An improved grey model optimized by multi-objective ant lion optimization algorithm for annual electricity consumption forecasting

Accepted Manuscript Title: An improved grey model optimized by multi-objective ant lion optimization algorithm for annual electricity consumption forecasting Authors: Jianzhou Wang, Pei Du, Haiyan Lu, Wendong Yang, Tong Niu PII: DOI: Reference:

S1568-4946(18)30410-1 https://doi.org/10.1016/j.asoc.2018.07.022 ASOC 4990

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

6-8-2017 11-6-2018 10-7-2018

Please cite this article as: Wang J, Du P, Lu H, Yang W, Niu T, An improved grey model optimized by multi-objective ant lion optimization algorithm for annual electricity consumption forecasting, Applied Soft Computing Journal (2018), https://doi.org/10.1016/j.asoc.2018.07.022 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.

An improved grey model optimized by multi-objective ant lion optimization algorithm for annual electricity consumption forecasting Jianzhou Wang a, Pei Du a, *, Haiyan Lu b, Wendong Yang a, Tong Niua a

School of Statistics, Dongbei University of Finance and Economics, Dalian 116025, China

b

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Department of Software Engineering, University of Technology, Sydney, Australia

*Corresponding author. Address: School of Statistics, Dongbei University of Finance and

Email adress : [email protected] or [email protected]

Highlights

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Economics, Dalian 116025, China

A dynamic choice rolling GM (1, 1) (DCRGM (1, 1)) is successfully developed.

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Proposed a novel hybrid forecasting model based on MOALO and DCRGM (1, 1).

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Small sample and extended versions of the grey model are discussed in this study.

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The proposed hybrid model demonstrates higher prediction accuracy.

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Abstract: Accurate and stable annual electricity consumption forecasting play vital role in modern social and economic development through providing effective planning and guaranteeing a reliable supply of sustainable electricity. However, establishing a robust method to improve prediction accuracy and stability simultaneously of electricity consumption forecasting has been proven to be a highly challenging task. Most previous researches only pay more attention to enhance prediction accuracy, which usually ignore the significant of forecasting stability, despite its importance to the effectiveness of forecasting models. Considering the characteristics of annual power consumption data as well as one criterion i.e. accuracy or stability is insufficient, in this study a novel hybrid forecasting model based on an improved grey forecasting mode optimized by multi-objective ant lion optimization algorithm is successfully developed, which can not only be utilized to dynamic choose the best input training sets, but also obtain satisfactory forecasting results with high accuracy and strong ability. Case studies of annual power consumption datasets from several regions in China are utilized as illustrative examples to estimate the effectiveness and efficiency of the proposed hybrid forecasting model. Finally, experimental results indicated that the proposed forecasting model is superior to the comparison models.

Keywords: Annual electricity consumption forecasting; Multi-objective ant lion 1

Nj

fitness of the antlions and ants random positions of antlions and ants number of inequality constraints kth-order forecasting effectiveness unit number of solutions

random number generated with uniform distribution

n1

number of the variables

b

grey input

o

bˆ

estimated values of b

obf1 ()

bt

the maximum of random movement of t-th variable

obf 2 ()

c

a constant bigger than 1

p

ctk

the minimum of t-th variable at k-th iteration

Pj

number of objective functions fitness function of objective 1 fitness function of objective 2 number of equality constraints probability of the antlions selected

d

number of dimension

Ps

Pareto optimal set

d tk

the maximum of t-th variable at k-th iteration,

Pf

Pareto optimal front

t

Forecasting errors

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f 

objective function

Qt

the random move towards the antlion discrete probability distribution

gt

the t-th inequality constraints

r ()

stochastic function

H ( x)

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optimization algorithm; Hybrid forecasting model.

H ( x, y)

single variable continuous function two-variable continuous function

ht

IterMax

r M OAL

m1

a

a developing coefficient

mk

aˆ

estimated values of



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a

Mr

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at

the j-th parameter's values of the i-th antlion and ant the minimum of random movement of t-th variable

Aijr

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Nomenclature

the t-th equality constraints

X (0)

non-negative time series

the maximum number of iteration

X (1)

increasing time series

k

the current iteration

YACT ( t )

the t-th actual values

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loss function of the prediction error lower boundary of the i-th variable

YPRE ( t )

the t-th forecasting value

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upper boundary of the i-th variable a constant

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1. Introduction 2

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Electricity consumption forecasting plays a fundamental and vital role in the sustainable energy management and economically efficient operation of power plants, which can help to minimize energy consumption, reduce environmental pollution, and improve the security, stability of the electrical power system. However, poor forecasting accuracy will pose a serious threat to the security, economy and quality of the power system [1]. The relevant research suggests that 10 million operating costs may increase if the forecasting error increases 1% [2]. On the one hand, the over-load forecasts will result in the start-up and fixed costs because of difficulties storing electricity [3]. On the other hand, the under-load forecasts will affect the quality of power supply which cannot satisfy the normal power demand and even endanger the safety and stability of the power system [4]. Clearly, if there is a smart early warning system on the strength of excellent prediction results, it can timely help decisionmakers enact suitable measures to avoid these above-mentioned. Therefore, developing a robust, accurate forecasting model and improving the prediction abilities of electricity consumption have become a top priority, especially for the annual power consumption prediction due to it can assist economic managers understanding a country's or a region's economic development trend in the future [5]. However, many variable factors, such as seasons, holidays, electricity policy, electricity price, population structure, social condition, economic and technological development level etc. [6], which may have a seriously impact on the abilities and precision of electricity consumption prediction [7]. Fortunately, in recent years, a great many forecasting techniques have been developed for enhancing prediction accuracy. However, due to the various features of annual electricity consumption, such as insufficient data sets and poor data information, autocorrelation and stochastic volatility etc. [8], the widely utilized methods i.e. the artificial neural networks (ANNs) [9-12, 38] limited by insufficient data [13], support vector Machine (SVM) [14-16] possessing a relatively complicated calculation process etc. [17], cannot always obtain the satisfactory forecasting results, although they are relatively mature methods. It’s worth noting that the grey forecasting model developed by Deng [18], is famous for predicting incomplete information and small sample, which have been widely applied into many fields [1921]. For instance, Zeng et al. [22] proposed a self-adapting intelligent grey model to forecast the natural gas demand in China, and finally obtained excellent results. Sun et al. [23] proposed a Grey-Markov model optimized by cuckoo search algorithm [24] to predict the annual foreign tourist arrivals to China and results showed that the presented model is more efficient and accurate than the conventional GM (1,1)— Markov-chain models. With the hope of improving the prediction performance, most previous researches are focused on the traditional intelligence algorithm i.e. single objective optimization algorithm, such as cuckoo search, particle swarm optimization, etc., which usually ignore the accuracy improvement or stability improvement. However, both accuracy and stability are very critical when evaluating the performance of a forecasting method. Moreover, when optimizing the parameters of the individual forecasting models by utilizing intelligence algorithms, both accuracy and stability should be taken into consideration at the same time. Therefore, it is no doubt that achieving two independent targets, simultaneously belongs to the multi-objective optimization (MOO) problems rather than the single-objective optimization problems. Different from the single-objective optimization algorithms owning single global optimum, MOO algorithms can help to discovery optimum solutions for engineering problems with more than one objective, which have been widely applied to solve real3

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world engineering problems, such as wind speed forecasts [25-27], electricity distribution network [28], and antenna design [29]. In the past few decades, a great number of MOO algorithms i.e. multi-objective dragonfly algorithm (MODA) [30], multi-objective water cycle algorithm (MOWCA) [31], multi-objective grey wolf optimizer (MOGWO) [32] etc., have been developed by many scientists. However, it is not easy to obtain a satisfactory result among a large solution space, where MOO algorithms need to explore and collect a set of optimal solutions at the same time. In order to obtain higher accuracy, a novel hybrid forecasting model based on the multi-objective ant lion optimizer (MOALO) developed by S. Mirjalili et al. [33] and an improved grey forecasting model—the dynamic choice RGM (1, 1) (DCRGM (1, 1)), which are utilized to optimize the parameters of DCRGM (1, 1) and dynamic choose the optimal training input sets, respectively, is developed in this study. Different from the traditional MOO algorithms, such as the multi-objective firefly algorithm (MOFA), Xiao et al. [34] used for short-term electrical load forecasting, whose coefficients of the objective functions need to preset. The MOALO, as one of the latest MOO algorithms, benefits from high convergence and coverage, which has been proved that its applicability can address challenging real-world problems as well [30]. Moreover, to further obtain high precision, the rolling mechanism was also employed to forecast annual electricity consumption time series in this study. The main contributions and novelty of this paper can be summarized as follows: (a) Focusing on small sample prediction. The traditional statistics analysis and prediction approaches pay more attention to large sample data, and many of forecasting models, such as autoregressive integrated moving average (ARIMA), ANNs, which have theoretical assurance all just under a large sample. However, the length of sample data is always restricted in most realistic cases remaining a difficult task for rarely mature method. It is necessary to pay more attention to the prediction of small sample. (b) Dynamic choice rolling–GM (1, 1). Traditional rolling–GM (1, 1) cannot always achieve the satisfactory forecasting results because of different trends or characteristics of samples. Hence, an improved grey forecasting model—the dynamic choice RGM (1, 1) (DCRGM (1, 1)) is proposed, which can dynamic choose input training sets to achieve better predictive effect. (c) Development of a hybrid forecasting model based on multi-objective ant lion optimization algorithm. Most previous articles of grey forecasting model focused only on single-objective optimization algorithm, which ignore the importance of multi-objective optimization. In this study, an improved grey model based on MOALO is successfully presented in this study, which can perform better than the comparison algorithms with high prediction accuracy; (d) Scientific and reasonable model evaluation system. Eight different regions’ samples in China, four widely utilized error measures DM test and forecasting effectiveness are all adopted to make a comprehensive evaluation of the proposed forecasting model. The experimental results show that the presented forecasting model performs its superior performance for generating forecasts in terms of forecasting accuracy. The remaining paper is organized as follows, Section 2 introduces the developed hybrid model and its relevant algorithms and theories; Section 3 illustrates the error measures and the data description; and empirical study is given in Section 4. Finally, the corresponding conclusions and conclusions of this study are given in sections 5 and 6, respectively. 4

2. Methods In this section, the proposed hybrid forecasting model and the related methods including the standard grey model (1, 1), the dynamic choice rolling grey model (1, 1) model and the multi-objective ant lion optimization algorithm are all introduced in this section.

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2.1 Standard GM (1, 1) model As one member of the grey system theory family, the first-order one-variable grey model, also named GM (1, 1), is an effective model and has been widely used in many forecasting issues because it does not need any sufficient and certain information. For example, when comes to forecasting issues, its input training set can be as few as four observations [8]. In this paper, the number of input training set is set bigger than four observations. Moreover, according to many previous literatures, the detailed steps of standard GM (1, 1) are listed as follows: Step 1: Given an initial non-negative time series X  0 including n  4 observed values, which can be expressed as: 0 0 (1) X    x   t   0, t  1, 2,..., n



0

t  ,

t  1, 2,..., n represents the t-th observed value.

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Where x



Step 2: By applying the first order accumulated generating operator (AGO) to

X   , the increasing series X   can be obtained as: 1



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0

 



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X    x  1 , x   2  ,..., x   n   x   t  , t  1, 2,..., n 1

1

1

(2)

1

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Where the increasing series: x   t  can be expresses by the following equation:

x1  t   x0 1  ...  x0  t  , t  1, 2,..., n

ED

(3)

Step 3: the standard GM (1, 1) model can be written as:

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x 0  t   az 1  t   b, t  1, 2,..., n

(4)

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Where a is developing coefficient and b is grey input, and the z    t  can be calculated using by: 1

x   t  1  x   t  z t   , t  2,..., n 2 1

1

1

(5)

Step 4: according to the least-square algorithm, the two estimated parameters aˆ

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and

bˆ can be obtained and written as: 1  aˆ bˆ    A A AYn   

 x 0  2     z 1  2  1      ... , Yn   ...  A   ...   0   1   x  n    z  n  1  Where aˆ and bˆ are estimated values of

a 5

and b , respectively.

(6)

(7)

Step 5: The whitenization equation of Eq. (4) is dx can be expressed by:

1

dt  ax   b , whose solution 1

 0 bˆ  bˆ 1 xˆ    t    x   0    e at  aˆ  aˆ 

(8)

Step 6: Finally, according to the inverse accumulated generating operator (IAGO), the forecasting equation can be expressed as: xˆ  0 1  x 0 1 (9)

xˆ  0  t   xˆ 1  t   xˆ 1  t  1 , t  2,..., n

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(10)

0 0 Where xˆ   1 and xˆ    t  , t  2,..., n are the forecasting values of the first and t-th observations, respectively.

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2.2 Dynamic choice Rolling–GM (1, 1) (DCRGM) model It’s well known that the rolling mechanism is an efficient method, which can be used to improve the prediction accuracy [35]. However, due to the datasets may exhibit different trends or characteristics, the traditional rolling–GM (1, 1) (RGM (1, 1)) cannot always achieve the satisfactory forecasting results. Moreover, it is worth pointing out that the traditional training sets of time series include the whole original data, but the whole original cannot always reflect the internal regularity of time series, sometimes a continuous original data segment can reflect it. Hence, in this subsection an improved grey forecasting model—the dynamic choice RGM (1, 1) (DCRGM (1, 1)) is proposed, which can achieve the best input training sets according to the minimum training error during the training process. Fig. 1 shows the two forecasting models i.e. RGM (1, 1) and DCRGM (1, 1), which can help readers understand the differences of the two forecasting models. Additionally, the definition of best input training set is given in Appendix A.

Fig. 1. The traditional RGM (1, 1) model and the proposed DCRGM (1, 1) model. 2.3 Multi-objective ant lion optimizer (MOALO) algorithm 6

The MOALO algorithm first developed by S. Mirjalili et al. [33], is an extended version of ALO. Thus, to better understand the MOALO algorithm, the basic theory of this algorithm should be introduced first.

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2.3.1 Ant lion optimizer (ALO) algorithm Inspired by the hunting mechanism of antlions in nature, a novel nature-inspired technique, namely ALO was first presented by S. Mirjalili et al. [33]. Two significant stages: the larvae stage and the adult stage exist in the lifecycle of the antlions, which are responsible for hunting prey and reproduction, respectively. Moreover, this algorithm consists of two populations: sets of antlions and ants. To change the abovementioned sets and eventually search for the global optimum, five main operations (i.e. random walks of ants, establishing trap, trapping in antlion’s pits, catching preys and reconstruction the pit) are performed in ALO algorithm, the details can be seen as follows.

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A12r A22r

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 f [ A11r  f [ A21r    r  f [ Am1

Amr 2

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r M OAL

(11)

i-th antlion and ant, respectively.

... A1rd ]   ... A2rd ]  , r  Antlion, Ant   r ... Amd ] 

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parameter's values of the

... A1rd   ... A2rd  , r  Antlion, Ant   r  Amr 2 ... Amd  i  1, 2,..., m; j  1, 2,..., d ) represent the j-th A12r A22r

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 A11r  r A M r   21   r  Am1 r Where Aij ,(r  Antlion, Ant;

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r The random positions M r and the fitness M OAL of the all the antlions and ants can be expressed by the two following equations, respectively.

(12)

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Where f  represents the objective function in this algorithm. 1) Random walks of ants Ants move stochastically in nature to search for food, and their random movement can be written as:

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X (k )  0, cumsum  2r  k1   1 , cumsum  2r  k2   1 ,..., cumsum  2r  km   1

(13)

Where the cumsum is the cumulative sum, whose detailed explanations has given in Appendix B, m represents the maximum number of iteration, k indicates the step

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of random walk (iteration in this paper), and the stochastic function r  k  is given as follows. 1 if   0.5 (14) r k    ,    0,1 0 if   0.5 Where  is the random number generated with uniform distribution. To make the random movement inside the search space, the min–max normalization is adopted, and the positions of ants can be obtained by 7

X

k t

X 

k t

 at  dtk  ctk 

 bt  at 

 ct

(15)

Where at and bt indicate the minimum and maximum of random movement of tth variable, respectively. ctk and d tk are the minimum and maximum of t-th variable at k-th iteration, respectively.

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2) Trapping in antlion’s pits The two following equations are used to mathematically model the random walks of ants, which are affected by antlions’ traps.

ctk  Antliontk  ck d  Antlion  d k t

k

k t

(16)

k

(17)

k

k

k

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where c (d ) is the minimum (maximum) of all variables at k-th iteration, ct (dt ) is the minimum (maximum) of all variables at for t-th ant, and Antliontk represents the position of the selected t-th antlion at k-th iteration.

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3) Establishing trap The fittest ant-lion is selected by applying the roulette wheel selection.

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4) Sliding ants towards antlion In order to model the sliding ants towards antlions, the scopes of the random movement should be decreased adaptively:

ck c  I dk k d  I k

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(18)

Where I  10 k K , k indicates the current iteration, K  iterationmax and represents a constant defined, whose details is given as follows: 2 if k  0.1K 3 if k  0.5K  w  4 if k  0.75K 5 if k  0.9 K  6 if k  0.95K

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w

(19)

w

(20)

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5) Catching prey and rebuilding the pit If an ant reaches the bottom of the pit and it will be caught and consumed by the antlions. After this, the antlions are waiting for catching new preys by updating their positions. The above-mentioned process can be expressed by the following equation. Antlionkj  Antik if f  Antik   f  Antlionkj  (21) 6) Elitism As a key trait of evolutionary algorithms, elitism is utilized to store the best solutions during the optimization process. In this algorithm, the best antlion obtained is regarded as an elite, which should have an impact on the whole ants in every stage. 8

Thus, affected by the roulette wheel and the elite at the same time, every ant randomly moves towards a selected antlion, which can be expressed as follows:

Antik 

RAk  REk 2

(22)

Where RA and RE indicate the random move towards the antlion selected by the roulette wheel and the elite, respectively. 2.3.1 MOALO

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Minimize : F ( x )   f1 ( x ), f 2 ( x ),..., f o ( x )

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Definition 1. As for MOO problems, the multiple objectives are often in conflict. In fact, the answer of MOO problems is a set of solutions called Pareto optimal solutions set (see Definition 4) which includes the best trade-offs between the objectives, also named the Pareto optimal solutions. In general, the multi-objective optimization problems can be expressed by a minimization problem.

t  1, 2,..., m1

ht ( x )  0,

t  1, 2,..., p

lt  xt  ut ,

t  1, 2,..., n1

(23)

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Subject to : gt ( x )  0,

n1 , o, m1 and p are the number of the variables, objective functions, inequality constraints and equality constraints, respectively. g t and ht represent

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the t-th inequality and equality constraints. [lt , ut ] indicate the boundaries of t-th variable.

y  iff:

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Definition 2. Pareto dominance: Different from the single-objective ALO, MOALO has more than one objective functions, which need to consider at the same time. Obviously, the traditional relational operators such as  ,  ,  ,  or = cannot be effectively utilized for comparing solutions of the multiple objectives. In this case, a crucial concept i.e. Pareto dominance is introduced to address this problem, whose detailed information is given as follows: Given two vectors x   x1 , x2 ,..., x p  and y   y1 , y2 ,..., y p  . x dominates y j  1, p  ,  f  x j   f  y j   j  1, p  : f ( x j ) 

(24)

Moreover, the definition of Pareto optimality, Pareto optimal set and Pareto optimal front can be expressed by the following three equations, respectively.

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Definition 3. Pareto optimality: A Pareto-optimal x  X iff:

y  X s.t. F ( y) F ( x)

Definition 4. Pareto optimal set: Ps :=  x, y  X F ( y)

F ( x )

(25)

(26)

Definition 5. Pareto optimal front: A set containing the value of objective functions for Pareto solutions set: (27) Pf := F ( x) x  Ps  9

Definition 6. As the above mentioned, the Pareto optimal solutions are the answers in MOALO. Thus, to search for the Pareto optimal solutions set with a high diversity, this algorithm adopts the leader selection and archive maintenance to store Pareto optimal solutions, and to select a non-dominated solution from the archive, a roulette wheel as well as equation (22) are also utilized. Furthermore, to enhance the distribution of the solutions in the archive, two mechanisms are also used in MOALO. Firstly, the probability of the antlions selected can be written as: c Pj  , c 1 (28) Nj

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where N j represents the number of solutions in the vicinity of the j-th solution, c represents a constant. Secondly, if the archive is full, the concentrated solutions will be removed so as to accommodate the new ones. And the removing probability can be used as 1 Pj . Finally, with the aim of making ALO solve multi-objective problems, the equation (21) should be modified in MOALO. (29) Antlionkj  Antik if f  Antik  f  Antlionkj  In conclusion, the remainder sections of MOALO are the same with ALO.

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2.4 MOALO-DCRGM (1, 1) To obtain high precision and strong stability simultaneously, a hybrid forecasting model—DCRGM (1, 1) optimized by MOALO is presented. The informative descriptions of the hybrid MOALO-DCRGM (1, 1) model can be given as the following steps. Moreover, the corresponding algorithm is also given in Algorithm 1.

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Step 1: Initialize the parameters in MOALO algorithm.

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Step 2: Determine the objective functions of MOALO algorithm. 1 N  obf ( y )  MSE  (YPRE (t )  YACT (t ) ) 2   1 N Minimize  i 1 obf ( y )  std (Y PRE ( t )  YACT ( t ) ), i  1, 2,..., N  2

(30)

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where N is the number of samples, YACT (t ) and YPRE (t ) are the t-th actual and predictive values of wind speed data, respectively. Step 3: Update the positions based on the obtained values of the objective function, next, run to next generation.

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Step 4: Set target conditions and begin optimizing. Specifically, search for the non-dominated solutions and update the archive during the process of iteration. If reaching the maximum number of iterations or expected error, then run the next step; otherwise, continue to run Step 4. Step 5: Optimize DCRGM (1, 1) using the obtained parameters in MOALO. Finally, input the historical data into forecasting model to obtain the forecasting value YPRE (t ) . Algorithm 1: MOALO-DCRGM (1, 1) 1 /*Set the parameters of MOALO. */ 10

2 WHILE the end condition is not met 3 FOR EACH ANT /*Choose a random antlion from the archive. */ 4 /*Choose the elite applying Roulette wheel from the archive. */ 5 /* Update c k and d k according to two following formulas. */ 6 7 ck  ck I

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Antik  ( RAk  REk ) 2

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END FOR /*Calculate the objective values of all ants. */ /*Find the non-dominated solutions. */ /*Update the archive in regard to the obtained non-dominated solutions. */ IF the archive is full DO /* Delete some solutions from the archive to hold the new solutions. */ Applying Roulette wheel and Pj  N j c , c  1 END IF IF any new added solutions to the archive are outside boundaries DO /* Update the boundaries to cover the new solution(s). */ END IF END WHILE RETURN archive Obtain X*=Select Leader(archive) Set parameters of DCRGM (1, 1) according to X*. Use xi to train and update the parameters of the DCRGM (1, 1) Input the historical data into DCRGM (1, 1) to obtain forecasting value YPRE (t ) .

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/*Update the position of ant. */

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X tk   X tk  at  dtk  ctk   bt  at   ct

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/* Create and normalize a random walk by following two formulas. */ X (k )  0, cumsum  2r  k1   1 ,..., cumsum  2r  km  1

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3. Data description and Error measures To test the performance of the forecasting models, samples from eight different regions in China as well as four widely utilized error measures are adopted in this section.

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3.1. Data description In this study, small samples of annual electricity consumption between 2000 and 2014 from several regions i.e. Liaoning, Inner Mongolia, Beijing and Shanghai etc. of China are adopted to test the performance of the proposed model. In addition, the traits i.e. maximum, minimum and average values etc. of the several small samples are listed in Table 1. Table 1 The traits of the small samples between 2000 and 2014 (Unit: 108kWh). Locations Min. Max. Average Std. 11

254.2 748.9 384.4 971.3 559.4 503.0 401.5 314.4

2416.7 2038.7 937.1 5012.5 1410.6 1656.5 1855.8 1308.0

1159.2 1363.4 656.0 2908.7 1022.4 1029.6 1037.9 746.1

735.8 462.5 190.2 1416.1 299.7 410.8 480.0 350.5

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Inner Mongolia Liaoning Beijing Jiangsu Shanghai Hubei Fujian Guangxi

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3.2. Error measures Four effective error measures including the percentage error (PE) [5], the mean absolute error (MAE) [36] reflecting the overall level of errors, the root mean square error (RMSE) [16], which can reflect the degree of differences between the observed and forecasted values, and the mean absolute percent error (MAPE) [37], as a measure of the prediction accuracy of a forecasting method in statistics. These measures are all applied to evaluate the performance of the forecasting models, which are given as following, where N is the number of test samples, YACT (t ) and YPRE (t ) are the

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actual and forecast values, respectively. Moreover, the criterion of MAPE is also listed in Table 2 in this study. YACT (t )  YPRE (t ) (31) PE  100% YACT (t )

1 N MAE   YACT (t )  YPRE (t ) N t 1

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(32)

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RMSE 

Table 2

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MAPE 

1 N

 Y N

ACT ( t )

t 1

 YPRE (t ) 

2

(33)

1 N YACT (t )  YPRE (t ) 100%  Y N t 1 ACT ( t )

(34)

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Criteria of MAPE [5, 38-39]

MAPE (%)

Forecasting power

<10

Excellent

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10—20

Good

20—50

Reasonable

>50

Incorrect

4. Empirical study It’s well known that reliable and accurate electricity consumption forecasting not only can assist decision-makers planning electricity management, arranging reasonable operation modes, but also help to reduce the loss of auxiliary power and electricity networks and enhance the security and stability of the electrical power 12

system. Hence, developing a robust, effective as well as accurate forecasting model is extremely desirable. In this section, the experimental details such as the experimental design, results and analysis are all introduced as follows. Moreover, all the experimental datasets are performed in the MATLAB R2014b environment running on Windows 7 with a 64-bit 3.30 GHz Intel Core i5 4590 CPU and 8.00 GB of RAM.

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4.1 Experimental design Owing to the historical relationship between the annual electricity consumption and the economic indexes, the development tendency of annual electricity consumption of China can be taken as a gradual increase. Moreover, according to its characteristic, in this study the grey theory system is utilized to predict annual electricity consumption. Nevertheless, the traditional GM (1, 1) cannot be satisfied. Thus, an improved grey forecasting model—DCRGM (1, 1), based on the MOALO algorithm, is proposed, which can enhance the forecasting accuracy and stability by estimating the optimal parameters and reducing the convergent error to a minimum. To demonstrate the effectiveness of the proposed forecasting model, the real case of annual electricity consumption in China is considered as an example. Moreover, to further verify the presented hybrid forecasting model, seven different forecasting models i.e. ARIMA, MLP, GM (1, 1), RGM (1, 1), back propagation neural network (BPNN), GM (1, 1) optimized by ALO (ALO-GM (1, 1)) and RGM (1, 1) optimized by ALO (ALO-RGM (1, 1)) proposed by Ref. [5], are also utilized as benchmark models in this paper. As for the variation and characteristics of the annual power consumption, it’s widely accepted that the length of the input training sets has a seriously impact on the accuracy of the GM (1, 1) prediction, and as there isn’t a clear theory or rule to assist researchers to set the number of the input subset. Therefore, based on trial and error [40], an improved grey forecasting model—DCRGM (1, 1) is developed, which can be employed to dynamic select the best input training sets. Moreover, Fig. 2 shows the flowchart of the proposed model, which including four parts: the optimization process of MOALO algorithm, eight different study areas in China, single forecasting model and a proposed hybrid model—DRCGM (1, 1) optimized by MOALO algorithm. Moreover, in this paper there are two objective functions in multi-objective optimization algorithm, the figures concerning solutions from Pareto-optimal sets of electricity consumption data using MOALO are also shown in part C, to equally evaluate these two objective functions of the proposed model, the solution at middle position of the Pareto-optimal sets is selected in this paper.

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Fig. 2. The flowchart of the proposed model

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4.2 Results analysis As mentioned above, in order to evaluate the performance of the proposed model as well as other different models for forecasting eight regions annual electricity consumption in China, two different kinds of forecasting mechanisms i.e. ARIMA, Multi-layer Perceptron ( MLP), GM (1, 1), BPNN and ALO-GM (1, 1) without rolling mechanism, RGM (1, 1), ALO-RGM (1, 1) and the developed model with rolling mechanism are adopted in this study, whose detailed forecasting results from 2011 to 2014 are listed in Table 3, where the values in bold represent the smallest values of MAE, RMSE and MAPE. Moreover, the corresponding optimal parameters (a, b) and the annual forecasting values of PE, which are given in Tables 4 and 5. As a result, the annual electricity consumption of the eight regions in China from 2011 to 2014 can be forecasted. According to the results of the above-mentioned forecasting models listed in Table 3 and Fig. 3, it can be observed that the proposed hybrid forecasting model acquires the highest prediction accuracy (via the MAE, RMSE, and MAPE criteria). Finally, it can make a conclusion that the developed MOAOL-DCRGM (1, 1) model 14

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significantly outperforms than the seven compared models, except for Liaoning when compared with ALO-RGM (1, 1), the MAPEs of ALO-RGM (1, 1) and the developed model are 4.4246% and 4.8470%, respectively. All in all, it is more suitable than the others to forecast annual power consumption. More detailed comparisons and analyses are given as follows. For the models without rolling mechanism i.e. GM (1, 1), ARIMA, MLP, BPNN and ALO-GM (1, 1), which possess poor prediction accuracy because of their unsatisfied forecasting results. Among the five mentioned forecasting methods, the forecasting results of ALO-GM (1, 1) reveal that it has higher prediction accuracy when compared with GM (1, 1) and BPNN, due to the parameters (a, b) of GM (1, 1) can be obtained by ALO. And ALO-GM (1, 1) performs better than ARIMA and MLP in most cases. However, when comes to the forecasting models with rolling mechanism, such as RGM (1, 1), ALO-RGM (1, 1) and MOALO-DCRGM (1, 1), ALO-GM (1, 1) still shows large forecasting errors, except for Shanghai, because the MAPEs of the ALOGM (1, 1), RGM (1, 1) and ALO-RGM (1, 1) are 5.1144%, 8.8998% and 8.0739%, respectively. For the forecasting models with rolling mechanism i.e. RGM (1, 1), ALO-RGM (1, 1) and MOALO-DCRGM (1, 1), which can obtain more reliable and accurate electricity consumption forecasting results than those compared forecasting models without rolling mechanism, because the forecasting results of RGM (1, 1), ALO-RGM (1, 1) and MOALO-DCRGM (1, 1) are more consistent with the actual values, especially for the developed MOALO-DCRGM (1, 1) model. Moreover, as an efficient method, the rolling mechanism has been widely used and proved that it can be used to improve the prediction accuracy because it can not only make full use of the most recent samples but also capture the latest development tendency and characteristic of datasets. With the aim of further demonstrating the satisfactory performance of the proposed model from various angles, the corresponding annual forecasting results and PEs of different models are also utilized in this study, which have been given in Table 5, from which it can be obviously found that the fitting precision of the presented model is better than another seven compared forecasting methods in most cases. However, sometimes the proposed model cannot perform well among the forecasting models, although it has smaller values of MAPEs. It is very interesting that BPNN model acquires better prediction performance when compared with the other models in some years such as in 2012, 2013 of Fujian and in 2012 of Guangxi, where the corresponding PEs are -1.76%, 3.53% and 7.08%, respectively. However, in the most cases the BPNN cannot achieves higher forecasting accuracy because it always requires more experience and several experiments and large training samples, which does not be suitable for forecasting annual electricity consumption. Furthermore, when compared with GM (1, 1), ALO-GM (1, 1) significantly outperforms GM (1, 1) model for its parameters can be obtained by ALO algorithm, which can perform better than that by using least square estimation method. Meanwhile, the forecasting results indicate that the values of MAE, RMSE and MAPE listed in Tables 3 and 5 of the proposed MOALO-DCRGM (1, 1) model are the smallest than that of another different forecasting methods, similarly to the PEs in most cases. Finally, it can safely make a conclusion that the developed model, MOALO-DCRGM (1, 1), is convincingly a robust, highly accurate and practical forecasting model for annual electricity consumption prediction, because it can obtain better forecasting results than BPNN, GM (1, 1) and ALO-GM (1, 1) whether they are using rolling mechanism or not. 15

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Table 3 Results of MAE, RMSE and MAPE for different forecasting models. Indices

ARIMA

MLP

Inner Mongolia

MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE MAE RMSE MAPE

420.7926 447.1931 19.3600 231.8384 242.5403 11.7551 36.8721 37.3071 4.1542 311.7345 384.8781 6.4764 108.0843 127.3710 7.8886 207.4948 220.1759 13.0927 201.4144 271.0233 11.6825 90.7700 122.4098 7.1586

248.2644 251.4864 11.9546 152.2257 161.9697 7.8439 44.5910 54.8944 4.9737 373.2452 394.7534 7.9228 98.8199 109.9025 7.2638 127.1591 143.8255 8.2264 140.5230 157.1841 8.3504 80.7558 90.3197 6.7714

Jiangsu

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Shanghai

Hubei

A

Fujian

Guangxi

M

ED

Beijing

PT

Liaoning

A

Region

GM (1,1)

BPNN

ALOGM (1,1)

RGM (1,1)

ALORGM (1,1)

MOALODCRGM (1,1)

497.9297 581.0020 22.3505 170.7507 213.0755 8.5299 99.9486 108.5041 11.0846 860.5193 1007.9878 17.7161 243.8649 276.7015 17.7459 139.9218 179.4212 8.6337 138.2438 163.3553 7.9654 148.3544 184.2386 11.8183

408.2975 495.0132 18.1714 164.9543 205.1085 8.2433 100.2911 108.8401 11.1230 816.4853 967.0137 16.7860 225.3515 257.7686 16.3969 141.8251 182.1655 8.7485 136.4661 160.7489 7.8667 153.2369 191.3902 12.1987

222.1782 249.8240 10.4844 148.3380 195.4643 7.6723 55.3437 58.4860 6.1711 400.8644 414.2267 8.6407 69.7630 73.7796 5.1144 92.3122 94.2019 5.9875 103.3558 121.3265 6.1177 94.6535 96.8089 7.8737

222.3978 231.7269 10.4068 106.8307 122.8329 5.3828 47.7692 48.2371 5.4086 397.8911 420.2273 8.3546 121.7362 126.4743 8.8998 92.3029 111.0381 5.7494 79.7897 86.7780 4.7421 96.1384 110.0262 7.7798

178.8615 194.6842 8.2699 88.2163 114.1716 4.4246 47.7943 48.2790 5.4120 361.8569 396.5586 7.6071 110.4497 112.6899 8.0739 86.2770 99.2263 5.3763 80.5595 87.1526 4.7946 94.3806 109.2625 7.6374

151.0534 166.6265 7.4075 94.3615 102.1711 4.8470 17.2973 20.4014 2.0103 221.5821 240.4744 4.6305 51.4969 55.8172 3.7608 63.8313 70.6544 4.0335 75.1231 87.1996 4.7319 63.3275 78.2483 5.3009

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IP T N U SC R A M ED PT CC E A Fig. 3. The Illustration of MAE, MSE and MAPE of benchmarks and proposed hybrid model for eight regions in China (Where GM, ALO-R*and MOALO-DCR* are GM (1, 1), ALO-RGM (1, 1) and MOALO-DCRGM (1, 1), respectively.)

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ALO-RGM (1,1) a b

MOALO-DCRGM (1,1) a b

-0.17236 -0.16649 -0.15182 -0.13477

269.48 339.48 446.82 580.03

-0.09470 -0.15260 -0.14771 -0.10789

986.24 893.80 1049.75 1381.28

2011 2012 2013 2014

1861.5 1899.9 2008.5 2038.7

-0.08681

678.29

-0.08681 -0.08678 -0.08026 -0.07351

678.29 745.64 849.05 960.06

-0.08597

683.65

-0.08676 -0.05483 -0.05560 -0.07700

678.71 979.54 1045.92 931.41

-0.07844 -0.09801 -0.07716 -0.04885

935.06 1189.18 942.75 1621.73

Beijing

2011 2012 2013 2014

821.7 874.3 913.1 937.1

-0.07635

363.76

-0.07635 -0.07060 -0.06639 -0.06057

363.76 406.97 446.97 496.56

-0.07637

363.79

-0.07637 -0.07060 -0.06641 -0.06058

363.79 406.91 446.83 496.50

-0.06756 -0.06036 -0.05118 -0.04263

518.04 636.26 695.80 752.40

Jiangsu

2011 2012 2013 2014

4281.6 4580.9 4956.6 5012.5

-0.12895

1046.62

-0.12895 -0.12089 -0.11026 -0.10088

1046.62 1261.49 1524.66 1800.64

-0.12886

1038.72

-0.12901 -0.12359 -0.10328 -0.10096

1037.23 1223.39 1619.24 1797.29

-0.09426 -0.11202 -0.10140 -0.08068

2269.17 2557.84 2643.91 3478.58

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Liaoning

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Table 4 The actual data and the optimal parameters values of different GM (1, 1) for 2011–2014. Actual data GM (1,1) RGM (1,1) ALO-GM (1,1) Region Year (Unit:108kWh) a b a b a b -0.17318 277.18 -0.17318 277.18 -0.17236 269.48 Inner Mongolia 2011 1864.1 -0.16653 347.61 2012 2016.8 -0.15183 453.54 2013 2181.9 -0.13567 580.34 2014 2416.7

2011 2012 2013 2014

1339.6 1353.5 1410.6 1369.0

-0.08038

566.85

-0.08038 -0.07338 -0.06321 -0.05563

566.85 643.28 736.48 818.33

-0.07813

577.80

-0.07972 -0.07343 -0.06322 -0.04804

569.69 642.70 736.18 871.16

-0.06340 -0.05803 -0.04987 -0.02657

848.13 976.69 1002.00 1248.78

Hubei

2011 2012 2013 2014

1450.8 1507.9 1629.8 1656.5

-0.10276

439.99

-0.10276 -0.10326 -0.09706 -0.09154

439.99 488.82 565.27 643.80

-0.10285

440.28

-0.09457 -0.09014 -0.10488 -0.08506

475.80 552.10 525.60 682.11

-0.09969 -0.10095 -0.09389 -0.06455

682.35 678.91 701.16 1223.65

Fujian

2011 2012 2013 2014

1515.9 1579.5 1700.7 1855.8

-0.11534

398.19

-0.11534 -0.11516 -0.10776 -0.10076

398.19 450.62 530.48 614.43

-0.11534

397.39

-0.11534 -0.11532 -0.10780 -0.10076

397.39 450.09 530.22 614.08

-0.09061 -0.12217 -0.10131 -0.07901

859.45 854.09 756.23 1209.58

Guangxi

2011 2012 2013 2014

1112.2 1153.9 1237.7 1308.0

-0.12568

255.68

-0.12568 -0.12705 -0.11840 -0.10951

255.68 289.62 350.38 415.07

-0.12593

256.43

-0.12593 -0.12724 -0.11833 -0.10906

256.43 290.22 350.51 415.72

-0.13147 -0.13077 -0.10923 -0.06849

418.77 617.94 592.21 918.72

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Shanghai

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PE comparison of different models (Unit: %)

RGM (1, 1)

ALORGM (1, 1)

MOALODCRGM (1,1)

ARIMA

MLP

GM (1, 1)

BPNN

ALOGM (1, 1)

RGM (1, 1)

ALORGM (1, 1)

MOALODCRGM (1,1)

1916.42 2276.92 2705.23 3214.12

1981.61 2269.01 2475.28 2643.19

1916.42 2221.03 2441.90 2615.59

1639.18 2129.97 2393.23 2471.50

10.47 18.80 23.55 24.62

14.51 15.02 9.51 8.78

-6.30 -16.83 -28.41 -37.85

16.58 -1.33 11.86 12.17

-2.81 -12.90 -23.99 -33.00

-6.30 -12.51 -13.45 -9.37

-2.81 -10.13 -11.92 -8.23

12.07 -5.61 -9.69 -2.27

1903.44 2264.58 1897.21 1963.26

1845.33 2010.98 2191.51 2388.25

1849.90 2020.39 2123.57 2218.86

1850.00 1816.82 1956.01 2244.49

1809.68 2056.30 2105.16 2111.26

7.72 9.53 14.35 15.41

8.42 9.07 10.74 3.14

0.62 -6.20 -9.57 -17.73

-2.25 -19.19 5.54 3.70

0.87 -5.85 -9.11 -17.15

0.62 -6.34 -5.73 -8.84

0.62 4.37 2.61 -10.09

2.78 -8.23 -4.81 -3.56

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Table 5 The forecasting results and PE comparison of different forecasting models. Forecasting results (Unit:108kWh) Region Year ALO-

876.55 946.10 1021.16 1102.18

862.68 841.53 843.34 859.24

876.83 946.42 1021.53 1102.60

876.55 913.54 956.31 990.87

876.83 913.46 956.20 990.89

857.30 887.78 920.53 949.77

-3.79 -4.82 -3.42 -4.58

-0.40 9.14 7.82 2.55

-6.68 -8.21 -11.83 -17.62

-4.99 3.75 7.64 8.31

-6.71 -8.25 -11.87 -17.66

-6.68 -4.49 -4.73 -5.74

-6.71 -4.48 -4.72 -5.74

-4.33 -1.54 -0.81 -1.35

3958.69 4221.38 4376.25 4782.29

4541.75 5166.86 5878.02 6687.05

4839.05 4273.55 5390.40 5317.36

4506.51 5126.31 5831.35 6633.37

4541.75 4955.90 5292.17 5633.35

4508.39 4969.91 5171.83 5628.90

4159.69 4784.37 5142.94 5387.10

-3.66 -5.37 -2.97 -13.90

7.54 7.85 11.71 4.59

-6.08 -12.79 -18.59 -33.41

-13.02 6.71 -8.75 -6.08

-5.25 -11.91 -17.65 -32.34

-6.08 -8.19 -6.77 -12.39

-5.30 -8.49 -4.34 -12.30

2.85 -4.44 -3.76 -7.47

1376.65 1452.40 1488.74 1587.25

1161.08 1267.92 1328.91 1418.50

1423.19 1542.30 1671.39 1811.28

1283.53 1435.40 1449.75 1470.93

1411.92 1526.65 1650.70 1784.84

1423.19 1484.43 1509.77 1542.26

1419.17 1483.89 1509.36 1502.07

1359.48 1423.14 1454.08 1441.98

-2.77 -7.31 -5.54 -15.94

13.33 6.32 5.79 -3.62

-6.24 -13.95 -18.49 -32.31

4.19 -6.05 -2.78 -7.45

-5.40 -12.79 -17.02 -30.38

-6.24 -9.67 -7.03 -12.66

-5.94 -9.63 -7.00 -9.72

-1.48 -5.15 -3.08 -5.33

2011 2012 2013 2014

1349.62 1322.47 1329.27 1413.66

1300.44 1389.88 1416.54 1629.50

1446.96 1603.55 1777.09 1969.41

1327.21 1596.63 1555.55 1573.82

1449.39 1606.40 1780.41 1973.28

1446.96 1606.88 1718.13 1834.55

1413.19 1545.18 1759.00 1797.52

1438.33 1598.73 1703.95 1734.37

6.97 12.30 18.44 14.66

10.36 7.83 13.08 1.63

0.26 -6.34 -9.04 -18.89

8.52 -5.88 4.56 4.99

0.10 -6.53 -9.24 -19.12

0.26 -6.56 -5.42 -10.75

2.59 -2.47 -7.93 -8.51

0.86 -6.02 -4.55 -4.70

Fujian

2011 2012 2013 2014

1672.59 1729.34 1698.37 2352.59

1343.74 1599.42 1528.66 1657.83

1492.99 1675.51 1880.34 2110.21

1380.47 1607.28 1640.65 1665.64

1490.42 1672.63 1877.13 2106.62

1492.99 1680.22 1811.45 1940.58

1490.42 1681.53 1811.55 1939.67

1410.69 1692.85 1781.30 1854.47

-10.34 -9.49 0.14 -26.77

11.36 -1.26 10.12 10.67

1.51 -6.08 -10.56 -13.71

8.93 -1.76 3.53 10.25

1.68 -5.90 -10.37 -13.52

1.51 -6.38 -6.51 -4.57

1.68 -6.46 -6.52 -4.52

6.94 -7.18 -4.74 0.07

Guangxi

2011 2012 2013 2014

1113.20 1194.41 1341.08 1526.19

975.56 1128.27 1171.14 1213.81

1105.35 1253.38 1421.23 1611.56

1026.65 1072.22 1107.97 1226.35

1111.36 1260.51 1429.67 1621.53

1105.35 1260.39 1361.99 1454.93

1111.37 1265.43 1361.37 1449.48

1126.26 1274.40 1333.97 1330.48

-0.09 -3.51 -8.35 -16.68

12.29 2.22 5.38 7.20

0.62 -8.62 -14.83 -23.21

7.69 7.08 10.48 6.24

0.08 -9.24 -15.51 -23.97

0.62 -9.23 -10.04 -11.23

0.07 -9.67 -9.99 -10.82

-1.26 -10.44 -7.78 -1.72

MLP

GM (1, 1)

BPNN

2011 2012 Inner Mongolia 2013 2014

1668.91 1637.70 1668.07 1821.64

1593.60 1713.93 1974.44 2204.47

1981.61 2356.28 2801.79 3331.54

1555.05 2043.66 1923.12 2122.67

Liaoning

2011 2012 2013 2014

1717.78 1718.77 1720.21 1724.49

1704.84 1727.49 1792.70 1974.66

1849.90 2017.66 2200.64 2400.20

Beijing

2011 2012 2013 2014

852.87 916.48 944.33 980.01

824.95 794.43 841.73 913.23

Jiangsu

2011 2012 2013 2014

4438.13 4827.01 5103.95 5709.45

Shanghai

2011 2012 2013 2014

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ARIMA

GM (1, 1)

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5. Discussions Although most previous developed grey forecasting methods as well as the proposed model are reported to be proper and helpful for the small sample forecasting, the detailed issues such as the effectiveness of proposed model, discussions of grey forecasting and the numbers of the input training sets, other multiobjective algorithms as well a small sample should be taken into consideration. The relative details are discussed as follows.

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5.1 Discussion of the effectiveness of proposed model In this subsection, three improvement percentage error criteria (listed in Table 6) are utilized to discussion the effectiveness of proposed model through comparing the forecasting results of the compared models i.e. ARIMA, MLP, BPNN, GM (1, 1), ALO-GM (1, 1) etc. In addition, Table 7 displays the results about the improvement percentages among the proposed model and the compared forecasting models. From Table 7, it can be found that when comparing with the other models, the improvement about the forecasting performance of the developed model is obvious. The details are clearly displayed in Table 7.

The improvement percentages of MAE

PRMSE

The improvement percentages of RMSE

PMAPE

PMAE 

MAE1  MAE2 MAE1

PRMSE 

RMSE1  RMSE2 RMSE1

The improvement percentages of MAPE

PMAPE 

MAPE1  MAPE2 MAPE1

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Table 6 The improvement percentages among different forecasting models Metric Definition Equation

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Table 7 Improvement percentages among the proposed model and the compared forecasting models. Inner Models Indices Liaoning Beijing Jiangsu Mongolia PMAE (%) 64.1026 59.2986 53.0884 28.9196 PRMSE (%) 62.7395 57.8746 45.3150 37.5193 Proposed vs. ARIMA PMPAE (%) 61.7381 58.7668 51.6080 28.5019

Shanghai

Hubei

Fujian

Guangxi

52.3549 56.1775 52.3261

69.2372 67.9100 69.1928

62.7022 67.8258 59.4958

30.2330 36.0768 25.9506

PMAE (%) PRMSE (%) PMPAE (%)

39.1562 33.7433 38.0364

38.0121 36.9196 38.2068

61.2090 62.8352 59.5814

40.6336 39.0824 41.5548

47.8881 49.2121 48.2254

49.8020 50.8749 50.9688

46.5404 44.5239 43.3333

21.5815 13.3652 21.7163

PMAE (%) PRMSE (%) PMPAE (%)

63.0041 66.3390 59.2354

42.7954 50.1868 41.2007

82.7529 81.2556 81.9266

72.8615 75.1323 72.4145

77.1482 78.3460 77.0640

54.9929 61.2142 53.8950

44.9511 45.7542 39.8490

58.6735 59.1158 56.5454

PMAE (%) PRMSE (%) PMPAE (%)

69.6637 71.3208 66.8576

44.7373 52.0493 43.1764

82.6938 81.1976 81.8640

74.2502 76.1431 73.8628

78.8830 79.8276 78.8075

54.3807 60.6209 53.2819

45.6590 46.6197 40.5943

57.3134 57.5288 55.1467

PMAE (%) PRMSE (%) PMPAE (%)

32.0796 28.0936 28.8206

11.6719 16.8211 9.9539

63.7898 57.7060 62.8314

44.3109 42.7752 44.5754

57.6980 55.8668 57.7429

30.8458 36.3692 29.8449

5.8486 0.4858 0.2151

34.1288 28.8821 31.8633

Proposed vs. ALO-GM (1, 1)

PMAE (%) PRMSE (%) PMPAE (%)

32.0125 33.3024 29.3474

36.3875 47.7290 36.8247

68.7457 65.1175 67.4240

44.7239 41.9462 46.4106

26.1831 24.3460 26.4664

30.8528 24.9968 32.6347

27.3160 28.1282 22.6523

33.0954 19.1724 32.6759

Proposed vs. ALO-RGM (1, 1)

PMAE (%) PRMSE (%) PMPAE (%)

15.5473 14.4119 10.4282

6.9661 10.5109 9.5466

63.8089 57.7427 62.8548

38.7653 39.3597 39.1292

53.3752 50.4683 53.4203

26.0159 28.7947 24.9763

6.7483 0.0539 1.3077

32.9020 28.3850 30.5929

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5.2 Testing based on the DM test and the forecasting effectiveness Concentrating on further comparing the forecasting performance of the proposed hybrid model with other compared models, this subsection adopts an effective hypothesis testing method—Diebold-Mariano (DM) test and forecasting effectiveness (FE) method to demonstrate the superiority of the developed hybrid model. Table 8 gives the results of the DM test and FE, from which it can be observed that the values of DM test are all bigger than the Z0.05 2  1.96 , indicating that the developed hybrid model is superior to the comparison models under 5% significance level. Additionally, the values of forecasting effectiveness also confirm the excellent performance of this proposed forecasting model. Finally, it can be concluded that the developed hybrid model has a more accurate forecasting ability than the other models adopted in this study. More information about DM test and FE method are shown in Appendix C.

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Table 8 Results for the DM test and the forecasting effectiveness. Models FE-1 FE-2 DM test 0.898040 0.848422 3.179203* ARIMA 0.920866 0.884326 2.298804** MLP 0.867695 0.792029 2.075640** GM (1, 1) 0.927423 0.893608 2.220480** BPNN 0.875581 0.799892 1.973598** ALO-GM (1, 1) 0.929095 0.900215 2.564742** RGM (1, 1) 0.935506 0.906119 1.997726** ALO-RGM (1, 1) Proposed model 0.954098 0.927736 — * is the 1% significance level Z 0.01 2  2.58 ; ** is the 5% significance level Z 0.05 2  1.96

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5.3 Input training number of grey forecasting Considering the input subset of traditional time series consists of the whole original data, but the whole original does not always reflect the internal regularity of time series. And there is no such general theory or rule that can be followed to determine the exact input training number for the power consumption forecasting. Although four observations are enough to design a grey model to conduct prediction, in this subsection it’s suggested that at least five data can be employed to establish DCRGM (1, 1). In this paper, an improved grey forecasting model DCRGM (1, 1) is presented, which can be utilized to dynamic choose the optimal input training sets. The corresponding input numbers of the optimal training data are listed in Table 9.

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Table 9 The number of the input training dataset of DCRGM (1, 1). Input training number of DCRGM (1, 1) Region Year 5 7 8 9 10 D00. 6 Inner Mongolia 2011 √ 2012 √ 2013 √ 2014 √ Liaoning 2011 √ 2012 √ 2013 √ 2014 √ 22

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Fujian

Guangxi

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√

√

√

√

√

√ √ √

√

√

√

√

√

√ 3

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√

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2011 2012 2013 2014 2011 2012 2013 2014 2011 2012 2013 2014 2011 2012 2013 2014 2011 2012 2013 2014 2011 2012 2013 2014

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From Table 9, some detailed discussions are as follows: (a) The total datasets are all from 2000 to 2014 in eight regions of China, if 2011(2014) is forecasted, the whole sample is from 2000 to 2010 (2013), the role of DCRGM (1, 1) is to determine the beginning point of its input training sample. (b) As an example, for Inner Mongolia in 2011, the input training number of DCRGM (1, 1) is 5, which represents that the sample from 2006 to 2010 is the best input training dataset for forecasting the 2011 electricity consumption in Inner Mongolia. (c) It can be observed form Table 9 that the most frequent number is 5 with appearing eighteen times, and the followed most frequent numbers is 7 with appearing six times. However, through further analyzing the above-mentioned cases, it can be found that the most frequent number 5 is most suitable for forecasting electricity consumption in 2014. Similar to the followed most frequent number: 7, which is most fit for predicting electricity consumption in 2011. (d) Based on the above-mentioned analysis as well as the characteristics of GM (1, 1), it can make a conclusion that as for different samples in different regions, making full use of the sufficient or certain information, sometimes does not obtain best forecasting results, due to different samples may have different tendency during different periods. Finally, the dynamic choice strategy adopted in this paper can be considered by scientists for prediction in many other fields.

5.4 Comparing with other multi-objective optimization algorithms This subsection is aimed at demonstrating the advantages of the multi-objective 23

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optimization algorithms applied to optimized grey models for small samples. It’s well known that considerable multi-objective optimization algorithms have been developed so far, as it cannot list the whole results of these algorithms, here four recent developed multi-objective algorithms i.e. MOALO, MODA, MOGWO and MOWCA are adopted to compare with each other and discuss the performances of multiobjective algorithms. And the corresponding results are given in Table 10 and Tables D1—D2, from which some comparisons and conclusions can be drawn as follows: (a) Based on the above analyses as well as the experimental forecasting results in Table 10, it can be observed that the four hybrid models using multi-objective algorithms obtains the large improvement compared with other forecasting models such as ARIMA, MLP, GM (1, 1), BPNN, ALO-GM (1, 1) etc., which indicate that the four models can perform better with their high prediction accuracy. (b) However, when comparing with each other among the four models in Table 10, it can be observed that MOALO-DCRGM (1, 1), MOGWO-DCRGM (1, 1) and MOWCA-DCRGM (1, 1) possess the similar prediction accuracy in most cases, which can perform better than MODA-DCRGM (1, 1) in most cases. For example, as for Shanghai (Guangxi), the MAPE values of the MOALO-DCRGM (1, 1), MODA-DCRGM (1, 1), MOGWO-DCRGM (1, 1) and MOWCA-DCRGM (1, 1) are 3.7608% (5.3009%), 5.2643% (7.1019%), 3.7636% (5.3056%), and 4.4341% (5.3106%), respectively. (c) Moreover, the four models i.e. MOALO-DCRGM (1, 1), MODA-DCRGM (1, 1), MOGWO-DCRGM (1, 1) and MOWCA-DCRGM (1, 1) possess higher prediction accuracy, which are all within 10%. According to Table 2, it can be found that they can obtain excellent forecasting results. (d) Overall, it can draw a conclusion that multi-objective optimization algorithms utilized for electricity consumption forecasting can really enhance forecasting performance of the models.

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Table 10 Results of MAE, RMSE and MAPE for the four hybrid forecasting models. MOALOMODARegion Indices DCRGM (1,1) DCRGM (1,1) 151.0534 140.4750 Inner Mongolia MAE 166.6265 159.6723 RMSE 7.4075 6.8067 MAPE 94.3615 101.4025 Liaoning MAE 102.1711 122.1173 RMSE 4.8470 5.1569 MAPE 17.2973 37.0775 Beijing MAE 20.4014 39.7395 RMSE 2.0103 4.2148 MAPE 221.5821 174.1475 Jiangsu MAE 240.4744 212.2427 RMSE 4.6305 3.6057 MAPE 51.4969 72.7100 Shanghai MAE 55.8172 86.1845 RMSE 3.7608 5.2643 MAPE 63.8313 71.3450 Hubei MAE 70.6544 79.2748 RMSE 4.0335 4.4968 MAPE 75.1231 68.6950 Fujian MAE 87.1996 73.7780 RMSE 4.7319 4.2665 MAPE 63.3275 83.0725 Guangxi MAE 78.2483 106.5151 RMSE 5.3009 7.1019 MAPE

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MOWCADCRGM (1,1)

151.3425 166.8918 7.4199 94.8150 102.4383 4.8707 17.2650 20.3514 2.0066 220.5425 239.5783 4.6086 51.5375 55.8189 3.7636 63.9325 70.8399

151.3875 166.9866 7.4230 94.3150 102.1232 4.8447 40.7825 57.0644 4.6964 205.0400 229.6470 4.2966 60.6100 68.6891 4.4341 63.8275 70.6525 4.0332 75.1125 87.2411 4.7315 63.4450 78.4098 5.3106

4.0384 75.0125 87.2819 4.7265 63.3825 78.2148 5.3056

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5.5 Small sample and grey forecasting It’s well known that excellent forecasting results play prerequisite and fundamental roles in the scientific development in modern economic and industrial fields. The traditional statistics methods and forecasting models such as ARIMA, ANNs and regression etc. are on the strength of many sample data. But due to the restriction of the statistical techniques and resources as well as the characters of data sets, sometimes it is difficult or impossible for scientists to collect incomplete information and large samples, in this situation it will limit the applications of traditional prediction methods [41]. Therefore, considering the length of sample data restricted in most realistic cases as well the importance of small sample forecasting such as the annual electricity consumption forecasting which can help to provide effective planning and ensure a reliable supply of sustainable electricity, it is necessary for researchers to pay more attention to the prediction of small sample. In other words, establishing robust forecasting methods with incomplete information and limited samples is extremely desirable and very vital to a business. Fortunately, grey system model is one kind of the forecasting models that is superior in terms of short-term predictions with small samples and poor information, which has been widely used in many fields. Moreover, by utilizing matrix perturbation theory, it has been proven that a grey prediction model is not suitable for large sample [42]. Over the past several decades, multifarious grey forecasting models and their improvement methods have been successfully developed and widely applied for predicting the demand and output of energy. To the best of our knowledge, these extended versions of the grey prediction models can be primarily divided into three types [22], whose details are given: (a) Grey models optimized by themselves for enhancing their forecasting performance. For instances, rolling grey forecasting models (RGM) utilized for predicting annual power load [35], grey model with optimizing initial condition [43] and background value and initial item for improving prediction precision of GM (1,1) model [44] etc. (b) Combining grey models with other modeling techniques, such as grey models combined with ARIMA [45], ANNs [46], chaotic theory [47], Markov approach [48], Genetic Algorithm (GA) [49] and so on. (c) Upgrading grey model structures and explore some novel methods for improving the accuracy. The following are some examples: the nonlinear grey Bernoulli model (NGBM) [50], kernel regularized nonhomogeneous grey model (KRNGM) [51], the discrete grey model, interval grey number prediction model [52], kernel-based nonlinear multivariate grey model [53] and non-homogenous discrete grey model [54]. Although there are many kinds of grey forecasting models successfully developed and used in many fields, but to further improve the prediction accuracy, scholars should develop other variations of GM (1, 1) encompassing the influence of different factors to provide accurate forecasts of the small samples and poor information, which can really help to achieve effective and efficient decision making in modern economic and industrial fields, even in a highly competitive business environment.

6. Conclusions Along with the rapid development of urbanization and industrialization, electricity consumption forecasting has been paid increasing attention to by many scientists and countries, due to it plays an irreplaceable role in the energy management and economic operation of power plants. Outstanding annual electricity consumption 26

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forecasts not only can help to build and adjust the structures of the national grid system according to the corresponding policy and the requirements of electricity demand, but also reduce generation costs, risks and improve the security of the electrical power system. On the contrary, poor forecasting accuracy will affect people’s daily life and probably result in huge economic losses, sometimes even leads to blackout events, such as the India’s power grids collapse happened on July 30 and 31, 2012, a large-area blackout event of Brazil and Paraguay blackouts etc. [55]. Thus, the ultimate issue of the forecasting models is to enhance forecasting accuracy as well as ensure getting strong and stable prediction results. For addressing this challenge, many useful methods were developed, which have been reported to be proper and advantageous for the power consumption forecasting. However, most of them are focus on improving prediction accuracy, which always ignore the stability of forecasting results, although it is important to the effectiveness of forecasting models. Aiming to develop robust methods for electricity consumption forecasting, which can simultaneously achieve high accuracy and stability, in this paper a novel hybrid forecasting model based on MOALO algorithm and DCRGM (1, 1) is successfully developed, which can not only be utilized to dynamic choose the best input training sets, but also obtain satisfactory forecasting results with high accuracy and strong ability. According to the forecasting results and analyses in section 4, it can be observed that the performance of the hybrid model is excellent by contrast to seven benchmarks. For instance, the MAPE values of the ARIMA, MLP, GM (1, 1), BPNN, ALO-GM (1, 1), RGM (1, 1), ALO-RGM (1, 1) and the proposed model in Jiangsu are 6.4764%, 7.9228%, 17.7161%, 16.7860%, 8.6407%, 8.3546%, 7.6071% and 4.6305%, respectively. Overall, this proposed model adds a novel viable option for electricity consumption forecasting, which can also be considered into another many forecasting issues such as the wave prediction, CO2 emissions, energy consumption and economic growth prediction, natural gas demand forecasting, and energy consumption prediction etc.

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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 71671029 and Grant No. 41475013).

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Appendix A Definition: Best input training sets A sample  yt  0, t  1, 2,..., N  15 , T0   yN 2 , yN 1, yN  is the testing subset, and the whole different training subset:

T1   y1 , y2 ,..., yN 3 , T2   y2 , y3 ,..., yN 3,..., TN 7   yN 7 , yN 6 ,..., yN 3

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(A1)

Where Tbest   ybest , ybest 1 ,..., yN 3 represents the best input training set iff:

Error Tbest   Min Error T1  ,..., Error Tbest  ,..., Error TN 7 

(A2)

Appendix B: Computational rules of "cumsum" in MATLAB. cumsum( A) returns the cumulative sum of A starting at the beginning of the first array dimension in A whose size does not equal 1. 27

(1) If A is a vector, then cumsum( A) returns a vector containing the cumulative sum of the elements of A . (2) If A is a matrix, then cumsum( A) returns a matrix containing the cumulative sums for each column of A . (3) If A is a multidimensional array, then cumsum( A) acts along the first nonsingleton dimension.

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Appendix C C.1. DM test Diebold-Mariano (DM) test [56, 57] is applied to compare the level predictive accuracy of different forecasting models from statistical perspective. Under the given significance level α, the null hypothesis H 0 is that the developed model and the

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H0 : E[ L( t1 )]  E[ L( t2 )], t  1,2

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H1 : E[ L( t1 )]  E[ L( t2 )], t  1,2

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(C3)

Where s 2 is an estimation for the variance of ( L( t )  L( t )). 2

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C.2. Forecasting effectiveness Forecasting effectiveness approach [58] can be applied to assess the performance of the presented hybrid model. A general discrete form of forecasting effectiveness is given as follows: The kth-order forecasting effectiveness unit can be expressed by: n (C4) mk   t 1 Qt Atk

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H (m1 , m2 ,..., mk ) can be written as:

m1 , H ( x)  x H (m , m ,..., m )   1 2 2  H (m , m ), H ( x, y)  x(1  y  x ) 1

2

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Where H ( x)  x is a single variable continuous function, the FE with 1st-order (FE1) is the expected forecasting accuracy, and H ( x, y)  x(1  y  x 2 ) is a two28

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variable continuous function, the FE with 2nd-order (FE-2) is the difference between the standard deviation and the expected value. Appendix D Table D1 and Table D2 are listed as follows.

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Table D1 The actual data and the optimal parameters values of different GM (1, 1) for 2011–2014. Actual data MOALO-DCRGM (1,1) MODA-DCRGM (1,1) Region Year (Unit:108kWh) a b a b -0.09470 986.24 -0.12144 849.31 Inner Mongolia 2011 1864.1 -0.15260 893.80 -0.09054 1203.07 2012 2016.8 -0.14771 1049.75 -0.15180 1026.87 2013 2181.9 -0.10789 1381.28 -0.11528 1337.32 2014 2416.7

MOGWO-DCRGM (1,1) a b

MOWAC-DCRGM (1,1) a b

-0.09476 -0.15241 -0.14764 -0.10778

986.14 895.06 1050.78 1382.24

-0.09470 -0.15261 -0.14766 -0.10790

986.25 893.95 1050.58 1381.18

2011 2012 2013 2014

1861.5 1899.9 2008.5 2038.7

-0.07844 -0.09801 -0.07716 -0.04885

935.06 1189.18 942.75 1621.73

-0.07850 -0.09060 -0.08589 -0.08097

938.33 1233.27 799.53 1429.31

-0.07842 -0.09798 -0.07714 -0.04881

934.49 1189.36 943.08 1622.45

-0.07845 -0.09801 -0.07717 -0.04885

934.96 1189.21 942.52 1621.74

Beijing

2011 2012 2013 2014

821.7 874.3 913.1 937.1

-0.06756 -0.06036 -0.05118 -0.04263

518.04 636.26 695.80 752.40

-0.03649 -0.04335 -0.06609 -0.05686

672.20 678.50 477.99 714.41

-0.06755 -0.06037 -0.05119 -0.04263

517.99 636.34 695.69 752.36

-0.06756 0.09692 -0.05118

518.01 1250.31 695.79 752.41

Jiangsu

2011 2012 2013 2014

4281.6 4580.9 4956.6 5012.5

-0.09426 -0.11202 -0.10140 -0.08068

2269.17 2557.84 2643.91 3478.58

-0.10323 -0.10216 -0.08874 -0.07963

2168.59 2696.83 2841.49 3497.98

-0.09426 -0.11197 -0.10140 -0.08058

2269.75 2558.13 2642.70 3480.19

-0.04263 -0.09427 -0.11203 -0.10386 -0.08068

2269.09 2557.44 2559.93 3478.57

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2011 2012 2013 2014

1339.6 1353.5 1410.6 1369.0

-0.06340 -0.05803 -0.04987 -0.02657

848.13 976.69 1002.00 1248.78

-0.07577 -0.04977 0.00309 0.01226

784.22 1033.18 1290.91 1489.07

-0.06342 -0.05794 -0.04985 -0.02658

848.03 977.14 1002.27 1248.75

-0.06340 -0.21347 -0.04987 -0.02657

848.13 238.95 1001.98 1248.78

Hubei

2011 2012 2013 2014

1450.8 1507.9 1629.8 1656.5

-0.09969 -0.10095 -0.09389 -0.06455

682.35 678.91 701.16 1223.65

-0.09811 -0.09474 -0.05887 -0.06338

689.69 717.40 1012.55 1228.98

-0.09969 -0.10101 -0.09375 -0.06469

682.59 678.39 702.49 1223.01

-0.09969 -0.10096 -0.09391 -0.06455

682.38 678.85 701.07 1223.64

Fujian

2011 2012 2013 2014

1515.9 1579.5 1700.7 1855.8

-0.09061 -0.12217 -0.10131 -0.07901

859.45 854.09 756.23 1209.58

-0.09599 -0.10623 -0.08547 -0.07044

839.69 920.74 872.95 1254.46

-0.09067 -0.12225 -0.10125 -0.07911

859.23 853.91 756.58 1209.36

-0.09060 -0.12211 -0.10130 -0.07902

859.62 854.58 756.24 1209.66

Guangxi

2011 2012 2013 2014

1112.2 1153.9 1237.7 1308.0

-0.13147 -0.13077 -0.10923 -0.06849

418.77 617.94 592.21 918.72

-0.00393 -0.08565 0.01338 -0.04893

920.69 739.61 1501.18 995.08

-0.13147 -0.13074 -0.10923 -0.06851

418.87 617.97 592.25 918.72

-0.13147 -0.13077 -0.10916 -0.06849

418.76 617.99 592.80 918.69

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Table D2 The forecasting results and PE comparison of different forecasting models. Forecasting results (Unit:108kWh) Region Year MOALOMODAMOGWOMOWCA-

PE comparison of different models (Unit: %)

DCRGM (1,1)

MOALODCRGM (1, 1)

MODADCRGM (1, 1)

MOGWODCRGM (1, 1)

MOWCADCRGM (1,1)

1639.64 2130.28 2394.33 2471.70

1639.28 2130.39 2394.24 2471.50

12.07 -5.61 -9.69 -2.27

11.30 2.48 -10.10 -3.34

12.04 -5.63 -9.74 -2.28

12.06 -5.63 -9.73 -2.27

1808.43 2056.27 2105.39 2111.63

1809.63 2056.32 2104.89 2111.28

2.78 -8.23 -4.81 -3.56

2.42 -7.37 1.37 -9.47

2.85 -8.23 -4.82 -3.58

2.79 -8.23 -4.80 -3.56

857.17 887.93 920.47 949.69

857.28 766.86 920.52 949.79

-4.33 -1.54 -0.81 -1.35

-6.58 1.65 -4.36 -4.27

-4.32 -1.56 -0.81 -1.34

-4.33 12.29 -0.81 -1.35

4227.49 4750.50 5057.60 5384.38

4160.48 4783.58 5141.04 5386.43

4159.62 4783.95 5077.15 5387.08

2.85 -4.44 -3.76 -7.47

1.26 -3.70 -2.04 -7.42

2.83 -4.42 -3.72 -7.46

2.85 -4.43 -2.43 -7.47

1359.48 1423.14 1454.08 1441.98

1385.43 1423.41 1262.25 1395.75

1359.61 1422.99 1454.26 1441.99

1359.48 1459.63 1454.05 1441.98

-1.48 -5.15 -3.08 -5.33

-3.42 -5.17 10.52 -1.95

-1.49 -5.13 -3.10 -5.33

-1.48 -7.84 -3.08 -5.33

2011 2012 2013 2014

1438.33 1598.73 1703.95 1734.37

1435.43 1595.37 1738.25 1730.59

1438.75 1598.39 1704.67 1734.82

1438.35 1598.69 1704.01 1734.36

0.86 -6.02 -4.55 -4.70

1.06 -5.80 -6.65 -4.47

0.83 -6.00 -4.59 -4.73

0.86 -6.02 -4.55 -4.70

Fujian

2011 2012 2013 2014

1410.69 1692.85 1781.30 1854.47

1422.06 1657.28 1780.53 1832.47

1410.78 1693.24 1781.23 1855.14

1410.87 1693.13 1781.32 1854.63

6.94 -7.18 -4.74 0.07

6.19 -4.92 -4.69 1.26

6.93 -7.20 -4.74 0.04

6.93 -7.19 -4.74 0.06

Guangxi

2011 2012 2013 2014

1126.26 1274.40 1333.97 1330.48

946.34 1173.54 1369.03 1292.54

1126.50 1274.21 1334.04 1330.58

1126.25 1274.46 1334.43 1330.44

-1.26 -10.44 -7.78 -1.72

14.91 -1.70 -10.61 1.18

-1.29 -10.43 -7.78 -1.73

-1.26 -10.45 -7.82 -1.72

DCRGM (1, 1)

2011 2012 2013 2014

1639.18 2129.97 2393.23 2471.50

1653.52 1966.69 2402.37 2497.44

Liaoning

2011 2012 2013 2014

1809.68 2056.30 2105.16 2111.26

1816.51 2039.95 1980.94 2231.71

Beijing

2011 2012 2013 2014

857.30 887.78 920.53 949.77

875.78 859.87 952.87 977.13

Jiangsu

2011 2012 2013 2014

4159.69 4784.37 5142.94 5387.10

Shanghai

2011 2012 2013 2014

M D

TE

EP

CC

A

Hubei

A

Inner Mongolia

N U SC R DCRGM (1, 1)

DCRGM (1, 1)

31

A

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PT

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