Forecasting China's electricity consumption using a new grey prediction model

Energy 149 (2018) 314e328

Contents lists available at ScienceDirect

Energy journal homepage: www.elsevier.com/locate/energy

Forecasting China's electricity consumption using a new grey prediction model Song Ding a, b, *, Keith W. Hipel b, Yao-guo Dang a a b

College of Economics and Management, Nanjing University of Aeronautics and Astronautics, 211106, China Department of System Designing Engineering, University of Waterloo, Waterloo, ON, N2L3G1, Canada

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 September 2017 Received in revised form 28 January 2018 Accepted 29 January 2018 Available online 6 February 2018

A modified grey prediction model is employed to accurately forecast China's overall and industrial electricity consumption. To this end, a novel optimized grey prediction model, combining a new initial condition and rolling mechanism, is designed on the principle of “new information priority”. The previous initial conditions possess the inherent deficiencies of having a fixed structure and poor adaptability to changing raw data. To overcome these deficiencies, the new initial condition, possessing alterable weighted coefficients, is proposed. Its generating parameters can be optimally determined by employing a particle swarm optimization algorithm according to various characteristics of the input data. In addition, to demonstrate its efficacy and applicability, the novel model is utilized to predict China's total and industrial electricity consumption from 2012 to 2014 and then compared to forecasts obtained from a range of benchmark models. The two empirical results illustrate that the novel initial condition with dynamic weighted coefficients can better adjust to the features of electricity consumption data than the previous initial conditions. They also show the superiority of the newly proposed model over the benchmark models. Within this paper, the new model is used for predicting the future values of China's total and industrial electricity consumption from 2015 to 2020. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Grey prediction model Initial condition optimization Rolling mechanism China's total and industrial electricity consumption prediction

1. Introduction Electricity consumption affects the operation, planning and maintenance of the power system [1]. Meanwhile, it is also considered as a vital driver of economic development and an indispensable part of people's daily activities [2]. Accurate and reliable projections are conducive to better operation and maintenance of power generators [3], and it also can help the electricity system operator regulate the schedule of the electricity grid [4]. What's more, the accurate electricity consumption prediction can mirror the developmental level of the economy to some extent, effectively contributing to understanding economic development trends of a country or a sector [5]. Since electricity has such a great influence on societies, it is imperative to have a reliable method for electricity consumption prediction. The accurate prediction of electricity consumption is influenced by a range of factors, such as population [6], economic growth [2],

* Corresponding author. College of Economics and Management, Nanjing University of Aeronautics and Astronautics, 211106, China. E-mail address: [email protected] (S. Ding). https://doi.org/10.1016/j.energy.2018.01.169 0360-5442/© 2018 Elsevier Ltd. All rights reserved.

power facilities [7] and climate factors [8], which make projections a challenging and complex task. To address such tough problems, many forecasting techniques have been put forward. Hernandez et al. [9] reviewed the most relevant studies on forecasting electricity demand over the last 40 years. These forecasting techniques could be generally divided into three categories: non-linear intelligent models, statistical analysis models and grey prediction models. Non-linear models mainly include Artificial Neural Network [9,10], Support Vector Machine [11,12] and Markov Chain [13,14]. These models possess some inherent limitations that may affect the accuracy of projections. One limitation is the strong reliance on the number of training data (generally more than 30), which are used to discover potentially predictive relationships in intelligent systems, machine learning, genetic programming and statistics. Furthermore, the experience level of practitioners may also have a great effect on the forecasting performance. In addition to the non-linear intelligent models mentioned above, statistical analysis models, such as regression analysis methods [15e17], Autoregressive Integrated Moving Average (ARIMA) [18], and Kalman filter-based techniques [19], have gained popularity in electricity consumption prediction. However, a limitation of the

S. Ding et al. / Energy 149 (2018) 314e328

315

2. Previous developments of GM(1,1) Nomenclature AGO Accumulated generating operation IAGO Inverse accumulated generating operation GM(1,1) Classical grey prediction model GM(1,1, x1(n)) GM(1,1) model with x1(n) as its initial condition [40]. OICGM(1,1) Optimized initial condition GM(1,1) model [42]. PSO Particle swarm optimization DE Differential Evolution Algorithm

statistical analysis models, similar to non-linear intelligent models, is that their performance is highly dependent on having sufficient data collections for calibrating the model parameters. Grey prediction models, proposed by Deng [20], enjoy high popularity in many forecasting applications owing to their capabilities to describe the characteristics of an uncertain system even in the face of having a small amount of data [21]. Besides, due to their applicability even in the presence of sparse data, grey prediction models appear to be more reliable, user friendly and practical than the other competitors. Like the non-linear intelligent models and the statistical analysis models, grey prediction models are also a collection of forecasting models, of which GMð1; 1Þ is the most popular one. As defined in Section 3.1, the GMð1; 1Þ model deals with the grey system issues characterized by insufficient information, uncertainty and small samples. Electricity consumption prediction can be considered as a grey system issue because it is influenced by an abundance of uncertainty. Numerous factors, such as urban population, economic growth, industrial structure and environmental factors, affect the forecasting accuracy, and one does not exactly know how these factors influence the electricity consumption. Moreover, the amount of electricity consumption grows rapidly in emerging countries like China, and the length of collected data is short. Therefore, the GMð1; 1Þ model offers another alternative forecasting approach to electricity consumption prediction. The literature for electricity consumption prediction by grey models is large and continues to increase rapidly. Previously, Hu [22] predicted the future trend of electricity consumption with a neural network-based grey forecasting method. Zhao et al. [23] developed Rolling  ALO  GMð1; 1Þ model to forecast China's and Shanghai's electricity loading. Bahrami et al. [24] integrated a grey model with the wavelet transform for forecasting short term electricity load. Xu et al. [25] used an optimized GMð1; 1Þ model for projections of China's electricity demand. Akav et al. [26] utilized a rolling-based grey model to forecast Tukey's overall and industrial electricity demand. In this paper, a new optimized GMð1; 1Þ model, combined with the rolling mechanism, abbreviated as rolling NOGMð1; 1Þ, is proposed on the basis of a novel initial condition. The remainder of this paper is organized as follows: Section 2 is dedicated to the previous developments of the GMð1; 1Þ model. Section 3 describes the basic theory underlining GMð1; 1Þ and presents three extended models having diverse optimized initial conditions. Then, the rolling NOGMð1; 1Þ model is clearly elaborated and, subsequently, the performance evaluation indices are introduced. Section 4 is devoted to conducting two experimental studies to verifying the competing models and selecting the model with the most accurate performance for predicting China's future total and industrial electricity demand from 2015 to 2020. Based on these findings, Section 5 contains the conclusions and suggestions for future work.

To improve the performance, many studies have been carried out from different perspectives: optimizing parameters, proposing hybrid GMð1; 1Þ models, extending GMð1; 1Þ models and modifying the initial conditions. Firstly, some researchers have been devoted to optimizing the estimation of parameters in the GMð1; 1Þ model. Tan et al. [27] adopted the chaotic co-evolutionary PSO method for estimating the parameters and obtained a good forecasting result of power loads. Lin et al. [28] developed an improved artificial fish swarm algorithm by minimizing the average relative errors to identify the parameters. Additionally, for the purpose of further improving a model's accuracy, a rolling mechanism is frequently used to build an optimized model. Zhao et al. [29] incorporated the rolling mechanism into the hybrid grey model with a differential evolution algorithm, and found that the novel model could significantly improve the forecasting precision in comparison with benchmark models. Secondly, some scholars recently have focused on hybrid GMð1; 1Þ models combined with other methods. Wang [30] integrated a data grouping approach and the GMð1; 1Þ model to forecast the hydropower production characterized by seasonal fluctuations. Thereafter, Lin [31] utilized a fuzzy membership function to establish the fuzzy grey optimization model. Chang [32] combined the data smoothing index with the modified GMð1; 1Þ model for forecasting short-term manufacturing demand. Thirdly, some researchers have extended the modelling form of the traditional GMð1; 1Þ model to further expand its application fields. Xie et al. [33] proposed a discrete grey forecasting model. To solve forecasting nonlinear problems, Chen et al. [34] constructed a Non-linear Grey Bernoulli Model. Xiao [35] proposed an improved seasonal rolling grey prediction model for accurately forecasting traffic flow. In addition, Wu et al. [36] developed fractional order accumulation techniques. Ma [37] designed a novel time-delayed polynomial grey model, which outperforms other competitors when forecasting China's natural gas consumption. Subsequently, utilizing fractional calculus, Yang et al. [38] put forward generalized fractional-order grey models. Finally, other researchers have put extensive efforts into optimizing the initial condition which is regarded as one of the most vital factors influencing the precision of grey prediction models. In the GMð1; 1Þ model, the first data point of a 1  AGO sequence is considered as the initial condition. However, in many cases, the predictive accuracy of this basic model is not always satisfactory, because it fails to incorporate the principle of “new information priority” proposed by Deng [39]. To tackle this challenge, Dang and Liu [40] regarded the last data point of the 1  AGO sequence as the initial condition and constructed new grey models based on this initial condition. To some extent, these optimized models can achieve satisfactory performance due to their consideration of the latest information. However, they exaggerate the influence of the newest information and ignore the former information learned from historical data points, which may reduce its forecasting precision. Subsequently, to further recognize the effect of the first and last data points on a prediction model, Wang [41] employed a weighted combination of these two data points as the initial condition. One of its disadvantages is ignoring the valuable information between the first and last data points of the 1  AGO sequence. Moreover, an explanation of how to obtain the time input parameter was not provided in that paper. In addition to the foregoing optimized initial conditions, Xiong et al. [42] used the weighted average value of each individual component of the 1  AGO sequence as the initial condition, which could not only take into account the influence of each data on forecasting precision, but also

316

S. Ding et al. / Energy 149 (2018) 314e328

conform to the principle of “new information priority” by giving greater weight to newer input data and smaller weight to older input data. A limitation of this model is that its fixed weighted value restrains its ability to adapt to characteristics of different types of data, potentially causing unacceptable forecasting errors. As can be appreciated from the above literature review, the initial condition plays an important role in influencing the predicting precision of GMð1; 1Þ. In order to tackle the inherent drawbacks in previous optimized initial conditions, it is necessary to establish a new grey prediction model with a novel initial condition, which will be clearly illustrated in following sections.

3. Methodology

dxð1Þ ðtÞ þ axð1Þ ðtÞ ¼ b: dt

(2)

Substituting parameters b r ¼ ½b a; b bT into Eq. (2) and solving the differential equation in Eq. (2), one can get the time response function:

. ð1Þ b b b b x ðtÞ ¼ Ce a t þ b a;

(3)

 ð1Þ where C is a constant. Then, selecting b x ðkÞk¼1 ¼ xð1Þ ð1Þ as the initial condition and substituting it into Eq. (3), one can obtain C ¼ xð1Þ ð1Þ  b b= b a . Thus, the time response function can be derived as

As the most popular grey model, GMð1; 1Þ can be considered as a complementary approach to find solutions to predict the behavior of uncertain systems which cannot be suitably explored by stochastic or fuzzy methods with sparse data [43]. Therefore, the following sections are devoted to defining the basic GMð1; 1Þ and its extended versions thereof having optimized initial conditions, and to providing a systematic introduction of the new proposed grey model.

h . i . ð1Þ b b b b a e a ðk1Þ þ b b b a; k x ðkÞ ¼ xð1Þ ð1Þ  b ¼ 2; 3; /; n; n þ 1; /

(4)

Step 4: Obtaining the fitted and predicted values in the original domain. The 1  AGO sequence is regarded as an intermediate sequence, and the restored function can be obtained by utilizing 1  IAGO ð0Þ ð1Þ ð1Þ with the expression b x ðkÞ ¼ b x ðkÞ  b x ðk  1Þ, namely

 h . i ð0Þ b b b b b a 1  e a e a ðk1Þ ; k x ðkÞ ¼ xð1Þ ð1Þ  b

3.1. GM(1,1) and its extensions 3.1.1. Basic GM(1,1) The GMð1; 1Þ model is the most widely used grey model which has gained popularity because of its numerous applications. Researchers such as Akay et al. [26], Lin et al. [28] and Wang et al. [41], have provided detailed modelling procedures for GMð1; 1Þ which are now outlined: Step 1: Transforming the original data. Assume that X ð0Þ ¼ ðxð0Þ ð1Þ; xð0Þ ð2Þ; /; xð0Þ ðnÞÞ, for which the data points are non-negative, equally spaced over time, and at least four in number ðn  4Þ. The 1  AGO sequence is given by X ð1Þ ¼ ðxð1Þ ð1Þ; xð1Þ ð2Þ; /; xð1Þ ðnÞÞ, for which the kth entry is P defined as xð1Þ ðkÞ ¼ ki¼1 xð0Þ ðiÞ; k ¼ 1; 2; /; n. Step 2: Estimating the model parameters. The first-order grey differential equation of GMð1; 1Þ is given by

xð0Þ ðkÞ þ azð1Þ ðkÞ ¼ b;

xð0Þ ð2Þ þ azð1Þ ð2Þ ¼ b xð0Þ ð3Þ þ azð1Þ ð3Þ ¼ b : « xð0Þ ðnÞ þ azð1Þ ðnÞ ¼ b In matrix form, Y ¼ B b r , where

zð0Þ ð2Þ 6 zð0Þ ð3Þ B¼6 4 « zð1Þ ðnÞ

ð0Þ b x ðkÞðk ¼ 1; 2; /; nÞ

where

(5) are

called

fitted

values,

and

ð0Þ b x ðkÞðk  n þ 1Þ are called predicted values.

3.1.2. Extensions of GM(1,1) Although the basic GMð1; 1Þ model is mathematically sound for obtaining predictions, it does not always produce satisfactory results due to its improper selection of an initial condition. To tackle this challenge, Dang and Liu [40] and Xiong et al. [42] proposed two extended grey models having different initial conditions. Dang and Liu [40] chose xð1Þ ðnÞ as the initial condition, namely  ð1Þ b ¼ xð1Þ ðnÞ, for which the resulting time response function x ðkÞ k¼n

(1)

where zð1Þ ðkÞ is called the background value, for which the kth entry is defined as zð1Þ ðkÞ ¼ 0:5xð1Þ ðkÞ þ 0:5xð1Þ ðk  1Þ; k ¼ 2; 3; /; n. Substituting the values of k in to Eq. (1), one can obtain

2

¼ 2; 3; /; n; n þ 1; /;

2 ð0Þ 3 3 x ð2Þ 1   ð0Þ 6 7 b a 17 7; Y ¼ 6 x ð3Þ 7; b r ¼ : b 4 5 b «5 « 1 xð0Þ ðnÞ

is given by

h . i . ð1Þ b b b b a e a ðknÞ þ b b b a; k x ðkÞ ¼ xð1Þ ðnÞ  b ¼ 2; 3; /; n; n þ 1; /

(6)

The restored function for the original sequence is obtained as:

 h . i ð0Þ b b b b b a 1  e a e a ðknÞ ; k x ðkÞ ¼ xð1Þ ðnÞ  b ¼ 2; 3; /; n; n þ 1; /

(7)

The above grey prediction model is called GMð1; 1Þ considering ð1Þ b x ðkÞjk¼n ¼ xð1Þ ðnÞ as its initial condition, which is more simply

written as GMð1; 1; xð1Þ ðnÞÞ. In order to take fully into account the impact of an individual component on the forecasting precision, Xiong et al. [42] employed the weighted average value of each component of the 1  AGO series as the initial condition, expressed as

By solving the above matrix form of the equation, the least squares estimation for a and b are b r ¼ ½b a; b bT ¼ ðBT BÞ1 BT Y.

 ð1Þ  b x ðkÞ

Step 3: Obtaining the time response function for forecasting. The whitened equation of GMð1; 1Þ is written as

P where ak ¼ k= nk¼1 k; k ¼ 1; 2; /; n and a1 þ a2 þ / þ an ¼ 1.

k¼b

¼ a1 xð1Þ ð1Þ þ a2 xð1Þ ð2Þ þ / þ an xð1Þ ðnÞ;

(8)

S. Ding et al. / Energy 149 (2018) 314e328

The time response function having the time input parameter b is given by

h . i . ð1Þ b b b b a e a ðkbÞ þ b b b a; k x ðkÞ ¼ xð1Þ ðbÞ  b ¼ 2; 3; /; n; n þ 1; /

(9)

By utilizing 1  IAGO, the restored function is obtained as follows:

 h . i ð0Þ ð1Þ b b b b b a 1  e a e a ðkbÞ ; k x ðkÞ ¼ b x ðbÞ  b ¼ 2; 3; /; n; n þ 1; /

(10)

where b ¼ lnðr=stÞ, s ¼ ½a1 xð1Þ ð1Þ þ a2 xð1Þ ð2Þ þ / þ an xð1Þ ðnÞ  b b= b a , ba Pn Pn b b b a  a k ð0Þ 2 a k x ðkÞ, and t ¼ k¼1 e . ð1  e Þ, r ¼ k¼1 e The model above is called GMð1; 1Þ considering  ð1Þ b ¼ a1 xð1Þ ð1Þ þ a2 xð1Þ ð2Þ þ / þ an xð1Þ ðnÞ as its initial x ðkÞ k¼b

condition, which is denoted as OICGMð1; 1Þ. 3.2. The new proposed rolling NOGM(1,1) model Despite the fact that three extended models having optimized initial conditions were proposed, they are still prone to produce unacceptable errors in practical applications [45]. To address this problem, a novel rolling-based grey prediction model is developed, namely the rolling NOGMð1; 1Þ model. The detailed procedures of the rolling NOGMð1; 1Þ model are elaborated in the following subsections. For comparison, the modelling mechanisms of the rolling GMð1; 1Þ, rolling GMð1; 1; xð1Þ ðnÞÞ, and rolling OICGMð1; 1Þ models are also introduced. 3.2.1. Rolling GM(1,1) and its extensions Since several scholars have elaborated on how the rolling mechanism works [22,28], here the authors outline its modelling procedure in a simplified and easily understandable way, as shown in Fig. 1. In this figure, for simplicity, each original input datum is represented by the time of its occurrence. For instance, 1; 2; /; n stand for the actual observations xð0Þ ð1Þ; xð0Þ ð2Þ; /; xð0Þ ðnÞ, respectively. Assume that,c data points are employed for constructing the GMð1; 1Þ, GMð1; 1; xð1Þ ðnÞÞ and OICGMð1; 1Þ models, and d data points are forecasted by using these three models at each rolling step. Using the rolling GMð1; 1Þ model for explanation purposes, it can be built as explained below in conjunction with the corresponding steps portrayed in Fig. 1: Step 1: Initially, utilize the original series xð0Þ ð1Þ; xð0Þ ð2Þ; /; xð0Þ ðcÞ to establish GMð1; 1Þ, and then produce ð0Þ ð0Þ ð0Þ the first forecasted series b x ðc þ 1Þ; b x ðc þ 2Þ; /; b x ðc þ dÞ as shown at the top of Fig. 1. Step 2: For predicting the future data points ð0Þ ð0Þ ð0Þ b x ðc þ d þ 1Þ; b x ðc þ d þ 2Þ; /; b x ðc þ 2dÞ, remove the oldest ð0Þ ð0Þ ð0Þ series x ð1Þ; x ð2Þ; /; x ðdÞ and employ the latest c data points

317

contribute to improving forecasting accuracy. However, deterministic methods may limit the model's adaptability to various data characteristics, and, as a consequence, even some excellent optimized models may produce undesirable results [45]. To tackle these challenges, the NOGMð1; 1Þ model is designed and supported by using a PSO algorithm to obtain optimal parameters. Considering the significant impact on the predictive precision, a dynamic weighted coefficient is proposed to reflect the influence level of each individual element constituting the 1  AGO sequence. The novel optimized initial condition for the whitened equation can be expressed as

  ð1Þ b x ðkÞ

k¼4

¼ ln1 xð1Þ ð1Þ þ ln2 xð1Þ ð2Þ þ / þ lnn xð1Þ ðnÞ Xn lnk xð1Þ ðkÞ; ¼ k¼1

(11)

where lnk ð0 < l < 1Þðk ¼ 1; 2; /; nÞ is the dynamic weighted coefficient, l is the weighted parameter, and 4 is the time input coefficient. b are provided by Eq. Theorem 1. If the values of parameters b a; b (1), namely ½ b a; b bT ¼ ðBT BÞ1 BT Y, one can obtain the time response and restored functions as follows: (1) The time response function possessing the novel initial condition can be determined using ð1Þ b x ðkÞ ¼

hXn i¼1

. i

.

lni xð1Þ ðiÞ  bb ba eba ðk4Þ þ bb ba ; k

¼ 2; 3; /; n; n þ 1; /

(12)

(2) The restored function is given by the following equation ð0Þ b x ðkÞ ¼

hXn i¼1

. i



lni xð1Þ ðiÞ  bb ba 1  eba eba ðk4Þ ; k

¼ 2; 3; /; n; n þ 1; /;

(13)

ð0Þ Where b x ðkÞ ðk ¼ 1; 2; /; nÞ are called the fitted values, and ð0Þ b x ðkÞðk  n þ 1Þ are referred to as the predicted values. Proof: (1) According to the solution to the whitened function in ð1Þ Eq. (3), namely b x ðtÞ ¼ Ceba t þ b b= b a, after substituting  P ð1Þ ni n b x ðkÞk¼4 ¼ i¼1 l xð1Þ ðiÞ into Eq. (3), one can obtain

C¼

hXn

. i

lni xð1Þ ðiÞ  bb ba ,eba 4 : i¼1

By substituting C into Eq. (3), the time response function is found to be ð1Þ b x ðkÞ ¼

hXn i¼1

. i

.

lni xð1Þ ðiÞ  bb ba eba ðk4Þ þ bb ba :

(2) The restored function is obtained as follows,

ð0Þ

x ðd þ cÞ as newly input items for xð0Þ ðd þ 1Þ; xð0Þ ðd þ 2Þ; /; b rebuilding GMð1; 1Þ, as portrayed in the second graph from the top in Fig. 1. Step 3: Repeat Step 2 by removing the oldest series and using the latest c data points for forecasting the next set of d data points. Step 4: Obtain the last d forecasted data points. 3.2.2. The novel optimized initial condition for GM(1,1) As discussed before, an optimized initial condition can usually

ð0Þ ð1Þ ð1Þ b x ðkÞ  b x ðk  1Þ x ðkÞ ¼ b  . i hXn lni xð1Þ ðiÞ  bb ba 1  eba eba ðk4Þ ; k ¼ i¼1

¼ 2; 3; /; n; n þ 1; /: Based on the principle of “new information priority”, the degree of xð1Þ ðkÞ being new information can be reflected by the corresponding

weighted

coefficient,

namely

lnk ð0 < l < 1; k ¼

318

S. Ding et al. / Energy 149 (2018) 314e328

Fig. 1. Forecasting procedures of the rolling GM(1,1), rolling GM(1,1,x(1)(n) and rolling OICGM(1,1) models.

1; 2; /; nÞ. Assuming that the weighted parameters l ¼ 0:2; 0:4; 0:6; 0:8 represent four kinds of input sequences possessing different features, and that the sample size n of the four input sequences equals 10, then the dynamic values of the weighed coefficients in the four initial conditions can be illustrated as in Fig. 2. Fig. 2 indicates that the varying values of l can create four increasing curves having different growth rates. This infers that the novel initial condition possessing adjustable weighted coefficients can be used to address various issues having diverse characteristics. By contrast, the initial condition having a fixed weighted value [42] or a constant value [40] can only address and analyze one kind of problem. Therefore, the novel initial condition has a broader range of applications than the old ones. What's more, the new proposed initial condition can also conform to the principle of “new information priority”. That is, the greater the value k, the larger the weighted value of the corresponding xð1Þ ðkÞ is. It can be intuitively illustrated by the following formula when l satisfies 0 < l < 1:

ln1 < ln2 < ln3 < / < l1 < lnn In summary, this model not only conforms to the principle of “new information priority,” but also can take fully into account the historical information influencing the accuracy of the model.

3.2.3. The solution to the generating coefficients of the NOGM(1,1) model As it can be seen from Section 3.2.2, the two generating coefficients, weighted coefficient l and time input coefficient 4, are unknown and need to be determined before projections. Reliable and precise estimation of the generating coefficients in the NOGMð1; 1Þ model is of vital importance for successful modelling and projecting. The optimum values of the generating coefficients are calculated through minimizing the mean absolute percentage error (MAPE) between the fitted and the actual data points. For this, the optimal generating coefficients in the optimized initial condition are determined by the subsequent objective function:

S. Ding et al. / Energy 149 (2018) 314e328

319

Fig. 2. Dynamic changing trend of the weighted value of each individual component.

  ðkÞ  xð0Þ ðkÞ 1 ; Min avgðeðl; 4ÞÞ ¼ n xð0Þ ðkÞ k¼1  hXn . i 8 ð0Þ b lni xð1Þ ðiÞ  bb ba 1  eba eba ðk4Þ x ðkÞ ¼ > > i¼1 > < h iT  1 b a; b b ¼ BT B BT Y > > > : l2ð0; 1Þ; k ¼ 1; 2; /; n 

n b x X

vkþ1 ¼ c1 ðpbestk  xk ÞR1 þ c2 ðgbestk  xk ÞR2 ;

ð0Þ

(14)

ð0Þ

where b x ðkÞ is the fitted data points, xð0Þ ðkÞ is the actual data points, and n is the number of the input data points. Approaching the minimum value of the objective function contributes to obtaining the optimal value of the generating coefficients. Owing to its nonlinear features, the objective function cannot be solved in the ordinary way. Hence, the PSO algorithm, which sets a group of particles in feasible space to find the best location, can work as a complementary methodology to determine the optimal values of vector ½l; 4T . The main procedure of the PSO algorithm is outlined as follows. Initially, denote vector Q ¼ ½l; 4T and build a fitness function of each particle according to the optimization function above. The expression of the fitness function can be calculated as

where c1 ; c2 are acceleration factors and R1 ; R2 2½0; 1 are random variables. Renew particles' positions by Q ði; j þ 1Þ ¼ Q ði; jÞ þ vk, and particle velocities in next iteration should be calculated as

vkþ1 ¼ wvk þ c1 ðpbestk  xk ÞR1 þ c2 ðgbestk  xk ÞR2 ; where w is an inertia weight, adjusting rate of convergence. l; 4 can be obtained from the corresponding Gbest when the fitness function approaches the minimal value mathematically. Subsequently, substitute the optimal l; 4 into the time restored function in Eq. (13), and the fitted values and predicted values are described as ð0Þ b x ðkÞ ¼

i¼1

. i



lni xð1Þ ðiÞ  bb ba 1  eba eba ðk4Þ ; k

¼ 2; 3; / ð0Þ ð0Þ When k  n, b x ð2Þ; /; b x ðnÞ represent fitted values to the actual data. As for the predicted values, extrapolating k > n generð0Þ ð0Þ x ðn þ 2Þ; /. ates the future series b x ðn þ 1Þ; b

3.2.4. Rolling NOGM(1,1)

 h P  . i   ni ð1Þ n n  x ðiÞ  b b b a 1  eba eba ðk4Þ  xð0Þ ðkÞ i¼1 l 1X : Fitness½Q ði; jÞ ¼ n xð0Þ ðkÞ T

hXn

(15)

k¼1

Subsequently, set parameters. In this case, Q ði; jÞ is regarded as the position of jth particle in ith iteration. PbestðiÞ records the minimal fitness of ith iteration and its position. Gbest represents the global best position in the search space, and renew its value when PbestðiÞ < Gbest. Then, initialize the particle velocity formula in first iteration as below:

The rolling NOGMð1; 1Þ model refers to combining the NOGMð1; 1Þ model with a rolling mechanism. The parameters l and 4 in this model are optimally obtained by employing the PSO algorithm at each rolling stage. Hence, the modelling process presents higher computational complexity than that of the NOGMð1; 1Þ, rolling GMð1; 1Þ, and its extended models. The procedure for the rolling NOGMð1; 1Þ model is schematically drawn in

320

S. Ding et al. / Energy 149 (2018) 314e328

MAPE ¼

n 1 X APEðkÞ n1

(17)

k¼2

RMSE ¼

Fig. 3. Flowchart of the rolling NOGM(1,1) model.

Fig. 3 and elaborated as follows: Step 1: Employ the actual observations xð0Þ ð1Þ; xð0Þ ð2Þ; /; xð0Þ ðcÞ as the input data for the NOGMð1; 1Þ model. According to the fitness function above, use the PSO algorithm to calculate the parameters l and 4. As a result, one can obtain the forecasted series ð0Þ ð0Þ ð0Þ b x ðc þ 2Þ; /; b x ðc þ dÞ. x ðc þ 1Þ; b Step 2: After removing the oldest d data points xð0Þ ð1Þ; xð0Þ ð2Þ; /; xð0Þ ðdÞ, update the modelling data with the latest ð0Þ

x ðd þ cÞ,. Then recalculate c data points xð0Þ ðd þ 1Þ; xð0Þ ðd þ 2Þ; /; b the new values of parameters l and 4 by using the PSO algorithm. Consequently, the new forecasting process can be performed again. Step 3: Repeat the steps above until all of the remaining data points of interest are predicted.

3.3. Evaluation of the modelling accuracy For the purpose of evaluating the predictive performance of the proposed forecasting techniques, the most important thing is to choose an appropriate evaluation index, which can effectively reveal the differences between the observed and predicted values presented by the competing forecasting models. In this study, three statistical indicators are determined, namely APE (Absolute Percentage Error), MAPE (Mean Absolute Percentage Error), and RMSE (Root Mean Squared Error). The APE, MAPE and RMSE are calculated by utilizing Equations (16)e(18), respectively.

APE ¼

 ð0Þ  b x ðkÞ  xð0Þ ðkÞ xð0Þ ðkÞ

 100%

(16)

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 Xn  ð0Þ b x ðkÞ  xð0Þ ðkÞ ; n  1 k¼2

(18)

ð0Þ where xð0Þ ðkÞ is the original value at time k, and the b x ðkÞ is the fitted or predicted value at time k. In general, two procedures to compare the performance of the proposed grey models using the above three indices can be followed. Firstly, the ability of each model to produce accurate fitted data points which reflect the observations used to calibrate the model, is examined. For the in-sample data points, the above three indices are employed to measure the differences between the original data and the fitted values for each competing model. The fitted performance of these competing models, can then be compared using the results calculated utilizing Equations (16)e(18). Experimentally, the two datasets consisting of the total and industrial electricity consumption from 2005 to 2011 are used for the parameter estimation of the rolling NOGMð1; 1Þ as well as the other prediction models. Then, one can measure the divergence in findings between the original data and the fitted values that are reproduced by using the calibrated models. Subsequently, a graph for comparing the historical data and the fitted data is provided in the experiments (see Section 4.1.1). Secondly, the calibrated model can be used to forecast one or more values of the data points not used to estimate the model parameters in what is called a split-sample forecasting experiment. Once the calibrated model is established, it can be employed for predicting the out-sample datasets. Then, the above three indices can be computed for each of the competing models to compare their accuracy in this split sample forecasting experiment. After completing the above two procedures, the model having the best fitted and predicted performance will be utilized for obtaining forecasts of the total and industrial electricity consumption from 2015 to 2020. All three indices reflect the levels of accuracy of each of the competing models. However, the RMSE in Eq. (18) possesses the theoretical properties that make it more attractive to use in practice [46].

4. Case studies on forecasting the electricity consumption in China Annual electricity consumption forecasting is not only important for the scheduling and operation of a power system, but it can also mirror the level of economic growth in a nation or a sector, such as industry, agriculture. Accordingly, two empirical cases are implemented: one is the total electricity consumption prediction, at the state level; the other is the industrial electricity consumption prediction within the industrial sector in China. In order to make successful and satisfactory projections for the total and industrial electricity consumption amounts, the rolling NOGMð1; 1Þ model, denoted as M10, is proposed in Section 3.2. To demonstrate its forecasting accuracy, the new model is compared to other

Table 1 Criteria of MAPE [22]. MAPE(%)

Forecasting power

<10 10e20 20e50 >50

Excellent Good Reasonable Incorrect

S. Ding et al. / Energy 149 (2018) 314e328

China's total and industrial electricity demand.

Table 2 Parameter values of the rolling NOGM(1,1) model for the total electricity consumption. Parameters

a

b

l

4

2012 2013 2014

0.095719 0.090804 0.091420

24940.15 27834.40 30077.98

0.161223 0.084567 0.079667

7.771895 7.469134 7.385206

4.1. Case one: projecting China's total electricity consumption 4.1.1. Calibration and forecasts of the competing models Predicting China's annual total electricity consumption is carried out to examine the efficacy of the newly proposed model. The small data sets were collected from the China Statistical Yearbook published by China's National Statistics Bureau (http://data.stats. gov.cn/english/easyquery.htm?cn¼C01). The small sample data sets include the total amount of electricity consumption from 2005 to 2014, as shown in Fig. 4. It can be graphically seen from Fig. 4 that the general characteristic of China's total electricity consumption is an increasing trend, although it might have a slight short-term fluctuation. Selecting the input data sets and determining their length are considered as prerequisite conditions, which may affect the accuracy of a prediction model [44]. In this paper, the optimum length of the input data sets is determined by using the optimized subset method introduced by Wang [45]. After using this method, the authors found that the forecasted results obtain the highest precision when the length of the data sets is set as seven. Therefore, the authors set c ¼ 7 and d ¼ 1, which implies that seven data points are employed as the input data points and the next is predicted. The flowchart for predicting China's total electricity consumption by using the rolling NOGMð1; 1Þ model is presented in Fig. 5. As shown,

Table 3 Parameter values of grey models without a rolling mechanism for the total electricity consumption. Parameters

a

b

b

M1 M2 M3 M4 M5

0.095718 0.095718 0.095718 0.095718 0.095719

24940.15 24940.15 24940.15 24940.15 24940.15

5.164535

l

4

0.161223 0.631234

7.771895 11.08254

321

competing models, consisting of the GMð1; 1Þ, GMð1; 1; xð1Þ ðnÞÞ, OICGMð1; 1Þ, NOGMð1; 1Þ, DE  NOGMð1; 1Þ, rolling GMð1; 1Þ, rolling GMð1; 1; xð1Þ ðnÞÞ, rolling OICGMð1; 1Þ and ARIMA models, which are denoted as M1,M2,M3,M4,M5,M6,M7,M8, and M9 from Tables 2e14, respectively. The following sections give detailed information about how to use these grey models for forecasting

Table 4 Parameter values of the rolling GMð1; 1Þ, rolling GMð1; 1; xð1Þ ðnÞÞ and rolling OICGMð1; 1Þ models for the total electricity consumption. Parameters

M6

2012 2013 2014

M7

M8

a

b

a

b

a

b

ak

b

0.095719 0.090804 0.091420

24940.15 27834.40 30077.98

0.095719 0.090804 0.091420

24940.15 27834.40 30077.98

0.095719 0.090804 0.091420

24940.15 27834.40 30077.98

k=28 k=28 k=28

5.164535 5.139365 5.098133

Table 5 Forecasted results by the competing models for the total electricity consumption [Unit:108 kWh]. Year

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

2012 2013 2014

50929.63 56045.49 61675.24

50958.66 56077.44 61710.4

50823.17 55928.35 61546.33

50769.73 55869.53 61481.61

50782.36 55883.44 61496.91

50929.63 54927.34 59929.01

50958.66 54951.43 59945.21

50823.17 54904.73 60133.38

49401.84 57858.02 62360.65

50769.73 54492.76 59780.44

Table 6 APE and MAPE determined by the competing models for the total electricity consumption [Unit: %]. Year

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

2012 2013 2014

2.345 3.398 9.385

2.404 3.457 9.447

2.131 3.182 9.156

2.024 3.074 9.041

2.049 3.100 9.069

2.345 1.336 6.288

2.404 1.380 6.317

2.131 1.294 6.650

0.725 6.742 10.600

2.024 0.534 6.024

MAPE

5.043

5.103

4.823

4.713

4.739

3.323

3.367

3.358

6.023

2.861

Note: the smallest gap between the predicted values and the observed values at each year is in bold.

Table 7 Residual error and RSME determined by the competing models for the total electricity consumption [Unit: 108 kWh]. Year

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

2012 2013 2014

1167.03 1842.09 5291.54

1196.06 1874.04 5326.70

1060.57 1724.95 5162.63

1007.13 1666.13 5097.91

1019.76 1680.04 5113.21

1167.03 723.94 3545.31

1196.06 748.03 3561.51

1060.57 701.33 3749.68

360.76 3654.62 5976.95

¡1007.13 ¡289.36 ¡3396.74

RMSE

3304.32

3332.48

3201.72

3150.61

3162.66

2195.09

2211.67

2285.95

4050.12

2052.31

Note: the smallest gap between the predicted values and the observed values at each year is in bold.

322

S. Ding et al. / Energy 149 (2018) 314e328

Table 8 Forecasted results (unit: 108 kWh) and optimal parameters by the Rolling NOGM(1,1) model for the total electricity consumption during 2015e2020. Year

2015

2016

2017

2018

2019

2020

a b 4

0.080300 34362.41 0.146303 7.698198

0.074842 38394.34 0.087844 7.384479

0.072530 41776.56 0.098144 7.407015

0.074700 44320.29 0.074604 7.320959

0.075226 47721.01 0.131517 7.616186

0.077290 50854.89 0.121191 7.584197

Forecasting Values

62652.95

67130.23

72180.07

77947.40

84125.87

91016.04

l

adjust to the characteristics of the input data in each rolling step, as explained in Section 3.2.1. Fig. 6 shows the best fitness values of the rolling NOGMð1; 1Þ model, found by searching the optimal parameters in the optimized initial conditions for forecasting the total electricity consumption from 2012 to 2014. It is obvious that the PSO algorithm is a sufficiently fast and effective approach to finding the optimal parameters at each rolling stage. In addition, Figs. 7e9 display the split sample experiments using the rolling NOGMð1; 1Þ model for predicting the total electricity consumption from 2012 to 2014. These figures intuitively show that the forecasts fit the original values well. This result can also be verified by utilizing the three indices mentioned in Section 3.2, which is explained in Section 4.1.2. To further effectively reflect the performance of this new model, nine alternative forecasting models are selected as benchmarks. Among these competing models, M1  M5 are rolling-free grey forecasting models. M6  M8 are rolling-based grey forecasting models. The ARIMA works as a non-grey model. In addition, because of the different modelling mechanisms of the nine grey models, their estimated parameters are separated in Tables 3 and 4 Specifically, Table 3 provides the estimated values of the parameters in the rolling-free grey forecasting models, while Table 4 presents the parameter values of the rolling-based grey models. The forecasted results for all competitors are provided in Table 5.

Table 9 Parameter values of the rolling NOGM(1,1) model for the industrial electricity consumption. Parameters

a

b

l

4

2012 2013 2014

0.094357 0.088968 0.089104

18467.03 20576.04 22188.76

0.103080 0.349984 0.149924

7.476573 9.006311 7.745657

Table 10 Parameter values of grey models without a rolling mechanism for the industrial electricity consumption. Parameters

a

b

b

M1 M2 M3 M4 M5

0.094357 0.094357 0.094357 0.094357 0.94357

18467.03 18467.03 18467.03 18467.03 18467.03

5.160236

l

4

0.103080 0.760721

7.476573 12.471856

the parameters (a,b,land4) are calculated three times for forecasting the total electricity consumption in 2012, 2013 and 2014. The values for the parameters in each rolling stage are presented in Table 2, which are estimated by using Equations (12)e(15). From Table 2, the authors find that the parameter values are different in each rolling step, which suggest that the newly proposed model can

Table 11 Parameter values of the rolling GMð1; 1Þ, rolling GMð1; 1; xð1Þ ðnÞÞ and rolling OICGMð1; 1Þ models for the industrial electricity consumption. Parameters

M6

2012 2013 2014

M7

M8

a

b

a

b

a

b

ak

b

0.094357 0.088968 0.089104

18467.03 20576.04 22188.76

0.094357 0.088968 0.089104

18467.03 20576.04 22188.76

0.094357 0.088968 0.089104

18467.03 20576.04 22188.76

k=28 k=28 k=28

5.160236 5.131385 5.087791

Table 12 Forecasting results of the industrial electricity consumption by the compared models [Unit:108 kWh]. Year

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

2012 2013 2014

37341.44 41036.46 45097.12

37364.48 41061.78 45124.95

37273.24 40961.52 45014.76

37283.78 40973.10 45027.49

37285.18 40974.64 45029.18

37341.44 40073.66 43474.30

37364.48 40090.67 43483.34

37273.24 40076.02 43645.64

35221.60 42557.02 44845.23

37283.78 39644.75 43333.43

Table 13 APE and MAPE determined by the compared models for the industrial electricity consumption [Unit: %]. Year

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

2012 2013 2014

3.061 4.586 10.525

3.125 4.651 10.593

2.873 4.395 10.323

2.902 4.425 10.354

2.906 4.429 10.358

3.061 2.133 6.548

3.125 2.176 6.570

2.873 2.139 6.968

2.789 8.462 9.908

2.902 1.039 6.202

MAPE

6.058

6.123

5.864

5.894

5.898

3.914

3.957

3.993

7.053

3.381

Note: the smallest gap between the predicted values and the observed values at each year is in bold.

S. Ding et al. / Energy 149 (2018) 314e328

323

Table 14 Residual error and RSME determined by the competing models for the industrial electricity consumption [Unit: 108 kWh]. Year

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

2012 2013 2014

1109.24 1799.56 4294.42

1132.28 1824.88 4322.25

¡1041.04 1724.62 4212.06

1051.58 1736.20 4224.79

1052.98 1737.74 4226.48

1109.24 836.76 2671.60

1132.28 853.77 2680.64

1041.04 839.12 2842.94

1010.60 3320.12 4042.53

1051.58 ¡407.85 ¡2530.73

RMSE

2763.50

2786.52

2695.65

2706.11

2707.50

1738.58

1750.89

1813.86

3076.07

1599.66

Note: the smallest gap between the predicted values and the observed values at each year is in bold.

Fig. 4. Total electricity consumption in China during 2005e2014.

4.1.2. Comparing the forecasting performance of the new model and its competitors To comprehensively evaluate the predictive performance of all

the selected models, evaluations from two different perspectives are considered: one is for the overall findings displayed graphically in figures and the other is to measure the index performances generated by these competing models. Fig. 10 illustrates the predicted results of the total electricity consumption from 2012 to 2014 by using the eight grey prediction models. From this figure, one can see that the rolling-free grey forecasting models obtain poorer forecasts because the predicted values are much larger than the real ones. Among these five models, the NOGMð1; 1Þ model achieves better performance than the other four models, which illustrates that the new optimized initial condition outperforms others and the PSO is a feasible method to select optimal parameters when comparing with the DE algorithm. However, although these rolling-free grey models performs well, a large gap still exists between its forecasted values and the actual observations. Compared with these rolling-free models, the rolling-based grey forecasting models perform better and their predicted results are much closer to the actual observations. Because these rolling-based grey models use the latest information for model calibration, they can grasp the most recent development trend and characteristics of the total electricity consumption in China. Among these rolling-based models, the rolling NOGMð1; 1Þ)

Fig. 5. Forecasting procedures of the total electrical consumption by the rolling NOGM(1,1) model.

324

S. Ding et al. / Energy 149 (2018) 314e328

Fig. 7. Split sample experiment using the rolling NOGM(1,1) model for predicting the total electricity consumption in 2012.

Fig. 8. Split sample experiment using the rolling NOGM(1,1) model for predicting the total electricity consumption in 2013.

Fig. 9. Split sample experiment using the rolling NOGM(1,1) model for predicting the total electricity consumption in 2014.

Fig. 6. The iterative fitness values evolution of the rolling NOGM(1,1) model searching for optimal parameters.

model is superior to the other three models, because its predicted values diverge the least from the actual values. As for the comparison between the ARIMA and newly proposed models, the novel model performs much better than this non-grey model. Accordingly, the rolling NOGMð1; 1Þ) model is the optimal methodology to predict the future demand for the total electricity in China. In addition to the above overall conclusion from Fig. 10, three performance evaluation indices (namelyAPE,MAPE and RMSE), introduced in Section 2.2.5, are selected to evaluate the forecasting precision of the competing models. The results of three indicators

S. Ding et al. / Energy 149 (2018) 314e328

325

Fig. 10. Forecasted results of the total electricity consumption from 2012 to 2014 by using the competing models.

are given in Tables 6 and 7 Table 6 shows that the rolling-based grey forecasting models achieve smaller APE values than those without a rolling mechanism. Of these rolling models, the APE values generated by the rolling NOGMð1; 1Þ for every forecasting year, from 2012 to 2014, are 2.024, 0.523, 6.024, respectively, which is the smallest annually among the ten competing models. In terms of MAPE, it is verified again that the rolling-based grey forecasting models perform better than the rolling-free ones. According to the standard of MAPE in Table 1, all of the eight grey forecasting models have excellent performance because their MAPE values are less than 10%. Although these models all perform quite well, one model is prominent. In particular, the rolling NOGMð1; 1Þ model is the best as a result of having the smallest MAPE of 2.861 among these competing models. Hence, the rolling NOGMð1; 1Þ model is the most appropriate forecasting model for projecting China's total electricity consumption. Table 7 shows a similar finding for the rolling NOGMð1; 1Þ model as that found from the MAPE values: the residual error for each forecasting year (1007.13, 289.36 and 3396.74, respectively) and the RMSE value (2052.31) of the rolling NOGMð1; 1Þ model are the smallest among the competing models. Therefore, the comparison further demonstrates the efficacy of the rolling NOGMð1; 1Þ model for improving the accuracy of the conventional and optimized GMð1; 1Þ models. In summary, the APE for each year forecasts are made, as well as MAPE and RMSE, of the rolling NOGMð1; 1Þ model are the smallest among the ten competing models. These values are directly related to the optimized initial condition having a dynamic weighted coefficient. As a result, the rolling NOGMð1; 1Þ model can flexibly capture the characteristics of the electricity consumption system and provide the best forecasting performance among the competing models. In addition, the NOGMð1; 1Þ model shows higher accuracy than the GMð1; 1Þ,GMð1; 1; xð1Þ ðnÞÞ, OICGMð1; 1Þ and DE  NOGMð1; 1Þ models, which also indicates that the novel proposed initial condition, optimally calculated by using the PSO algorithm, contributes significantly to improving the forecasting precision of grey forecasting models. Equally important to the success of the optimized initial condition is the rolling mechanism. Because this algorithm, which embodies the latest information and development trend of a variable, employs the most recent data sets as the input, this rolling-based model is superior to those without

rolling mechanisms. What's more, the PSO algorithm is a more appropriate approach to determining optimal parameters when comparing with the DE algorithm. The two foregoing findings confirm that the rolling NOGMð1; 1Þ model furnishes more accurate projections. Therefore, the authors utilize this model as the optimal choice to quantitatively characterize the future trends of the total electricity consumption up to 2020 in China. 4.1.3. Forecasting the future total electricity demand from 2015 to 2020 Because of its demonstrated capability to provide accurate forecasts, the rolling NOGMð1; 1Þ model is employed to predict China's future total electricity consumption from 2015 to 2020. The parameters and forecasted results are presented in Table 8. It can be seen that the total electricity consumption will maintain an upward growth and, by 2020, it is expected to increase from 563.84 million kWh in 2014 to more than 910 million kWh. This result is of great importance for energy planning and policy making. Thus, the government needs to take appropriate actions to satisfy high energy demands in the future. 4.2. Case two: forecasting China's industrial electricity consumption 4.2.1. Calibration of the competing models and calculating forecasts The available data sets of China's industrial electricity consumption from 2005 to 2014 come from the China Statistical Yearbook (http://data.stats.gov.cn/english/easyquery.htm? cn¼C01), which are plotted in Fig. 11. This clearly shows an upward trend as was also the situation for the total electricity consumption. In this case, by setting c ¼ 7 and d ¼ 1, the authors still use seven data points as the input data to predict the eighth data point. For the ten competing models, their forecasting procedures for the industrial consumption are the same as those for the total electricity consumption. Their parameters for predicting the industrial electricity consumption from 2012 to 2014 are listed in Tables 9e11, for which the projected results are presented in Table 12. 4.2.2. Comparing the forecasting performance of the ten competing models Because the approaches to calibrating these models are similar

326

S. Ding et al. / Energy 149 (2018) 314e328

Fig. 11. Industrial electricity consumption in China during 2005e2014.

to those for the total electricity consumption (see Section 4.1.1), a detailed explanation for the competing models will not be provided in this section. As for the forecasting performance evaluation indices, the authors still use the most popular indices, namely APE, MAPE and RMSE. The estimated parameter values in the rolling-free and rolling based models are given in Tables 10 and 11, respectively. Subsequently, the forecasted results of the ten competing models are listed in Table 12 and Fig. 12. Table 12 and Fig. 12 both indicate that the rolling-based grey forecasting models have better forecasting performance than the rolling-free except for the year 2012. In 2012, the forecasted values of the OICGMð1; 1Þ and rolling OICGMð1; 1Þ models have the smallest difference from the actual ones, while the NOGMð1; 1Þ and rolling NOGMð1; 1Þ models posse the second smallest deviation. However, it is worth mentioning that in the subsequent two years, the NOGMð1; 1Þ and rolling NOGMð1; 1Þ models show the greatest forecasting accuracy, with the predicted values being closest to the actual ones. The APE s of the ten forecasting models in Table 13 reveal that the rolling NOGMð1; 1Þ model has the best forecasting performance in 2013 and 2014, and the OICGMð1; 1Þ model is the best model in 2012. The initial conditions of these two models emphasizes the effect of recent information on the forecasting precision, which implies that applying the latest data points of the industrial

electricity consumption to the construction of an initial condition can significantly improve a model's accuracy. As for MAPE, the rolling NOGMð1; 1Þ model has the smallest value compared to the other nine models, which verifies again that the new proposed model outperforms the remaining models in terms of predicting the industrial electricity consumption. The RMSE values for the industrial electricity demand are shown in Table 14. The findings are the same as those for APE and MAPE values. The rolling NOGMð1; 1Þ model has the smallest RMSE and is, therefore, recommended for predicting the future demand for the industrial electricity. In conclusion, the rolling-based grey forecasting models perform better than the rolling-free grey forecasting models. The three evaluation indices also confirm that the rolling NOGMð1; 1Þ model with the optimized initial condition is most suitable for forecasting purposes. Therefore, this novel model will be utilized for forecasting China's industrial electricity consumption from 2015 to 2020. 4.2.3. Forecasting the future total electricity demand during 2015e2020 Because of its forecasting accuracy, the rolling NOGMð1; 1Þ model is employed to predict future industrial electricity demand from 2015 to 2020. The parameters and forecasted results are

Fig. 12. Forecasted results of the industrial electricity consumption from 2012 to 2014 by using the competing models.

S. Ding et al. / Energy 149 (2018) 314e328

327

Table 15 Forecasted results (unit: 108 kWh) and optimal parameters by the rolling NOGM(1,1) model for the industrial electricity demand during 2015e2020. Year

2015

2016

2017

2018

2019

2020

a b 4

0.077593 25338.36 0.263860 8.323698

0.071417 28416.55 0.033234 7.169939

0.068582 30826.69 0.125245 7.579487

0.071359 32437.67 0.025991 7.103372

0.071598 34876.16 0.017488 7.065316

0.073013 37162.37 0.073890 7.353512

Forecasting Values

45504.40

48212.35

51621.93

55608.35

59764.29

64351.45

l

shown in Table 15. It can be seen that industrial electricity consumption will keep growing and by 2020 it could potentially see an increase of more than 643 million kWh from 408.03 million in 2014, which will create a heavy pressure on the operation, planning and maintenance of the electric power system for the industrial sector.

5. Discussion and conclusions Electricity consumption prediction is of great importance for developing and expanding a power system, fostering economic development and supporting people's daily activities. However, due to increasing demand in many areas and higher complexity, an accurate prediction for electricity consumption becomes more and more difficult. Therefore, designing an appropriate methodology for short-term prediction with limited data makes a great deal of sense. For the purpose of solving certain forecasting problems, this paper proposes a rolling NOGMð1; 1Þ model, namely GMð1; 1Þ combined with a novel optimized initial condition and a rolling mechanism, and utilizes a PSO algorithm to automatically determine the optimal values of the generating coefficients in the new optimized initial condition (elaborated in Section 3.2.3). Two case studies based on China's total and industrial electricity consumption observations from 2005 to 2014 are presented to demonstrate the reliability and efficacy of the newly proposed model. The empirical results suggest that the rolling NOGMð1; 1Þ model outperforms the other competing models. Accordingly, this proposed model is employed for future projections of China's total and industrial electricity demand up to 2020. Based on the previous findings, the authors can further draw some conclusions as follows: (1) By analyzing two empirical results of China's total and industrial electricity consumption, it can be concluded that the novel model possessing a new initial condition provides higher precision than the other competing models that have different optimized initial conditions. (2) The novel optimized initial condition having dynamic weighted coefficients can improve the forecasting performance of a grey model, compared to the former optimized initial conditions possessing fixed weighted values. The novel optimized initial condition not only conforms to the principle of “new information priority” put forward by Deng [24], but also can take into account the effect of the individual component in the 1-AGO sequence on predictive precision. Meanwhile, the novel optimized method can flexibly and intelligently adapt to various characteristics of the official statistics for China's overall and industrial electricity consumption with the help of the PSO algorithm. (3) The introduction of a rolling mechanism can further improve forecasting accuracy, owing to its ability to utilize the latest information and account for the development trend of input data.

(4) The modelling procedure of the rolling NOGMð1; 1Þ model is easily operationalized due to its simple structure, which makes it readily applicable for obtaining forecasts in many fields. (5) The total and industrial electricity consumption are important indicators, not only reflecting economic development and industrial productivity, but also for developing energy strategies and associated environmental protection policies. Forecasted results indicate that China's total and industrial electricity consumption will continue to have a strong increasing trend in the upcoming years. As the future work, the authors will expand the novel optimized initial condition into a multivariable grey model, such as GMð1; nÞ, which can incorporate influencing factors into the model construction to predict electricity consumption. What's more, the novel model in this paper can also be used in other sectors, such as the agricultural electricity and residential electricity consumption areas. In addition, by using this newly proposed model, different forecasting studies for a provincial electricity consumption can be carried out as well. Acknowledgments This work is financially supported by the National Natural Science Foundation of China (71371098, 71771119); Funding for Outstanding Doctoral Dissertation in Nanjing University of Aeronautics and Astronautics (BCXJ16-09); and Key Project of Social Science Fund In Jiangsu Province (16GLA001). References [1] Kavousi A, Samet H, Marzbani F. A new hybrid modified firefly algorithm and support vector regression model for accurate short term load forecasting. Expert Syst Appl 2014;41(13):6047e56. [2] Lin B, Liu C. Why is electricity consumption inconsistent with economic growth in China? Energy Pol 2016;88:310e6. [3] Kandil MS, El-Debeiky SM, Hasanien NE. Long-term load forecasting for fast developing utility using a knowledge-based expert system. IEEE Trans Power Syst 2002;17(2):491e6. [4] Al-Hamadi HM, Soliman SA. Long-term/mid-term electric load forecasting based on short-term correlation and annual growth. Elec Power Syst Res 2005;74(3):353e61. [5] Pao HT. Forecast of electricity consumption and economic growth in Taiwan by state space modeling. Energy 2009;34(11):1779e91. [6] Hussain A, Rahman M, Memon JA. Forecasting electricity consumption in Pakistan: the way forward. Energy Pol 2016;90:73e80. [7] Khosravi A, Nahavandi S, Creighton D, et al. Interval type-2 fuzzy logic systems for load forecasting: a comparative study. IEEE Trans Power Syst 2012;27(3): 1274e82. ndez L, Baladro n C, Aguiar JM, et al. Experimental analysis of the input [8] Herna variables' relevance to forecast next Day's aggregated electric demand using neural networks. Energies 2013;6(6):2927e48. ndez L, Baladro n C, Aguiar JM, et al. A survey on electric power demand [9] Herna forecasting: future trends in smart grids, microgrids and smart buildings. IEEE Commun Surv Tutorials 2014;16(3):1460e95. [10] Ekonomou L. Greek long-term energy consumption prediction using artificial neural networks. Energy 2010;35(2):512e7. [11] Bouzerdoum M, Mellit A, Pavan AM. A hybrid model (SARIMAeSVM) for short-term power forecasting of a small-scale grid-connected photovoltaic plant. Sol Energy 2013;98:226e35.

328

S. Ding et al. / Energy 149 (2018) 314e328

[12] Pai PF, Hong WC. Support vector machines with simulated annealing algorithms in electricity load forecasting. Energy Convers Manag 2005;46(17): 2669e88. [13] Kumar U, Jain VK. Time series models (Grey-Markov, Grey Model with rolling mechanism and singular spectrum analysis) to forecast energy consumption in India. Energy 2010;35(4):1709e16. [14] Zhao W, Wang J, Lu H. Combining forecasts of electricity consumption in China with time-varying weights updated by a high-order Markov chain model. Omega 2014;45:80e91. [15] Chui F, Elkamel A, Surit R, et al. Long-term electricity demand forecasting for power system planning using economic, demographic and climatic variables. Eur J Ind Eng 2009;3(3):277e304. [16] Mohamed Z, Bodger P. Forecasting electricity consumption in New Zealand using economic and demographic variables. Energy 2005;30(10):1833e43. [17] Kaytez F, Taplamacioglu MC, Cam E, et al. Forecasting electricity consumption: a comparison of regression analysis, neural networks and least squares support vector machines. Int J Electr Power Energy Syst 2015;67:431e8. [18] Yuan CQ, Liu SF, Fang ZG. Comparison of China's primary energy consumption forecasting by using ARIMA (the autoregressive integrated moving average) model and GM (1, 1) model. Energy 2016;100:384e90. [19] Al-Hamadi HM, Soliman SA. Short-term electric load forecasting based on Kalman filtering algorithm with moving window weather and load model. Elec Power Syst Res 2004;68(1):47e59. [20] Deng JL. Control problems of grey systems. Syst Contr Lett 1982;1(5):288e94. [21] Deng JL. Introduction to grey system theory. J Grey Syst 1989;1(1):1e24. [22] Hu YC. Electricity consumption prediction using a neural-network-based grey forecasting approach. J Oper Res Soc 2017;68:1259e64. [23] Zhao H, Guo S. An optimized grey model for annual power load forecasting. Energy 2016;107:272e86. [24] Bahrami S, Hooshmand R, Parastegari M. Short term electric load forecasting by wavelet transform and grey model improved by PSO (particle swarm optimization) algorithm. Energy 2014;72:434e42. [25] Xu N, Dang YG, Gong YG. Novel grey prediction model with nonlinear optimized time response method for forecasting of electricity consumption in China. Energy 2017;118:473e80. [26] Akay D, Atak M. Grey prediction with rolling mechanism for electricity demand forecasting of Turkey. Energy 2007;32(9):1670e5. [27] Tan LZ, Ouyang AJ, Peng XY, et al. A fast and stable forecasting model to forecast power load. Int J Pattern Recogn Artif Intell 2015;29(03):1559e64. [28] Lin Z, Zhang Q, Liu H. Parameters optimization of GM (1, 1) model based on artificial fish swarm algorithm. Grey Syst Theor Appl 2012;2(2):166e77. [29] Zhao Z, Wang J, Zhao J, et al. Using a grey model optimized by differential evolution algorithm to forecast the per capita annual net income of rural households in China. Omega 2012;40(5):525e32.

[30] Wang ZX, Li Q, Pei LL. Grey forecasting method of quarterly hydropower production in China based on a data grouping approach. Appl Math Model 2017;51:302e16. [31] Lin YH, Chiu CC, Lee PC, et al. Applying fuzzy grey modification model on inflow forecasting. Eng Appl Artif Intell 2012;25(4):734e43. [32] Chang CJ, Lin JY, Jin P. A grey modeling procedure based on the data smoothing index for short-term manufacturing demand forecast. Comput Math Organ Theor 2017;23:409e22. [33] Xie NM, Liu SF. Discrete grey forecasting model and its optimization. Appl Math Model 2009;33(2):1173e86. [34] Chen CI, Chen HL, Chen SP. Forecasting of foreign exchange rates of Taiwan's major trading partners by novel nonlinear Grey Bernoulli model NGBM (1, 1). Commun Nonlinear Sci Numer Simulat 2008;13(6):1194e204. [35] Xiao XP, Yang J, Mao SH, et al. An improved seasonal rolling grey forecasting model using a cycle truncation accumulated generating operation for traffic flow. Appl Math Model 2017;51:386e404. [36] Wu LF, Liu SF, Yao L, et al. Grey system model with the fractional order accumulation. Commun Nonlinear Sci Numer Simulat 2013;18(7):1775e85. [37] Ma X, Liu ZB. Application of a novel time-delayed polynomial grey model to predict the natural gas consumption in China. J Comput Appl Math 2017;324: 17e24. [38] Yang Y, Xue D. Continuous fractional-order grey model and electricity prediction research based on the observation error feedback. Energy 2016;115: 722e33. [39] Deng JL. Basis on grey system theory. Wuhan. Huazhong University of Science and Technology Press; 2002. [40] Dang YG, Liu SF. The GM models that x (n) be taken as initial value. Kybernetes 2004;33(2):247e54. [41] Wang YH, Dang YG, Li Y, et al. An approach to increase prediction precision of GM (1, 1) model based on optimization of the initial condition. Expert Syst Appl 2010;37(8):5640e4. [42] Xiong PP, Dang YG, Yao TX, et al. Optimal modeling and forecasting of the energy consumption and production in China. Energy 2014;77:623e34. [43] Hamzacebi C, Es HA. Forecasting the annual electricity consumption of Turkey using an optimized grey model. Energy 2014;70:165e71. [44] Ding S, Dang YG, Li XM, et al. Forecasting Chinese CO2 emissions from fuel combustion using a novel grey multivariable model. J Clean Prod 2017;162: 1527e38. [45] Wang J, Zhu S, Zhao W, et al. Optimal parameters estimation and input subset for grey model based on chaotic particle swarm optimization algorithm. Expert Syst Appl 2011;38(7):8151e8. [46] Hipel KW, McLeod AI. Time series modelling of water resources and environmental systems. Elsevier; 1994.