A Hybrid Seasonal Mechanism with a Chaotic Cuckoo Search Algorithm with a Support Vector Regression Model for Electric Load Forecasting

: Providing accurate electric load forecasting results plays a crucial role in daily energy management of the power supply system. Due to superior forecasting performance, the hybridizing support vector regression (SVR) model with evolutionary algorithms has received attention and deserves to continue being explored widely. The cuckoo search (CS) algorithm has the potential to contribute more satisfactory electric load forecasting results. However, the original CS algorithm suffers from its inherent drawbacks, such as parameters that require accurate setting, loss of population diversity, and easy trapping in local optima (i.e., premature convergence). Therefore, proposing some critical improvement mechanisms and employing an improved CS algorithm to determine suitable parameter combinations for an SVR model is essential. This paper proposes the SVR with chaotic cuckoo search (SVRCCS) model based on using a tent chaotic mapping function to enrich the cuckoo search space and diversify the population to avoid trapping in local optima. In addition, to deal with the cyclic nature of electric loads, a seasonal mechanism is combined with the SVRCCS model, namely giving a seasonal SVR with chaotic cuckoo search (SSVRCCS) model, to produce more accurate forecasting performances. The numerical results, tested by using the datasets from the National Electricity Market (NEM, Queensland, Australia) and the New York Independent System Operator (NYISO, NY, USA), show that the proposed SSVRCCS model outperforms other alternative models.


Introduction
Accurate electric load forecasting is important to facilitate the decision-making process for power unit commitment, economic load dispatch, power system operation and security, contingency scheduling, and so on [1,2]. As indicated in existing papers, a 1% electric load forecasting error increase would lead to a £10 million additional operational cost [3], on the contrary, decreasing forecasting errors by 1% would produce appreciable operation benefits [2]. Therefore, looking for more accurate forecasting models or applying novel intelligent algorithms to achieve satisfactory load forecasting results, to optimize the decisions of electricity supplies and load plans, to improve the efficiency of the power system operations, eventually, reduces the system risks to within a controllable range. However, due to lots of factors, such as energy policy, urban population, socio-economical activities, weather conditions, holidays, and so on [4], the electric load data display seasonality, non-linearity, and a chaotic nature, which complicates electric load forecasting work [5].

Support Vector Regression (SVR) Model
The modeling details of an SVR model are presented briefly as follows. The training data set, , is mapped into a high dimensional feature space by a non-linear mapping function, ϕ(x). Then, in the high dimensional feature space, the SVR function, f, is theoretically used to formulate the nonlinear relationships between the input training data (x i ) and the output data (y i ). This can be shown as Equation (1): where f (x) represents the forecasted values; the weight, w, and the coefficient, b, are computed along with minimizing the empirical risk, as shown in Equation (2): where Θ ε (y, f (x)) is so-called ε-insensitive loss function, as shown in Equation (3). It is used to determine the optimal hyperplane to separate the training data into two subsets with maximal distance, i.e., minimizing the training errors between these two separated training data subsets and Θ ε (y, f (x)), respectively. C is a parameter to penalize the training errors. The second term, 1 2 w T w, is then used to represent the maximal distance between mentioned two separated data subsets, meanwhile, it also determines the steepness and the flatness of f (x).
Then, the SVR modeling problem could be demonstrated as minimizing the total training errors. It is a quadratic programming problem with two slack variables, ξ and ξ * , to measure the distance between the training data values and the edge values of ε-tube. Training errors under ε are denoted as ξ * , whereas training errors above ε are denoted as ξ, as shown in Equation (4): Min w,ξ,ξ * R(w, ξ, ξ * ) = with the constraints: The solution of Equation (4) is optimized by using Lagrange multipliers, β * i , and β i , the weight vector, w, in Equation (1) is computed as Equation (5): Eventually, the SVR forecasting function is calculated as Equation (6): where K x i , x j is the so-called kernel function, and its value could be computed by the inner product of ϕ(x i ) and ϕ x j , i.e., K x i , x j = ϕ(x i ) × ϕ x j . The are several kinds of kernel function, such as Gaussian function (Equation (7)) and the polynomial kernel function. Due to its superior ability to map nonlinear data into high dimensional space, a Gaussian function is used in this paper: Therefore, determining the three parameters, σ, C, and ε of an SVR model would play the critical role to achieve more accurate forecasting performances [5,28,29]. The parameter ε decides the number of support vectors. If ε is large enough, it implies few support vectors with low forecasting accuracy; if ε has a value that is too small, it would increase the forecasting accuracy but be too complex to adopt. Parameter C, as mentioned, penalizes the training errors. If C is large enough, it would increase the forecasting accuracy but suffer from being difficult to adopt; if C has a too small value, the model would suffer from large training errors. Parameter σ represents the relationships among data and the correlations among support vectors. If σ is large enough, the correlations among support vectors are strong and we can obtain accurate forecasting results, but if the value of σ is small, the correlations among support vectors are weak, and adoption is difficult.
However, structural methods to determine the SVR parameters are lacking. Hong and his colleagues have pointed out the advanced exploration way by hybridizing chaotic mapping functions with evolutionary algorithms to overcome the embedded premature convergence problem, to select suitable parameter combination, to achieve highly accurate forecasting performances. To continue this valuable exploration, the chaotic cuckoo search algorithm, the CCS algorithm, is proposed to be hybridized with an SVR model to determine an appropriate parameter combination.

Tent Chaotic Mapping Function
The chaotic mapping function is an optimization technique to map the original data series to show sensitive dependence on the initial conditions and infinite different periodic responses (chaotic ergodicity), to maintain the diversity of population in the whole optimization procedures, to enrich the search behavior, and to avoid premature convergence. The most popular chaotic mapping function is the logistic function, however, based on the analysis on the chaotic characteristics of the different mapping functions, a tent chaotic mapping function [39] demonstrates a range of dynamical behavior ranging from predictable to chaos, i.e., with good ergodic uniformity [40]. This paper thus applies the tent chaotic mapping function to be hybridized with the CS algorithm to determine the three parameters of an SVR model.
The tent chaotic mapping function is shown as Equation (8): where x n is the iterative value of the variable x in the nth step, and n is the number of iteration steps.

Cuckoo Search (CS) Algorithm
The CS algorithm is a novel meta-heuristic optimization algorithm, inspired by cuckoo birds' obligate brood parasitic behavior of laying their eggs in the nests of other host birds. Meanwhile, by applying Lévy flight behaviors, the search speed is much faster than that of the normal random walk. Therefore, cuckoo birds can reduce the number of iterations and thus speed up the local search efficiency. For CS algorithm implementation, each egg in a nest represents a potential solution. The cuckoo birds could choose, by Lévy flight behaviors, recently-spawned nests to lay their eggs in the host nests to ensure their eggs could hatch first due to the natural phenomenon that cuckoo eggs usually hatch before the host birds' eggs. It takes times for the host birds to discover that the eggs in their nests do not belong to them, based on the probability, p a . When these "stranger" eggs are discovered, they either throw out those eggs or abandon the whole nest to build a new nest in a new location. The cuckoo birds would continuously lay new eggs (solutions), and they would choose the nest, by Lévy flight behaviors, around the current best solutions.
The CS algorithm contains three famous idealized rules [31]: (1) each cuckoo lays one egg at a time in a randomly selected host; (2) high-quality eggs and their host nests would survive to the next generation; (3) the number of available host nests is fixed, and the host bird detects the "stranger" egg with a probability p a ∈ [0, 1]. In this case, the host bird can either throw away the egg or abandon the nest, and build a completely new nest. The last rule can be approximated by a fraction (p a ) of the n host nests that are replaced by new nests (with new random solutions). The value of p a is often set as 0.25 [37].
The CS algorithm could maintain the balance between two kinds of search (random walks), the local search and the global search, by a switching parameter, p a . The switching parameter p a determines the cuckoo birds to abandon a fraction of the worst nests and build new ones for discovering new and more promising regions in the search space. These two random walks are defined by Equations (9) and (10), respectively: where x t j and x t k are current positions randomly selected; α is the positive Lévy flight step size scaling factor; s is the step size; H(·) is the Heavy-side function; δ is a random number from uniform distribution; ⊗ represents the entry-wise product of two vectors; L(s, λ) is the Lévy distribution and is used to define the step size of random walk, it is defined as Equation (11): where λ is the standard deviation of step size; the gamma function, Γ(λ), is defined as movements, it is also capable to find out the global optimum, i.e., it could ensure that the system will not be trapped in a local optimum [41].

Implementation Steps of CCS Algorithm
The procedure of the hybrid CCS algorithm with an SVR model is illustrated as followings. The relevant flowchart is shown in Figure 1.
where and are the minima and the maxima of the three parameters, respectively.
1. Initial the locations of random n nests for the three parameters 2. Map the three parameters into chaotic variables .
Evaluate the fitness value to find out the best nest position , by MAPE index.

Yes
Using global search (Eq. (10)) and Lévy flight distribution (Eq. (11)) to obtain a new set of nest positions ) 1 ( , Turn to discover the nests in with lower probability, while is lower than to a random number r

Fitness Evaluation
Step 5: Determine New Nest Position Step 6: Cuckoo Local Search Step 2: Chaotic Mapping and Transferring Apply Tent chaotic mapping function to obtain the next iteration chaotic variables ) 1 ( , to obtain three parameters for the next iteration, Step 4:

Cuckoo Global Search
Determine the new nest position, , with a better fitness value.   Step 1: Initialization.
The locations of random n nests for the three parameters of an SVR model as , k = C, σ, ε; i represents the iteration number; j represents the number of nests. Let i = 0, and normalize the parameters as chaotic variables, cx (i) k,j , within the interval [0, 1] by Equation (12): where Min k and Max k are the minima and the maxima of the three parameters, respectively.
Step 2: Chaotic Mapping and Transferring.
Apply the tent chaotic mapping function, defined as Equation (8), to obtain the next iteration of chaotic variables, cx , as shown in Equation (13): Then, transform cx (i+1) k,j to obtain three parameters for the next iteration, x (i+1) k,j , by the following Equation (14): Step 3: Fitness Evaluation.
Evaluate the fitness value with x (i+1) k,j for all nests to find out the best nest position, x k,best , in terms of smaller forecasting accuracy index value. In this paper, the forecasting error is calculated as the fitness value by the mean absolute percentage error (MAPE), as shown in Equation (15): where N is the total number of data; a i is the actual electric load value at point i; f i is the forecasted electric load value at point i.
Step 4: Cuckoo Global Search. Implement a cuckoo global search, i.e., Equation (10), by using the best nest position, x (i+1) k,best , and update other nest positions by Lévy flight distribution (Equation (11)) to obtain a new set of nest positions, then, compute the fitness value.
Step 5: Determine New Nest Position.
Compare the fitness value of the new nest positions with the fitness value of the previous iteration, and update the nest position with a better one. Then determine the new nest position as Step 6: Cuckoo Local Search.
If p a is lower than to a random number r, then turn to discover the nests in x  Step 8: Stop Criteria.
If the number of search iterations are greater than a given maximum search iterations, then, the best nest position, x (t) k,best , among the current population is determined as parameters (C, σ, ε) of an SVR model; otherwise, go back to Step 2 and continue searching the next iteration.

Seasonal Mechanism
As indicated in existing papers [5,28,29] the short term electric load data often display cyclic tendencies due to the cyclic nature of economic activities (production, transportation, operation, etc.) or the seasonal climate in Nature (air conditioners and heaters in summer and winter, respectively). It is useful to increase the forecasting accuracy by calculating these seasonal effects (or seasonal indexes) to adjust the seasonal biases. Several researchers have proposed seasonal adjustment approaches to determine the seasonal effects, such as Koc and Altinay [42], Goh and Law [43], and Wang et al. [44], who all apply regression models to decompose the seasonal component. Martens et al. [45] apply a flexible Fourier transform to estimate the daily variation of the stock exchange, and compute a seasonal estimator. Deo et al. [46] composed two Fourier transforms in a cyclic period to further identify the seasonal estimator. Comparing these seasonal adjustment models, Deo's model extends Martens's model for application to general cycle-length data, particularly for hour-based or other shorter cycle-length data. Considering that this paper deals with half-hour based short term electric load data, this paper would like to employ the seasonal mechanism proposed by Hong and his colleagues in [5,28,29]. That is, firstly apply the ARIMA model to identify the seasonal length of the target time series data set; secondly, calculate these seasonal indexes to adjust cyclic effects to receive more satisfied forecasting performances, as shown in Equation (16): where q = j, l + j, 2l + j, . . . , (m − 1)l + j with m seasonal (cyclic) periods and l seasonal length in each period. Thirdly, the seasonal index (SI) for each seasonal point j in each period is calculated as Equation (17): where j = 1,2, . . . l. The seasonal mechanism is demonstrated in Figure 2. or the seasonal climate in Nature (air conditioners and heaters in summer and winter, respectively). It is useful to increase the forecasting accuracy by calculating these seasonal effects (or seasonal indexes) to adjust the seasonal biases. Several researchers have proposed seasonal adjustment approaches to determine the seasonal effects, such as Koc and Altinay [42], Goh and Law [43], and Wang et al. [44], who all apply regression models to decompose the seasonal component. Martens et al. [45] apply a flexible Fourier transform to estimate the daily variation of the stock exchange, and compute a seasonal estimator. Deo et al. [46] composed two Fourier transforms in a cyclic period to further identify the seasonal estimator. Comparing these seasonal adjustment models, Deo's model extends Martens's model for application to general cycle-length data, particularly for hour-based or other shorter cycle-length data. Considering that this paper deals with half-hour based short term electric load data, this paper would like to employ the seasonal mechanism proposed by Hong and his colleagues in [5,28,29]. That is, firstly apply the ARIMA model to identify the seasonal length of the target time series data set; secondly, calculate these seasonal indexes to adjust cyclic effects to receive more satisfied forecasting performances, as shown in Equation (16): where q = j, l + j, 2l + j,…, (m − 1)l + j with m seasonal (cyclic) periods and l seasonal length in each period. Thirdly, the seasonal index (SI) for each seasonal point j in each period is calculated as Equation (17): where j = 1,2,…l. The seasonal mechanism is demonstrated in Figure 2.

Data Set of Numerical Examples
To demonstrate the superiorities of the tent chaotic mapping function and seasonal mechanism of the proposed SSVRCCS model, this paper uses the half-hour electric load data from the Queensland regional market of the National Electricity Market (NEM, Queensland, Australia) [47], named Example 1, and the New York Independent System Operator (NYISO, New York, NY, USA)

Data Set of Numerical Examples
To demonstrate the superiorities of the tent chaotic mapping function and seasonal mechanism of the proposed SSVRCCS model, this paper uses the half-hour electric load data from the Queensland regional market of the National Electricity Market (NEM, Queensland, Australia) [47], named Example 1, and the New York Independent System Operator (NYISO, New York, NY, USA) [48], named Example 2. The employed electric load data contains a total of 768 half-hour electric load values in Example 1, i.e., from 00:30 01 October 2017 to 00:00 17 October 2017. Based on Schalkoff's [49] recommendation that the ratio of validation data set to training data set should be approximately one to four, therefore, the electric load data set is divided into three sub-sets. The training set has 432 half-hour electric load values (i.e., from 00: During the modeling processes, in the training stage, the rolling-based procedure, proposed by Hong [28], is also applied to assist CCS algorithm to implement well searching for an appropriate parameter combination (σ, C, ε) of an SVR model. Specifically, the CCS algorithm minimizes the empirical risk, as shown in Equation (4), to obtain the potential parameter combination by employing the first n electric load data in the training set; then, it receives the first forecasted electric load by the SVR model with these potential parameter combination, i.e., the (n + 1)th forecasting electric load. For the second round, the next n electric load data, from 2nd to (n + 1)th electric load values, are then used by the SVR model to obtain new potential parameter combination, then, similarly, the (n + 2)th forecasting electric load is receive. This procedure would never be stopped till the totally 432 forecasting electric load are computed. The training error and the validation error are also calculated in each iteration.
Only with the smallest validation and testing errors, a potential parameter combination could be finalized as the determined parameter combination of an SVR model. Then, the never used testing data set would be employed to demonstrate the forecasting performances, i.e., eventually, the 192 half-hour/hourly electric load would be forecasted by the proposed SSVRCCS model.

Embedded Parameter Settings of the CCS Algorithm
The embedded parameters of CCS algorithm for modeling are set as follows: the number of host nests is set to be 50; the maximum number of iterations is set as 500; the initial probability parameter p a is set as 0.25. During the parameter optimizing process of an SVR model, the searching feasible ranges of the three parameters are set as following, σ ∈ [0.01, 5], ε ∈ [0.01, 1], and C ∈ [0.01, 60,000]. In addition, considering that the iteration time would affect the performance of each model, the given optimization time for each model with an evolutionary algorithm is set at the same inasmuch as possible.

Forecasting Accuracy Indexes
Three forecasting accuracy evaluation indexes are used to compare the forecasting performances for each model: (1) the MAPE mentioned in Equation (5); (2) the root mean square error (RMSE); and (3) the mean absolute error (MAE). The latter two indexes could be calculated by Equations (18) and (19), respectively: where N is the total number of data; a i is the actual electric load value at point i; f i is the forecasted electric load value at point i.

Forecasting Accuracy Significance Tests
To demonstrate the significant superiority of the proposed SSVRCCS model in terms of forecasting accuracy, some famous statistical tests are implemented. Based on Diebold and Mariano's [50] and Derrac et al. [51] research suggestions, the Wilcoxon signed-rank test [52] and Friedman test [53] are simultaneously applied in this paper.
The Wilcoxon signed-rank test is used to compare the significant differences in terms of central tendency between two data set with the same size. Let d i represent the i-th pair difference of the i-th forecasting errors from any two forecasting models, the differences are ranked according to their absolute values. Let r + represent the sum of ranks that the first model larger than the second one; r − represent the sum of ranks that the second model larger than the first one. In case of d j = 0, then, exclude the j-th pair and reduce sample size. The statistic W of the Wilcoxon signed-rank test is shown as Equation (20): If W meets the criterion of the Wilcoxon distribution under N degrees of freedom, then, the null hypothesis of equal performance of these two compared models cannot be accepted. It also implies that the proposed model is significantly superior to the other model. Of course, if the comparison size is larger than the critical size, the sampling distribution of W would approximate to the normal distribution instead of Wilcoxon distribution, and the associated p-value would also be provided.
On the other hand, due to the non-parametric statistical test in the ANOVA analysis procedure, the Friedman test is devoted to compare the significant differences among two or more models. The statistic F of the Friedman test is shown as Equation (21): where N is the total number of forecasting results; k is the number of compared models; R j is the average rank sum obtained in each forecasting value for each compared model as shown in Equation (22), where r j i is the rank sum from 1 (the smallest forecasting error) to k (the worst forecasting error) for ith forecasting result, for jth compared model.
Similarly, if the associated p-value of F meets the criterion of not acceptance, the null hypothesis, equal performance among all compared models, could also not be held.

Forecasting Results and Analysis for Example 1
To compare the improved forecasting performance of the tent chaotic mapping function, a SVR with the original CS algorithm (without the tent chaotic mapping function), namely the SVRCS model, will also be taken into comparison. Therefore, according to the rolling-based procedure mentioned above, by using the training data set from Example 1 (mentioned in Section 3.1) to conduct the training work, and the parameters for SVRCS and SVRCCS models are eventually determined. These trained models are further used to forecast the electric load. Then, the forecasting results and the suitable parameters of SVRCS and SVRCCS models are listed in Table 1. It is clearly indicated that the proposed SVRCCS model has achieved smaller forecasting performances in terms of the forecasting accuracy indexes, MAPE, RMSE, and MAE. As shown in Figure 3, the employed electric load data demonstrates seasonal/cyclic changing tendency in Example 1. In addition, the data recording frequency is on a half-hour basis, therefore, to comprehensively reveal the electric load changing tendency, the seasonal length is set as 48. Therefore, there are 48 seasonal indexes for the proposed SVRCCS and SVRCS models. The seasonal indexes for each half-hour are computed based on the 576 forecasting values of the SVRCCS and SVRCS models in the training (432 forecasting values) and validation (144 forecasting values) processes. The 48 seasonal indexes for the SVRCCS and SVRCS models are listed in Table 2, respectively. The forecasting comparison curves of six models, including the SARIMA (9,1,8)×(4,1,4) , GRNN (σ = 0.04), SSVRCCS, SSVRCS, SVRCCS, and SVRCS models mentioned above and actual values are shown in Figure 4. It illustrates that the proposed SSVRCCS model is closer to the actual electric load values than other compared models. To further illustrate the tendency capturing capability of the proposed SSVRCCS model during the electric peak loads, Figures 5-8 are enlargements from four peaks in Figure 4 to clearly demonstrate how closer the SSVRCCS model matches to the actual electric load values than other alternative models. For example, for each peak, the red real line (SSVRCCS model) always follows closely with the black real line (actual electric load), whether climbing up the peak or climbing down the hill.                    Table 3 illustrates the forecasting accuracy indexes for the proposed SSVRCCS model and other alternative compared models. It is clearly to see that the MAPE, RMSE, and MAE of the proposed SSVRCCS model are 0.70%, 56.90, and 40.79, respectively, which are superior to the other five alternative models. It also implies that the proposed SSVRCCS model contributes great improvements in terms of load forecasting accuracy.  Table 3 illustrates the forecasting accuracy indexes for the proposed SSVRCCS model and other alternative compared models. It is clearly to see that the MAPE, RMSE, and MAE of the proposed SSVRCCS model are 0.70%, 56.90, and 40.79, respectively, which are superior to the other five alternative models. It also implies that the proposed SSVRCCS model contributes great improvements in terms of load forecasting accuracy. Finally, to ensure the significant contribution in terms of forecasting accuracy improvement for the proposed SSVRCCS model, the Wilcoxon signed-rank test and the Friedman test are conducted. Where Wilcoxon signed-rank test is implemented under two significance levels, α = 0.025 and α = 0.05, by two-tail test; the Friedman test is then implemented under only one significance level, α = 0.05. The test results in Table 4 show that the proposed SSVRCCS model almost reaches a significance level in terms of forecasting performance than other alternative compared models.

Forecasting Results and Analysis for Example 2
Similar to Example 1, SVRCS and SVRCCS models are also trained based on the rolling-based procedure by using the training data set from Example 2 (mentioned in Section 3.1). The forecasting results and the suitable parameters of SVRCS and SVRCCS models are shown in Table 5. It is also obviously that the proposed SVRCCS model has achieved a smaller forecasting performance in terms of forecasting accuracy indexes, MAPE, RMSE, and MAE.  Figure 9 also demonstrates the seasonal/cyclic changing tendency from the used electric load data in Example 2. Based on the hourly recording frequency, to completely address the changing tendency of the employed data, the seasonal length is set as 24. Therefore, there are 24 seasonal indexes for the proposed SVRCCS and SVRCS models. The seasonal indexes for each hour are computed based on the 576 forecasting values of the SVRCCS and SVRCS models in the training (432 forecasting values) and validation (144 forecasting values) processes. The 24 seasonal indexes for the SVRCCS and SVRCS models are listed in Table 6, respectively. data in Example 2. Based on the hourly recording frequency, to completely address the changing tendency of the employed data, the seasonal length is set as 24. Therefore, there are 24 seasonal indexes for the proposed SVRCCS and SVRCS models. The seasonal indexes for each hour are computed based on the 576 forecasting values of the SVRCCS and SVRCS models in the training (432 forecasting values) and validation (144 forecasting values) processes. The 24 seasonal indexes for the SVRCCS and SVRCS models are listed in Table 6, respectively.   The forecasting comparison curves of six models in Example 2, including SARIMA (9,1,10)×(4,1,4) , GRNN (σ = 0.07), SSVRCCS, SSVRCS, SVRCCS, and SVRCS models and actual values are shown as in Figure 10. It indicates that the proposed SSVRCCS model is closer to the actual electric load values than the other compared models. Similarly, the enlarged figures, Figures 11-14, from eight peaks in Figure 10 are provided to demonstrate the tendency capturing capability of the proposed SSVRCCS model and how closer the SSVRCCS model matches the actual electric load values than other alternative models. It is clear that for each peak, the red real line (SSVRCCS model) always follows closely with the black real line (actual electric load), whether climbing up the peak or climbing down the hill. in Figure 10. It indicates that the proposed SSVRCCS model is closer to the actual electric load values than the other compared models. Similarly, the enlarged figures, Figures 11-14, from eight peaks in Figure 10 are provided to demonstrate the tendency capturing capability of the proposed SSVRCCS model and how closer the SSVRCCS model matches the actual electric load values than other alternative models. It is clear that for each peak, the red real line (SSVRCCS model) always follows closely with the black real line (actual electric load), whether climbing up the peak or climbing down the hill.            For comparison with other alternative models, Table 7 demonstrates the forecasting accuracy indexes for each compared model. Obviously, the proposed SSVRCCS model almost achieves the smallest index values in terms of the MAPE (0.46%), RMSE (126.10), and MAE (80.85), respectively. It is superior to the other five compared models. Once again, it indicates that the proposed SSVRCCS model could produce more accurate forecasting performances. Finally, two statistical tests are also conducted to ensure the significant contribution in terms of forecasting accuracy improvement for the proposed SSVRCCS model. The test results are illustrated in Table 8 that the proposed SSVRCCS model almost reaches significance level in terms of forecasting performance than other alternative compared models.  For comparison with other alternative models, Table 7 demonstrates the forecasting accuracy indexes for each compared model. Obviously, the proposed SSVRCCS model almost achieves the smallest index values in terms of the MAPE (0.46%), RMSE (126.10), and MAE (80.85), respectively. It is superior to the other five compared models. Once again, it indicates that the proposed SSVRCCS model could produce more accurate forecasting performances. Finally, two statistical tests are also conducted to ensure the significant contribution in terms of forecasting accuracy improvement for the proposed SSVRCCS model. The test results are illustrated in Table 8 that the proposed SSVRCCS model almost reaches significance level in terms of forecasting performance than other alternative compared models.

Discussions
To learn about the effects of the tent chaotic mapping function in both Examples 1 and 2, comparing the forecasting performances (the values of MAPE, RMSE, and MAE in Tables 3 and 7) between SVRCS and SVRCCS models, the forecasting accuracy of SVRCCS model is superior to that of SVRCS model. It reveals that the CCS algorithm could determine more appropriate parameter combinations for an SVR model by introducing the tent chaotic mapping function to enrich the cuckoo search space and the diversity of the population when the CS algorithm is going to be trapped in the local optima. In Example 1, as shown in Table 1 Table 5, the CCS algorithm also helps to improve the result by 1.12% (=3.42% − 2.30%). These two examples both reveal the great contributions from the tent chaotic mapping function. In future research, it would be worth applying another chaotic mapping function to help to avoid trapping into local optima.
Furthermore, the seasonal mechanism can successfully help to deal with the seasonal/cyclic tendency changes of the electric load data to improve the forecasting accuracy, by determining seasonal length and calculating associate seasonal indexes (per half-hour for Example 1, and per hour for Example 2) from training and validation stages for each seasonal point. In this paper, authors hybridize the seasonal mechanism with SVRCS and SVRCCS models, namely SSVRCS and SSVRCCS models, respectively, by using their associate seasonal indexes, as shown in Tables 2 and 6, respectively. Based on these seasonal indexes, the forecasting results (in terms of MAPE) of the SVRCS and SVRCCS models for Example 1 are further revised from 2.63% and 1.51%, respectively, to achieve more acceptable forecasting accuracy, 0.99% and 0.70%, respectively. They almost improve 1.64% (=2.63% − 0.99%) and 0.81% (=1.51% − 0.70%) forecasting accuracy by applying seasonal mechanism. The same in Example 2, as shown in Table 7, the seasonal mechanism also improves 2.56% (=3.42% − 0.86%) and 1.84% (=2.30% − 0.46%) for SVRCS and SVRCCS models, respectively. In the meanwhile, based on Wilcoxon signed-rank test and Friedman test, as shown in Tables 4 and 8 for Examples 1 and 2, respectively, the SSVRCCS models also achieve statistical significance among other alternative models. Based on above discussions, this seasonal mechanism is also a considerable contribution, and it is worth the time cost to deal with the seasonal/cyclic information during modeling processes.
Therefore, it could be remarked that by hybridizing novel intelligent technologies, such as chaotic mapping functions, advanced searching mechanism, seasonal mechanism, and so on, to overcome some inherent drawbacks of the existing evolutionary algorithms could significantly improve forecasting accuracy. This kind of research paradigm also inspires some interesting future research.

Conclusions
This paper proposes a novel SVR-based hybrid electric load forecasting model, by hybridizing the seasonal mechanism, the tent chaotic mapping function, and the CS algorithm with an SVR model, namely the SSVRCCS model. The experimental results indicate that the proposed SSVRCCS model significantly outperforms other alternative compared forecasting models. This paper continues to overcome some inherent shortcomings of the CS algorithm, by actions such as enriching the search space and the diversity of the population by using the tent chaotic mapping function to avoid premature convergence problems and applying seasonal mechanism to provide useful adjustments caused from seasonal/cyclic effects of the employed data set. Eventually, the proposed SSVRCCS model achieves significant accurate forecasting performances.
This paper concludes some important findings. Firstly, by applying appropriate chaotic mapping functions it could help empower the search variables to possess ergodicity characteristics, to enrich the searching space, then, determine well appropriate parameter combinations of an SVR model, to eventually improve the forecasting accuracy. Therefore, any novel hybridizations of existed evolutionary algorithms with other optimization methods or mechanisms which could consider those actions mentioned above during modeling process are all deserving to take a trial to achieve more interesting results. Secondly, only hybridizing different single evolutionary algorithm with an SVR model could contribute minor forecasting accuracy improvements. It is more worthwhile to hybridize different novel intelligent technologies with single evolutionary algorithms to achieve more high forecasting accurate levels. This could be an interesting future research tendency in the SVR-based electric load forecasting field.