A seasonal GM(1,1) model for forecasting the electricity consumption of the primary economic sectors

Accepted Manuscript A seasonal GM(1,1) model for forecasting the electricity consumption of the primary economic sectors Zheng-Xin Wang, Qin Li, Ling-Ling Pei PII:

S0360-5442(18)30771-0

DOI:

10.1016/j.energy.2018.04.155

Reference:

EGY 12794

To appear in:

Energy

Received Date: 18 July 2017 Revised Date:

12 March 2018

Accepted Date: 25 April 2018

Please cite this article as: Wang Z-X, Li Q, Pei L-L, A seasonal GM(1,1) model for forecasting the electricity consumption of the primary economic sectors, Energy (2018), doi: 10.1016/ j.energy.2018.04.155. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A seasonal GM(1,1) model for forecasting the electricity consumption of the primary economic sectors

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Zheng-Xin Wanga,b, Qin Li a, Ling-Ling Peic a. School of Economics, Zhejiang University of Finance & Economics, Hangzhou 310018, China b. Center for Research of Regulation & Policy, Zhejiang University of Finance & Economics, Hangzhou 310018, China c. School of Business Administration, Zhejiang University of Finance & Economics, Hangzhou 310018, China E-mail: [email protected] (Zheng-Xin Wang, Corresponding author); [email protected] (Qin Li); [email protected] (Ling-Ling Pei). Abstract: To accurately predict the seasonal fluctuations of the electricity consumption of the primary economic sectors, we propose a seasonal grey model (SGM(1,1) model ) based on the accumulation operators generated by seasonal factors. We use the proposed model to carry out an empirical analysis based on the seasonal electricity consumption data of the primary industries in China from 2010 to 2016. The results from the SGM (1,1) model are compared with those obtained using the grey model (GM(1,1)), the particle swarm optimization algorithm combines with the grey model (PSO-GM(1,1) model), and the adaptive parameter learning mechanism based seasonal fluctuation GM (1,1) model (APL-SFGM(1,1) model). The results of the comparison show that the SGM(1,1) model can effectively identify seasonal fluctuations in the electricity consumption of the primary industries and its prediction accuracy is significantly higher than those of the GM(1,1), PSO-GM(1,1) and APL-SFGM(1,1) models. The forecast results for China from 2017 to 2020 obtained using the SGM(1,1) model suggest that the electricity consumption of the primary industries is expected to increase slightly, but obvious seasonal fluctuations will still be present. It is forecasted that the annual electricity consumption in 2020 will be 107.645 TWh with an annual growth rate of 2.83%. This prediction can provide the basis for power-supply planning to ensure supply and demand balance in the electricity markets. Graphical abstract:

ACCEPTED MANUSCRIPT Keywords: electricity consumption prediction; GM(1,1) model; PSO-GM(1,1) model; SGM(1,1) model.

1. Introduction

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1.1 Motivation The primary industries are the basic industries necessary to ensure the health of the people and steady development of the economy. The production activity of these industries is intimately linked with the support they receive in the form of electricity. Due to the influence of natural climate and social factors, the production activity of the primary industries experiences seasonality. As a result, their electricity consumption also shows significant seasonal characteristics [1]. As electricity cannot be stored, and is consumed in real time, the demand for electricity and electricity supplied to the primary industries are very likely to be unbalanced, which is not conducive to their production activities. According to data published in the Statistical Communique of the People’s Republic of China on the National Economic and Social Development issued by the National Bureau of Statistics, 2.514378×1012 Yuan of fixed assets were invested in national electricity construction in 2016 (corresponding to a year-on-year growth of 11.3%). Compared to the corresponding figure of 1.287937×1012 Yuan in 2010, the increase is 95.2%. The State Grid Corporation of China has also forecasted that the scale of investment in the power sector will be maintained at the current rate of increase in the foreseeable future. Therefore, accurately forecasting the electricity demand of the primary industries has important practical significance. Forecasts are used to guide power-supply enterprises to improve the dispatching ability of the power grid and maintain the balance of supply and demand in the electricity market. As China’s statistical departments did not release data relating to the electricity consumption of the primary industries before 2009, the sample size is limited and so statistical modeling methods based on large sets of data are not applicable. Grey system theory is an effective method of studying and modeling systems consisting of small sample sizes that contain a limited amount of information and is widely used in many fields [2]. The valuable information is extracted by processing the known information. This is further used to explore the evolution laws of the system and thus establish a prediction model. As there are many factors influencing the electricity consumption of the primary industries, e.g. economic structure, climate variation, and policy changes, it can be regarded as a grey system. Thus, it can be described using a grey model (GM). The GM(1,1) model is the most generally used grey model. The time-response function in the GM(1,1) model is approximated using an exponential function. However, the actual electricity consumption cannot strictly change in an exponential manner due to seasonal fluctuations. Therefore, ideal forecast results cannot be achieved by directly applying the GM(1,1) model to forecast the electricity consumption of the primary industries. By considering the seasonal changes in the electricity consumption of the primary industries, this study puts forward the seasonal grey model (SGM(1,1)model). In doing so, we try to eliminate the bias caused by seasonal fluctuations in the original data. 1.2 Literature review 1.2.1 Research progress on forecasting electricity generation In recent years, the prediction of energy has become a hot issue for scholars. Karadede et al. [3] proposed a breeder hybrid algorithm, which consisting of the simulated annealing and constitution of nonlinear regression-based breeder genetic algorithm, it was used to forecast the natural gas

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demand with a smaller error rate. A novel model combines the Wavelet Transform (WT), Genetic Algorithm (GA), Adaptive Neuro-Fuzzy Inference System(ANFIS) and Feed-Forward Neural Network (FFNN)was proposed by Panapakidis et al[4], and the novel model was applied to the natural gas grid of a country with all distribution points. Kamyab and Bahrami[5] proposed a linear time EMS scheduling algorithm to determine the competitive equilibrium for the energy hubs. Then they performed the simulations in a competitive electricity market, and the results shown that the dynamic pricing scheme is efficient for the energy hubs to modify their daily operation. At present, a large number of research methods have been applied to forecasting electricity consumption. Traditional prediction methods are based on establishing a mathematical model to use for forecasting by analyzing the historical changes in the electricity consumption data and deriving the relationships between the relevant factors that influence the data. The methods used generally include exponential smoothing [6], use of linear models [7–8], regression analysis [9–13], auto-regressive integrated moving-average models [14–20], auto-regressive conditional density models [21], and support vector machines (SVMs) [22–23]. For time series undergoing stable changes and obvious trends, time-series methods have high prediction accuracies. However, the prediction accuracy is not ideal when they are used to directly forecast monthly electricity sale series which have large fluctuations. In view of this, time-series models have been established based on adjusting the original data on a quarterly basis. High prediction accuracies were subsequently obtained [24–28]. Regression analysis requires a large number of high-quality data samples to be effective. In light of this, the forecasting errors produced by fitted regression models are generally large when the data follows atypical distributions or all the factors of relevance have not been taken into account. Therefore, it is very difficult to establish a universal load prediction model using regression analysis. With the development of computer technology, scholars have constantly updated the forecast methods used for predicting electricity consumption in recent years. By exploring the intrinsic mathematical characteristics of the historical electricity-consumption data, the established model can be corrected in real time using such methods. This effectively avoids the disadvantage of using a single model to describe the seasonal trends in electricity consumption and greatly improves prediction accuracy. Various methods have been employed, including adaptive PSO [29– 31], the ‘cuckoo’ search algorithm [32], artificial neural network model-based approaches [33–39], chaos theory [40–41], combined methods [42–46], fuzzy logic [47–48], and hybrid energy systems [49–50]. Using these methods, the inherent laws of the system can be found from the chaotic data with no need to construct predetermined functions. This effectively realizes the progression from training to high-accuracy prediction. However, these methods need a large number of samples to support them and they take a long time to run the complex calculations required. Moreover, the calculations cannot be realized when the data is incomplete. 1.2.2 Application of GM models in electricity prediction The simple grey prediction method needs only a few samples and shows high accuracy. Grey theory was first applied to predicting electricity loads by Morita in 1995 [51]. Since then, much more research has been undertaken which has greatly enriched the application of grey theory to electricity prediction. In view of the problem that traditional GM (1,1) model has a low modeling accuracy for seasonal time series, scholars, after conducting much research, proposed using quarterly time series. There are two main approaches employed. One is to carry out seasonal adjustment on the original data series and modeling is based on

ACCEPTED MANUSCRIPT strengthening the stability of series. Wang et al. [52] presented indices for seasonal adjustment and introduced a parameter ⊗ H to deal with nonlinear quarterly data series. In addition, they

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established the SFGM(1,1) model and optimized the parameters, finally obtaining the APL-SFGM(1,1) model, which has high prediction accuracy. Zhang et al. [53] used particle swarms to optimize the parameters of their model and reduced the fluctuations in the original series by utilizing the K-nearest neighbor algorithm to further improve the prediction accuracy. Bao et al. [54] combined the GM(1,1) model with linear regression and constituted the data (which had a large degree of ‘jumping’) into independent sequences. The aberrant and corresponding values were fitted and predicted by employing the GM(1,1) model, and the unchanged values were fitted and predicted using a linear regression model. This nicely overcomes the defects of the linear and GM(1,1) models. Based on a moving-average method, Song et al. [55] first enhanced the smoothness of the original series and then determined the combination weights of the GM(1,1) and SVM by utilizing the ‘leapfrog’ algorithm. They constructed a combined-prediction model based on leapfrog optimization and the resulting model has been well applied. Sun et al. [56], using a quarterly-average method, proposed a method employing a dynamic seasonal index. In their work, calculated runoff states were used to enhance the smoothness of the original data in combination with a Markov chain. This improved the GM(1,1) model and produced highly accurate predictions for the generation ability of small hydropower stations. Qian et al. [57] applied an accelerated translation transformation to the original data and thus changed a non-monotonic oscillatory sequence into a monotonic sequence. As a result, the modeling and forecasting processes were subject to reduced fitting errors. The other approach employed, is to improve the model itself and to increase adaptation of the model. By utilizing particle swarms and neural networks, Wu et al. [58] and Huang et al. [59] optimized and improved the development and coordination coefficients of the GM (1,1) model, respectively. Although these methods improve the prediction accuracy (to some extent), they are not particularly applicable to the model. Niu et al. [60] dealt with the prediction problems associated with complex seasonal time series possessing nonlinear growth and fluctuation by establishing a grey neutral model, which effectively improved the prediction accuracy. Ding et al. [61] designed a novel optimized grey prediction model based on the principle of “new information priority”, which combining the rolling mechanism to improve the initial conditions, the proposed model’s high efficacy and applicability has been confirmed by predicting China's total and industrial electricity consumption from 2012 to 2014.Coskun et al. [62] proposed an optimized grey model based on a rolling mechanism (in which the forecasted value obtained for the next first period is used to predict the value for the next second period). Their results show that the optimized grey model is capable of obtaining better forecast results. 1.3 Contribution and Organization The prediction accuracy improvements made using the two approaches mentioned above are not realized via the grey model itself. For this reason, an SGM(1,1) model is proposed here by introducing seasonal factors into the accumulating generation process. Thus, it integrates the seasonal fluctuations of the time series further into the widely used GM(1,1) model. Our aim is to essentially improve the adaptation of the GM(1,1) model to the seasonal time series. The new model is verified to be effective (and superior to previous models) by applying it as a test case to predict the quarterly electricity consumption of China’s primary industries. The contributions of this paper include the following three aspects:

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(1) To accurately identify the characteristics of seasonal fluctuations in the time series of electricity consumption of the primary industries, we proposed a method to calculate the seasonal factors. (2) We introduced the seasonal factors into the grey accumulation generation, and proposed the SGM (1,1) model to predict the seasonality and tendency of the electricity consumption of the primary industries. (3) The results from the SGM (1,1) model are compared with those obtained using the traditional grey model (GM(1,1)), the particle swarm optimization algorithm combines with the grey model (PSO-GM(1,1) model), and the adaptive parameter learning mechanism based seasonal fluctuation GM (1,1) model (APL-SFGM(1,1) model). The rest of this paper is arranged as follows. Section 2 gives a review of the relevant literature, while Section 3 lays down the foundations of the SGM(1,1) model. In Section 4, we empirically analyze the electricity consumption of China’s primary industries and compare the forecast results with those obtained using the traditional GM(1,1), PSO-GM(1,1) and APL-SFGM(1,1) models. Our conclusions are presented in Section 5.

2. Methods

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2.1 The traditional GM(1,1) model

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Grey system theory is a relatively important method of studying discrete data series with small numbers of samples and incomplete information [63]. By fully developing and utilizing the explicit and implicit information in the existing data, the randomness that is present in the series is cumulatively weakened. The laws governing the changes in the system are thus generated and can be used to research the future time distributions for specific time intervals. The traditional GM(1,1) modeling process is as follows. First, first-order accumulation generation is conducted on the original data series X (0) = ( x (0) (1), x (0) (2),⋯ , x (0) (n) ) to obtain the sequence

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X (1) = ( x (1) (1), x (1) (2),⋯ , x (1) ( n) )

k

where x(1) (k ) = ∑ x(0) (i) .

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i =1

Then, the mean series is calculated:

z (1) ( k ) = 0.5 x(1) ( k ) + 0.5 x(1) ( k − 1) , k = 2,3,…, n .

(1)

Using this series, the first-order differential equation based on a single variable is established and used as the prediction model (that is, the GM(1,1) model). The standard form of the grey difference equation is:

x(0) ( k ) + az (1) ( k ) = b, k = 2,3,…, n .

(2)

The corresponding whitening differential equation is: dx (1) (t ) + a x (1) (t ) = b dt

(3)

ACCEPTED MANUSCRIPT where a and b are the development coefficient of the system and endogenous control grey scale, respectively. The estimation formula for the parameter vector αˆ can be written in the form

αˆ = ( a, b ) = ( BT B) BTY T

−1

(4)

 − z (1) (2)  (1) − z (3) B=  ⋮   − z (1) (n)

1  x (0) (2)    (0)  x (3)  1 , and Y =  .   ⋮  ⋮   (0)  1  x ( n ) 

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The time-response function of the GM(1,1) model is:

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where

b b  xˆ (1) ( k ) =  x(0) (1) −  e− a ( k −1) + , k = 2,3,…, n . a a 

(5)

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Finally, through a single inverse accumulated generating operation (IAGO), the predicted

⌢ values of the original series, x (0) (k ) , can be obtained:

b  1 xˆ (0) ( k ) = xˆ (1) ( k ) − xˆ ( ) ( k − 1) =  x(0) (1) −  (1 − ea ) e− a ( k −1) . k = 2,3,… , n a 

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2.2 The PSO-GM(1,1) model

(6)

Niu et al. [64] showed that the main reason for low accuracy of the traditional GM (1,1) model is the method used to calculate the background value z (1) (k ) . Furthermore, the prediction accuracy can be expected to be better if the following formula is used instead (7)

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z (1) ( k ) = ex(1) ( k − 1) + (1 − e) x(1) ( k − 1) , e ∈ [ 0,1]

The forecasted values xˆ (0) ( k ) can be calculated according to the same method given above for the GM(1,1) model once parameter e has been determined. However, finding the optimal e

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value to use is difficult using traditional methods of optimization. In the PSO-GM(1,1) model, the optimal value of parameter e is accomplished using the PSO algorithm, which yields improved prediction accuracy.

The PSO algorithm is aimed at stimulating the foraging behavior of birds. That is, we

imagine a flock of birds randomly searching for food in a fixed field and mimic their behavior by randomly changing flight velocity. In the actual algorithm, the solution to each optimization problem is equivalent to a bird, or more generally, a ‘particle’. Each particle has a fitness that is determined according to the fitness function that is to be optimized. In addition, each particle has its own flight velocity which is dynamically adjusted according to its own or its partners’ flight experiences. A particle’s individual flight experience produces its own optimal solution which is labeled pbest . Similarly, the optimal solution experienced by any of the particles, the global optimal solution, is denoted by gbest . A brief mathematical description of the calculation process

ACCEPTED MANUSCRIPT is given below. Assume there are m particles in a D-dimensional space. The position of the ith particle is denoted by X i = ( xi1 , xi2 ,..., xiD ) , where i = 1, 2,… , m , and its velocity by Vi = (vi1 , vi2 ,..., viD ) . The position of each particle is a potential solution. Moreover, the fitness of the particle is calculated by substituting X i into the objective (fitness) function and its quality is judged. Its best position is recorded as Pi = ( pi1 , pi2 ,..., piD ) and the best position of the whole swarm is recorded as particle are given by: vi +1 = wvid + c1r1 ( pid − xid ) + c2 r2 ( p gd − xid )

xid+1 = xid + α vid

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Pg = ( p1g , pg2 ,..., pgD ) . The formulae for calculating the change in velocity and position of the ith

(8)

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(9)

where w represents an inertia factor, c1 and c2 are learning factors (and generally c1 =c2 =2 ), and d = 1, 2,..., D . Furthermore, r1 and r2 are random numbers that change in the range [0, 1]

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and α is a constraining factor that controls the velocity weights. The conditions set to terminate the iteration process are chosen according to the specific problem at hand. In general, however, the termination condition is usually the reaching of a pre-set maximum number of iteration times or the current optimal position meeting a pre-set minimum threshold.

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2.3 The SGM(1,1) model As Eq. (6) shows, the prediction formula in the traditional GM(1,1) model is essentially an exponential function of time. Even though the background values of the GM(1,1) model can be improved by using the PSO algorithm, the form of the prediction formula is unchanged. Therefore, the traditional GM(1,1) and PSO-GM(1,1) models cannot be used to identify and predict the seasonal fluctuations in the time series. To address this issue, various modifications have been made to the GM(1,1) model to improve its prediction accuracy. For example, Wang et al. [52] proposed a seasonal adjustment index to establish the SFGM(1,1) model. Zhang et al. [53] combined the K-nearest neighbor algorithm with the GM(1,1) model. Based on Markov chain, Sun et al. [56] built an improved GM(1,1) model by putting forward a dynamic seasonal index. These researchers all carried out seasonal adjustment of the external data used in the models but did not change the modeling mechanism. In comparison, in this research we improve the internal mechanism of the model and establish the SGM(1,1) model by defining a seasonal accumulating generation operator. Definition: Suppose that X (0) = ( x (0) (1), x (0) (2),⋯ , x (0) (n) ) is a seasonally-affected original series and S is a series operator such that X s (1) = X (0) S = ( x (1) (1) s, x (1) (2) s ,⋯ , x (1) (n) s )

with k

x(1) (k )s = ∑ x(0) (i) f s (i ), k = 1, 2,⋯, n i =1

(10)

ACCEPTED MANUSCRIPT where f s (i) is the seasonal factor acting at the ith point in time of the original series. Then, S is known as the first-order seasonal accumulating generation operator and is denoted by 1-SAGO. The seasonal factor f s (i) is a dimensionless parameter that reflects the average degree to which the actual value deviates from the trend value due to seasonal effects in the current period. It may be calculated using the expression

xM (0) (i ) xMN (0) (i )

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f s (i ) =

(11)

where M represents the number of season cycles in a year (for a quarterly season cycle M = 4 , while a monthly cycle would correspond to M = 12 ) and N the year of the ith time point. The

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quantities xM (0) (i) and xMN (0) (i ) indicate the average value for the month (or quarter, etc.) at the ith time point and the total average value for all seasons or months, respectively.

f s (i ) = 1 for each i ∈ K = {1, 2,⋯ , k} then the seasonal

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accumulating generation operator is identical to the classic accumulating generation operator. Therefore, the seasonal accumulating generation operator can be seen as an extension of the classic accumulating generation operator. On this basis, the mean series is calculated, as before

zs(1) (k ) = 0.5xs(1) ( k ) + 0.5 xs(1) ( k − 1) , k = 2,3,..., n

(12)

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The grey difference equation is established as:

x (0) (k ) f s (k ) + as zs(1) (k ) = bs , k = 2,3,..., n

(13)

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The corresponding whitening differential equation is: dxs(1) (t ) + as xs(1) (t ) = bs . dt

(14)

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In Eq. (13), parameter vector αˆ s can be estimated by using the following formula:

αˆ s = ( as , bs ) = ( Bs T Bs ) −1 Bs TYs T

(15)

where

 − zs (1) (2)  (1) − z (3) Bs =  s ⋮  (1)  − zs (n)

1  x(0) (2) f s ( 2 )    (0)  1  x (3) f s ( 3)  , Y = s   ⋮ ⋮   (0)  1  x (n) f s ( n ) 

The prediction formula is:  b  b xˆs (1) (k ) =  x (0) (1) / f s (1) − s  e − as ( k −1) + s , k = 2,3,..., n . as  as 

(16)

ACCEPTED MANUSCRIPT Finally, through seasonal IAGO of xˆs (1) (k ) , the forecasted value of original series X (0) is obtained

xˆ(0) (k ) = fs (k ) ( xˆs(1) (k ) − xˆs(1) (k −1)) , k = 2,3,..., n .

(17)

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The steps involved in the modeling process can be illustrated using a flow chart (Fig. 1).

Fig. 1. The steps involved in the prediction process using the proposed SGM(1,1) model.

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2.4 Evaluation criteria To investigate the effectiveness of the model, its accuracy has to be determined. Here, we use three measures: the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), which can be found using the formulae:

RMSE =

MAE =

MAPE =

1 n ( 2) ∑ e (i ) , n i =1

(18)

1 n ∑ e (i ) , n i =1

(19)

1 n e(i) ×100% . ∑ n i =1 x( 0) ( i )

(20)

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xˆ

( 0)

(i )

0)

( i ) − xˆ(0) ( i )

is the error, x(

0)

(i )

the actual data value, and

is the predicted value at the ith time point, respectively. The prediction accuracy can also

be ‘graded’, as we demonstrate in Table 1 for the MAPE evaluation criterion.

Table 1. Assessing accuracy using the MAPE criterion [65]. Forecasting ability

MAPE (%)

Forecasting ability

< 10

High ability

20–50

Reasonable ability

10–20

Good ability

> 50

Weak ability

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MAPE (%)

3. Predicting the electricity consumption of China’s primary industries

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In order to verify the effectiveness (and superiority) of the SGM(1,1) model, we establish models for the seasonal electricity consumption of China’s primary industries from 2010 to 2016. First, the original data to use in the empirical analysis is divided into two groups. One group (from 2010 to 2014) is used as the training set and the other (from 2015 to 2016) forms the test set. The three models discussed in this work (traditional GM(1,1), PSO-GM(1,1), APL-SFGM(1,1) and SGM(1,1) models) are established and the forecasts produced by them are compared and analyzed. The CEInet statistics database (http://db.cei.gov.cn/page/Login.aspx) was used to extract data for the electricity consumption of China’s primary industries from 2010 to 2016. The data was sorted (Table 3) and used to highlight the quarterly (seasonal) variation in the data (Fig. 2). 40

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Q1

Q2

Q3

Q4

Electricity consumption of china's primary industries(TWh) Means by Season

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Fig. 2. Quarterly variation in the electricity consumption of the primary industries (data for 2010 to 2016 from left to right in each set).

Fig. 2 shows that the mean values of the data in the four quarters are significantly different. The electricity consumption varies such that Q3 > Q2 > Q4 > Q1 as the production activities of the basic economic sectors (agriculture, forestry, fisheries, mining, etc.) are influenced by natural and social factors. Agriculture, forestry, and fisheries have significantly increased production activities in the summer, resulting in increased electricity consumption. Spring is the season associated with seeding, so agriculture consumes more electricity in this season. In the autumn and winter, the activities of the agriculture and fishery industries are much lower due to the gradual change to cold weather — hence, electricity consumption decreases.

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deviation in the third quarter is the most significant. This is because the production activities of the primary industries in this quarter are raised due to seasonal effects resulting in a significantly larger electricity consumption compared to the other quarters. The second-largest deviation occurs in the first quarter. Compared with the other quarters, the electricity consumption in this quarter is clearly the least. The deviations are smaller in the fourth and second quarters and, in particular, the deviation occurring in the second quarter is the smallest observed.

Table 2. The values of the seasonal indices for the training set.

0.7474

1.0749

Q3

Q4

1.2972

0.8805

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Seasonal index

Q2

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Quarters Q1

Subsequently, the 1-SAGO is generated using the seasonal indices and the SGM(1,1) model established for prediction purposes T

a  where αˆ s =  s  = [ −0.000036, 25.0735] .  bs 

The resulting time-response function is xˆs (1) (k ) = 69007.937e0.000036( k −1) − 69005.395 . Using

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xˆs (1) (k ) , the forecasted value xˆs (0) (k ) can be obtained using the first-order IAGO. The training and test results obtained using the SGM(1,1) approach are shown in Table 5.

T

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3.1.2 The GM(1,1) model The same data in the training set was directly utilized to establish a GM(1,1) model according to the steps outlined in Section 3.1. The electricity consumption of the primary industries in each quarter from 2015 to 2016 were then predicted and compared with the actual data. The results obtained using a traditional GM(1,1) model were acquired here using the corresponding MATLAB™ toolbox

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a  where αˆ =   = [ 0.0010, 25.6843] . b  The resulting time-response function is xˆ (1) ( k ) = −2546.7821e−0.001( k −1) + 2548.682 . The

forecasted values xˆ (0) (k ) were the obtained using first-order IAGO on xˆ (1) (k ) . The training and test results for the traditional GM(1,1) model are also displayed in Table 5. 3.1.3 The PSO-GM(1,1) model The main steps involved in optimizing the traditional GM(1,1) model using particle swarms are:

Step 1: Initialize m = 100 particles. The range of values of parameter e that we need to calculate in this study is small. Considering this, small learning factors are set ( c1 = c2 = 1.5 ) in order to avoid missing the optimal solution due to the use of too fast a learning speed. Furthermore,

ACCEPTED MANUSCRIPT the inertia weight is set as w = 0.8 based on the study carried out by Shi and Eberhar [66]. The largest number of iterations set for ending the calculation is Tmax = 200 and the particles are set to be one dimensional; e representing the parameter of the solution. The positions and velocities of the particles in the initialized particle swarm are as follows:

E = (e1 , e2 ,…, ei ) , V = (v1 , v2 ,…, vi ) , i = 1,2,..., m .

MAPEmin

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Step 2: Calculate the fitness of each particle. Minimizing the value of the MAPE measure of error is used as the objective function in this research

 1 n x( 0) ( i ) − xˆ ( 0) ( i )  = Min ( f ( ei ) ) = Min  ∑ × 100%  x ( 0) ( i )  n i =1 

(21)

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Step 3: Parameters a and b are obtained by substituting ei in E = (e1 , e2 ,…, ei ) into Eq. (7) and the forecasted value xˆ (1) ( k ) is obtained in accordance with Eq. (5). Finally, the fitness f of each particle is calculated using Eq. (21) and compared with the best position pbest of the particle

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and the best global position g best . If f ( ei ) < f ( pbest ) , then f ( ei ) = pbest ; if f ( ei ) < f ( gbest ) ,

then f ( ei ) = gbest .

Step 4: Optimizing velocity and position. The velocities and positions are changed according to Eqs. (8) and (9) with the additional constraint that velocity remains below Vmax .

Step 5: Stopping criteria: the maximum iteration times Tmax is set to be 200. If the iteration

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time is less than 200, then the calculation returns back to Step 2.

In this work, MATLAB 2016a was utilized in the solution process. We obtained the relationship between the e parameter and MAPE value (Fig. 3) and also mapped the MAPE convergence process as the iterations were carried out (Fig. 4). The global optimal solution is

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obtained when e = 8.3350 (where MAPEmin = 16.98% ). Fig. 3 shows that the MAPE value decreases as e increases in the range [0, 1]. Thus, when e is equal to 1, the MAPE value is the smallest in the range [0, 1].

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Therefore, as e ∈ [ 0,1] , the overall result of the PSO optimization process is that we must T

a  set e = 1 and so αˆ =   = [ 0.001721, 25.84885] is calculated. b  The resulting time-response function is xˆ (1) ( k ) = −1499.887e−0.001721( k −1) + 1501.787 . The training and test results for the PSO-GM(1,1) approach are also listed in Table 5.

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Fig. 3. The relationship between MAPE value and parameter e for the PSO-GM(1,1) model.

Fig. 4. Typical variation in MAPE during the iteration process.

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3.1.4 APL-SFGM(1,1) model Wang et al. [48] proposed the SFGM(1,1) model in 2012 aimed at data subject to seasonal fluctuations. On this basis, an optimized APL-SFGM(1,1) model was built by updating the data. Our study first predicts the training set using the SFGM(1,1) model, which can then be used to obtain the seasonal index values, as shown in Table 3.

Table 3. Seasonal index values in SFGM(1,1) model.

Value

Seasonal index I1

I2

I3

I4

0.7474

1.0747

1.2975

0.8804

T

a  −5 At the same time, the parameter αˆ =   =  −3.53 × 10 , 25.074  is attained. b  Next, the APL-SFGM(1,1) model is used to forecast the test set using the adjustment parameters determined in previous research [48], i.e. α = 0.33, β = 0.5, γ = 0.18 . The intermediate and final results obtained using the model are displayed in Table 4. A comparison of the final

ACCEPTED MANUSCRIPT values forecasted using the APL-SFGM(1,1) model and the actual results is shown in Table 5.

Table 4. The intermediate and final results from the APL-SFGM(1,1) model. Item

t = 20

t = 21

t = 22

t = 23

t = 24

t = 25

t = 26

t = 27

X t (TWh)

26.128

24.706

28.955

30.480

23.051

24.533

34.106

37.089

xˆ( t + j ) (TWh)

0.944

0.846

0.798

0.794

0.763

0.707

0.735

0.735

0.735

0.735

0.727

0.727

0.727

j=2

1.075

1.075

1.072

1.072

1.072

1.072

1.074

1.074

j=3

1.298

1.298

1.298

1.320

1.320

1.320

1.320

1.328

j=4

0.880

0.880 a

21.270

0.880 a

22.390

0.880 a

18.021

0.874

a

18.465

j=2

26.527

32.969

38.632

30.952

33.125

j=3

31.996

42.171

55.278

47.727

51.361

j=4

21.689

30.315

44.342

39.874

42.813

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The final forecasting value.

16.933

0.874 a

j=1

24.810

0.874

a

26.980

0.874

a

21.595 a

47.950

56.316

45.827

77.413

97.915

81.506

91.652

77.092

67.190

3.2 Comparison of prediction accuracies 3.2.1Model comparison In order to compare the prediction accuracies of the four models, the errors incurred using them are first analyzed (Table 5 and Fig. 5). Ignoring the initial value x (0) (1) , the minimum

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absolute errors of the models GM(1,1), PSO-GM(1,1), APL-SFGM(1,1), and SGM(1,1) for the training set are 1.72, 1.17, 0.37, and 0.38, and the maximum absolute errors are 45.2, 44.5, 7.76, and 7.77, respectively. For the test set, their minimum absolute errors are 6.89, 7.24, 2.16, and 0.1 and their maximum absolute errors are 47.2, 46.0, 38.7, and 12.0, respectively. For both the maximum and minimum absolute error estimates, the GM(1,1) and PSO-GM(1,1) models clearly show much lower prediction accuracies than the SGM(1,1) model. Compared to the SGM(1,1) model, the APL-SFGM(1,1) model is marginally more accurate with respect to the training set but much less accurate when it comes to the test set. Next, we compare three different error indices (RMSE, MAE, and MAPE). The calculated errors are displayed in Table 6. The values of the RMSE, MAE, and MAPE of the four models in the test set are shown in Fig.6. It can be seen from Table 6 and Fig. 6 that the RMSE, MAE, and MAPE values of the GM(1,1), PSO-GM(1,1), and APL-SFGM(1,1) models are much larger than those of the SGM(1,1) model for both the training and test sets. Thus, their prediction accuracies are lower. The MAPE values of the GM(1,1), PSO-GM(1,1), and APL-SFGM(1,1) models for the training set are 17.86%, 17.81%, and 2.97%, respectively, increasing to 24.21%, 23.79%, and 17.34% in the test set. The APL-SFGM(1,1) model shows a high prediction accuracy for the training set while its prediction accuracy declines to below that of the SGM(1,1) model when forecasting the out-of-sample data of the test set. Meanwhile, although the prediction accuracy of the PSO-GM(1,1) model is greater than that of the traditional GM(1,1) model, neither model falls into the high prediction accuracy grade, according to Table 1. This means that the widely-used intelligent optimization methods cannot properly reflect the seasonal fluctuations of such series. The MAPE values of the SGM(1,1) model in the training and test sets are 2.90% and 6.13%, respectively. According to the evaluation standards shown in Table 1, this level of prediction

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a

1.001

0.747

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I (t) j

0.980 j=1

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e

at

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accuracy does reach the high-accuracy prediction grade. In addition, as shown in Fig. 7, the GM(1,1) and PSO-GM(1,1) models only reveal the trends in the change in electricity consumption of China’s primary industries; they cannot forecast the seasonal fluctuations. The APL-SFGM(1,1) model can adequately describe the fluctuation in the training set data, while it merely reflects the undulation of the data in the test set and fails to embody the variation trends. Hence, the SGM(1,1) model presents a significantly higher prediction accuracy compared to the other three models, on the whole. This is because of the seasonal indices that were introduced to reflect the degree of deviation between the actual data and average trend. This means that the model established using the first-order seasonal accumulating generation operator is able to effectively identify and predict the seasonal fluctuations in the time series. Thus, our results clearly indicate that the new SGM(1,1) model is more appropriate for time series that contain seasonal fluctuations. At the same time, the prediction accuracy of the models generally grows as the predictive period is prolonged. In this research, eight quarters (2015 and 2016) have been used to form the test set for forecasting. As shown by the prediction accuracies attained for the test set (Table 6), the prediction accuracy of the SGM(1,1) model for the out-of-sample prediction of eight periods still falls into the high-accuracy category. Therefore, the model has favorable accuracy over short- and medium-term prediction periods (5–10 periods), so the prediction results have practical reference value.

Table 5. Forecasted values and errors incurred using the four different models. GM(1,1) Time

PSO-GM(1,1)

Actual value

APL-SFGM(1,1)

SGM(1,1)

Forecasted value

Error (%)

Forecasted value

Error (%)

Forecasted value

Error (%)

Forecasted value

Error (%)

19.00

0.00

18.74

1.37

19.00

0.00

25.79

1.17

26.95

-3.25

26.95

-3.27

Training stage 19.00

19.00

0.00

2010Q2

26.10

25.65

1.72

2010Q3

31.50

25.63

18.65

25.75

18.26

32.54

-3.29

32.53

-3.27

2010Q4

21.80

25.60

-17.43

25.71

-17.91

22.08

-1.27

22.08

-1.28

2011Q1

19.50

25.57

-31.15

25.66

-31.60

18.74

3.88

18.74

3.88

2011Q2

28.00

25.55

8.75

25.62

8.51

26.95

3.74

26.96

3.73

2011Q3

32.00

25.52

20.24

25.57

20.08

32.54

-1.69

32.53

-1.67

2011Q4

22.00

2012Q1

19.00

2012Q2

27.50

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2010Q1

25.50

-15.90

25.53

-16.04

22.08

-0.37

22.08

-0.38

25.47

-34.06

25.49

-34.13

18.75

1.34

18.75

1.34

25.44

7.49

26.96

1.98

26.96

1.96

33.00

25.45 25.42

22.97

25.40

23.04

32.55

1.37

32.54

1.40

2012Q4

21.80

25.40

-16.49

25.35

-16.30

22.08

-1.30

22.09

-1.31

2013Q1

18.90

25.37

-34.23

25.31

-33.92

18.75

0.80

18.75

0.80

2013Q2

27.20

25.34

6.82

25.27

7.11

26.96

0.88

26.96

0.86

2013Q3

31.50

25.32

19.62

25.22

19.93

32.55

-3.34

32.54

-3.31

2013Q4

23.80

25.29

-6.27

25.18

-5.80

22.09

7.20

22.09

7.19

2014Q1

17.40

25.27

-45.22

25.14

-44.46

18.75

-7.76

18.75

-7.77

2014Q2

26.10

25.24

3.29

25.09

3.86

26.96

-3.31

26.97

-3.33

2014Q3

34.80

25.22

27.54

25.05

28.02

32.56

6.45

32.55

6.47

2014Q4

21.10

25.19

-19.39

25.01

-18.52

22.09

-4.69

22.09

-4.70

25.17

-47.17

24.96

-45.99

18.46

-7.98

18.75

-9.67

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7.47

2012Q3

Verification stage 2015Q1

17.10

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25.14

6.89

24.92

7.70

21.27

21.22

26.97

0.10

2015Q3

36.50

25.12

31.19

24.88

31.84

22.39

38.66

32.55

10.82

2015Q4

21.40

25.09

-17.24

24.84

-16.05

16.93

20.87

22.10

-3.25

2016Q1

18.40

25.06

-36.22

24.79

-34.74

18.02

2.06

18.76

-1.94

2016Q2

29.10

25.04

13.95

24.75

14.95

24.81

14.74

26.98

7.30

2016Q3

37.00

25.01

32.39

24.71

33.22

26.98

27.08

32.56

12.01

2016Q4

23.00

24.99

-8.65

24.67

-7.24

21.59

6.11

22.10

3.92

0 0 0 Percentage error, calculated according to: Error = 100% ×  x ( ) (i ) − xˆ ( ) (i )  x ( ) (i ) .

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a

2015Q2

Table 6. Comparison of the prediction accuracies of the four models (2010Q1–2016Q4). RMSE (TWh)

MAE (TWh)

GM(1,1)

5.097

4.336107

PSO-GM(1,1)

5.098

4.328119

17.81

APL-SFGM(1,1)

0.929

0.749593

2.97

SGM(1,1)

0.927

0.736155

2.90

GM(1,1)

7.248

6.212513

24.21

PSO-GM(1,1)

7.303

6.212537

23.97

APL-SFGM(1,1)

6.844

5.221

17.34

SGM(1,1)

2.345

1.768684

6.13

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Verification stage:

17.86

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Training stage:

MAPE (%)

Fig. 5. Comparison of the percentage errors incurred using the four models to predict the electricity consumption of China’s primary industries from 2010 to 2016.

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25

PSO-GM(1,1)

APL-SFGM(1,1)

SGM(1,1)

20

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15 10 5 0 MAE (TWh)

MAPE (%)

SC

RMSE (TWh)

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Fig. 6. Bar figures of the errors for these four models

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Fig. 7. Comparison of forecasted and actual values from 2010 to 2016. 3.2.2 Further analysis: Mean comparison In this section, mean value of the actual values and forecasted values by the four models are compared to give a further analysis to the advantages and disadvantages of these models. The method of mean comparison is implemented by multiple comparison of mean values among the forecasted values by these four models and the actual values. The statistical results of the forecasted values of the four models and the meanings of the groups are shown in Table 7-8. Table 7. Statistical results of the forecasted values from 2010-2016 by the four models Model

Actual values

GM(1,1)

PSO-GM(1,1)

APL-SFGM(1,1)

SGM(1,1)

Lower quartile(TWH)

20.30

25.11

24.86

18.75

20.54

Median (TWH)

26.10

25.29

25.18

22.09

25.10

Upper quartile(TWH)

30.30

25.49

25.51

26.96

29.76

The smallest outlier(TWH)

17.10

24.99

24.67

16.93

18.75

The largest outlier(TWH)

37.00

25.17

24.96

26.98

32.56

ACCEPTED MANUSCRIPT Table 8. Meaning of group. Meaning

Group 1

Actual electricity consumption time series

Group 2

Forecasted electricity consumption time series by GM(1,1) model

Group 3

Forecasted electricity consumption time series by PSO-GM(1,1) model

Group 4

Forecasted electricity consumption time series by SGM(1,1) model

Group 5

Forecasted electricity consumption time series by APL-SFGM(1,1)model

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Group number

Mean of this group

Group 5

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Group 4 Group 3

Group 1

16

21

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Group 2

26

31

36

No groups have means significantly different from Group 1

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Fig.8. Mean multiple comparison figure Fig.8 shows the mean value of the five groups and the range of variation around that. The distance between the value of maximum and minimum for Group 1 is the largest, and the variation range of Group 2 is the smallest, followed by Group 3, Group 5, and Group 4. Among the five horizontal lines, the more parts they overlap, the less difference they have. As shown in Fig. 8, the overlap between Group 4 and Group 1 is the largest, and the mean value of Group 4 is closer to Group 1. Therefore, the forecasted accuracy of SGM (1,1) model is superior to traditional GM(1,1), PSO-GM(1,1) and APL-SFGM(1,1) models, and SGM(1,1) model is more suitable for predicting the electricity consumption of the primary industries with the seasonal fluctuations.

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3.3 Out-of-sample predictions and discussion Our tests suggest that the SGM(1,1) model is more suitable for predicting the electricity consumption of China’s primary industries than the traditional GM(1,1) and PSO-GM(1,1) models. Therefore, the SGM(1,1) model is selected in this section to make out-of-sample predictions of the electricity consumption from 2017 to 2018. First we note that when we made out-of-sample predictions when we tested the SGM(1,1) model that, although the prediction accuracy is high, the forecasted values from 2015 to 2016 are clearly less accurate than those from 2010 to 2014 (Fig. 6). This is because we assume that the seasonal factor is stable so that its variational trend is not effectively utilized. As a result, prediction accuracy is greatly affected if the seasonal factor changes significantly during the out-of-sample predictive period. In order to more accurately forecast the seasonal fluctuations of the electricity consumption in 2017–18, all the seasonal data from 2010 to 2016 should be employed to calculate the xMN (0) (i ) values. By calculating xM (0) (i)

ACCEPTED MANUSCRIPT values using the seasonal data for 2015–16, rather than 2010–16, amended seasonal indices for the four quarters can finally be calculated (Table 9).

Table 9. The values of the adjusted seasonal indices. Q1

Q2

Q3

Q4

Seasonal index

0.6985

1.1039

1.4462

0.8736

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Comparing the indices in Tables 9 and 2, the Q1 seasonal index is seen to decrease from 0.7474 to 0.6985, while that in Q3 increases from 1.2972 to 1.4462. This suggests that the seasonal effects of these two quarters have been made relatively larger. This results in an increase in the size of the deviations from the average electricity consumption trend. For clarity, the original model with unadjusted seasonal indices will be referred to as the SGM(1,1)-1 model. The SGM(1,1) model established via 1-SAGO applied to the original data using the dynamically adjusted seasonal indices will be referred to as the SGM(1,1)-2 model. The predictions made using these two variants can be compared and their accuracies further investigated. A comparison of their accuracies is given in Table 10.

Table 10. Comparison of the prediction accuracies of the two SGM(1,1) models (2015Q1–2016Q4). Model

RMSE (TWh)

MAE (TWh)

MAPE (%)

SGM(1, 1)-1

2.345

1.769

6.13

0.680

0.600

2.57

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SGM(1, 1)-2

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Table 10 shows that the SGM (1,1)-2 model (with dynamically adjusted seasonal indices) produces significantly smaller errors than the SGM(1,1)-1 model. This suggests that dynamically adjusting the seasonal indices can well-reflect the average deviations from the underlying trend due to seasonal influences. In other words, the SGM(1,1) model, after dynamically adjusting the seasonal indices, is more suited to making out-of-sample predictions. Table 11 presents the forecasts made for the electricity consumption of China’s primary industries from 2017 to 2020 and associated errors from 2010 to 2016.

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Table 11. Forecasted values and errors associated with the SGM(1,1)-2 model. Actual

Forecasted

Actual

Forecasted

Error

value

value

value

value

(%)

2010Q1

19.0

2015Q3

36.5

36.483

0.05

2010Q2

-2.81

2015Q4

21.4

22.077

-3.17

35.219

-11.81

2016Q1

18.4

17.683

3.90

21.8

21.313

2.24

2016Q2

29.1

27.994

3.80

2011Q1

19.5

17.071

12.46

2016Q3

37.0

36.741

0.70

2011Q2

28.0

27.024

3.49

2016Q4

23.0

22.234

3.33

2011Q3

32.0

35.468

-10.84

2017Q1

17.808

2011Q4

22.0

21.463

2.44

2017Q2

28.192

2012Q1

19.0

17.191

9.52

2017Q3

37.001

Time

Error (%)

Time

19.000

0.00

26.1

26.834

2010Q3

31.5

2010Q4

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27.215

1.04

2017Q4

22.391

2012Q3

33.0

35.719

-8.24

2018Q1

17.934

2012Q4

21.8

21.615

0.85

2018Q2

28.391

2013Q1

18.9

17.313

8.40

2018Q3

37.263

2013Q2

27.2

27.408

-0.76

2018Q4

22.549

2013Q3

31.5

35.972

-14.20

2019Q1

18.061

2013Q4

23.8

21.768

8.54

2019Q2

28.592

2014Q1

17.4

17.435

-0.20

2019Q3

2014Q2

26.1

27.602

-5.75

2019Q4

2014Q3

34.8

36.226

-4.10

2020Q1

2014Q4

21.1

21.922

-3.90

2020Q2

2015Q1

17.1

17.559

-2.68

2020Q3

2015Q2

27.0

27.797

-2.95

2020Q4

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2012Q2

37.526 22.709 18.189 28.794

SC

37.792

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22.870

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Fig. 9. The variation of the forecasted and actual values obtained using the SGM(1,1)-2 model from 2010 to 2020.

As shown in Fig. 9, the electricity consumed by China’s primary industries is forecasted to continue to fluctuate seasonally (and slightly increase, on the whole, at a growth rate of 2.83%) over the next four years according to the SGM(1,1)-2 model. On November 7, 2016, the National Development and Reform Commission and National Energy Administration published Planning for Electric Power Development as part of ‘The 13th Five-Year Plan’ which forecasted that the electricity consumption of the whole society in 2020 is expected to reach 6800–7200 TWh (an annual growth rate of 3.6–4.8%). It can be seen from our forecasted results for the electricity consumption of the primary industries that its annual growth rate is expected to be lower than that of the society as a whole. China is currently in a state of developing structural adjustment and steady growth. The manufacturing and service sectors constitute the pillars of the economy, while agriculture, forestry,

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fishing, and animal husbandry are developing relatively slowly. Electricity consumption is also increasing slowly. Therefore, in formulating their plans, the government can appropriately reduce the construction of power grid facilities serving the primary industries, and increase construction with respect to secondary and tertiary industries. Furthermore, with respect to power dispatch, the quarterly characteristics of the electricity consumption of the primary industries cannot be ignored. The power-supply enterprises should transmit electrical power reasonably according to the actual production taking place, so as to realize a balance between supply and demand in the electricity market and ensure stable and orderly development of the primary industries.

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4. Conclusions In this study, we have highlighted the seasonal changes occurring in the electricity consumption of China’s primary industries. We carried out a smoothing transformation by adjusting newly introduced seasonal indices and then generated a seasonal accumulating generation operator to establish a new GM(1,1) model that incorporates seasonal variation. We obtain the following conclusions. (1) The new model was empirically compared with traditional GM(1,1), PSO-GM(1,1), and APL-SFGM(1,1) models. The results showed that the problem of large prediction errors (resulting from seasonal fluctuations in the original data) can be effectively solved by using the SGM(1,1) model. Thus, more accurate prediction results can be obtained. (2) Our study showed that although the introduction of intelligent optimization improves the prediction accuracy of the traditional GM(1,1) model, the improvement is essentially insignificant and does not allow the seasonal fluctuations to be identified and taken into account. On the other

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hand, the seasonal indices introduced in the SGM(1,1) model can reflect the average amount by which the actual value deviates from the trending value due to seasonal variations. The seasonal accumulating generation operator proposed can be seen as an extension of the traditional accumulating generation operator. When f s (i ) = 1 , the seasonal accumulating generation operator has the same form as the traditional operator.

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(3) A short-term forecast of the electricity consumption of China’s primary industries made using the SGM(1,1) model showed that the electricity consumption is expected to continue to

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increase slightly and show significant seasonal variation.

Acknowledgements

The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions. This research is supported by the National Natural Science Foundation of China (71571157; 71742001).

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Highlights

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A seasonal grey model is proposed to predict quarterly electricity consumption. The proposed model can accurately identify and predict the seasonal fluctuations. The prediction accuracy is significantly higher than those of traditional models. The electricity consumption of China’s primary economic sectors is predicted.

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