WANG Zilong()， XIA Chenxia()

College of Economics and Management， Nanjing University of Aeronautics and Astronautics， Nanjing 211106， China

Abstract： Electricity is the guarantee of economic development and daily life. Thus， accurate monthly electricity consumption forecasting can provide reliable guidance for power construction planning. In this paper， a hybrid model in combination of least squares support vector machine(LSSVM) model with fruit fly optimization algorithm(FOA) and the seasonal index adjustment is constructed to predict monthly electricity consumption. The monthly electricity consumption demonstrates a nonlinear characteristic and seasonal tendency. The LSSVM has a good fit for nonlinear data， so it has been widely applied to handling nonlinear time series prediction. However， there is no unified selection method for key parameters and no unified method to deal with the effect of seasonal tendency. Therefore， the FOA was hybridized with the LSSVM and the seasonal index adjustment to solve this problem. In order to evaluate the forecasting performance of hybrid model， two samples of monthly electricity consumption of China and the United States were employed， besides several different models were applied to forecast the two empirical time series. The results of the two samples all show that, for seasonal data， the adjusted model with seasonal indexes has better forecasting performance. The forecasting performance is better than the models without seasonal indexes. The fruit fly optimized LSSVM model outperforms other alternative models. In other words， the proposed hybrid model is a feasible method for the electricity consumption forecasting.

Key words： forecasting； fruit fly optimization algorithm(FOA)； least squares support vector machine(LSSVM)； seasonal index

Introduction

The electric power industry is the important basic industry in the energy field. Particularly， it is the essential pillar industry for the economic development[1]. Electricity consumption is vital because of the rapid development of electric power industry and the deepening of electric power system. The improvement of prediction precision is of great significance for making economic power generation plan and ensuring daily electricity consumption. The forecasting methods of seasonal data mainly include Holt-Winters model[2]， autoregressive moving average model[3]， regression model[4]，etc. The time series of monthly electricity consumption is a nonlinear time series containing both dynamic trend and stochastic fluctuation. At the same time， it is also affected by seasonality. Therefore， it is difficult to grasp the features of time series， and it is more difficult to predict. The linear models above have poor ability to fit nonlinear data， which can cause inaccurate results.

A lot of efforts have been placed on the nonlinear models such as neural network model[5-6]， least squares support vector machine(LSSVM)[7]， back propagation neural network(BPNN) model[8]and grey model[9]. However， the neural network model have several shortcomings， such as too dependent on the initial value， requiring large training sample， risk of over-fitting， and slow convergence rate. Also， the grey model has some advantages that data dispersion should not be too large. In recent years， the LSSVM model is proposed to be used in forecasting. It has the following advantages. It can select different kernel functions according to different nonlinear functions； it has a tunable parameter to deal with data defects， and has a higher global optimization feature. Nevertheless， inappropriate regularization parameters or kernel parameters of LSSVM may lead to over-learning or under-learning， which affect prediction accuracy.

In recent years， the application of intelligent optimization algorithms including genetic algorithm[10]， ant colony optimization algorithm[11]， and particle swarm optimization algorithm[12-13]is becoming much common. These algorithms can not only be used to optimize the LSSVM model， but also be employed to search the suitable parameters of LSSVM. Pan proposed a fruit fly optimization algorithm(FOA) for seeking global optimization[14]， which is widely used in knapsack problem[15]， continuous function optimization[16]， semiconductor scheduling[17]， medical problem[18]，etc. This optimization algorithm has the advantages of simple calculation process， easy to convert its ideas into program codes， easy to be understood and fast convergence speed. In order to predict annual power load， Huetal. combined FOA and generalized regression neural network(GRNN) to search the optimal spread value[19]. The empirical results showed that the GRNN model optimized by FOA had better prediction performance. Guoetal. proposed an improved FOA model[20]， and the simulation results indicated that the proposed model was effective and practical for electricity price forecasting， which could improve the electricity price forecasting accuracy.

As electricity consumption is also affected by seasonality， not considering seasonal factors will lead to large error values between the predicted results and the actual values. Therefore， the treatment of seasonal variables of electricity consumption is crucial. Traditional seasonal adjustment methods include seasonal autoregressive moving average model(SARIMA)， X-11-ARIMA and X-12-ARIMA models， which are often used for seasonal adjustment and prediction of seasonal time series， but these models require presupposition data forms. Therefore， first， it is impossible to know the real formation process of the data and to reflect the true nature of the data. Second， the differencing processing is effective to eliminate seasonality for linear trend， but it is limited for a nonlinear trend.

The FOA is applied to searching the regularization parameters and kernel parameters of LSSVM， and then the seasonal index is employed to deal with the seasonal tread and dedicated to improve the forecast performance. The proposed model of LSSVM combined with FOA and seasonal index adjustment (SFOALSSVM) has less application in electricity consumption. In order to verify the prediction effect of the proposed model， the following models are constructed to test the prediction results. They are GRNN combined with FOA with seasonal index adjustment (SFOAGRNN)， LSSVM combined with particle swarm optimization algorithm with seasonal index adjustment (SPSOLSSVM)， generalized regression neural network optimized by FOA (FOAGRNN)， least squares support vector machine optimized by particle swarm optimization algorithm (PSOLSSVM)， least squares support vector machine optimized by FOA (FOALSSVM)， and LSSVM.

A new model， SFOALSSVM， is constructed by combining FOA with LSSVM and seasonal index adjustment， which has a certain theoretical significance. The application of this method to monthly electricity consumption prediction is a new prediction method， which provides information for power planning policy， so it is of practical significance. The rest of this paper is organized as follows. Section 1 introduces the basic model briefly， including LSSVM， FOA， LSSVM optimized by FOA， and seasonal index adjustment. Other intelligent optimization models with and without seasonal index adjustment are constructed in section 2. Electricity consumption data of China and the United States are applied to testing the prediction performance of SFOALSSVM model and other models. Section 3 draws the conclusions.

Many researchers applied LSSSVM model in prediction. Shayeghietal. presented a novel hybrid algorithm which consists of generalized mutual information， wavelet packet transform， LSSVM and a modified artificial bee colony algorithm to forecast day-ahead prices[21]. Lietal. proposed an LSSVM model with parameter optimization in the short-term wind speed forecast[22]. Razaketal. developed a multistage optimization for hybrid LSSVM and genetic algorithm model to predict electricity price[23]. Yeetal. presented a forecasting model which combines modified ensemble empirical mode decomposition and LSSVM to forecast hunting instability state of high-speed train bogie[24]. Yang and Wang set up immune particle swarm optimized LSSVM to predict traffic safety[25]. Yang and Xue proposed a combination model of fuzzy C mean clustering and LSSVM to forecast load[26]. Geng and Chen proposed a railway freight volume forecasting method based on FOA and mixed-kernel LSSVM[27].

There are many researches on the application of LSSVM model in prediction， but few literatures have used FOA to optimize LSSVM model and few researches apply the LSSVM model to the electricity consumption. Few literatures apply the FOA optimized LSSVM to the electricity consumption prediction. What’s more， monthly electricity consumption is seasonal， and few literatures combine the three methods and apply it to electricity consumption.

1 Methodology

1.1 LSSVM

The support vector machine(SVM) is a nonlinear mapping that maps the data of the input space to a high dimensional feature space and solves the optimal regression function in the high dimensional feature space. The actual problem is transformed into a quadratic programming problem with inequality constraints. The LSSVM is an improved algorithm of SVM. Suykens proposed the LSSVM， in which the quadratic square of the training error is used to replace the relaxation variable in the optimization object， and the equality constraint is used to replace the inequality constraint[28]. The problem is transformed into a linear matrix solution problem， avoiding the time-consuming quadratic programming problem.

f(x)=wTφ(x)+b.

(1)

According to the principle of structural risk minimization， inequality constraints in support vector machines are replaced with equality constraints， so the objective function is transformed into

(2)

whereJis the optimization objective function andCis the regularization parameter.

By introducing the Lagrange multiplier， it is obtained that

(3)

whereα=[α1，α2， …，αl]Tis the Lagrange multiplier.

The partial derivation is obtained as

(4)

Thus， the optimization problem is transformed into solving the following linear equations

(5)

where，Q=[1， 1， …， 1]T，y=y1，y2， …，ylT，α=α1，α2， …，αmT，Iis a unit matrix， andΩi， l=φ(xi)Tφ(xj)，i，j=1， 2， …，l.

The regression estimation function of SLLVM is

(6)

Radial basis function(RBF) can reduce the number of parameters in the training process and improve the convergence rate. In this paper， the radial basis function is chosen as the kernel function. The expression of RBF function is

(7)

whereσis kernel parameter.

1.2 Fundamentals of FOA

FOA was proposed by Pan[28]to search a global optimum solution based on the food foraging behavior of fruit fly， which is superior to other species in terms of sight and smell. The olfactory organ helps them to collect smell floating in the air， and the keen vision organ makes them fly in the direction with companions after finding the location of food. The global search of the population is combined with the information exchange of individual in this algorithm. On the basis of global search， the global optimal solution is updated through the exchange of individual information. Finally， the algorithm is terminated under the condition of maximum iteration times or satisfying the precision of the convergence target. Figure 1 shows the food finding iterative process of fruit fly swarm.

Fig. 1 Food finding iterative process of fruit fly swarm

The procedure steps of FOA can be described as follows[9].

(1)Parameters initialization. The parameters include populationSp， the maximum iteration numberMg.

(2) Random direction and distance. The random direction and distance are initialized for searching food by using the sense of smell by individual fruit fly，RVis the search distance，i.e. an iterative step value：

(8)

(3) Preliminary calculations. Firstly， the distance between the estimate and the originDistishould be estimated. And then the smell concentration judgment value (Si) needs to be calculated， andSiis the reciprocal of distance：

(9)

(4) Smell concentration determination function. In order to calculate the smell concentration(Sl) of the location of fruit fly individual， the smell concentration determination value is substituted into the smell concentration determination function(Ff)：

Sl=Ff(Si).

(10)

(5) Search for the best smell concentration. The fruit fly with the lowest smell concentration in this fruit fly population is the best individual：

bestSmellbestIndex=min(Smelli).

(11)

(6) Retain the best smell concentration. Retain the best taste concentrationbestSmelland coordinatesXandY， where the fruit fly population flies out of the position with vision：

(12)

(7) Repeat steps (2)-(6) and determine whether the taste concentration is superior to the previous taste concentration. And determine whether the number of iterations is less than the maximum number of iterationsMg， if step (6) is executed， otherwise the algorithm will terminate.

1.3 LSSVM with FOA

Regularization parameters and kernel parameters in LSSVM are needed to be optimized by FOA， and the specific optimization steps are as follows[30].

(1) Initialize parameter. The maximum number of iterations， population size， initial position of fruit fly(Xa，Ya)， the range of random flight distance(FR) are determined. The population position of fruit fly is initialized at random. Each location value corresponds to a set of parameters (C，σ) of the LSSVM model. The initial position of fruit fly individual is initialized according to the variation range of parameters.

(2) Evolution starting. Setgen=0， and the random flight direction and distance of fruit fly are given.

X(i，：)=Xa+rand(FR)，
Y(i，：)=Ya+rand(FR).

(13)

(3) Preliminary calculations. Calculate the distance(Disti) between the fruit fly(Flyi) and the initial position， thus the smell concentration judgmet value(Si) is got.

(14)

(4) InputSito the LSSVM prediction model. SetC=20*S(i， 1)，σ=S(i， 2)， and the parameters of the LSSVM are represented by [S(i， 1)，S(i， 2)]. The flavor concentration functionSmelli， the fitness function， is expressed by mean square error.

(15)

(5) The lowest smell concentration of fruit fly in population is found.

min(Sl(i，：))=[(bestSmell，bestIndex)].

(6) Keeping the optimum concentration and coordinates， fruit fly flies to this position with vision.

X=X(bestIndex)，Y=Y(bestIndex)，

Smellbest=bestSmell.

(16)

Generate descendants. Setgen=gen+1. The offspring are generated by steps (1)-(5)， and the offspring are inputted into the LSSVM to calculate the smell concentration value.

(7) The iteration calculation stops. Whengenreaches the maximum number of iterations， the stopping criterion is satisfied and the optimal parameters of the LSSVM model are obtained. Otherwise， return to step 2.

1.4 Seasonal adjustment

The monthly or quarterly time series affected by seasonality will show regular periodic changes in the normal year， which will have a certain impact on the prediction of time series. Seasonal index is to modify the seasonality in the time series to reduce the prediction error value. Many scholars make seasonal adjustment through different methods. In this paper， the method in Ref.[31] is used for seasonal adjustment.

(17)

where，t=j，l+j， 2l+j， …，(m-1)l+j, and is the same time point for each period;atis the actual value， andftis the predicted value. The seasonal index at each pointSIjcan be represented by the following formula：

(18)

where，j=1， 2， …，l. According to the seasonal index， the predicted value will be adjusted to

fN+K=

(19)

where，k=1， 2， …，l, and is the time point of the predicted value.

2 Prediction and Result Analysis of Monthly Electricity Consumption in China

In this paper， the monthly China’s electricity consumption from January 2010 to December 2016 and the monthly electricity consumption of the United States from January 2008 to December 2016 are selected as the research samples to verify the validity of SFOALSSVM model. Since the two sample data will be predicted by same models， the forecasting process of China’s electricity consumption is described in detail in this paper.

2.1 Data set

China is the benchmark country and the data on electricity consumption are obtained from the China Energy Statistical Yearbook and cover the period from January 2010 to December 2016， in which 72 data points(from January 2010 to December 2015) are used as the training set， and the data points from January 2016 to December 2016(12 data points) are used as the test set to verify the prediction performance of the established model. The development trend of China’s electricity consumption is shown in Fig. 2(the picture is drawn by the author according to the data of electricity consumption).

Fig. 2 Development trend of electricity consumption in China

From Fig. 2， it can be seen that China’s electricity consumption shows a growing trend， and has obvious seasonal and nonlinear characteristics. In this paper， SFOALSSVM， SPSOLSSVM， SFOAGRNN， LSSVM， FOAGRNN， FOALSSVM， and PSOLSSVM are used for prediction. The characteristics of this study are：(1) the influence of different intelligent optimization methods on prediction accuracy is examined. SPSOLSSVM and LSSVM are used to predict the sample data set， and the prediction results are compared with the predicted results of the model SFOALSSVM；(2) the effect of seasonal indexes on the prediction performance of seasonal data is verified. Non-seasonal adjustment models, such as FOAGRNN， FOALSSVM and PSOLSSVM are adopted to predict the sample data set and the results are compared with the prediction result of the corresponding seasonal adjustment model；(3) the prediction accuracy of the generalized neural network and support vector machine optimized by seasonal adjustment is studied. The prediction result of SFOALSSVM model is compared with the result of SFOAGRNN， and the prediction accuracy of generalized neural network optimized by FOA and support vector machine with seasonal adjustment is studied.

2.2 Prediction of electricity consumption based on SFOALSSVM model

2.2.1ParameterselectionofFOAalgorithm

(1) In order to improve the prediction accuracy and the iterative speed， the original data are normalized. In this paper， the following formula is applied to normalizing the original data to [0， 1].

(20)

whereximaxandximinrepresent the maximum and minimum values in the original data， respectively.

(2) In this paper， the training data(from January 2010 to December 2015) is divided into inflow subset and outflow subset， and the rolling prediction is used in the forecasting. In this paper， twenty-four data points(from January 2010 to December 2011) are selected as inflow subset to minimize the training error， and a one-step-ahead predicted electricity consumption is gained. And then the next twenty-four data points(from February 2010 to January 2012) are selected as a new inflow subset to forecast electricity consumption in February 2012[32]， and the second one-step-ahead forecasted electricity consumption is received. The rolling-based prediction procedure is repeated till the electricity consumption data in the training set are all obtained. The most appropriate parameter values are chosen as the adjusted parameters where the minimum training error points occur.

(3) Forecasting of monthly electricity consumption in China. Through previous research and repeated verification， the FOA parameters are set toMg=100，Sp=10，LR=[0， 1]，F(xiàn)R=[-10， 10]. Because RBF function has strong generalization ability， RBF is selected as kernel function in this paper. The fruit fly swarm flying route and the iterativeMSEtrend of the FOALSSVM model searching of optimization parameters after 100 times iterative are shown in Fig. 3. It can be seen that the convergence stopped in generation 87 with theMSEvalue at 0.041. According to the results of FOA， the optimum kernel parameterσand regularization parameterCof LSSVM are 1.202 2 and 36.805， respectively.

2.2.2SeasonalindexofFOALSSVM

Since China’s monthly electricity consumption has its own regularity among the months of each year， a

(a) Fruit fly swarm flying route

(b) Convergence of MSE after iterative adjustment

Fig. 3 Trace of fruit fly search and the curve of minimal value of iterative search function

seasonal training period of 12 is set in this paper. Monthly seasonal indexes calculated by the forecasting value and the actual value are shown in Table 1. The seasonal indexes (greater than 1) indicate that the predicted values of FOALSSVM are overvalued. On the contrary， the seasonal indexes (less than 1) mean that the predicted values of FOALSSVM are undervalued. The predicted values of the test set are adjusted according to the seasonal indexes， so that the predicted values are closer to the actual values. In other words， the final predicted values of SFOALSSVM can be obtained by adjusting the predicted values of FOALSSVM with seasonal indexes.

Table1Seasonal indexes for each month of the

SFOALSSVM model

MonthSeasonal indexMonthSeasonal indexJanuary0.982July1.100February0.839August1.118March1.004September1.009April0.991October1.010May1.011November1.030June1.026December1.063

2.3 Parameter optimization and prediction of other models

In the FOAGRNN model， the FOA parameters are set toMg=100，Sp=10，LR=[0， 1]，F(xiàn)R=[-10， 10]， and the spread value is set to[0.01， 1]. The convergence can be got in generation 62 with theMSEvalue at 0.103， and the best spread value is 0.151 at coordinate(-0.829， 5.550). In the PSOLSSVM model， the particle swarm optimization algorithm parameters are set toc1=c2=1，Mg=100，Sp=10，C=[0.1， 1 000]， andσ=[0.01， 400]. The best parameter valueC=19.047 andσ=9.586 are obtained through particle swarm optimization. In LSSVM， the optimum parameter valuesC=58.241 andσ=1.375 are obtained by grid method.

2.4 Forecasting results and discussion

(21)

(22)

TheRMSEandMAPEvalues of the eight models employed in this paper are shown in Table 2.

As shown in Table 2， the results are as follows.(1) The error values of hybrid model adjusted by seasonal indexes are smaller than those of the corresponding hybrid model without seasonal indexes adjustment. TheMAPEandRMSEvalues of SFOALSSVM， SPSOLSSVM and SFOAGRNN models are(2.740%， 194.569)，(2.855%， 163.471)， (5.570%， 348.380) respectively， which are obviously smaller than those values without seasonal

Table 2 Error values of monthly electricity consumption forecasting result in China

indexes adjustment， indicating that seasonal adjustment helps to narrow the deviations of the predicted values and has a great improvement in the prediction accuracy of seasonal data.(2) The prediction accuracy of the model optimized by intelligent algorithm is higher. The error values of both FOALSSVM and PSOLSSVM models are smaller than those of LSSVM model， which indicate that the intelligent optimization method is superior to the grid optimization method. Moreover， the error value of FOALSSVM model is smaller than that of PSOLSSVM model. (3) The prediction performances of SFOALSSVM model and SPSOLSSVM model need further study. TheMAPEvalue of SFOALSSVM model is 2.740%， which is less than that of SPSOLSSVM model (2.855%). However， theRMSEvalue of SFOALSSVM is 194.569， which is larger than that of SPSOLSSVM， indicating that the prediction performances of SFOALSSVM and SPSOLSSVM need further study.(4) The prediction ability of LSSVM model is better than that of FOAGRNN model. It shows that FOAGRNN has poor prediction performance for small samples and is suitable for large sample.

The prediction performances of the three models adjusted by seasonal indexes are shown in Fig. 4.

Fig. 4 Forecasting results of SFOALSSVM， SPSOLSSVM and SFOAGRNN models

2.5 Prediction and result analysis of monthly electricity consumption in the United States

In this part， the United States is the benchmark country， and the data on monthly electricity consumption are selected from the United States Energy Information Administration between January 2008 and December 2016. The monthly electricity consumption data come from January 2008 to December 2015 are applied as a training set and those data from January 2016 to December 2016 are employed as a test set. The trend of electricity consumption in the United States is shown in Fig. 5(the picture is drawn by the author according to the data of electricity consumption).

The parameters of the FOA are set toMg=200，Sp=10，LR=[0， 1]， andFR=[-10， 10]， and RBF is selected as kernel function. The fruit fly swarm flying route and the iterativeMSEtrend of the FOALSSVM model searing after 200 times are shown in Fig. 6. The convergence can be seen in generation 118 with theMSEvalue at 0.049 96. According to the results of FOA， the optimum kernel parameterσand regularization parameterCof LSSVM are 2.626 and 22.162, respectively.

Fig. 5 Trend of electricity consumption in the United States

In the PSOLSSVM model， the optimal parameterCis 316.948 andσis 61.260 whenMSEis 0.845. In the FOAGRNN model， the optimal spread value is 0.160 when theRMSEreaches 0.067 and the coordinate is (0.234， -2.173). The grid optimization method is applied in the LSSVM model， and the optimal parametersCandσare 8.789 and 1.946 respectively. TheRMSEandMAPEvalues of the eight models employed in this paper are shown in Table 3.

(a) Fruit fly swarm flying route

(b) Convergence of MSE after iterative adjustment

Fig. 6 Trace of fruit fly search and the curve of minimal value of iterative search function of electricity consumption in the United States

(1) The error values of the SFOALSSVM model are less than the other hybrid models of seasonal adjustment. TheMAPEandRMSEerror values of SFOALSSVM model

Table 3 Error values of monthly electricity consumption forecasting results in the United States

are smaller than the error values of SPSOLSSVM and SFOAGRNN models， which indicate that the prediction performance of SFOALSSVM is superior to other models.(2) The error values of hybrid model adjusted by seasonal indexes are smaller than those of the corresponding hybrid model without seasonal indexes adjustment. TheMAPEandRMSEvalues of the SFOALSSVM， SPSOLSSVM and SFOAGRNN models are(4.24%， 162.599)，(4.79%， 170.457) and(5.46%， 224.41)， which are obviously smaller than the corresponding models without seasonal index adjustment. It shows that the seasonal index adjustment can greatly improve the prediction accuracy of seasonal data and help to narrow the deviation of the predicted values.(3) The prediction accuracy of the model optimized by intelligent algorithm is higher. Both the LSSVM models optimized by FOA and the LSSVM models optimized by PSO are smaller than those of the LSSVM， which indicates that parameter optimization method of FOA and PSO can improve the prediction accuracy of LSSVM model. The error value of FOALSSVM without seasonal index judgment is slightly larger than that of PSOLSSVM model， which is contrary to the result of the study above. The findings show that the optimization performance of these two models need further study.(4) The prediction accuracy of the hybrid model based on GRNN is low. The prediction accuracy of hybrid model based on GRNN is the lowest among the eight models. Because GRNN has complex input layer， pattern layer， sum layer and output layer， it needs a large number of input samples to determine the parameters of each layer. However， rolling prediction is applied in this method and the inflow data are 24， whether the input sample will improve the prediction accuracy needs further study.

In general， the FOA optimization method has some advantages. SFOALSSVM model can improve the prediction accuracy of seasonal data such as electricity consumption， and has a certain application prospect.

3 Conclusions

The forecasting of electricity consumption is helpful for government departments to formulate power policy， and the improvement of prediction accuracy of electricity consumption is helpful to provide reliable policy basis for power planning. The main works of this paper are as follows. Firstly， the FOA is used to select the parameters of LSSVM. Secondly， the electricity consumption of China and United States are taken as sample data. And then， seasonal indexes are employed to adjust the predicted results. At last， in order to compare the prediction performance of SFOALSSVM， the influence of different optimization methods and seasonal index adjustment on prediction accuracy is detected by using SPSOLSSVM， SFOAGRNN， LSSVM， FOAGRNN， FOALSSVM and PSOLSSVM models.

By predicting the two sample data， the following conclusions are drawn.

Firstly， SFOALSSVM model has the highest prediction accuracy for nonlinear and seasonal data. TheRMSEof the electricity consumption in China is 194.569 and theMAPEis 2.74%， in addition， theRMSEof electricity consumption in the United States. is 162.599 and theMAPEis 4.24%. The result indicates that the SFOALSSVM model is superior to other models used in this paper. The sample data of China’s electricity consumption are not larger than those of the United States， so the prediction error values of SFOALSSVM model and SPSOLSSVM model are small. However， when the amount of sample data are increased in the prediction of electricity consumption in the United States， the prediction error value of the SFOALSSVM model is obviously smaller than those of the SPSOLSSVM model.

Secondly， the prediction error values of seasonal index adjustment models are smaller than that of non-seasonal adjustment models. Seasonal index adjustment of seasonal data can reduce the prediction error value. Taking the value ofMAPEas an example， in China’s energy consumption， theMAPEof the seasonal index adjustment models SFOALSSVM， SPSOLSSVM and SFOAGRNN are 2.740%， 2.855% and 5.570% respectively， and theMAPEof the non-seasonal index adjustment models FOALSSVM， PSOLSSVM and FOAGRNN are 5.29%， 5.80% and 7.42% respectively. In energy consumption of the United States， theMAPEof the seasonal index adjustment models SFOALSSVM， SPSOLSSVM and SFOAGRNN are 4.24%， 4.79% and 5.46% respectively， while theMAPEof the non-seasonal adjustment models FOALSSVM， PSOLSSVM and the FOAGRNN model are 5.47%， 5.27% and 7.33%， respectively. With the above error values， we can see that the error values of seasonal adjustment models are smaller than those of the corresponding non-seasonal index adjustment models.

Finally， intelligent parameter optimization method improves the accuracy of prediction precision. The error values of FOALSSVM and PSOLSSVM models are smaller than those of the LSSVM model. In China’s electricity consumption， theMAPEerror values of the FOALSSVM and PSOLSSVM models are 5.29% and 5.80% respectively， while the error value of the LSSVM model is 6.33%. In the electricity consumption of the United State， theMAPEerror value of FOALSSVM model is 5.47%， the error value of the PSOLSSVM model is 5.27%， and the error value of the LSSVM model is 5.73%. In a word， the prediction accuracy of the models selected parameters by intelligent optimization algorithm is much higher than that of non-intelligent optimization models.

In future research， FOA can be applied to different models to solve different problems in more fields. Since electricity consumption is affected by economic development， power supply capacity， industrial structure，etc.， multivariate prediction can be carried out to study the different effects of single variable and multivariable on prediction accuracy. Electricity consumption is also affected by external policies， economic events or natural disasters， which can have an impact on the process of data generation. Therefore， models need to be further developed to take into account the impact of structural breaks on forecasting results and predict structural breaks in advance in future research.