A Hybrid Multi-step Rolling Forecasting Model Based on Ssa and Simulated Annealing—adaptive Particle Swarm Optimization for Wind Speed

With the limitations of conventional energy becoming increasing distinct, wind energy is emerging as a promising renewable energy source that plays a critical role in the modern electric and economic fields. However, how to select optimization algorithms to forecast wind speed series and improve prediction performance is still a highly challenging problem. Traditional single algorithms are widely utilized to select and optimize parameters of neural network algorithms, but these algorithms usually ignore the significance of parameter optimization, precise searching, and the application of accurate data, which results in poor forecasting performance. With the aim of overcoming the weaknesses of individual algorithms, a novel hybrid algorithm was created, which can not only easily obtain the real and effective wind speed series by using singular spectrum analysis, but also possesses stronger adaptive search and optimization capabilities than the other algorithms: it is faster, has fewer parameters, and is less expensive. For the purpose of estimating the forecasting ability of the proposed combined model, 10-min wind speed series from three wind farms in Shandong Province, eastern China, are employed as a case study. The experimental results were considerably more accurately predicted by the presented algorithm than the comparison algorithms.


Introduction
Energy plays a vital part in modern social and economic development.Along with the rapid development of technology in the last few decades, energy demands continue to increase rapidly [1].In accordance with the IEA World Energy Outlook 2010, China and India will be responsible for approximately 50% of the growth in global energy demand by 2050.The consumption of energy in China will be close to 70% greater than the energy consumed by the United States today.Second only to America, China will become the second leading energy-consuming country in the world.Nevertheless, China's per capita energy consumption will remain lower than 50% of that of the USA [2].Since conventional energy sources such as coal, natural gas, and oil for electricity generation are being quickly depleted, sufficient energy reserves and sustainable energy problems are garnering increased attention.Additionally, using traditional resources produces large amounts of carbon dioxide, which may lead to global warming, and is considered an international security threat.Therefore, it not only affects the environment, it also threatens the safety of individuals and the planet [3].can enhance the prediction accuracy and convergence of the basic PSO algorithm.Moreover, APSOSA is able to avoid falling into local extreme points so that the parameters of the Back Propagation neural network (BPNN) can be better optimized.Finally, to achieve better forecasting performances, the wind speed data after noise elimination are input into the BPNN.In addition, four commonly used error criteria (AE, MAE, MAPE, and MSE) are applied to assess the performance of the raised hybrid algorithm.The main aspects of the model are introduced as follows: (1) data preprocessing; (2) the best forecasting method, BPNN, is selected and its parameters are tuned by an artificial intelligence (APSOSA) model; (3) forecasting; and (4) comparison and analysis.
The main contributions of this paper are summarized as follows: (1) With the aim of reducing the randomness and instability of wind speed, the Singular Spectrum Analysis technique is applied to decompose the wind series data, revealing real and useful signals from the wind series.(2) The best prediction system, BPNN, is selected from the different methods, including WNN, GRNN, and BPNN.(3) In view of the shortcomings of the PSO algorithm, APSOSA is developed to optimize parameters, which can assist the individual PSO in jumping out of the local optimum.Ultimately, parameters are selected and optimized by combining their respective advantages.(4) To examine the stability and accuracy of the new combined forecasting algorithm, 10-min wind speed series from three different stations are used in the experimental simulations.
The experimental results indicate that the novel hybrid algorithm has a higher performance, significantly outperforming the other forecasting algorithms.(5) Giving full consideration to the other influential factors in the experiments, such as the seasonal factors, the geographical factors, etc., according to the results, this action proves that the new combined algorithm has a powerful adaptive capacity, which can be widely applied to the prediction field.(6) The Bias-Variance Framework and statistical hypothesis testing are employed to further illustrate the stability and performance of the proposed algorithm.
The remainder of this paper is designed as follows.The methodology is described specifically in Section 2. To verify the prediction accuracy of the raised algorithm, a case study is examined in Section 3. Next, the wind farms area and datasets are introduced in Section 3.1, whereas Section 3.2 displays the performance criteria of the forecast results.In Section 3.3, the results of the different algorithms are listed and compared with the proposed algorithm.In order to further illustrate the stability and performance of the proposed algorithm, in Section 4, the Bias-Variance Framework and statistical hypothesis testing are employed.Finally, the conclusions are provided in Section 5.

Methodology
In this section, all of the algorithms involved in this work are described.The full list of algorithms to be discussed in this section is as follows: the singular spectrum analysis algorithm, an efficient technology for time series analysis; the particle swarm optimization algorithm; the simulated annealing algorithm, which overcomes PSO falling into the local minima, and the back propagation neural network.The hybrid algorithm-APSOSA, raised to search for the optimal parameters of BPNN, will also be introduced in detail.

Singular Spectrum Analysis
Compared with other nonparametric approaches, such as EMD, which exhibits a potential mode-mixing problem, and EEMD, which does not completely neutralize the added white noise, singular spectrum analysis (SSA), which overcomes the traditional analysis methods' (such as Fourier analysis) shortcomings, has been proven to be one of the most effective and powerful methods in time series analysis.It was developed by Broomhead and King in 1986 [43]. Figure 1

Intelligent Optimization Algorithms
In this section, several optimization algorithms will be introduced.

Particle Swarm Optimization
Inspired by imitating the social behavior of flocks of bird and schools of fish, an effective approach for optimization, Particle swarm optimization (PSO), was first developed by Dr. Eberharts and Dr. Kennedy in 1995 [44].It is a stochastic, population-based evolutionary algorithm, which involves searching for solutions [39] (details in Appendix B.1).

Back Propagation Neural Network
First proposed by Rumelhart and McCelland (1986) [45], the Back Propagation Neural Network (BPNN) is one of the most widely employed artificial neural network (ANN) models (details in Appendix B.2).It is not only capable of learning and storing a large amount of input-output mode mappings without needing to reveal the mathematical equations of the mapping relationship, but can also apply the steepest descent method, by back propagation, to constantly adjust the network weights and thresholds, resulting in the minimum square error.In addition, BPNN [46] consists of three layers: the input layer, the hidden layer, and the output layer.The experimental parameters are listed in Table 1.

Intelligent Optimization Algorithms
In this section, several optimization algorithms will be introduced.

Particle Swarm Optimization
Inspired by imitating the social behavior of flocks of bird and schools of fish, an effective approach for optimization, Particle swarm optimization (PSO), was first developed by Dr. Eberharts and Dr. Kennedy in 1995 [44].It is a stochastic, population-based evolutionary algorithm, which involves searching for solutions [39] (details in Appendix B.1).

Back Propagation Neural Network
First proposed by Rumelhart and McCelland (1986) [45], the Back Propagation Neural Network (BPNN) is one of the most widely employed artificial neural network (ANN) models (details in Appendix B.2).It is not only capable of learning and storing a large amount of input-output mode mappings without needing to reveal the mathematical equations of the mapping relationship, but can also apply the steepest descent method, by back propagation, to constantly adjust the network weights and thresholds, resulting in the minimum square error.In addition, BPNN [46] consists of three layers: the input layer, the hidden layer, and the output layer.The experimental parameters are listed in Table 1.[47].Based on the Monte-Carlo iteration solving method, currently the SA algorithm has become one of the most popular heuristic random search methods.Unlike the PSO algorithm, SA can jump out of the trap of local minima in a timely manner to update the solutions and obtain the global optimum (details in Appendix B.3).

The Proposed Optimization Algorithm, APSOSA
This paper proposes a hybrid APSO algorithm by employing the SA algorithm to prevent the PSO from falling into local minima (the algorithm of SA-APSO is shown in Table 2).The major steps of the hybrid optimization algorithm are as follows.First, enhance the accuracy and convergence rate of the basic PSO algorithm by applying a compression factor and selecting suitable parameters.Next, by combining the Simulated Annealing characteristics, PSO can more easily and quickly obtain the optimal solution in a larger search space.SA can also cancel the restrictions on speed border.Moreover, the Roulette Wheel Selection Strategy is chosen, shown in Figure 2, in this algorithm.
PSO can obtain better results in a faster setting with fewer parameters and is cheaper than other methods.It is also currently being widely used for promising results in continuous problems.However, the movement directionality of the particles is not certain, and particles are likely to jump out to obtain near-optimal solutions and their local search ability is relatively weak and easily trapped by the local optimum.Therefore, PSO is combined with the simulated annealing algorithm.The annealing algorithm is used when poor quality is probable to temporarily accept some solution features to construct a particle swarm algorithm, based on simulated annealing.A multitude of papers have verified that the improved particle swarm optimization algorithm obtains better results and have documented the effectiveness of the method through experimental simulation results.The velocity and position updating formula are as follows: v i,j pk `1q " χ rv i,j pkq `c1 r 1 pp i,j pkq ´xi,j pkqq `c2 r 2 pp g,j pkq ´xi,j pkqqs x i,j pk `1q " x i,j pkq `vi,j pk `1q, j " 1, ..., n where r 1 and r 2 are set randomly between 0 and 1, and the learning factors c 1 and c 2 are positive numbers, where χ is computed by the following formula: In Equation (3), applying the best group positions, all particles fly to the best group positions, and then tend to the local minima solution if the best group positions are in the local minimum.Accordingly, this situation will cause the search dispersability and ability to become worse.To overcome this weakness, a new position p 1 g will be selected from among the p i to replace p g .Finally, in this paper, the Equation (1) is rewritten as the following formula, Equation (4): v i,j pk `1q " χ rv i,j pkq `c1 r 1 pp i,j pkq ´xi,j pkqq `c2 r 2 pp 1 i,j pkq ´xi,j pkqqs However, how to address the suitable position p i is one of the most critical steps of the combined algorithm.Clearly, better performance of p i shall be considered a higher priority.Under the characteristics of the SA algorithm, the best solutions of every particle p i should be taken as the special one, which may be worse than the global optimal solution p g .Therefore, in the case of temperature T, we can calculate the leap probability using Equation (5): P l p pp i q " e ´pFpp i q´Fpp g qq{KT (5) where F is the objective function value of the particle position.
The leap probability can be computed by the following formula, Equation (6): where N is the population, on the side; considering the Roulette Wheel Selection strategy, we can randomly choose the p i , which will be regarded as p 1 g .The chief steps of the APSOSA algorithm are as follows, and a flowchart is depicted in Figure 2.
Step 1: Set the initial temperature, and initialize the population along with every particle velocity and position.
Step 3: Update the present position and fitness value of each particle by using p i , Fpp i q and p g , Fpp g q, respectively.
Step 4: Compute the initial temperature using Equation (7): Step 5: Update the particle position and velocity and compute Ppp i q.
Step 6: Through the roulette wheel selection strategy, the new global optimal solution p 1 g is not regarded as p g until Ppp i q ą rand p q.
Step 7: Update every particle velocity and position by the pre-set update formula.
Step 8: Compute every particle Fpp 1 i q, and then do not apply p i Fpp i q to update the current global position p g and optimal fitness value Fpp g q, respectively, until Fpp i q>Fpp g q.
Step 9: By applying the pre-set rules, the temperature reduces slowly.
Step 10: Analyze whether the pre-set conditions are met; if they have been met, output the information of p g and then end running.Otherwise, repeat the above steps, beginning with Step 5. where F is the objective function value of the particle position.
The leap probability can be computed by the following formula, Equation ( 6): where N is the population, on the side; considering the Roulette Wheel Selection strategy, we can randomly choose the i p , which will be regarded as g p .
The chief steps of the APSOSA algorithm are as follows, and a flowchart is depicted in Figure 2.
Step 1: Set the initial temperature, and initialize the population along with every particle velocity and position.
Step 2: Compute , where N is the updated population.
Step 3: Update the present position and fitness value of each particle by using i p , ( ) Step 4: Compute the initial temperature using Equation ( 7): Step 5: Update the particle position and velocity and compute ( ) Step 6: Through the roulette wheel selection strategy, the new global optimal solution g p  is not regarded as g p until ( ) rand ( ) i p  P .
Step 7: Update every particle velocity and position by the pre-set update formula.
Step 8: Compute every particle ( ) Step 9: By applying the pre-set rules, the temperature reduces slowly.
Step 10: Analyze whether the pre-set conditions are met; if they have been met, output the information of g p and then end running.Otherwise, repeat the above steps, beginning with Step 5.   Algorithm: SA-APSO Input: x p0q s " ´xp0q p1q , x p0q p2q , . . ., x p0q plq ¯-a sequence o f training data.x p0q p " ´xp0q pl `1q , x p0q pl `2q , . . ., x p0q pl `nq ¯-a sequence o f veri f ying data.

Output:
P g -the value of x with the best fitness value in population of particles Parameters: Iter max -the maximum number of iterations n-the number of particles F i -the fitness function of particle i x i -particle i g-the current iteration number d-the number of dimension

SSA-APSOSA-BPNN Algorithm
In this section, we will introduce the hybrid model (SSA-APSOSA-BPNN) more clearly.The flowchart of the model is described in Figure 3.And the experimental parameters of APSOSA are given in Table 3.The hybrid algorithm contains four main stages.Here, the SSA-APSOSA-BPNN algorithm is utilized for wind speed forecasting.Additionally, the detailed rules are given as below: Step 1: Determine and initialize the parameters of APSOSA.
Step 2: Set the fitness function; the mean absolute error (MAE) of validation is taken as the fitness of the particles: where N is the number of validation sets and ŷi and y i stand for the predictive value and the observed value, respectively.
Step 3: Update the historical extremum p j of every particle and the global extremum p g and then repeat the above rules for the next particles.
Step 4: Set the conditions and judge whether the fitness value meets the conditions; if it does, save the corresponding optimal parameters and then stop running.Otherwise, run Step 3 again and continue to run.
Stage IV: Forecasting.In this stage, the optimal parameters from Step 4 of Stage III will be applied into the BPNN model to forecast.Finally, the wind speed forecasting data will be obtained by completing all of the above steps.
Stage IV: Forecasting.In this stage, the optimal parameters from Step 4 of Stage III will be applied into the BPNN model to forecast.Finally, the wind speed forecasting data will be obtained by completing all of the above steps.

Case Study
To examine the accuracy of the novel combined algorithm, four different multi-step forecasting algorithms are compared by analyzing the three-step-ahead-prediction (half-1-h-ahead) and the sixstep-ahead-prediction (1-h-ahead) of a 10-min wind speed series at three different wind power stations.

Study Area and Datasets
Shandong, located on the east coast of China, is not only one of the provinces with the largest economy, but also one of the biggest energy consumers.However, 99% of the electrical energy comes from coal power generation.As a result, Shandong faces enormous energy pressures.
However, as a coastal province, Shandong possesses one of China's largest wind farms, with an installed capacity of 58 million kilowatts.A simple map of the research area is depicted in Figure 4.With the aim of satisfying social development, achieving energy conservation, and protecting the environment, Shandong has begun developing wind power stations.Due to the area's unique geographical advantages, capacity reached 260 billion KWH in 2007.In addition, the Shandong Province Bureau of Meteorology assessment notes that the entire output of wind energy resources in Shandong province is 67 million kilowatts, which is equivalent to the installed capacity of 3.68 times the capacity of the Three Gorges Hydro-power Station (18.20 million kilowatts), which ranks in the top three.To actively build wind power green energy bases, promote wind energy development, and protect the environment, Shandong has been focusing on building large-scale wind farms in Weihai, Yantai, Dongying, Weifang, Qingdao, and other coastal areas, and is gradually developing offshore wind power projects.
In this work, Penglai, which is located north of Shandong and lies north of the Yellow Sea and the Bohai Sea, was chosen as the area of study.It has tremendous, potentially valuable wind

Case Study
To examine the accuracy of the novel combined algorithm, four different multi-step forecasting algorithms are compared by analyzing the three-step-ahead-prediction (half-1-h-ahead) and the six-step-ahead-prediction (1-h-ahead) of a 10-min wind speed series at three different wind power stations.

Study Area and Datasets
Shandong, located on the east coast of China, is not only one of the provinces with the largest economy, but also one of the biggest energy consumers.However, 99% of the electrical energy comes from coal power generation.As a result, Shandong faces enormous energy pressures.
However, as a coastal province, Shandong possesses one of China's largest wind farms, with an installed capacity of 58 million kilowatts.A simple map of the research area is depicted in Figure 4.With the aim of satisfying social development, achieving energy conservation, and protecting the environment, Shandong has begun developing wind power stations.Due to the area's unique geographical advantages, capacity reached 260 billion KWH in 2007.In addition, the Shandong Province Bureau of Meteorology assessment notes that the entire output of wind energy resources in Shandong province is 67 million kilowatts, which is equivalent to the installed capacity of 3.68 times the capacity of the Three Gorges Hydro-power Station (18.20 million kilowatts), which ranks in the top three.To actively build wind power green energy bases, promote wind energy development, and protect the environment, Shandong has been focusing on building large-scale wind farms in Weihai, Yantai, Dongying, Weifang, Qingdao, and other coastal areas, and is gradually developing offshore wind power projects.
resources.The specific advantages are as follows: (i) higher elevation but relatively flat hilltops, ridges, and a special terrain that has much potential as an air strip; (ii) longer cycle of efficient power generation; (iii) suitable climatic conditions that are conducive to the normal operation of wind turbines; and (iv) small diurnal and seasonal variations of wind speed, which can reduce the impact on power.In this study, the 10-min wind speed data from Penglai wind farms are used to obtain a detailed example for evaluating the performance of the proposed model.First, the wind speed data are divided into four parts according to the seasons, so that the impact of seasonal variations can be considered to increase the stability of the proposed model.Next, every seasonal wind speed dataset is divided into three parts: a training set, a validation set, and a testing set.Additionally, the noise is removed from the data by using SSA.Finally, the processed data are entered into the model and, judging from the forecasting results, we determine whether the raised algorithm can be widely employed for real-world farm use.In this study, the experiment is applied to three different sites (Site 1, 2, and 3).The above-described experiment is scientific and is used to validate the performance of the proposed model.

Performance Criteria of Forecast Accuracy
To evaluate the prediction accuracy of the raised hybrid algorithm, four indexes are applied to measure the quality of the forecasting methods: absolute error (AE), mean absolute error (MAE), root mean square error (RMSE), and mean absolute percent error (MAPE), shown in Table 4 (here N is the number of test samples, and ˆi y and i y represent the real and forecast values, respectively).Here, the absolute error (AE) and the mean absolute error (MAE) are both selected so that the level of error can be more clearly reflected.RMSE is chosen because it can easily reflect the degree of changes between the actual and forecasted value.Additionally, MAPE is chosen because of its ability to reveal the credibility of the forecasting model.Wind speed forecasting errors are related to not only the forecasting models but the selected samples.Consequently, the forecasting errors within a certain In this work, Penglai, which is located north of Shandong and lies north of the Yellow Sea and the Bohai Sea, was chosen as the area of study.It has tremendous, potentially valuable wind resources.The specific advantages are as follows: (i) higher elevation but relatively flat hilltops, ridges, and a special terrain that has much potential as an air strip; (ii) longer cycle of efficient power generation; (iii) suitable climatic conditions that are conducive to the normal operation of wind turbines; and (iv) small diurnal and seasonal variations of wind speed, which can reduce the impact on power.
In this study, the 10-min wind speed data from Penglai wind farms are used to obtain a detailed example for evaluating the performance of the proposed model.First, the wind speed data are divided into four parts according to the seasons, so that the impact of seasonal variations can be considered to increase the stability of the proposed model.Next, every seasonal wind speed dataset is divided into three parts: a training set, a validation set, and a testing set.Additionally, the noise is removed from the data by using SSA.Finally, the processed data are entered into the model and, judging from the forecasting results, we determine whether the raised algorithm can be widely employed for real-world farm use.In this study, the experiment is applied to three different sites (Site 1, 2, and 3).The above-described experiment is scientific and is used to validate the performance of the proposed model.

Performance Criteria of Forecast Accuracy
To evaluate the prediction accuracy of the raised hybrid algorithm, four indexes are applied to measure the quality of the forecasting methods: absolute error (AE), mean absolute error (MAE), root mean square error (RMSE), and mean absolute percent error (MAPE), shown in Table 4 (here N is the number of test samples, and ŷi and y i represent the real and forecast values, respectively).Here, the absolute error (AE) and the mean absolute error (MAE) are both selected so that the level of error can be more clearly reflected.RMSE is chosen because it can easily reflect the degree of changes between the actual and forecasted value.Additionally, MAPE is chosen because of its ability to reveal the credibility of the forecasting model.Wind speed forecasting errors are related to not only the forecasting models but the selected samples.Consequently, the forecasting errors within a certain scientific range can be accepted.Moreover, in order to better evaluate performance, four percentage error criterions are also applied in this study, listed in Table 5.

AE
The average forecast error of i times forecast results

MAE
The average absolute forecast error of i times forecast results

RMSE
The root average of the prediction error squares RMSE " The average of absolute error Table 5.Four metric rules.

Q AE
The percentage error of AE

Experimental Simulations
In this subsection, three single models, WNN, GRNN, and BPNN, are compared to obtain the best prediction approach.As a result, whether for half-hour (rolling three-step) or one-hour (rolling six-step) predictions, BPNN gives the best prediction accuracy (see Tables 6 and 7) of the four proposed models.Next, the APSOSA-BPNN is selected from BPNN, PSO-BPNN as the best prediction algorithm.Finally, the hybrid SSA-APSOSA-BP algorithm was proposed as our best prediction model.
The BPNN, PSO-BPNN, and APSOSA-BPNN hybrid algorithms were selected to compare the forecasting results.Three sites from the Shandong-Penglai wind farms were selected, and then a sample (the 10-min wind speed series) of every season from each site was selected and entered into the above algorithms.Next, the multi-step predicted results were displayed.The specific results of the three sites are shown in Tables 6 and 7, respectively.Additionally, the results from Site 1 are displayed in Figure 5 and the absolute errors of the three sites are shown in Tables 6 and 7. Using the four percentage error criteria, the improvement percentages between each set of algorithms are shown in Tables 8 and 9.   From Tables 6 and 7, we can see following: (a) Different forecasting algorithms have different forecasting results; (b) All the algorithms' forecasting results from the three sites are effective.Examples are included in Figure 5; (c) For different seasons at the same site, the hybrid algorithms show strong forecasting stability; (d) Among the algorithms studied, the hybrid SSA-PSOSA-BP algorithm (see in Figure 6) obtained better accuracy than the others.Moreover, to further illustrate the quality of the proposed hybrid algorithm, four percentage error criterions are used in Table 5.
It can be analyzed in detail that: (a) When comparing the hybrid PSO-BP algorithm with the single BP algorithm, we can make a conclusion that the PSO selects excellent parameters to run BP model, but the prediction accuracy of PSO-BP is increased only slightly.From Tables 6 and 7, in the spring, the three-step MAPE results of the PSO-BP and the BP are 7.2490% and 7.4083%, respectively.For the six-step, they are 9.0061% and 9.3530%, respectively.(b) When comparing the hybrid PSOSA-BP algorithm with the combined PSO-BP algorithm, the former combines the advantages of simulated annealing, and further optimizes the parameters; as a result, with respect to (a), the predicted quality rises again, but not particularly clearly.
The specific upgrade percentages are provided in Tables 8 and 9. (c) When comparing the hybrid SSA-APSOSA-BP algorithm with the hybrid PSOSA-BP algorithm, the former MAPE results are better than the latter.In other words, the forecasting quality of the new combined algorithm is better because of the higher accuracy when comparing it with the BP, PSO-BP, and PSOSA-BP algorithms.(d) The forecasting quality of the hybrid SSA-APSOSA-BP algorithm is better than that of the hybrid PSO-BP algorithm.The decreases in MAPE results in comparison with the PSO-BP and SSA-APSOSA-BP algorithm of three-step and six-step forecasts are 37.4439% and 36.9998% in Tables 8 and 9 for the spring season, respectively.(e) When comparing the hybrid SSA-APSOSA-BP algorithm with the single BP algorithm, the accuracy of the wind speed forecasting, is improved more obviously.As an example, in Table 6, the three-step forecasting MAPE results for the latter are 9.4392%, 13.8388%, 13.0224%, and 11.4632%, respectively.However, for the former, the three-step forecasting MAPE results are 6.4101%, 9.0027%, 9.7494%, and 6.7077%, respectively.(f) From (a) to (e), the reasons include: (1) The combination of the SA algorithm and the PSO algorithm has increased the forecasting ability and accuracy of the single BP algorithm effectively.(2) The SSA algorithm removes the noise signal from the original wind speed data and, due to the APSOSA algorithm, the best initial weights and thresholds are given to optimize the BP algorithm, which can lead to high-precision forecasting results.(3) The scientific and rational data selection used in this paper is also one of the paramount reasons for the outstanding performance achieved.
In different seasons at the same site, the proposed algorithms' forecasting qualities can also be different.This phenomenon indicates that wind speed can be affected by seasonal factors.In this paper, we also consider this factor, and the different seasons' forecasting results are listed in Tables 6 and 7. Tables 10 and 11 are chosen as examples, and the detailed descriptions of this phenomenon are as follows: (a) Different sites can give different results.In Table 10, for the hybrid SSA-PSOSA-BP algorithm, the MAPE results of the three-step and six-step are 8.9477% and 10.8290%, respectively, at Site 1.However, for Sites 2 and 3, they are 7.9093% and 7.0741% vs. 7.9675% and 9.5219%, respectively.
(b) In Table 11, for the hybrid SSA-APSOSA-BP algorithm in spring and winter, the MAPE of the three-step and six-step are 5.9362% and 6.8909% vs. 7.0652% and 8.8046%, respectively.However, in summer and autumn, they are 8.9720% and 10.7722% vs. 11.1259% and 12.7657%, respectively.Obviously, in spring and winter, the MAPE results are less than 9%, but in summer and autumn, the wind speed forecasting errors are all more than 10%, especially in autumn when they are close to 12%.(c) As can be clearly observed from the circumstances described above, geographical and seasonal factors must be considered in the wind speed prediction.From (b), it can be concluded that the prediction accuracy in spring and winter is better than that in summer and autumn.However, comparing with the other proposed algorithms, the forecasting errors of the hybrid algorithm are still effectively less.

Discussion
In this section, the bias-variance framework and the Diebold-Mariano (DM) are used to examine the accuracy, the stability, and the forecasting performance of the forecast models.

Bias-Variance Framework
In order to evaluate the different models' accuracy and stability, in this subsection, we utilize the bias-variance framework, which includes bias and variance.The bias-variance framework can be shown as follows: var " Erp ŷ ´yq ´Ep ŷ ´yqs 2 (9) Bias 2 " Er ŷ ´ys 2 ´Er ŷ ´Ep ŷq 2 s (10) where y and ŷ are the observed values and the forecasting values respectively.The higher the bias, the lower the forecasting accuracy.Similarly, the bigger the variance, the worse the prediction performance.The results of the models are shown in Tables 12 and 13.Obviously, the bias and variance of the proposed hybrid model are smaller than the comparison models (GRNN, WNN, APSOSA-BPNN, etc.).In other words, the developed model possesses a higher accuracy and stability in wind-speed forecasting and performs much better than the comparison models in forecasting.

Statistical Hypothesis Testing
Hypothesis testing is a basic method of statistical inference, also called confirmatory data analysis.Its basic principle can be described as below: firstly, making some assumptions, then statistical reasoning, and lastly determining whether to reject or accept the hypothesis under a level of significance that is defined beforehand [4].There are many commonly used methods of hypothesis testing such as T-test, F-test, rank and inspection, etc.In this subsection, another hypothesis testing approach is employed to assess the models' efficiency, called the Diebold-Mariano test [10].The concrete content is described as follows: where the Loss function L is the function of the prediction error, e 1 t and e 2 t are the forecasting errors of the two comparison models.
Establishing the DM Statistics: where 2π fd p0q represents a consistent estimator of the asymptotic variance, fd p0q is the zero spectral density, and N is the length of forecasting results.
Comparing the calculated DM with the Z α{2 , which can be found in the normal distribution table, the null hypothesis will be rejected if ˇˇDM ˇˇą ˇˇZ α{2 ˇˇ; this means that under the significance level α, there is a significant difference between the two models (the proposed model and the compared models including WNN, GRNN, BPNN, etc.) in terms of their prediction performance.The concrete results are shown in Tables 12 and 13.

Analysis
From Tables 12 and 13, we can see that: (a) No matter the bias or the variance, the values of the proposed model are far smaller than those of the other five models, which means that the hybrid model has a higher accuracy and stability than the other five models.(b) The smallest value of the |DM| in both tables is 12.080501, which is much larger than the Z α{2 (Z 0.005 " 2.58, Z 0.025 " 1.96); as a consequence, the null hypothesis can be rejected and the hybrid model observably outperforms the other five models.

Conclusions
With the conventional energy for electricity generation being quickly depleted, wind energy has become the most significant new type of green renewable energy available, and contains enormous power.However, due to the uncertainty of meteorological factors, it is still an extremely challenging task to forecast wind speed.In this paper, we put forward a novel hybrid SSA-APSOSA-BP model based on SSA and simulated annealing-adaptive particle swarm optimization algorithm (the specific process is given in Figure 6).From the above discussion and analysis, the conclusions are expressed as follows: (1) Among the three single prediction methods (WNN, GRNN, and BPNN), the best one is BPNN, which possesses a stronger prediction performance than the others (see Tables 6 and 7).(2) In summer and autumn, wind speed forecasting errors are larger than in another two seasons because of the more complex features of wind speed in Penglai.(3) The experimental simulations indicate that the hybrid SSA-APSOSA-BP algorithm can perform better than the other five algorithms.There is no difference between the means of the forecasting series and the real series, and the accuracy of the wind speed forecasting results can be acceptable and credible within a reasonable range.The detailed reasons are provided in the above experimental simulations Section 3.3.
Overall, the proposed hybrid model adds a new viable option for wind speed forecasting, and the excellent performance and reasonable prediction accuracy reveal that they can be employed for time series forecasting, especially for wind-speed forecasting in some cases.
Step 1: Calculate the outputs of all hidden layer nodes.Based on the input vector X, the weight ω ij , which is between the input layer and the hidden layer, and the hidden layer threshold O, compute outputs H of the whole hidden layer node in Equation (B4).S is the number of the hidden layer nodes.

Appendix B.3 Simulated Annealing (SA)
Definition 1.The main steps of simulated annealing are given as follows: Step 1: Parameter initialization.Set the initialization temperature T 0 as high as feasible and randomly generate initial solution x 0 .
Step 2: Repeat the following until equilibrium temperature is reached: Tpkq pk " 1, .., Lq pL is the number of iteration).
(1) Generating the new solution x 1 in the range of the solution X, set objective function Fpxq and calculate Fpxq and Fpx 1 q: ∆F " Fpx 1 q ´Fpxq (B12) (2) If ∆F ă 0, accept x 1 as the new solution, else accept the worse solution x 1 as the new one with the probability in Equation (B13): P " e ´∆Fpxq{KT (B13) where K is the Boltzmann Constant.
Step 3: Repeat step 2 until the declining temperature reaches zero or the pre-set temperature T.

26 Figure 1 .
Figure 1.Original wind speed series and the decomposed series by SSA.

Figure 1 .
Figure 1.Original wind speed series and the decomposed series by SSA.

Figure 2 .
Figure 2. The conceptual diagram of the Roulette Wheel Selection Strategy.Figure 2. The conceptual diagram of the Roulette Wheel Selection Strategy.

Figure 2 .
Figure 2. The conceptual diagram of the Roulette Wheel Selection Strategy.Figure 2. The conceptual diagram of the Roulette Wheel Selection Strategy.

Figure 3 .
Figure 3.The flowchart of the proposed hybrid algorithm.

Figure 3 .
Figure 3.The flowchart of the proposed hybrid algorithm.

Figure 4 .
Figure 4.The study area, Penglai in Shandong province, eastern China.

Figure 4 .
Figure 4.The study area, Penglai in Shandong province, eastern China.

Figure 5 .
Figure 5. Rolling forecasting results of different models at the three study sites.

Figure 5 .
Figure 5. Rolling forecasting results of different models at the three study sites.

Figure 6 .
Figure 6.The flowchart of the proposed model.Figure 6.The flowchart of the proposed model.

Figure 6 .
Figure 6.The flowchart of the proposed model.Figure 6.The flowchart of the proposed model.

Table 1 .
The experimental parameters of BPNN.
2.2.3.Simulated Annealing The concept of Simulated Annealing was first introduced by N. Metropolis et al. in 1953.In 1983, Kirkpatrick et al. succeeded in introducing SA in the field of combinatorial optimization [47].Based

Table 1 .
The experimental parameters of BPNN.The concept of Simulated Annealing was first introduced by N. Metropolis et al. in 1953.In 1983, Kirkpatrick et al. succeeded in introducing SA in the field of combinatorial optimization

Table 2 .
A rudimentary SA-APSO algorithm is outlined as follows.

Table 3 .
The experimental parameters of APSOSA.
Stage I: Data prepossessing.Utilize the SSA method to process the original wind speed signals; as a result, the noise signals are removed, and the real and effective signals, which are shown in Figure1, can be preserved.Lastly, the useful processed signal will be fed into the abovementioned hybrid model.Stage II: Data selection.The processed valid data from Stage I is classified into three parts: the training set, the validation set, and the test set for model training, validation, and testing, respectively.Stage III: Algorithm training and validation.

Table 6 .
Comparison of errors of rolling three-step (half an hour ahead) forecasts.

Table 7 .
Comparison of errors of rolling six-step (one hour ahead) forecasts.

Table 8 .
Improvement percentages among different forecasting models of rolling three-step (half an hour ahead) forecasts.

Table 9 .
Improvement percentages among different forecasting models of rolling six-step (one hour ahead) forecasts.

Table 10 .
Average errors of the different rolling forecasting models at the three sites.

Table 11 .
Average errors of the different rolling forecasting models in different seasons.

Table 12 .
Bias-variance and Diebold-Mariano test of different models (half an-hour ahead).

Table 13 .
Bias-variance and Diebold-Mariano test of different models (one hour ahead).