Using a self-adaptive grey fractional weighted model to forecast Jiangsu’s electricity consumption in China

Journal Pre-proof Using a self-adaptive grey fractional weighted model to forecast Jiangsu’s electricity consumption in China

Xiaoyue Zhu, Yaoguo Dang, Song Ding PII:

S0360-5442(19)32112-7

DOI:

https://doi.org/10.1016/j.energy.2019.116417

Reference:

EGY 116417

To appear in:

Energy

Received Date:

02 July 2019

Accepted Date:

21 October 2019

Please cite this article as: Xiaoyue Zhu, Yaoguo Dang, Song Ding, Using a self-adaptive grey fractional weighted model to forecast Jiangsu’s electricity consumption in China, Energy (2019), https://doi.org/10.1016/j.energy.2019.116417

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Journal Pre-proof

Using a self-adaptive grey fractional weighted model to forecast Jiangsu’s electricity consumption in China Xiaoyue Zhu a, Yaoguo Dang a, Song Ding b a College of Economics and Management, Nanjing University of Aeronautics and Astronautics, 211106, China b School of Economics, Zhejiang University of Finance & Economics, Hangzhou 310018, China abstract The remarkable prediction performance of electricity consumption has always assumed particular importance for electric power utility planning and economic development. On account of the complexity and uncertainty of the electricity system, this paper establishes a self-adaptive grey fractional weighted model to predict Jiangsu’s electricity consumption, which efficiently enhances the prediction quality of electricity consumption. This newly constructed grey model introduces the fractional weighted coefficients to design a novel initial condition. Compared with the old one in the conventional grey models, the newly optimized initial condition has a flexible structure, which has advantages in capturing the dynamic characteristics of the electricity consumption observations. In addition, to further promote the forecasting precision, the adjustable fractional weighted coefficients and corresponding time parameter of the initial condition are estimated by utilizing the Particle Swarm Algorithm (PSO). Furthermore, five competing models are employed to forecast Jiangsu’s electricity consumption in China, which certifies the validity of the established model. Experimental results illustrate that the newly designed model has significant advantages over other five competing models. According to the forecasted results, electricity consumption in Jiangsu Province is expected to reach 6778 billion kilowatt-hours in 2020, while the growth rate will fall down by 1.11%. Therefore, several proposals are made for decision-makers. Keywords: Grey system; Grey prediction model; Novel initial condition; Fractional grey model; Electricity consumption prediction 1. Introduction As many studies reveal, electricity consumption has close contact with local economic development, which is described as a barometer. As the most important secondary energy, electric power occupies a higher percentage of terminal energy consumption. Considering the non-storage of power resources, excessive supply will 

Corresponding author: Song Ding ([email protected])

Journal Pre-proof generate huge investment waste and energy dissipation, while insufficient supply will negatively affect economic development. Hence, reliable forecasts of electricity consumption are of utmost significance for making correct energy development plan. Jiangsu’s electricity consumption intensity has reached a high level in China. According to the China statistical yearbook published in 2018, Jiangsu ranks the second among all provinces in electricity consumption, reaching 5808 billion kilowatt-hours of electricity. Since 2001, Jiangsu Province has experienced several power supply shortages. There is no doubt that predicting the growth trends of electricity consumption accurately is extremely crucial for Jiangsu Province, whether in the economic development or in the reasonable power planning. As electricity consumption is affected by various uncertain factors, accurate forecasts are facing a challenge. To tackle such tough problems, several methods have been applied to construct the prediction models. Bouzerdoum et al. combined the seasonal auto-regressive integrated moving average method and the support vector machines method to establish a novel compounded mode for short-term power prediction [1]. Zhou et al. constructed an optimized fuzzy clustering method to dig out the characteristics of household electricity consumption [2]. Aiming at estimating the effects of oil prices and policy tools on the reproducible energy plan, Lee and Huh extended the logistic growth model [3]. Aiming to forecast the yearly residential electricity consumption, Chen et al. employ ensemble learning technique to develop a novel data-drive methods Error! Reference source not found.. In brief terms, these approaches can be categried into three types, time series methods, statistical models and machine learning methods [1]. Time series models are among the most fundamental and widely used methods in handling electricity consumption prediction, which predict future trends based on historical sequence. Oliveira et al. utilized the exponential smoothing method and bagging ARIMA to estimate the electric energy consumption [6]. Consider the seasonal effects, Yukseltan et al. generated an hourly demand model to dip out the electricity development trend [7]. In view of the seasonal effects, Elamin and Fukushige designed a novel autoregressive integrated moving average model to evaluate the hourly load requirement [8]. Cabral et al. presented a spatial ARIMA model, in which Moran's I test was implemented [9]. When the number of observations is large enough, time series methods can achieve higher prediction accuracies However, they rely too heavily on past data and suffer from a lack of interpretability. Statistical models gain advantages over other models in their interpretability, simplicity and ease of deployment. Linear regression models are one of the most popular statistical models. Satre-Meloy utilized the regression models, optimized by five regularization means, to forecast the household electricity consumption in

Journal Pre-proof California [10]. Fan et al. proposed a novel electricity load forecasting model by hybridizing PSR algorithm with BSK regression model, namely PSR-BSK model [11]. Liu et al. integrated the Analytic Hierarchy Process and interval linear programming to optimize energy prediction [12]. Nevertheless, their limitations are also obvious. Firstly, they sacrifice the prediction accuracy for the privilege of higher interpretability. Secondly, they required expert-knowledge to identify relationships between different variable. Machine learning methods are more advanced methods for handling electricity consumption problems. They can solve more complicated calculations. Cao and Wu hybridized the Support Sector Regression model with the adjusted seasonal indicator to predict monthly electricity consumption [13]. Taking advantage of the differential evolution algorithm, Wang et al. produced an efficient model called ESN-DE to predict the electricity consumption [14]. To estimate the annual electricity consumption in the households, Chen et al. employed the ensemble learning technique to construct a novel framework driven by data [15]. Yaslan and Bican incorporated the pattern disintegration and the Support Vector Regression Method to forecast electricity load demand [16]. Yu et al. employed the sparse coding to forecast electricity loads in the households [17]. Aowabin et al. established a recurrent neural network model to forecast the hourly electricity consumption, which contained blocks of missing data [18]. Dong et al. proposed the SVR with chaotic cuckoo search (SVRCCS) model based on using a tent chaotic mapping function to enrich the cuckoo search space and diversify the population to avoid trapping in local optima [19]. Making the long-term estimation of energy consumption in Greek, Ekonomou addressed the Artificial Neural Networks [20]. Kaytez et al. implemented the LS-SVM to estimate the development trend of electricity consumption [21]. The main shortcoming of machine learning methods lies in that their internal operating mechanism is unknown. Noteworthy is that they obtaine satisfactory prediction results only when large-scale data is available, which require considerably extensive efforts in data compilation to obtain the satisfying forecasting accuracy. In many electrical scenarios, adequate information is unavailable from the analyzed system in light of the limitation for time and cost. Espeicially for Jiangsu Province in China, fundamental reforms from the industrial structure deepen the complexity of the electricity consumption prediction. And the data of electricity consumption is limited. Grey forecasting theory shows superior performance in forecasting the development trend of an uncertain system. In comparison with the conventional methods, the grey forecasting models possess a clear superiority of generating more accurate predictions of scarce information system. In virtue of their outstanding performance, the grey forecasting models have gained widespread attention in electricity consumption. Making use of the ant lion optimization

Journal Pre-proof algorithm, Wang et al. developed a hybrid grey method to promote the robustness and stability of power consumption prognostication [22]. Taking seasonal features of electricity consumption into consideration, a seasonal grey model was proposed by Wang et al [23]. Regarding the significant factors, Wu et al. constructed a novel grey model driven by several variables to predict Shandong’s electricity consumption [24]. On account of the effects that the developing coefficient and control variables have on the grey background sequence, Hu introduced the neural network [25]. The GM (1,1) model is an important research branch of the grey prediction method. For its convenience and better predicting effects, the GM (1,1) model has enjoyed wide popularity. Since the time response function was usually derived according to the fixed time input, Xu et al. put forward an optimized grey model with flexible time parameter [26]. Through fractional order accumulating operator, Zeng and Meng designed a new SAIGM model [27]. Zhao and Guo hybridized the Ant Lion Optimizer and the rolling mechanism to propose a rolling- ALO- GM (1,1) model [28]. Considering the time-delayed influence, Ma and Liu implemented a polynomial forecasting method [29]. Seeking out the optimal parameter by genetic algorithms, Hsu suggested an optimized nonlinear grey Bernoulli model [30]. Bahrami et al. provided a new grey wavelet-transform method, in which the parameters were computed by PSO [31]. Introducing the rolling mechanism approach, Akay et al. predicted the total industrial electricity consumption of Turkeys [32]. Although the above methods optimize the GM (1,1) model from various perspectives, the initial conditions remain the constraints. To repair this deficiency, subsequent researches strengthened new information priority principle to establish the weighted initial condition in the modeling process to improve precision. From the perspective of the new information priority, the new development trend is more indicative in predicting an uncertain system behavior sequence. Dang et al. set the last cumulative data as the initial condition to establish the grey model, which consolidated the effects of new data. However, the other time points’ effects on the system were ignored, which was prone to give rise to information loss [33]. Wang et al. applied the first and last accumulation data to generate the initial condition with weighted coefficients, whereas the time parameter of the weighted initial conditions was not given [34]. Xiong et al. unified the ordinal sequence of the first-order accumulation generated sequence, weighted its components, and took the weighted average as the initial condition. Nevertheless, it took a fixed value for each component weight, which constrained its adaptability to the sequence [35]. Ding et al. introduced the dynamic weighting coefficients for the

Journal Pre-proof first time. Simultaneously, these weight parameters were solved through the Artificial Intelligence Optimization Algorithm. Nevertheless, it still existed a problem that the sum of the cumulative sequence weight coefficient did not equal to one, which meant that the normalization of the model needed to be improved [36]. In light of the aforementioned methods, the accuracy of forecasting ability was higher through strengthening the correction effect of new information on the development trend despite some imperfections. By putting more weight to short-term information and less weight to long-term information, the fractional order accumulation operator can effectively promote the modeling accuracy of the grey model. Aiming to attain more accurate forecasts of electricity consumption, this paper constructs the dynamic fractional order weighted coefficients to design the initial condition. The newly self-adaptive fractional weighted grey model is established, abbreviated as SFOGM (1,1) . Accordingly, the self-adaptive fractional weighted coefficients and corresponding time parameter are calculated through the particle swarm optimization, which further promotes the adaptability and prediction quality The major contributions of this paper are in three aspects.  Firstly, the flexible and dynamic fractional order weighted coefficients are designed to construct a novel initial condition on the basis of the new information priority.  Secondly, the self-adaptive fractional weighted grey model is developed. And the newly proposed SFOGM (1,1)

model can better utilize the recent

information hiding in the time series, whereas adequate data is unavailable. In addition, the self-adaptive fractional weighted coefficients and the time parameter are estimated by Particle Swarm Optimization.  Thirdly, the self-adaptive fractional weighted grey model is implemented to research the development trend of Jiangsu’s electricity consumption. Furthermore, the potential development trend is forecasted until 2020. The rest of the study can be divided into the following sections. In Section 2, the modeling mechanisms of conventional GM (1,1) models are investigated. The new self-adaptive fractional weighted grey model is established. Later, the solution of the two parameters is given. In Section 3, the empirical analysis of electricity consumption prediction in Jiangsu is performed. And the prediction results are compared with the other five competing models. In Section 4, essential conclusions are drawn. 2. Methodology This section is committed to offering the detailed introduction of the

Journal Pre-proof self-adaptive fractional weighted grey model. Firstly, the mechanism of the novel model is interpreted. Afterthat, the fractional dynamic weighted coefficient vector is constructed to optimize the initial condition. Secondly, the calculation method of the parameters of the initial condition are given. Lastly, the modeling procedures for the self-adaptive fractional weighted grey model are interpreted through detailed steps and flowchart. 2.1 The newly proposed SFOGM(1,1) model The grey prediction model having one order and one variable, namely GM (1,1) , is the most widely used model in the grey system theory, connecting with various fields, such as electricity[23, 36], energy [35], and high-tech industries [36]. Many studies have introduced its mechanism in detail. Then, we will not provide its detailed procedures. Although many researchers have optimized this model, there still exist some shortcomings that reduce the forecasting precision. The initial condition in the GM (1,1)

model plays an essential role in influencing the

predicting accuracy. Although Dang et al. [33], Xiong et al. [35], and Ding et al. [36] have done some works to modify the initial condition, their models still have their drawbacks, which is explained in Section 1. In this paper, we will put forward a novel fractional weighted initial condition that can intelligently adapt to various characteristics of original data. Then, based on such new initial condition, the propsed model, namely SFOGM (1,1) , is designed and its theoretical novelties can be presented as follows. Definition 1 The original data is denoted as X (0)   x (0) (1), x (0) (2), , x (0) (n) ,

n  4 , where x (0) (k ) (k  1, 2, , n) is nonnegative. Based on the original data, the

first

order

cumulative

sequence

can

be

defined

as

k

(1) (0) X (1)   x (1) (1), x (1) (2), , x (1) (n) , where x (k )   x (i ) . i 1

Accordingly, we generate the background value Z (1) =  z (1) (2), z (1) (3), , z (1) (n) through calculating

the

average

values

of

adjacent

cumulative

values,

z (1) (k )  0.5 x (1) (k  1)  0.5 x (1) (k ), k  2,3, , n .

Definition 2 The differential formula of the GM (1,1) model can be expressed via Eq. (1):

Journal Pre-proof x (0) (k )  az (1) (k )  b

(1)

where a, b are the equation coefficients. Parameter a denotes the development coefficient while parameter b represents the grey input. The whitening differential equation is defined as Eq. (2):

dx (1) (t ) dt

 ax (1) (t )  b

(2)

In the aforementioned formula, x (1) (t ) is 1-AGO data sequence, among which

t is the sequence number of the data, while equation coefficients.

a and b represent the unknown

  z (0) (2)  x (0) (2)   (0)  (0)  x (3)    z (3)  B In contrast, let Y   ,        (0) (0)   z ( n)  x ( n) 

1 



1 

a  , and P    . Obviously, the  1 b   1 

calculation of vector P is the key of equation solving to perform the grey equation. On the basis of the least square method, the solution of Eq. (1) satisfies the following equation: Pˆ  ( BT B ) 1 BT  Y

(3)

When the prepared parameters are given, the continuous time response function could be derived as follows: xˆ (1) (t )  ce  atˆ 

bˆ . aˆ

(4)

c denotes the initial condition. In this paper, the initial condition is optimized by introducing the dynamic adjustable fractional order weighted coefficients. Subsequently, the newly self-adaptive fractional weighted grey model, abbreviated as SFOGM (1,1) , is put forward.

Definition 3 Assuming that parameter,

and

the

fractional

r (0 ＜ r ＜ 1)

dynamic

 =[1 ,  2 , ,  n ] ,  j is defines as Eq. (5):

is the self-adaptive fractional

weighted

coefficient

vector

is

Journal Pre-proof

j 

j n

 j

(5)

,

j 1

where  j 

(r  n  j  1)! . It is obvious that the newly constructed dynamic weighted (r  1)!(n  j )!

coefficients satisfy the normalization condition, 1 + 2 +  n  1 . During the generation period of  j , the self-adaptive fractional parameter r has the flexibility to fit the changing trends of the sequence by updating the weights of new and old information. Subsequently, the newly constructed initial condition denotes as:

x (1) (t0 )  1 x (1) (1)   2 x (1) (2)   n x (1) (n) ,

(6)

where t0 is the input time parameter, and x (1) (t0 ) is its corresponding initial condition. Theorem

1

The

newly

constructed

initial

condition

x (1) (t0 )  1 x (1) (1)   2 x (1) (2)   n x (1) (n) satisfies the new information priority.

Proof: (r  n  j  1)(r  n  j  2) (r  1)r , (n  j )! (r  n  j  2)(r  n  j  3) (r  1)r ,  j 1  (n  j  1)!

j 

when 0  r  1 , (r  n  j  2)(r  n  j  3) (r  1)r (r  n  j  1)(r  n  j  2) (r  1)r (n  j  1)! (n  j )! (r  n  j  2)(r  n  j  3) (r  1)r = [(n  j )  (r  n  j  1)]＞0. (n  j )!

 j 1 - j 

j 

j

 j 1

Therefore,



satisfies

the

.

n

j

inequality 1 ＜  2 ＜  3 ＜＜  n 1 ＜  n .

The

short-term information is given more weight while the long-term information is given less weight. Theorem 2. Setting

x (1) (t0 )  1 x (1) (1)   2 x (1) (2)  n x (1) (n) as the initial

condition, t0 is time input parameter, the discrete time response function and

Journal Pre-proof the simulated formula are given as Eq. (7), Eq. (8). b  aˆ ( k t ) bˆ xˆ (1）( k )  [ 1 x (1) (1)   2 x (1) (2)   n x (1) ( n)  ]e  , a aˆ 0

b xˆ (0) (k )=xˆ (1) (k )  xˆ (1) (k  1)=[ 1 x (1) (1)   2 x (1) (2)  n x (1) (n)  ](1  e aˆ ) e  a ( k t0 ) . a

(7) (8)

The model above is named the newly self-adaptive fractional weighted grey model, which is abbreviated as the SFOGM (1,1) model. Different from the former optimized models, the GM (1,1)  x (1) (n) model proposed by Dang et al. [33], the OICGM (1,1) model developed by Xiong et al. [35], the SFOGM (1,1) model better interprets new information priority and remedy the limitations generated by them. 2.2 The solution to the parameters of the initial condition Investigaitng the mechanism of the SFOGM (1,1) model, it could be found that the solution to the parameters of the initial condition, r and t0 , are critical and could affect the forecasting performance in large part. In this section, the parameters of the initial condition are decided by Particle Swarm Optimization (PSO) through minimizing the mean absolute percentage error between the simulation value and the actual value. As a kind of evolutionary computation technology, the PSO algorithm was firstly proposed in 1995. The algorithm was initially inspired by the regularity of birds swarm activity. After that, a simplified model was established by swarm intelligence. On the basis of observing the behavior of animal cluster, PSO makes use of the information sharing of individuals in the group to generate the iteration process to seek out the optimal solution. Since its introduction, PSO has gained an effective application. Therefore, PSO is implemented to retain the parameters [r , t0 ] of the initial condition The primary steps of the PSO algorithm proceed as follows. Firstly, define vector

[r , t0 ] and establish the fitness function. Secondly, the process is initialized. Denote M as the particles number and L represents the maximum iteration. To seek the minimum MAPE, the fitness function formula is acquired, n b fitness[Q(i, j )]   | x (0) (k ) [ 1 x (1) (1)   2 x (1) (2)  n x (1) (n)  ](1  e aˆ ) e  a ( k t0 ) , a k 1

where Q(i, j ) records the position of the ith generation’s jth particle. And Pbest (i )

Journal Pre-proof is utilized to keep track of the ith generation’s optimal fitness. Furthermore, Gbest denotes the optimal position and renews its value under the condition of

Pbest (i )  Gbest . After that, the formula of the particle velocity is proposed, Vk 1  c1 ( pbestk  x k ) R1  c2 ( gbestk  x k ) R2 ,

where c1 , c2 represent the acceleration factors, and R1 , R2  [0,1] denote the random variables.

In

addition,

the

locations

of

the

particles

are

renewed

via

Q(i, j  1） =Q(i, j )  vk . The velocities of the particle in the following generation are

expressed as below:

Vk 1  wvk  c1 ( pbestk  x k ) R1  c2 ( gbestk  x k ) R2 , among which, w denotes an inertia weight. Lastly, the parameters [r , t0 ] can be acquired from the corresponding Gbest, while it approached the minimal. 2.3 Modeling procedures for the SFOGM(1,1) model Based on the theoretical explanations in the previous two subsections, the detailed procedures for the new proposed model can be outlined below, which mainly contains five steps (seen in Fig. 1). Step 1: Obtain the raw data and calculate the values of the intermediate variables. According to definition 1, the cumulative sequence and background values, namely

X (1) and Z (1) , are the intermediate variable for modeling the grey model. By using these two processed sequences, the randomness hidden in the raw data can be further weaken, which enables the grey model to achieve accurate prediction. Step 2: Build the SFOGM (1,1) model and estimate its parameters

Pˆ . As

introduced in the grey theory, after substituting the cumulative sequence and background values into the grey function in Eq. (1), we can obtain a group of functions, followed as x (0) (2)  az (1) (2)  b x (0) (3)  az (1) (3)  b .  x (0) (n)  az (1) (n)  b

In matrix form, Y  B Pˆ , where

Journal Pre-proof   z (0) (2)  (0)  z (3) B  M  (1)   z (n)

 x (0) (2)  1   (0)  1 x (3)  ˆ  aˆ  ,Y   , P . ˆ  M  M b    (0)  1  x (n) 

By solving the above matrix form of the equation, the least squares estimation for a and b are Pˆ  [aˆ , bˆ]T  ( BT B)1 BT Y . Step 3: Construct the optimized initial condition and obtain the forecasting function by solving the whitening differential equation in Eq. (2). To address the drawbacks of the previous initial condition, a novel initial condition having flexible and dynamic fractional order weighted coefficients, seen in Eq. (5), is put forward. This new initial condition gives the new datapoints higher weights compared with the old datapoints, which enable the grey model to provide better forecasting performance. Subsequently, once the optimized initial condition is obtained, the time (0)

(1) response function xˆ (k ) and the simulated formula xˆ (k ) , seen in Eq. (7) and

(8), will be deduced by solving the whitening grey differential equation. Step 4: Determine the optimal values of the self-adaptive fractional parameter

r and the time parameter t0 by using the Particle Swarm Optimization (PSO). As section 2.2 introduced, the fitness function can be obtained as below: min f (r ) 

1 n | x (0) (k )  xˆ (0) (k )|  n  1 k 2 x (0) (k )

   ˆ  ( BT B) 1 BT Y   xˆ (0) (k )=xˆ (1) (k )  xˆ (1) (k  1)=[ 1 x (1) (1)   2 x (1) (2)  n x (1) (n)  b ](1  e aˆ ) e  a ( k t0 )  a  (r  n  j  1)(r  n  j  2) (r  1)r  s.t.  j  (n  j )!   j  j  n , j  1, 2, , n    j  j 1   r  (0,1), t0  [1, n]

Subsequently, by usting the PSO algorithm, the optimal values of the parameters

r and t0 will be determined. Step 5: Generate the forecasted values by substituting r and t0 into the (0)

formulas xˆ ( k ) . Assuming that the predicted length of the main system behavior sequence is m , the predicted values ( x ( n  1), x ( n  2),  , x ( n  m)) are obtained. ( 0)

( 0)

( 0)

Journal Pre-proof 2.4 Comparative analysis of the newly proposed and existing models In this section, the advantages of the SFOGM (1,1) model over the existing optimized grey models, including the GM (1,1)  x (1) (1) model, the GM (1,1)  x (1) (n) model and the OICGM (1,1) model, are demonstrated by forecasting the energy consumption in China. Analyzing the development trend and the length of Chinese energy consumption data, it could be found that the data owns the characteristics of uncertainty and small sample. Therefore, making the most of the known and new information in time series is especially important. The newly proposed SFOGM (1,1) model could adapt well to this situation. In order to evaluate the predicition performance difference between these models, three precision indicators are calculated, consisting of the absolute percentage error, the mean absolute percentage error and the root mean square error. The absolute percentage error is applied to qualify each data’s prediction precision, while the mean absolute percentage error (MAPE) and the root mean square error (RMSE) are employed to evaluate the overall precision. They are denoted as below:

| x (0) (k )  xˆ (0) (k )| APE  100%  , k  1, 2, , n. x (0) (k )

MAPE  N

RMSE 

1 n  APE (k ) n k 1

 (x

(0)

(k )  xˆ (0) (k )) 2

i 1

N

Following the modeling steps 1-2, the least squares estimation for a and b are

a  0.06964 obtained as  . Determine the best values of r and t0 by PSO, the b  200200.73 optimal values of the parameters r and t0 are acquired as [0.40524, 5.42902]. Then the restored time response function could be described as

xˆ (0）( k )  211997.9e 0.06964( k 5.42902) -2874724.7 . Once the restored time response function is acquired, the simulative values are produced. The forecasting precision of the four models, the GM(1,1)-x(1) model, the GM(1,1)-x(1)(n) model, the OICGM(1,1) model and the SFOGM(1,1) model, are calculated and listed in Table 1. Three precision indicators show that the newly

Journal Pre-proof proposed model achieves the best overall results. The MAPE of the SFOGM (1,1) model is 4.03%, while others have reached 7.42%, 7.67%, 4.22%, separately. Similarly, the RMSE of the SFOGM (1,1) model is 15309.97, far below other three models. For all the former initial condition optimization methods strengthened the correction effect of new information to some extent, they are supposed to be improved. The SFOGM(1,1) model changes the condition of fixed time parameter and cumulative order of the original model, thus improving the flexibility of the forecasting models. 3. Prognosticating Jiangsu’s electricity consumption Entering a stage of rapid economic development, Jiangsu’s electricity consumption has grown up at a high speed. At the same time, the population in Jiangsu Province is larger and the resources are relatively insufficient. Hence, the supply-demand conflicts of electricity become increasingly prominent. As an important economic district in China, it’s unquestionable that the higher forecasting accuracy of Jiangsu’s electricity consumption has always been of vital importance. In this section, Jiangsu’s electricity consumption are forecasted by the SFOGM (1,1) model. And the superiority of the SFOGM (1,1) model is verified by comparing to other five models, including the GM (1,1)  x (1) (1) model, the GM (1,1)  x (1) (n) model, the OICGM (1,1) model, the exponential smoothing method and the autoregressive integrated moving average model. 3.1 Data description The primary data of Jiangsu’s electricity consumption derives from the statistical yearbook of Jiangsu Province in 2018 (http://tj.jiangsu.gov.cn/2018/nj09/nj0909.htm). The statistical data of Jiangsu’s electricity consumption is drawn in Fig. 2. As the picture shows, the electricity consumption in Jiangsu has experienced huge growth. In 2001, its electricity consumption was 1078 billion kilowatt-hours. However, the electricity consumption has reached 5808 billion kilowatt-hours in 2017, which is a fourfold increase. In view of the composition of electricity consumption, the total industrial electricity consumption in 2017 was 5123.57, accounting for 88.22% of the total consumption. The second highest was the household electricity consumption, which accounted for 11.78% of the total electricity consumption. The growth of electricity consumption is closely related to the industrial development. Analyzing Jiangsu’s electricity consumption from the angle of industry, as shown in Fig. 3, it can be found that the secondary industry has been the main force of electricity consumption for a long time, occupying more than 80% of the total

Journal Pre-proof industrial electricity consumption. The second highest was the tertiary industry, which had a share of between 10% and 15%. And its share of the electricity consumption continued to grow with the development of the tertiary industry. The primary industry's share of electricity consumption remained the lowest, with a share of 4.6% in 2001, and continued to decline. In 2017, its share has fallen to 1.3%. Population size is also a decisive factor that affects the electricity consumption. The higher total population is bound to increase the electricity. From 2001 to 2017, the number of resident populations in urban and rural areas in Jiangsu province experienced significant growth, rising from 73.5852 million to 80.293 million. On the one hand, because of the differences between the urban and rural development, the rural population flowed into the city. On the other hand, uneven development in the eastern and western parts of China has also led to a massive influx of people into the east. In terms of electricity consumption and population quantity in urban and rural areas, the proportion of the urban population in Jiangsu’s total population was 68.8% in 2017. Nevertheless, the proportion of urban residents' electricity consumption only reached 49.72%, shown in Fig. 4, indicating that there were great differences in electricity consumption between the urban and rural households. Although electricity consumption in Jiangsu Province has achieved tremendous growth during the past 17 years, the growth rate of electricity consumption has decreased. From 2001 to 2007, Jiangsu’s electricity consumption continued to grow with an annual growth rate of more than 10%. The average growth rate in 2003-2006 even exceeded 20%. Since 2008, although Jiangsu’s electricity consumption was still in the growth phase, the growth rate has dropped significantly, basically below 10%. In 2014, the growth rate of electricity consumption ushered in another decline. The growth rates in 2014 and 2015 were both below 2%. 3.2 Model calibration for the SFOGM(1,1) Model To certify the validity of the established model, the SFOGM (1,1) model is carried out to forecast Jiangsu’s electricity consumption, abbreviated as Model 6. The advantages of the SFOGM (1,1) model over the existing models are validated by comparing

to

the

other

five

competing

models,

consisting

of

the

GM (1,1)  x (1) (1), GM (1,1)  x (1) (n), OICGM (1,1) , the exponential smoothing method

and the autoregressive integrated moving average model, namely Model 1, Model 2, Model 3, Model 4, Model 5. These competing models can be described as two different categories, namely other grey models (Model 1, Model 2, and Model 3), and non-grey time series models (Model 4 and Model 5). By comparing with different kinds of benchmark models, the effectiveness and applicability can be further

Journal Pre-proof validated. As shown in the Fig. 2, Jiangsu’s electricity consumption also has nonlinearity and periodicity. Subsequently, through several sets of experiments, the authors find that the best precision performance could be acquired when the number of the simulative data is five. Consequently, five data points are used for model calibration so as to estimate the unknown parameters, and the next three are predicted. Step 1: Acquire original data Afterwards, the 1  AGO

sequence

X (0) =[4693.40, 4941.41, 5202.52, 5477.43, 5766.86] .

X (1) and the background value Z (1) are

calculated.

a  T 1 T Step 2: Compute the vector P    , according to Pˆ  ( B B) B  Y , and b   a  0.05149 obtain the values  . b  4560.05 (0)

(1) Step 3: The time response function xˆ (k ) and the formula xˆ (k ) are

obtained: 0.05149( k  t ) xˆ (1）( k )  [ 1 x (1) (1)   2 x (1) (2)   n x (1) ( n)+88557.8]e -88557.8 , 0

xˆ (0) (k )=xˆ (1) (k )  xˆ (1) (k  1)=[ 1 x (1) (1)   2 x (1) (2)  n x (1) (n)+88557.8](1  e-0.05149 ) e0.05149( k t0 ) .

Step 4: Seek out the best values of r and t0 by using PSO. The fitness function could be expressed as:   n fit (r , t0 )  min  [ x (0) (k )  ( 1 x (1) (1)   2 x (1) (2)  n x (1) (n)  88557.8)(1  e 0.05149 ) e0.05149( k t0 )  .  k 1 

Initialize unknown parameters. Then, PSO is applied to obtain the optimal values that have the best fitness value. Subsequently, the unknown parameters r , t0 can be acquired as [0.031758, 4.747287]. Fig. 5 exhibits the track of the optimal fitting position during the procedure. From the view of the track path in each iteration, it can be seen that the PSO algorithm has good convergence. And the optimal values are obtained quickly, within 100 iterations. Step 5: Finally, substituting the optimial parameters into the restored time response function in Step 3, the simulated formula is expressed as: xˆ (0）( k )  4734.322e 0.05149( k  4.747287) -88557.8, k  1, 2, , n, n  1, , n  l .

3.3 Comparative analysis

(18)

Journal Pre-proof The analysis of the case is described in detail below. In comparison, five alternative models are constructed as benchmark models, and their forecasts from 2013 to 2017 are listed in Table 2. Table 3 demonstrates the APE, MAPE and RMSE determined by six models. Fig. 6 shows the simulated error distribution. From Table 2, it can be found that the MAPE of the SFOGM (1,1) model is 1.77%, while the MAPE of the other three grey models are bigger, reaching 1.90%, 18.76%, 1.86%, respectively. As for the RMSE, the SFOGM (1,1) model also behaves better, realizing the RMSE of 120.83. The RMSE of the other three grey models are 129.68, 990.52, 123.99, respectively. Obviously, the

SFOGM (1,1)

model is the optimal model, while the OICGM (1,1) model obtains the suboptimal precision. Considering the modeling mechanism, the SFOGM (1,1) model and the OICGM (1,1) model both set time parameters as unknown parameters. To further

improve their forecasting accuracy, several intelligent optimization algorithms are applied to seek the best parameters. These intelligent algorithms increase the flexibility and drop the simulation error. In addition, the SFOGM (1,1) model has the self-adaptive weighted coefficients and the factional accumulation operator, which is different from the OICGM (1,1) model. On the one hand, the self-adaptive weighted coefficients enable the SFOGM (1,1)

model to flexibly adapt to the

development trend of the data. On the other hand, the factional accumulation makes the accumulation order of the old information fall from the integer to the fraction, better interpreting the principle of new information priority. Compared to the non-grey time series models, the predictive performance of the SFOGM (1,1)

model is also more prominent. The MAPE of the exponential

smoothing method and ARIMA are 2.86% and 3.21%, both higher than the SFOGM (1,1) model. Meanwhile, the RMSE of them are 183.09 and 225.58, much

higher than that of the SFOGM (1,1) model. In 2014, the APE of the exponential smoothing method reaches 5.90%. The APE of ARIMA in 2013 even reaches 7.79%. Analyzing the modeling mechanism, the exponential smoothing method and ARIMA are both traditional time series methods, which is built by mining the time trend of historical data to predict. The large sample size is usually required. Therefore, their

Journal Pre-proof modeling effects are not so satisfactory in small samples. Besides, Fig.6 shows the simulated error profile, demonstrating that the optimized SFOGM (1,1) model is the most efficient while the GM (1,1)  x (1) (1) model performs the worst. Unlike the previous three models, the GM (1,1)  x (1) (1) model is constructed upon setting the first data as the initial condition, which pays less attention to utilizing the new information. Although the simple modelling mechanism makes the GM (1,1)  x (1) (1) model used widely, the existed defects are also noteworthy, which makes it difficult to provide a more accurate reference for energy decision-making. Comprehensively, the

SFOGM (1,1)

model shows its

excellent performance in predicting electricity consumption in Jiangsu Province. It makes better use of the information from all data points. By using the SFOGM (1,1) model as Eq. (8) introduces, the predicted values of Jiangsu’s electricity consumption, ranging from 2018 to 2020, are drawn in Fig. 7 Jiangsu’s electricity consumption is expected to keep sustainable growth. It’s estimated to reach 6778 billion kilowatt-hours in 2020, while the growth rate will slow, decreasing from 6.39% in 2017 to 5.28% in 2020. The reasons are mainly explained in two aspects. On the one hand, energy-saving technologies are widely promoted. On the other hand, with the strengthening of environmental protection awareness, people's consumption of electricity has been reduced. 4.Conclusions and suggestions 4.1 Conclusions Accurate prediction of electricity consumption always has a great influence on the energy plan and economic development. Accordingly, proposing an appropriate technique for short-term prediction makes a lot of sense. Although the existing studies promote the prediction performance to some degree, their adaptability to various cases is restricted due to their strict dependence on data quantity and quality. To tackle this situation, a self-adaptive grey fractional weighted model is put forward to forecast Jiangsu’s electricity consumption, which is characteristized by uncertainty, insufficient information, and sparse data. Through two empirical cases, the authors can further obtain the following conclusions. (1) The novel proposed model was empirically compared with grey models ( GM (1,1)  x (1) (1) , GM (1,1)  x (1) (n) and OICGM (1,1) ) and non-grey models (the exponential smoothing method and the ARIMA model). The results illustrate that the proposed model has better adaptibility and reliability in terms of forecasting

Journal Pre-proof sequences characterized by uncertainty, insufficient information, and sparse data, compared with other five competitors. (2) The fractional weighted initial condition can significantly improve the predictive performance of a grey model, in comparison with the previous optimized initial conditions. This initial condition highlights the significant influence of the latest data points so as to account for the evolving trends of a system sequence. Equally important, the PSO algorithm helps to automatically determine the optimal parameters in the new initial condition (elaborated in Section 2.2), which further enhances the forecasting accuracy. (3) The new proposed model is easily operationalized and readily applicable for generating predictions in various areas because of the adjustable parameters in the new initial condition. This property enables the proposed model to flexibly and intelligently adapt to diverse features of the original data under the support of the PSO algorithm. (4) A short-term prediction of Jiangsu’s electricity demand by using the proposed model illustrates that Jiangsu’s electricity demand will remain a strong upward trends in the future coming years, reaching 6778 billion kilowatt-hours in 2020. Suggestions for facing such challenges are urgent for decision makers. 4.2 Suggestions With the advancement of economic growth and urbanization, electricity consumption in Jiangsu Province has achieved leapfrog growth over the past decade. Although the growth rate is in a downward trend, the electricity consumption in Jiangsu Province is estimated to increase at 6778 billion kilowatt-hours in 2020, which remains a huge challenge. In order to cope with this challenge, this paper proposes to provide countermeasures from the following aspects. (1) From the analysis of the distribution of three industries of Jiangsu’s electricity consumption, the conclusion could be drawn that the secondary industry’s electricity consumption occupies a large proportion. Supporting the rapid growth of the secondary industry is the high-energy-consuming heavy industry, which leads to higher energy consumption and generates severe environmental pollutants. Therefore, in future industrial development, the percentage of the heavy industry should be reduced. And the current industrial structure characterized by a high proportion of the secondary industry in Jiangsu Province should be gradually changed. More efforts are supposed to focus on putting more energy into the development of the first and third industries, improving the automation level of agriculture, forestry, animal husbandry and sideline fishing, and taking the new industrialization road to increase the proportion of the tertiary industry. Only in this way can we reduce the high demand for electricity from the economic development and improve the electricity efficiency. (2) Residential electricity consumption has been a basilic embranchment of

Journal Pre-proof Jiangsu's total electricity consumption. The new round of power system reform, including the reduction of general industrial and commercial electricity prices, has also spurred the enthusiasm of the entire society for electricity use. Under this circumstance, the promotion of residents' awareness of environmental protection will greatly reduce unnecessary waste of domestic electricity consumption and have certain social and economic benefits. Through the previous analysis, we know that the proportions of permanent residents and electricity consumption in urban and rural areas are not positively related. In 2017, the resident population of rural areas in Jiangsu occupied 31.2% of the total population, while the electricity consumption reached 50.28% of the total population. Taking into account the difference between urban and rural residents’ consumption, more energy and resources need to be placed on raising the environmental awareness of the rural families. (3) With the deepening of the supply-side structural reform and the good start of the three major battles, the annual growth rate of electricity consumption in Jiangsu province has dropped significantly and has maintained below 10% from 2008, indicating that Jiangsu’s electricity consumption has entered a new stage. During the process, the government's macro-control and policy planning played a very important role. In 2016, Jiangsu Province issued the “Special Plan for the Development of Power in the 13th Five-Year Plan of Jiangsu Province” to propose the development goal of power development. At present, the demand for electricity consumption is still large. To ensure sufficient transportation capacity in various regions, the government needs to further develop its functions, optimize the layout of the power facility,, and improve the peaking capacity. Acknowledgment This work is supported by National Natural Science Foundation of China (Grant NO. 71901191, 71771119), Youth Fund Project for Humanities and Social Science Research of the Ministry of Education (Grant numbers 19YJC630167), Social Science Fund Key Project of Jiangsu Province (Grant numbers 16GLA001), Projects Funded by Special Funds for Basic Scientific Research Operating Expenses of Central Universities (Grant number NW2019001, and NP2017301), Research on the Improvement of Teaching Ability of Teachers in Nanjing University of Aeronautics and Astronautics (Grant numbers 1611JF0901Z) References [1] Bouzerdoum M, Mellit A, MassiPavan A. A hybrid model for short-term power forecasting of a small-scale grid-connected photovoltaic plant. Sol Energy 2013; 98: 226-235. [2] Zhou K L, Yang S L, Shao Z. Household monthly electricity consumption pattern mining: A fuzzy clustering-based model and a case study. Journal of Cleaner Production 2017; 141: 900-908. [3] Lee C Y, Huh S Y. Forecasting the diffusion of renewable electricity considering the impact of

Journal Pre-proof policy and oil prices: The case of South Korea. Applied Energy 2017; 197: 29-39.

[4] Chen K L, Jiang J C, Zheng F D. A novel data-driven approach for residential electricity consumption prediction based on ensemble learning. Energy 2018; 150: 49-60. [5] Bahrami S, Hooshmand R A, Parastegari M. Short term electric load forecasting by wavelet transform and grey model improved by PSO algorithm. Energy 2014; 72(7): 434-442. [6] Oliveira E M D, Oliveira F L C. Forecasting mid-long term electric energy consumption through bagging ARIMA and exponential smoothing methods. Energy 2018; 144: 776-788. [7] Yukseltan E, Yucekaya A, Bilge A H. Forecasting electricity demand for Turkey: Modeling periodic variations and demand segregation. Applied Energy 2017; 193: 287-296. [8] Elamin N, Fukushige M. Modeling and forecasting hourly electricity demand by SARIMAX with interactions. Energy 2018; 165: 257-268. [9] Cabral J D, Legey L F L, Cabral M V D. Electricity consumption forecasting in Brazil: A spatial econometrics approach. Energy 2017; 126: 124-131. [10] Satre-Meloy, Aven. Investigating structural and occupant drivers of annual residential electricity consumption using regularization in regression models. [11] Fan G F, Peng L L, Hong W C. Short term load forecasting based on phase space reconstruction algorithm and bi-square kernel regression model. Applied Energy 2018; 224:13-33. [12] Liu, Y, He L, Shen, J. Optimization-based provincial hybrid renewable and non-renewable energy planning - A case study of Shanxi, China. Energy 2017; 128: 839-856. [13] Cao G H, Wu L J. Support vector regression with fruit fly optimization algorithm for seasonal electricity consumption forecasting. Energy 2016; 115: 734-745. [14] Wang L, Hu H L, Ai X Y, et al. Effective electricity energy consumption forecasting using echo state network improved by differential evolution algorithm. Energy 2018; 153: 801-815. [15] Chen K L, Jiang J C, Zheng F D. A novel data-driven approach for residential electricity consumption prediction based on ensemble learning. Energy 2018; 150: 49-60. [16] Yaslan Y, Bican B. Empirical mode decomposition based denoising method with support vector regression for time series prediction: A case study for electricity load forecasting. Measurement 2017; 103: 52-61. [17] Yu C N, Mirowski P, Ho T K. A sparse coding approach to household electricity demand forecasting in Smart Grids. IEEE Transactions on Smart Grid 2017; 8(2): 738-748. Energy 2019; 174: 148-168. [18] Aowabin R, Vivek S, Amanda D. Predicting electricity consumption for commercial and residential buildings using deep recurrent neural networks. Applied Energy 2018; 212: 372–385. [19] Dong Y Q, Zhang Z C, Hong W C. A Hybrid Seasonal Mechanism with a Chaotic Cuckoo Search Algorithm with a Support Vector Regression Model for Electric Load Forecasting. Energies 2018; 11(4). [20] Ekonomou L. Greek long-term energy consumption prediction using artificial neural networks. Energy 2010; 35(2): 512-517. [21] Kaytez F, Taplamacioglu M, Cam E, et al. Forecasting electricity consumption: A comparison of regression analysis, neural networks and least squares support vector machines. International Journal of Electrical Power & Energy Systems 2015; 67: 431-438. [22] Wang J Z, Du P, Lu H Y, et al. An improved grey model optimized by multi-objective ant lion optimization algorithm for annual electricity consumption forecasting. Applied Soft Computing 2018; 72: 321-337.

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Journal Pre-proof AUTHOR DECLARATION TEMPLATE We wish to draw the attention of the Editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing, we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected] Signed by all authors as follows: Xiaoyue Zhu, [email protected] Yaoguo Dang, [email protected] Song Ding, [email protected]

Journal Pre-proof List of figures Fig. 1. The procedure of the SFOGM (1,1) model. Fig. 2. Statistical data of Jiangsu’s electricity consumption Fig. 3. The industry distribution of Jiangsu’s electricity consumption Fig. 4. The electricity consumption proportion of the residents in Jiangsu province Fig. 5. Track of the seeking process of PSO Fig. 6. The simulated error profile. Fig. 7. The development trend of electricity consumption in Jiangsu

STEP 1

Input raw data

Predict values

STEP 2

Establish the prediction expression SFOGM(1,1)

Calculate and

Build the opimization model and determine and

Substitute and into the formula

STEP 4

STEP 5

Fig. 1. The procedure of the

1

SFOGM (1,1) model.

derive the formula of the parameters

Obtain the equations of and

STEP 3

Journal Pre-proof 6500

5808

6000

5459

5500 5000

4597

5013

5115

2013

2014

2015

electricity consumption

4581 4500

4282 3864

4000 3500

2952

3000

3314

3118

2570 2500

2193 1820

2000

1505

1500

1078

1245

1000 500 0

2001

2002

2003

2004

2005

2006

2007

2008

2009 year

2010

2011

2012

2016

2017

Fig. 2. Statistical data of Jiangsu’s electricity consumption 1.00

9.47

9.10

8.80

8.55

8.54

8.56

8.64

9.55

10.18

10.39

10.73

11.39

11.83

0.50 85.93

87.79

88.66

89.37

89.99

90.37

90.45

89.62

88.97

88.79

88.41

87.68

87.19

1.07

0.91

0.82

0.85

12.01

12.66

13.45

14.42

86.96

86.20

85.27

84.28

1.03

1.15

1.28

0.90 0.80 0.70 0.60

0.40 0.30 0.20 0.10 0.00

4.60

3.10

2.53

2.08

1.46

0.85

0.83

0.92

0.98

1.30

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 the primary industry

the secondary industry

the tertiary industry

Fig. 3. The industry distribution of Jiangsu’s electricity consumption

2

Journal Pre-proof 100.00 90.00 80.00

44.12

47.31

48.74

49.44

49.09

49.41

55.88

52.69

51.26

50.56

50.91

50.58

70.00

50.57

50.84

50.14

50.68

50.81

51.65

51.86

51.35

50.94

50.28

49.43

49.16

49.86

49.32

49.19

48.35

48.14

48.65

49.06

49.72

60.00 50.00 40.00 30.00 20.00

49.80

10.00 0.00

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 the proportion of electricity used by urban residents

the proportion of electricity used by rural residents

Fig. 4. The electricity consumption proportion of the residents in Jiangsu province 0.09

0.08

Fitness Values

0.07

0.06

0.05

0.04

0.03

0

50

100

150

200 250 300 Iteration Number

350

400

Fig. 5. Track of the seeking process of PSO

3

450

500

Journal Pre-proof 10.00 5.00 0.00 2013

2014

2015

2016

2017

-5.00 -10.00 -15.00 -20.00 -25.00 Model 1

Model 2

Model 3

Model 4

Model 5

Model 6

Fig. 6. The simulated error profile 6115 6778 6438 7000 6800 6600 6400 6200 6000 5800 5600 5400 5200 5000 4800 4600 4400 4200 4000

5,808 5,516 5,240 4,977 4,727

2013

2014

2015

2016

Simulation value

2017

2018

2019

Prediction value

Fig. 7. The development trend of electricity consumption in Jiangsu

4

2020

Journal Pre-proof Highlights: 

A self-adaptive fractional weighted grey model is developed for electricity demand forecast.



The novel initial condition having dynamic fractional order weighted coefficients is proposed.



The PSO algorithm is applied to estimate the weighted coefficients and time parameter.



Jiangsu’s electricity consumption is predicted from 2018 to 2020.

Journal Pre-proof List of tables Table 1 The APE, MAPE and RMS of four models Model

APE(%)

GM 1,1  x 1

GM 1,1  x  n 

OICGM (1,1)

SFOGM (1,1)

5.28 6.33 10.65

5.53 6.58 10.91

2.14 3.16 7.35

2.88 2.99 6.23

7.42 27467.07

7.67 28312.93

4.22 17087.42

4.03 15309.97

2010 2011 2012

MAPE(%) RMSE

Table 2 The simulation values of six models. Year　

Actual

Model 1

Model 2

Model 3

Model 4

Model 5

Model 6

2013 4956.62

4693.40

3854.54

4726.83

4820.3

5342.91

4736.80

2014 5012.54

4941.41

4058.21

4987.13

5308.1

5009.34

4976.57

2015 5114.70

5202.52

4272.65

5250.65

5128.9

5104.42

5239.53

2016 5458.95

5477.43

4498.43

5528.10

5216.8

5184.02

5516.40

2017 5807.89

5766.86

4736.13

5820.21

5754.6

5636.11

5807.89

value　

Table 3 The APE, MAPE and RMSE of six models Year

Model1

Model2

Model3

Model4

Model5

Model6

2013

5.31

22.23

4.64

2.75

7.79

4.64

2014

1.42

19.04

0.51

5.90

0.06

0.72

2015

1.72

16.46

2.66

0.28

0.2

2.44

2016

0.34

17.6

1.27

4.44

5.04

1.05

2017

0.71

18.45

0.21

0.92

2.96

0

MAPE(%)

1.9

18.76

1.86

2.86

3.21

1.77

RMSE

129.68

990.52

123.99

183.09

225.58

120.83

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