting performance on the basis of the decomposition methods applied, singular spectrum analysis (SSA) was employed, to extract the dominant and residuary ingredients from all sub-series, which has been proven as an effective technique to improve the capabilities of the prediction models when they are combined with time-frequency signal decomposition methods [
23,
24]. Furthermore, in order to exploit the inherent laws of chaotic systems for the preprocessed sub series, phase space reconstruction (PSR) [
25], which is considered to be a powerful tool for chaotic time series analysis, is employed to deduce the inputs and outputs of the forecasting engine. Nevertheless, the reconstruction performance for the chaotic system would be affected by the parameters in PSR to some extent, with which the forecasting performance would be restricted [
9,
26].
In order to achieve better parameter optimization for the approaches mentioned above, an improved hybrid grey wolf optimizer-sine cosine algorithm (IHGWOSCA) is proposed, to handle such problems. Hence, to enhance the accuracy of the multi-step short-term wind speed prediction, a novel hybrid model based on multi-scale dominant ingredient chaotic analysis, kernel extreme learning machine (KELM), and IHGWOSCA-based synchronous optimization strategy is proposed in this paper. The proposed multi-scale dominant ingredient chaotic analysis, combining VMD, SSA, and PSR is implemented to preprocess the raw wind speed data, with which the non-stationarity of the wind speed time series could be significantly weakened. In this phase, the residual of VMD would be accumulated with the residuary ingredients obtained by SSA, to generate an additional forecasting component. Meanwhile, the inputs and outputs of KELM could be deduced by PSR effectively, after which the predictors for all the components could be constructed. Finally, the ultimate prediction values of the original wind speed are calculated by integrating the predicted results of all the components. The optimal parameters of each module could be obtained by repeating the whole process introduced above, in the IHGWOSCA optimizer, with which the best performance could be achieved. Furthermore, the superiority and effectiveness of the proposed model has been testified by comparative experiments, of which four sets of wind speed time series were collected from Sotavento Galicia (SG) as well as other six relevant single and combined models were used for comparative analysis.
The remaining parts of this paper are organized as follows:
Section 2 detailed presents the base knowledge for VMD, SSA, PSR and KELM.
Section 3 introduces multi-scale dominant ingredient chaotic analysis, the proposed IHGWOSCA algorithm, optimization strategies, and the specific procedure of the proposed model.
Section 4 denotes the effectiveness of the proposed model through experimental results and analysis.
Section 5 details the perspectives about further investigation directions. The conclusions are summarized in
Section 6. The abbreviations of technical terms are listed in Abbreviations.
3. The Proposed Approach
3.1. Multi-Scale Dominant Ingredient Chaotic Analysis
Due to the non-linearity, non-stationarity, and random fluctuation characteristics within the wind speed time series, the forecasting performance would be severely restricted. Therefore, VMD is employed to preliminarily decompose the collected wind speed data into several sub-series with various frequency scales, of which the decomposition efficiency of VMD and the prediction accuracy are greatly affected by the parameters
K,
α and
γ [
21,
22]. In order to further reduce the non-stationarity of the decomposed sub-series, SSA is implemented to extract the dominant ingredients and residuary ingredients from the sub-series, thus achieving multi-scale dominant ingredient analysis effectively. In this study, the set of indices
Z = {1, 2, …,
l} in the grouping phase of SSA is divided into two discrete subsets, that is,
Z1 = {1, 2, …,
s} and
Z2 = {
s + 1,
s + 2, …,
l}, with which the matrix
HZ could be represented as
HZ =
HZ1 +
HZ2. 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.,
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
The hybrid grey wolf optimizer-sine cosine algorithm (HGWO-SCA) is proposed by Singh et al. [
38], of which both the grey wolf optimizer (GWO) and the sine cosine algorithm (SCA) are developed by Mirjalili et al. [
39,
40]. Four categories of grey wolves,
α,
β,
δ, and
ω are defined for simulating the leadership hierarchy in normal GWO, which are determined by the top three best positions (fitness value). Then, the equations that could be utilized for mathematically model encircling behavior are defined as below:
where
t denotes the
t-th iteration,
represents the position of the prey,
indicates the position vector of the grey wolf, and
and
are coefficient vectors calculated as follows:
where
r1 and
r2 are random vectors in the scopes of [0, 1]; the components of
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
and
, of which half of the iterations are divided toward exploration, when
and the remaining iterations are assigned for exploitation when
[
39]. Hence, to better improve the corresponding abilities of these two phases, a cosine function-based decreasing formula for updating the components of
is proposed in this study:
where
t and
T represent the current iteration and the maximum number of iterations, respectively. In addition, the comparison of the proposed function and the original one for updating the components of
over the course of the iterations is intuitively exhibited in
Figure 1. As can be seen from
Figure 1, the values of
are generally larger in the proposed function during the first half of the iterations, which could contribute to improving the exploration ability of the algorithm at this stage. The corresponding effects in the exploitation phase would be obtained with smaller values of
, compared to the original ones.
The other stage of the whole algorithm, namely hunting, is usually guided by the
α wolf, while the
β and
δ wolves participate in hunting occasionally [
39]. To effectively mathematically simulate the hunting behavior, the
α,
β, and
δ wolves are assumed to possess more knowledge about the potential location of prey. In addition, the position updating functions of the
α wolf in HGWO-SCA are modified by applying the position updating equations of SCA, thus enhancing the convergence capabilities of GWO [
38]. The corresponding updating equations of the
α,
β, and
δ wolves are defined as follows:
where
,
and
indicate the positional information owned by the
α,
β, and
δ wolves so far. Due to the fact that individuals in HGWO-SCA are merely updated by simple averaging
,
and
, some scholars have focused on the updating approaches for the individuals [
41]. In this study, a weighted averaging strategy is proposed to iterate the individuals, in which
α,
β, and
δ wolves are separately assigned a weight value that is deduced by inversing the corresponding fitness values of the wolves. The detailed calculations are as follows:
where
fit denotes the fitness of the corresponding individual. Furthermore, the pseudo code of the proposed IHGWOSCA algorithm is exhibited in Algorithm 1.
Algorithm 1. The pseudo code of the proposed IHGWOSCA algorithm |
1: | Initialization the population (i = 1, 2, …, N) |
2: | Initialize a, and |
3: | Calculate the fitness of each search member |
4: | : the best search agents, : the second-best search agent, : 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, , and 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 | = rand () × sin (0.5⋅π⋅rand ()) × | × − | |
15: | Else: |
16: | = rand () × cos (0.5⋅π⋅rand ()) × | × − | |
17: | = − · |
18: | End if |
19: | End else |
20: | t = t+1 |
21: | End while |
22: | Return |
3.3. Optimization Strategy
In order to effectively optimize the parameters in various modules, as well as construct the hybrid forecasting model, the proposed IHGWOSCA algorithm is adopted to handle this problem. To begin with, the parameters of SSA, PSR, and KELM for all the sub-series are considered to be the same in this study, which could facilitate fast convergence as well as reduce computation. Hence, the total number of the variables is eight, while the corresponding coding strategy of the agents in the proposed IHGWOSCA is described in
Figure 2. Additionally, the metric root-mean-square error (
RMSE) represented in Equation (27) is adopted as the objective function for parameter optimization.
3.4. Specific Procedures
The main procedures of the proposed novel hybrid wind speed forecasting model, combined with VMD, SSA, PSR, KELM, and IHGWOSCA-based synchronous optimization strategies are described as follows:
Step 1: Collect the original wind speed data and initialize the population of IHGWOSCA;
Step 2: Calculate the fitness value for each agent;
Step 2.1: Decode the population and assign the corresponding parameters for each module, i.e., the parameters K, α, γ for VMD, s for SSA, τ, d for PSR as well as C, and σ2 for KELM;
Step 2.2: Decompose the collected wind speed data into K modes by utilizing VMD, then calculate the residual mr of VMD;
Step 2.3: Implement SSA for all the sub-series, then extract the dominant ingredients as well as accumulate all of the residuary ingredients with mr.
Step 2.4: For the k-th (k = 1, …, K, K + 1) component, construct the input and output matrixes for the k-th (k = 1, …, K, K + 1) KELM, applying PSR;
Step 2.5: Model the k-th (k = 1, …, K, K + 1) KELM with parameters C and σ2. 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).
Step 2.6: Repeat Steps 2.1 to 2.5 until the fitness values for all the agents are generated;
Step 3: Execute the operators of IHGWOSCA.
Step 4: Repeat Step 2 to 3 until the maximum number of iterations is reached;
Step 5: Obtain the optimal parameters for all of the modules by decoding the best individual, as well as calculate the final forecasting values of the collected wind speed data by repeating Steps 2.1 to 2.5.
The overall process of the proposed wind speed forecasting model is depicted in
Figure 3.
6. Conclusions
To improve multi-step prediction performance, a novel hybrid based on a multi-scale dominant ingredient chaotic analysis, the KELM- and IHGWOSCA-based synchronous optimization strategy, is proposed in this paper. Specifically, the proposed model possesses a structure of VMD-SSA-PSR-KELM, of which the parameters in each module was synchronously optimized by the proposed IHGWOSCA algorithm. Firstly, VMD was applied to decompose the raw non-stationary wind speed data into several sub-series, while the residual of VMD was calculated concurrently. Then, SSA was employed to extract the dominant and residuary ingredients for each sub-series, while the residuary ones were integrated with the residual of VMD to be an additional forecasting component. Later, PSR was executed, to deduce the inputs and outputs of KELM for all of the forecasting components. Finally, the prediction models for all of the components were constructed by KELM, as well as the forecasting results that were accumulated to obtain the final forecasting values of the raw wind speed data. The whole procedure of the proposed VMD-SSA-PSR-KELM structure was iterated in the parameters searching phase, with which the parameters of each module would be optimized effectively. In the experimental stage, six relevant comparative models were employed to compare with the proposed model. Through an intensive analysis of the prediction results for the one-step and multi-step predictions, it can be concluded that the forecasting performance could be enhanced through an implementation of the proposed dominant ingredient chaotic analysis with appropriate parameters, from which the metrics obtained by EMD-SSA-PSR-KELM in a one-step prediction were decreased to an average of 32.46% compared with EMD-KELM. Besides, the proposed VMD-SSA-PSR-KELM structure achieved satisfactory results compared with other combined models, of which the indicators of such models achieved the second-lowest minimum in all of the experimental cases, and decreased by an average of 78.13%, 73.28% and 63.61%, in various prediction horizons compared to EMD-SSA-PSR-KELM. Nevertheless, the synchronous optimization strategy based on the proposed IHGWOSCA algorithm could maximize the performance of the proposed hybrid structure; i.e., the parameter optimizations for each module could be effectively implemented. The performance improvement brought about by the synchronous optimization strategy achieved an average by 25%, compared with the separated optimized VMD-SSA-PSR-KELM model. In consequence, the proposed novel hybrid approach could be considered as a credible tool for multi-step short-term wind speed forecasting.