# A Novel Power-Driven Grey Model with Whale Optimization Algorithm and Its Application in Forecasting the Residential Energy Consumption in China

Authors: Peng Zhang; Xin Ma; Kun She
Journal: Complexity
 (2019)

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## Abstract

Along with the improvement of Chinese people’s living standard, the proportion of residential energy consumption in total energy consumption is rapidly increasing in China year by year. Accurately forecasting the residential energy consumption is conducive to making energy programming and supply plan for the administrative departments or energy companies. By improving the grey action quantity of traditional grey model with an exponential time term, a novel power-driven grey model is proposed to forecast energy consumption as reference data for decision makers. The nonlinear parameter of power-driven grey action quantity is a crucial factor to influence the prediction precision. To promote the prediction accuracy of the power-driven grey model, whale optimization algorithm is adopted to seek for the optimal value of the nonlinear parameter. Two validations on real-world datasets are conducted, and the results indicate that the power-driven grey model has significant advantages on the aspect of prediction performance compared with the other seven classical grey prediction methods. Finally, the power-driven grey model is applied in forecasting the total residential energy and the thermal energy consumption of China.


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## Body

## 1. Introduction

The residential energy, which includes the energy consumed by urban and rural residents and public facilities, accounts for a large percentage of total energy and continues to expand in China [1]. Meeting people’s residential energy demand is always an important part of the energy supply in China. Therefore, it is significantly crucial to predict the energy consumption accurately for energy programming and supply plan of governments or energy companies. Numerous studies have been conducted to predict total energy consumption or other various energy consumptions, e.g., natural gas consumption [2], oil consumption [3], electricity consumption [4], nuclear energy consumption [5], wind energy and renewable energy consumption prediction [6], and so on. For obtaining better prediction result, lots of conventional statistical models and machine learning models were adopted to predict energy consumption, such as ridge regression [7], autoregressive integrated moving average model (ARIMA) [8], support vector regression (SVR) [9], and artificial neural network (ANN) [10]. Unfortunately, machine learning models often need enough training samples to construct models, while the aforementioned statistical models require more available and reliable historical data. There are still difficulties to solve the prediction problems with poor information or small samples. Therefore, the grey prediction method (GM) becomes one of the inevitable choices to handle these problems.The grey prediction theory was initially put forward to study the prediction problem with inadequate information or small samples by Deng in 1980s [11]. Traditional GM(1,1) model has exhibited excellent ability for homogeneous exponential datasets in the aspect of prediction. However, it cannot always provide a satisfactory result for inhomogeneous exponential sequence. For tackling the challenge, many scholars have engaged in optimization of the traditional grey prediction model. Wang et al. extended the existing grey model by using the exponential preprocessing method and applied it to forecast Beijing’s tertiary industry [12]. Xu et al. optimized the initial value of time response function to boost the stability of grey model and predict China’s electricity consumption [13]. Wang et al. analyzed the general analytic solution of the grey model’s whitening equation and presented an improved grey model by optimizing the initial condition which consisted of the first and last items of the accumulated generating sequence [14]. Meanwhile, optimization of the background value is another crucial aspect to boost the forecasting power of the classical grey prediction methods. Wang et al. utilized the finite integral of the accumulative generating sequence within the interval k−1,k as background value to estimate the parameters of the grey model by the least-squares method [15]. Integrating the optimization of background value and the triangular whitening weight function, Ye et al. established a modified Grey-Markov model to handle the fluctuating sequences [16]. Zeng and Li modified the multivariate grey model based on dynamic background-value coefficient whose optimum value was sought out by PSO [17]. Chang et al. designed an adaptive grey prediction model to deal with the non-equigap sequence by optimizing background value coefficient [18]. The hybrid optimization is also an effective method to increase the grey model’s prediction performance. Li et al. enhanced the accuracy and application fields of the classical grey model by improving the grey model with the joint optimization of the initial condition and background values [19]. Besides, some scholars made efforts to enhance the adaptation ability of the grey model. Zeng et al. designed a grey predictive framework with a series of various grey structures which can intelligently select the most suitable model to predict the electricity consumption [20]. With the sum of weighted first-order accumulative generating values as an initial condition, Ding designed a self-adapting grey model called NSGM(1,1) to enhance the adaptation ability for various original sequences. Also, the tunable weighted parameters of NSGM(1,1) are automatically sought out by using the ant lion optimizer [21]. Zeng et al. optimized the structure compatibility of a multivariable grey model with adding a random term, a linear term, and a dependent variable lag term [22]. Besides, the fractional-order accumulation is also significantly valid for increasing the prediction capacity of the grey model [23]. Ma et al. presented a fractional time delayed grey model to boost the precision and applicability of the traditional fractional grey model [24]. At the same time, Ma et al. proposed an unbiased fractional discrete grey model (FDGM) in which the order was intelligently sought out by using the grey wolf optimizer to deal with the multivariate time series [25]. By eliminating the inconsistency between its grey difference equation for modeling and discrete function for forecasting, Ma and Liu designed an improved GMC(1,n) model to promote the accuracy of the classical GM(1,n) with convolution integral [26]. The optimization of the accumulated operator is also an effective measure to enhance the ability of the grey prediction model. Ma et al. presented a new fractional accumulated operator and designed a comfortable grey prediction model which obtained better accuracy than the classical fractional grey model [27]. Meanwhile, the optimization of the grey action quantity is also an important method used to boost the prediction performance and applicability of the grey model. Shaikh et al. utilized the GVM(1, 1) [28] and NGBM(1, 1) [29] models to handle the prediction problem of China’s natural gas demand with the characteristics of S-shaped data [30]. Li et al. proposed a full-order time power grey model to increase the structure adaptability of the grey model and adopted it to forecast the production of clean energy [31]. The grey action quantity affects the prediction performance and the applicability of the grey model. More details of the grey models with optimization of the grey action quantity is presented in Section 2. However, these improved grey models cannot solve all prediction problems. It is indispensable to continue improving the grey model and extending the application range of the grey model.Therefore, a novel grey model is proposed to predict the residential energy consumption of China in this paper. There are two aspects of contribution as follows: (1) A novel power-driven grey model is proposed by optimizing the grey action quantity of traditional GM(1,1) model with an exponential term of time. The nonlinear parameter of the exponential term is determined by the whale optimization algorithm (WOA) to promote prediction accuracy. (2) The GM(1,1,eαt) model is used to predict China’s total residential energy and thermal energy consumption, in which the prediction performance is significantly superior to the other seven contrast grey models.The rest of the paper is organized as follows. Firstly, an overview of the traditional GM(1,1) model and its extension models is introduced in Section2. Then, the power-driven grey model is proposed by substituting an exponential time grey action quantity for the constant grey input of traditional GM(1,1) in Section 3. In Section 4, a nonlinear programming problem with equality constraint is established to seek the optimum value of the nonlinear parameter by using the whale optimization algorithm. Meanwhile, the overall algorithm flowchart of GM(1,1,eαt) is presented. In Section 5, the validations of GM(1,1,eαt) is performed on two real-world datasets. Compared with the other seven existing grey prediction methods, the predicted results show that the proposed model has the most excellent prediction accuracy. In Section 6, the power-driven grey model is used to forecast China’s total residential energy and residential thermal energy consumption. At last, the conclusions are drawn in Section 7.


## 2. GM(1,1) Model and Its Extension Models

The grey prediction method is one of the most popular and effective models to deal with time series prediction. It has been widely employed in many application areas and has obtained remarkable achievement in prediction problems with small samples. To boost the prediction accuracy of the classical grey prediction model, scholars have conducted numerous studies, such as improving the grey accumulation generation, optimizing grey background value, optimizing the initial condition, and so on. This section presents a survey of the classic grey prediction model and its extension models with optimizing grey action quantity.Definition 1.
LetX0=x01,x02,…,x0n be the raw sequence. The first-order accumulated sequence generated from the raw sequence is defined as(1)X1=x11,x12,…,x1n,where x1k=∑i=1kx0i. The background value sequence generated from consecutive neighbors of first-order accumulated sequence is defined as(2)Z1=z12,z13,…,z1n,where z1k=0.5x1k+x1k−1.
Then, the definition of the classical grey model for dealing with univariate time series prediction is presented as follows. The equation(3)x0k+az1k=b,is the definition equation of GM(1,1), in which a and b are development coefficient and grey input of the grey model.
The differential equation(4)dx1tdt+ax1t=b,is the whitening equation of GM(1,1).
In order to perform a prediction task, the key issue is to resolve the optimal value of linear parametersa and b. By using the least-squares method, the optimal parameters u^=a^,b^T can be calculated as follows:(5)u^=a^,b^T=ATA−1ATY,where(6)A=−z121−z131⋮⋮−z1n1,Y=x02x03⋮x0n.
Once the linear parameters of the grey model are estimated, the time response function of the grey model can be obtained by solving the whitening equation (4). By substituting the initial condition into the solution of the whitening equation (4), the discrete time response sequence of the grey model is formulated as(7)x^1k=x01−b^a^e−a^k−1+b^a^.
However, the sequence is not the final predicted result of the grey model. By using inverse accumulated generating operator, the restored value of the grey model is calculated as(8)x^0k=x^1k−x^1k−1.
Then, the sequence produced by the grey model is represented as(9)x^0k=x01−b^a^1−ea^e−a^k−1,where k=2,3,…,n, and x^01=x01.
By analyzing the above stored sequence (9), it can be noticed that the GM(1,1) model has an ideal performance for the univariate time series with homogeneous exponential characteristics. Nevertheless, there are many time series data with nonhomogeneous exponential characteristics. Many improved grey models have been studied and designed to enhance the prediction accuracy for the nonhomogeneous exponential data sequence. One of the boosting strategies is to optimize the grey action quantity. A series of grey models with optimization of grey action quantity were proposed as follows.
By replacing the grey inputb of the original GM(1,1) model with the term bk, the NGM model [32] can be obtained with the following formula:(10)x0k+az1k=bk.
By replacing the grey inputb of the original GM(1,1) model with the term bk+c, the SAIGM model [20] can be obtained with the following formula:(11)x0k+az1k=bk+c.
By replacing the grey inputb of the whitening equation of the original GM(1,1) model with the term bx1tn, the whitening equation of the NGBM model [29] can be obtained as follow:(12)dx1tdt+ax1t=bx1tn.
Whenn=2, the NGBM model can be degenerated into the GVM(1,1) [28] model.
By replacing the grey inputb of the original GM(1,1) model with the term ∑i=1hbiki, the FOTP-GM(1,1,k) model [31] can be obtained with the following equation:(13)x0k+az1k=∑i=1hbiki,h≥1.
By replacing the grey inputb of the whitening equation of the original GM(1,1) model with the term btα+c, GM(1,1,tα) is obtained with the whitening equation:(14)dx1tdt+ax1t=btα+c.
By replacing the grey inputb of the original GM(1,1) model with the term ∑j=2Nbjxj1k, the GM(1,N) model [33] is obtained with the following form:(15)x10k+az11k=∑j=2Nbjxj1k.
These research results show that the performance and accuracy of these grey models are significantly improved as well as the application range is also expanded into more fields by improving the grey action quality.


## 3. The New Proposed Power-Driven Grey Model

Obviously, the optimization of the grey action quantity is an effective means to increase the performance and applicability of the grey model from the previous section. This section proposes a novel power-driven grey model in which a natural exponential function of time is considered as the grey action quantity.


### 3.1. The Power-Driven Grey Model

Definition 2.
Assume thatX0,X1,Z1 are defined as the same in Definition 1. The differential equation(16)dx1tdt+ax1t=beαt+c,is defined as the whitening equation of the power-driven grey model (GM(1,1,eαt)). The parameter a denotes the development coefficient. The term beαt+c denotes the power-driven grey input in which the coefficient α is a tunable parameter.By integrating the both sides of whitening equation (16) within k−1,k, the discrete formulation of the GM(1,1,eαt) model can be represented as follows.Definition 3.
The grey differential equation(17)x0k+az1k=beα−1αeαk−1+c,is called the discrete form of the GM(1,1,eαt) model.


### 3.2. Parameter Estimation of the Power-Driven Grey Model

For the traditional GM(1,1) model, the parametersa and b can be directly estimated by using the least-squares method because they are linear parameters. From Definition 2, it can be clearly noticed that the parameters a, b, and c of the power-driven grey model are linear parameters while the parameter α is a nonlinear parameter. It is difficult to estimate the nonlinear parameter by using the least-squares method directly. A two-stage strategy is adopted to gain the optimum parameters of the proposed model. In the first stage, the equality equation between the linear parameters and nonlinear parameter is obtained by using the least-squares method under the hypothetical condition that the nonlinear parameter α is given. Then, the optimal nonlinear parameter α is determined by solving an established nonlinear programming problem with equality constraint by using an intelligence algorithm (e.g., whale optimization algorithm [34]). In the second stage, the linear parameters are determined by the least-squares method after seeking out the optimum value of the nonlinear parameter. The process of determining the nonlinear parameter is presented in Section 4, while the linear parameters are estimated as follows.Assuming the nonlinear parameterα is given, the parameters u^=a^,b^,c^T of the power-driven grey model can be determined by employing the least-square method, and it satisfies(18)u^=a^,b^,c^T=BTB−1BTY,where(19)Y=x02x03⋮x0v,B=−z12βe2α1−z13βe3α1⋮⋮⋮−z1vβevα1,in which β=α−1eα−1 and v denotes the number of samples used for constructing model. The detailed proof process of the linear parameter estimation is omitted here because it is similar to the classical GM(1,1) model.


### 3.3. The Time Response Function and Restored Response Sequence

After the parameters of the proposed grey model are determined, the time response and restored value sequence can be obtained as follows.Theorem 1.
The time response function of the power-driven grey model is defined as(20)x^1k=x01−b^a^+αeα−c^a^e−a^k−1+b^a^+αeαk+c^a^.The restored response sequence of the power-driven grey model can be obtained by(21)x^0k=1−ea^x01−b^a^+αeα−c^a^e−a^k−1+b^1−eαa^+αeαk.Proof.
Assume thatμt is an arbitrary function and satisfies(22)μ′t=aμt.
Multiply both sides of equation (16) by μt and obtain(23)μtdx1tdt+μtax1t=μtbeαt+c.
Equation (22) is substituted into equation (23) to get the following formula:(24)μtdx1tdt+μ′tx1t=μtbeαt+c.
Rearrange equation (24) and obtain(25)μtx1t′=μtbeαt+c.
Integrate both sides of equation (25) and obtain(26)x1t=∫μtbeαt+cdt−C​μt.
Solving equation (22), the solution is obtained as(27)μt=keat,where k is an arbitrary real number. Substitute equation (27) into equation (26) and obtain(28)x1t=Ce−at+ba+αeαt+ca.
Substituting the initial condition and the estimated parameters calculated by (18) into equation (28), the time response function is obtained as(29)x^1k=x01−b^a^+αeα−c^a^e−a^k−1+b^a^+αeαk+c^a^.
By using inverse accumulation generating operator, the stored value can be calculated as follows:(30)x^0k=x^1k−x^1k−1.
Then, the stored value is obtained as(31)x^0k=1−ea^x01−b^a^+αeα−c^a^e−a^k−1+b^1−eαa^+αeαk.This completes the proof.According to Maclaurin’s formula, the expansion of the termeαt can be obtained as(32)eat=1+αt+αt22!+αt33!+⋯+αtnn!+Rn,where Rn is known as the error term. If some higher-order term of equation (32) is ignored, the power-driven grey model can be degenerated into other existing grey models. If the higher-order terms other than first-order terms are ignored, the term eat=1+αt can be obtained. Then, the GM(1,1,eαt) model can be degenerated into the grey SAIGM [20] with whitening equation (11). In a similar way, the GM(1,1,eαt) model can be degenerated to a kind of FOTP-GM(1,1,k) model [31] with special whitening equation (13) in which the parameter bi=bαi/i!. When bα=0, the GM(1,1,eαt) model can be degenerated into the traditional grey model with whitening equation (4).


## 3.1. The Power-Driven Grey Model

Definition 2.
Assume thatX0,X1,Z1 are defined as the same in Definition 1. The differential equation(16)dx1tdt+ax1t=beαt+c,is defined as the whitening equation of the power-driven grey model (GM(1,1,eαt)). The parameter a denotes the development coefficient. The term beαt+c denotes the power-driven grey input in which the coefficient α is a tunable parameter.By integrating the both sides of whitening equation (16) within k−1,k, the discrete formulation of the GM(1,1,eαt) model can be represented as follows.Definition 3.
The grey differential equation(17)x0k+az1k=beα−1αeαk−1+c,is called the discrete form of the GM(1,1,eαt) model.


## 3.2. Parameter Estimation of the Power-Driven Grey Model

For the traditional GM(1,1) model, the parametersa and b can be directly estimated by using the least-squares method because they are linear parameters. From Definition 2, it can be clearly noticed that the parameters a, b, and c of the power-driven grey model are linear parameters while the parameter α is a nonlinear parameter. It is difficult to estimate the nonlinear parameter by using the least-squares method directly. A two-stage strategy is adopted to gain the optimum parameters of the proposed model. In the first stage, the equality equation between the linear parameters and nonlinear parameter is obtained by using the least-squares method under the hypothetical condition that the nonlinear parameter α is given. Then, the optimal nonlinear parameter α is determined by solving an established nonlinear programming problem with equality constraint by using an intelligence algorithm (e.g., whale optimization algorithm [34]). In the second stage, the linear parameters are determined by the least-squares method after seeking out the optimum value of the nonlinear parameter. The process of determining the nonlinear parameter is presented in Section 4, while the linear parameters are estimated as follows.Assuming the nonlinear parameterα is given, the parameters u^=a^,b^,c^T of the power-driven grey model can be determined by employing the least-square method, and it satisfies(18)u^=a^,b^,c^T=BTB−1BTY,where(19)Y=x02x03⋮x0v,B=−z12βe2α1−z13βe3α1⋮⋮⋮−z1vβevα1,in which β=α−1eα−1 and v denotes the number of samples used for constructing model. The detailed proof process of the linear parameter estimation is omitted here because it is similar to the classical GM(1,1) model.


## 3.3. The Time Response Function and Restored Response Sequence

After the parameters of the proposed grey model are determined, the time response and restored value sequence can be obtained as follows.Theorem 1.
The time response function of the power-driven grey model is defined as(20)x^1k=x01−b^a^+αeα−c^a^e−a^k−1+b^a^+αeαk+c^a^.The restored response sequence of the power-driven grey model can be obtained by(21)x^0k=1−ea^x01−b^a^+αeα−c^a^e−a^k−1+b^1−eαa^+αeαk.Proof.
Assume thatμt is an arbitrary function and satisfies(22)μ′t=aμt.
Multiply both sides of equation (16) by μt and obtain(23)μtdx1tdt+μtax1t=μtbeαt+c.
Equation (22) is substituted into equation (23) to get the following formula:(24)μtdx1tdt+μ′tx1t=μtbeαt+c.
Rearrange equation (24) and obtain(25)μtx1t′=μtbeαt+c.
Integrate both sides of equation (25) and obtain(26)x1t=∫μtbeαt+cdt−C​μt.
Solving equation (22), the solution is obtained as(27)μt=keat,where k is an arbitrary real number. Substitute equation (27) into equation (26) and obtain(28)x1t=Ce−at+ba+αeαt+ca.
Substituting the initial condition and the estimated parameters calculated by (18) into equation (28), the time response function is obtained as(29)x^1k=x01−b^a^+αeα−c^a^e−a^k−1+b^a^+αeαk+c^a^.
By using inverse accumulation generating operator, the stored value can be calculated as follows:(30)x^0k=x^1k−x^1k−1.
Then, the stored value is obtained as(31)x^0k=1−ea^x01−b^a^+αeα−c^a^e−a^k−1+b^1−eαa^+αeαk.This completes the proof.According to Maclaurin’s formula, the expansion of the termeαt can be obtained as(32)eat=1+αt+αt22!+αt33!+⋯+αtnn!+Rn,where Rn is known as the error term. If some higher-order term of equation (32) is ignored, the power-driven grey model can be degenerated into other existing grey models. If the higher-order terms other than first-order terms are ignored, the term eat=1+αt can be obtained. Then, the GM(1,1,eαt) model can be degenerated into the grey SAIGM [20] with whitening equation (11). In a similar way, the GM(1,1,eαt) model can be degenerated to a kind of FOTP-GM(1,1,k) model [31] with special whitening equation (13) in which the parameter bi=bαi/i!. When bα=0, the GM(1,1,eαt) model can be degenerated into the traditional grey model with whitening equation (4).


## 4. Determining the Nonlinear Parameter of the Power-Driven Grey Model with Whale Optimization Algorithm

From the previous section, the linear parameters of the proposed model are determined by using the least-squares approach under the assumption that the nonlinear parameter is given. However, the nonlinear parameter cannot be directly calculated by the ordinary least-squares method because it is an exponential coefficient of grey action quantity. In fact, the nonlinear parameterα plays an indispensable role in promoting the prediction performance of the power-driven grey model. In this section, an intelligent nature-inspired optimization method called whale optimization algorithm is employed to seek for the optimal value of nonlinear parameter α.


### 4.1. Constructing the Optimization Problem for the Power-Driven Grey Model

Actually, an optimum value of nonlinear parameterα can make the power-driven grey model obtain the best prediction performance because the parameter not only directly affects the grey action quantity but also can control the development coefficient. Therefore, an optimization problem with constraint is built to obtain the optimum value of α, in which the objective function is to minimize the fit error of the power-driven grey prediction model. The equality constraints of the optimization problem are formulated in the previous modeling process. Mathematically, the optimization problem for seeking out optimal nonlinear parameter is formulated as follows:(33)minα1n−v∑m=v+1nx0m−x^0mx0m×100%s.t.Y=x02x03⋮x0v,B=−z12βe2α1−z13βe3α1⋮⋮⋮−z1vβevα1,x^1m=x01−b^a^+αeα−c^a^e−a^m−1+b^a^+αeαm+c^a^,x^0m=x^1m−x^1m−1,m=2,3,…,v,…,n,u^=a^,b^,c^T=BTB−1BTY.In this paper, a different strategy which is similar to nest cross validation in machine learning [35] is utilized to seek the optimal value of the coefficient α.. During the simulation stage, the samples are partitioned into two subsets. The first set, including the first v samples (from 1 to v samples), is used for establishing the equality constraint equation (19) between the linear parameters and the nonlinear parameter. The second set, including the last n−v samples (from v+1 to n samples), is utilized to compute the fitness value of the established optimization problem. The value of v satisfies v=n−2 in Sections 5 and 6. The optimum value of nonlinear parameter α is sought out by solving the optimization problem equation (33). In the meanwhile, the linear parameters are also obtained when the optimal value α is substituted into equation (19). This strategy has been utilized to search for the optimum order of the fractional grey prediction model [25, 36].


### 4.2. Whale Optimization Algorithm

Motivated by the social behavior of humpback group, an intelligent nature-based optimization approach called whale optimization algorithm (WOA) was originated by Mirijalili and Lewis in 2016 [34]. In recent years, WOA has been widely employed to settle the optimization problems in many fields such as image retrieval [37], classification [38], bioinformatics [39], feature selection [40], image processing [41], and so on. Meanwhile, it is also effective to solve the optimization problems like training multilayer perceptron neural network which involves a complex nonlinear optimization problem [42] and is more complicated than problem (33). This paper adopts the WOA algorithm to solve the nonlinear optimization problem (33). The main idea and model of WOA are mathematically described as follows.The main idea of WOA is to imitate the predation behaviors of humpback group, for example, bubble-net feeding for catching fish. When the humpback whales catch fish, they usually encircle the fish school whose position is considered as the current best candidate target. Then, these whales update their positions based on the candidate target. Mathematically, the encircling behavior is represented as follows:(34)D⇀=2r⇀⋅P⇀∗i−P⇀i,C⇀=2fi⋅r⇀−fi,fi=2−2iT,P⇀i+1=P⇀∗i−C⇀⋅D⇀,where P⇀i denotes the current position of the humpback whale, P⇀∗i denotes the best current position of the humpback whale, the vector r⇀ is randomly generated in the interval 0,1, and T denotes the maximum number of iterations. Furthermore, humpback whales move in spirals when they catch the prey. To simulate the helix-shaped movement, the spiral updating position is represented as follows:(35)P⇀i+1=P⇀∗i−P⇀i⋅eβl⋅cos2πl+P⇀∗i,where the coefficient l is a stochastic number in the interval −1,1 and β is an arbitrary constant which determines the shape of the spiral movement. However, encircling and spiral moving behaviors happen simultaneously in the real world. For keeping it simple in this model, the entire predation movement of humpbacks is mathematically represented as follows:(36)P⇀i+1=P⇀∗i−2fi⋅r⇀−fi⋅D⇀,ifξ<0.5a,P⇀∗i−P⇀i⋅eβl⋅cos2πl+P⇀∗i,ifξ≥0.5b,where ξ is a probability to choose a movement strategy from encircling and spiral moving behaviors. When the norm of C⇀ is greater than 1, the position of all whales is updated based on the position of a whale randomly selected, not on optimal ones. Mathematically, the model can be formulated as follows:(37)P⇀i+1=P⇀ri−C⇀⋅2r⇀⋅P⇀ri−P⇀i,where P⇀r is the position of a whale randomly chosen from humpback group. Based on the principle of humpback’s predation behavior, it is iterative to update the position of each whale until the stop criteria are met.


### 4.3. Implementation of WOA for Searching the Optimal Nonlinear Parameter

In the nonlinear programming problem (33), the main purpose is to find out the optimum value of the nonlinear parameter to obtain the highest performance of the proposed grey model. From section 4.2, it can be noticed that the original WOA is initially designed for unconstrained optimization and cannot directly solve the optimization problem with constraint. Therefore, the original WOA needs to be revised based on equation (33). Primarily, the fitness function needs to be established to calculate the fitness of each whale agent. According to equation (33), the fitness function can be represented as follows:(38)fitness=1n−v∑m=v+1nx0m−x^0mx0md×100%.The revised WOA is presented in detail in Algorithm1.Algorithm 1: Algorithm of WOA to search for the nonlinear parameter α of the power-driven grey model.
Input: The raw data X0, lower and upper bound of α.
Output: The optimal value of the nonlinear parameter α.
(1)
Initialize the maximum number of iterationsT and the number of humpback whales;
(2)
Initialize the locationsP⇀ of the humpback population;
(3)
Compute the fitness of each humpback by equation (38);
(4)
Determine the best candidateP⇀∗ based on fitness of each whale agent;
(5)
for
k
=
1
;
k
<
T
;
k
=
k
+
1
do
(6)
foreach humpback whaledo
(7)
Update the parametersr,p,l,β;
(8)
if ξ<0.5then
(9)
if C⇀<1then
(10)
Update the location of each humpback by equation (36);
(11)
else
(12)
DetermineP⇀r by randomly choosing a whale;
(13)
Update the location of each humpback by equation (37);
(14)
end
(15)
else
(16)
Update the location of each humpback by equation (36);
(17)
end
(18)
Compute the fitness of each humpback by equation (38);
(19)
end
(20)
UpdateP⇀∗ if a better solution exists;
(21)
end
(22)
return the optimum valueP⇀∗;


### 4.4. Modeling Procedure of the Power-Driven Grey Model

Based on the modeling process of the power-driven grey model and WOA for seeking out the optimal nonlinear parameter, the overall computational steps of GM(1,1,eαt) with WOA is depicted in the flowchart shown in Figure 1. In the proposed model, the critical issue is to search optimal nonlinear parameter α and estimate the linear parameters to construct the model for achieving a better prediction performance. Firstly, the equality equation of the optimization problem is formulated under the assumption that the nonlinear parameter is given. Then, the optimal nonlinear parameter is sought out through solving the optimization problem by nature-inspired optimization algorithm WOA, and the estimated linear parameters are calculated by the least-squares method. Finally, the power-driven grey model with optimal parameters is used to forecast the future value in case study.Figure 1
The flowchart of the power-driven grey model.


## 4.1. Constructing the Optimization Problem for the Power-Driven Grey Model

Actually, an optimum value of nonlinear parameterα can make the power-driven grey model obtain the best prediction performance because the parameter not only directly affects the grey action quantity but also can control the development coefficient. Therefore, an optimization problem with constraint is built to obtain the optimum value of α, in which the objective function is to minimize the fit error of the power-driven grey prediction model. The equality constraints of the optimization problem are formulated in the previous modeling process. Mathematically, the optimization problem for seeking out optimal nonlinear parameter is formulated as follows:(33)minα1n−v∑m=v+1nx0m−x^0mx0m×100%s.t.Y=x02x03⋮x0v,B=−z12βe2α1−z13βe3α1⋮⋮⋮−z1vβevα1,x^1m=x01−b^a^+αeα−c^a^e−a^m−1+b^a^+αeαm+c^a^,x^0m=x^1m−x^1m−1,m=2,3,…,v,…,n,u^=a^,b^,c^T=BTB−1BTY.In this paper, a different strategy which is similar to nest cross validation in machine learning [35] is utilized to seek the optimal value of the coefficient α.. During the simulation stage, the samples are partitioned into two subsets. The first set, including the first v samples (from 1 to v samples), is used for establishing the equality constraint equation (19) between the linear parameters and the nonlinear parameter. The second set, including the last n−v samples (from v+1 to n samples), is utilized to compute the fitness value of the established optimization problem. The value of v satisfies v=n−2 in Sections 5 and 6. The optimum value of nonlinear parameter α is sought out by solving the optimization problem equation (33). In the meanwhile, the linear parameters are also obtained when the optimal value α is substituted into equation (19). This strategy has been utilized to search for the optimum order of the fractional grey prediction model [25, 36].


## 4.2. Whale Optimization Algorithm

Motivated by the social behavior of humpback group, an intelligent nature-based optimization approach called whale optimization algorithm (WOA) was originated by Mirijalili and Lewis in 2016 [34]. In recent years, WOA has been widely employed to settle the optimization problems in many fields such as image retrieval [37], classification [38], bioinformatics [39], feature selection [40], image processing [41], and so on. Meanwhile, it is also effective to solve the optimization problems like training multilayer perceptron neural network which involves a complex nonlinear optimization problem [42] and is more complicated than problem (33). This paper adopts the WOA algorithm to solve the nonlinear optimization problem (33). The main idea and model of WOA are mathematically described as follows.The main idea of WOA is to imitate the predation behaviors of humpback group, for example, bubble-net feeding for catching fish. When the humpback whales catch fish, they usually encircle the fish school whose position is considered as the current best candidate target. Then, these whales update their positions based on the candidate target. Mathematically, the encircling behavior is represented as follows:(34)D⇀=2r⇀⋅P⇀∗i−P⇀i,C⇀=2fi⋅r⇀−fi,fi=2−2iT,P⇀i+1=P⇀∗i−C⇀⋅D⇀,where P⇀i denotes the current position of the humpback whale, P⇀∗i denotes the best current position of the humpback whale, the vector r⇀ is randomly generated in the interval 0,1, and T denotes the maximum number of iterations. Furthermore, humpback whales move in spirals when they catch the prey. To simulate the helix-shaped movement, the spiral updating position is represented as follows:(35)P⇀i+1=P⇀∗i−P⇀i⋅eβl⋅cos2πl+P⇀∗i,where the coefficient l is a stochastic number in the interval −1,1 and β is an arbitrary constant which determines the shape of the spiral movement. However, encircling and spiral moving behaviors happen simultaneously in the real world. For keeping it simple in this model, the entire predation movement of humpbacks is mathematically represented as follows:(36)P⇀i+1=P⇀∗i−2fi⋅r⇀−fi⋅D⇀,ifξ<0.5a,P⇀∗i−P⇀i⋅eβl⋅cos2πl+P⇀∗i,ifξ≥0.5b,where ξ is a probability to choose a movement strategy from encircling and spiral moving behaviors. When the norm of C⇀ is greater than 1, the position of all whales is updated based on the position of a whale randomly selected, not on optimal ones. Mathematically, the model can be formulated as follows:(37)P⇀i+1=P⇀ri−C⇀⋅2r⇀⋅P⇀ri−P⇀i,where P⇀r is the position of a whale randomly chosen from humpback group. Based on the principle of humpback’s predation behavior, it is iterative to update the position of each whale until the stop criteria are met.


## 4.3. Implementation of WOA for Searching the Optimal Nonlinear Parameter

In the nonlinear programming problem (33), the main purpose is to find out the optimum value of the nonlinear parameter to obtain the highest performance of the proposed grey model. From section 4.2, it can be noticed that the original WOA is initially designed for unconstrained optimization and cannot directly solve the optimization problem with constraint. Therefore, the original WOA needs to be revised based on equation (33). Primarily, the fitness function needs to be established to calculate the fitness of each whale agent. According to equation (33), the fitness function can be represented as follows:(38)fitness=1n−v∑m=v+1nx0m−x^0mx0md×100%.The revised WOA is presented in detail in Algorithm1.Algorithm 1: Algorithm of WOA to search for the nonlinear parameter α of the power-driven grey model.
Input: The raw data X0, lower and upper bound of α.
Output: The optimal value of the nonlinear parameter α.
(1)
Initialize the maximum number of iterationsT and the number of humpback whales;
(2)
Initialize the locationsP⇀ of the humpback population;
(3)
Compute the fitness of each humpback by equation (38);
(4)
Determine the best candidateP⇀∗ based on fitness of each whale agent;
(5)
for
k
=
1
;
k
<
T
;
k
=
k
+
1
do
(6)
foreach humpback whaledo
(7)
Update the parametersr,p,l,β;
(8)
if ξ<0.5then
(9)
if C⇀<1then
(10)
Update the location of each humpback by equation (36);
(11)
else
(12)
DetermineP⇀r by randomly choosing a whale;
(13)
Update the location of each humpback by equation (37);
(14)
end
(15)
else
(16)
Update the location of each humpback by equation (36);
(17)
end
(18)
Compute the fitness of each humpback by equation (38);
(19)
end
(20)
UpdateP⇀∗ if a better solution exists;
(21)
end
(22)
return the optimum valueP⇀∗;


## 4.4. Modeling Procedure of the Power-Driven Grey Model

Based on the modeling process of the power-driven grey model and WOA for seeking out the optimal nonlinear parameter, the overall computational steps of GM(1,1,eαt) with WOA is depicted in the flowchart shown in Figure 1. In the proposed model, the critical issue is to search optimal nonlinear parameter α and estimate the linear parameters to construct the model for achieving a better prediction performance. Firstly, the equality equation of the optimization problem is formulated under the assumption that the nonlinear parameter is given. Then, the optimal nonlinear parameter is sought out through solving the optimization problem by nature-inspired optimization algorithm WOA, and the estimated linear parameters are calculated by the least-squares method. Finally, the power-driven grey model with optimal parameters is used to forecast the future value in case study.Figure 1
The flowchart of the power-driven grey model.


## 5. Validation of the Power-Driven Grey Model

In this section, the validations are performed to examine the forecasting superiority of the proposed GM(1,1,eαt) model through two real-world examples.


### 5.1. Contrast Grey Models and Performance Criteria

To illustrate the advantages of the power-driven grey model with WOA in the aspect of prediction performance, the numerical validation study is conducted on two real-world data sequences. A series of classic existing grey models listed in Table1 are used to compare with the power-driven grey model. The power-driven grey model and the contrastive grey models are all realized by MATLAB. Then, all validation experiments and case studies are performed on the MATLAB platform 2019a.Table 1
The definition of the comparative grey models used.
No.
Name
Model
Reference
1
GM(1,1)
d
x
1
t
/
d
t
+
a
x
1
t
=
b
[11]
2
ARGM
x
0
k
=
a
x
0
k
−
1
+
b
[43]
3
DGM
x
1
k
=
a
x
1
k
−
1
+
b
[44]
4
NGM
x
0
k
+
a
z
1
k
=
b
k
[32]
5
NIGM
∑
i
=
k
n
x
0
i
=
a
∑
i
=
k
n
x
0
i
−
1
+
n
−
k
+
1
b
[43]
6
FGM(1,1)
d
x
r
t
/
d
t
+
a
x
r
t
=
b
[45]
7
SAIGM
x
0
k
+
a
z
1
k
=
b
k
+
c
[20]Nine evaluation criteria tabulated in Table2 are adopted to evaluate the prediction ability of the aforementioned grey prediction models. Meanwhile, Lewis’ criteria [47]shown in Table 3 are also adopted to illustrate the prediction power of grey models.Table 2
The definition of performance metrics used.
No.
Name
Formula
Reference
1
RMSE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
0.5
[24]
2
MAE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
[24]
3
NRMSE
1
/
x
¯
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
×
100
[24]
4
MAPE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
x
0
m
−
1
×
100
[24]
5
RMSPE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
x
0
m
−
1
2
0.5
×
100
[24]
6
MSE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
[24]
7
IA
1
−
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
∑
m
=
1
v
x
0
m
−
x
¯
+
x
^
0
m
−
x
¯
2
−
1
[5]
8
U1
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
0.5
∑
m
=
1
v
x
0
m
2
0.5
+
∑
m
=
1
v
x
^
0
m
2
0.5
−
1
[46]
9
U2
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
0.5
∑
m
=
1
v
x
0
m
2
−
0.5
[46]Table 3
Lewis’ criterion for model evaluation.
MAPE (%)
Prediction performance
<
10
Excellent
10
−
20
Good
20
−
50
Reasonable
>
50
IncorrectFor seeking out the optimal nonlinear parameter of the proposed model, the necessary parameters of WOA are set to the same values in validation experiments and applications as follows. The population size of the humpbacks is set to 30. Maximum iteration is set to 200. The minimum value and maximum value of the nonlinear parameter are set to −10 and 10, respectively.


### 5.2. Example A: Predicting the Natural Gas Consumption of China

In this section, the validation experiment is to study and analyze China’s natural gas consumption (NGC). The original sequence of the natural gas consumption during 2005–2015 is listed in Table4, which was collected from China’s National Bureau of Statistics. To construct a power-driven grey model and validate its superiority, the dataset is partitioned into two sub-datasets, including training set and test set. The test set, including the natural gas consumption in the last 3 years, is employed to check the prediction performance of grey models. The training set, including the natural gas consumption from 2005 to 2012, is utilized to build models of the eight grey models.Table 4
The original data of China’s natural gas consumption (NGC) (2005–2015).
Year
NGC
2005
467.63
2006
561.41
2007
705.23
2008
812.94
2009
895.2
2010
1069.41
2011
1305.3
2012
1463
2013
1705.37
2014
1868.94
2015
1931.75For achieving better prediction accuracy, the WOA method is used to seek the optimum value of the proposed model’s nonlinear parameter. Figure2 shows the convergence curve. It can be noticed that the fitness function converges to a constant after dozens of iterations. Then, the linear parameters a,b,andc and nonlinear parameter α are obtained and are equal to −0.175843, −4.869699, 463.613598, and 0.469429, respectively. The order of FGM(1,1) is equal to 0.862399 obtained by the whale optimization algorithm. All established models are used to predict the consumption from 2013 to 2015. For each grey model, the produced results and their absolute percentage error (APE) [12] are tabulated in Table 5. The evaluation values of the eight models are given in Table 6. Based on Lewis’ criteria, GM(1,1,eαt), GM(1,1), ARGM, SAIGM, NIGM, and FGM models exhibit excellent prediction performance while the other grey models only show good prediction performance. Furthermore, the GM(1,1,eαt) model obtains the lowest MAPE of prediction. From Figure 3, it can be noticed that the results produced by GM(1,1,eαt) are more approximate to real values than those of the other seven contrast models. Overall, the proposed model can more accurately predict natural gas consumption, though its fitted performance is not better than the other comparative models.Figure 2
Convergence curve of WOA in Example A.Table 5
Fitted and predicted results of various grey models in Example A.
Year
Raw data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2005
467.63
467.63
0
467.63
0
467.63
0
467.63
0
469.7263546
0.448293434
467.63
2.43E − 14
467.63
0
467.63
0
2006
561.41
581.0296492
3.494709602
570.8747479
1.69E + 00
582.5467989
3.764948767
326.4241266
−41.85637473
566.9866424
0.993327942
561.41
−1.44E − 09
578.1643099
2.984326951
585.984921
4.377357191
2007
705.23
678.504639
−3.789595023
685.1861578
−2.842170945
680.4246572
−3.517340834
541.2409827
−23.25326735
676.4109333
−4.086477709
675.7856868
−4.175136232
678.9353041
−3.728527697
692.1766157
−1.850940014
2008
812.94
792.3322774
−2.534962309
811.7504501
−0.146326902
794.7476753
−2.237843466
735.3614988
−9.542955343
799.5205363
−1.650732361
798.9734068
−1.718034933
795.5276016
−2.141904496
814.9110141
0.242455057
2009
895.2
925.2559256
3.357453712
951.8809942
6.331657078
928.2789222
3.695143228
910.7796456
1.740353617
938.0270257
4.784073466
935.7082618
4.525051583
930.4251873
3.934895813
955.0483743
6.685475232
2010
1069.41
1080.479178
1.035073373
1107.031938
3.518008797
1084.24571
1.387279884
1069.297286
−0.010539815
1093.856036
2.28593671
1089.675641
1.895030039
1086.502038
1.598268044
1112.224417
4.00E + 00
2011
1305.3
1261.743072
−3.336928487
1278.813297
−2.029165926
1266.417594
−2.978809925
1212.542685
−7.106206627
1269.174034
−2.767637051
1264.347874
−3.137372698
1267.083353
−2.927805624
1283.782232
−1.648492178
2012
1463
1473.416252
0.711978939
1469.007663
0.410640007
1479.197481
1.107141546
1341.987232
−8.271549391
1466.418438
0.233659495
1463.349784
0.023908691
1476.016395
0.88970577
1463
2.62E − 08
2013
1705.37
1720.600254
0.89307624
1679.588701
−1.511771576
1727.72804
1.311037472
1458.96056
−14.44903103
1688.331509
−0.999108165
1690.650448
−0.863129502
1717.752459
0.726086333
1636.180316
−4.057165562
2014
1868.94
2009.252464
7.507595978
1912.74163
2.343661633
2018.015997
7.976499897
1564.664195
−16.28066204
1937.998469
3.695060789
1950.695202
4.374415544
1997.4417
6.875646112
1777.904354
−4.870977476
2015
1931.75
2346.329692
21.4613533
2170.885899
12.37923639
2357.077313
22.01772034
1660.183907
−14.05803509
2218.8904
14.86426298
2248.518858
16.3980255
2321.042867
20.15234203
1843.333931
−4.576993327Table 6
Evaluation result of different grey models in Example A.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
23.86274134
27.06135354
23.66839461
118.1206638
24.48123827
24.4715338
23.55366157
28.84466814
MAE
20.25637699
19.68692977
20.89112313
88.25197924
19.59112421
18.18583986
20.50145901
20.47248493
NRMSE
0.007664991
0.024109321
0.00130884
0.262191084
3.09E − 16
0.009028807
6.38E − 05
0.036768004
MAPE
2.28258768
2.120482298
2.336063456
11.47265586
2.156267271
1.934316772
2.275679299
2.351034331
RMSPE
2.668940276
2.90890819
2.674700195
17.69775476
2.656105278
2.605146741
2.617187635
3.280388261
MSE
569.4304242
732.3168555
560.1929035
13952.49123
599.3310274
598.8559665
554.7749735
832.01488
IA
0.998657193
0.998291523
0.998685903
0.969110255
0.998594898
0.998594002
0.998699542
0.998042174
U1
0.01235891
0.013937773
0.012237389
0.063448569
0.012659895
0.012673831
0.012179981
0.014834273
U2
0.024676152
0.027983796
0.024475181
0.122147051
0.025315732
0.025305697
0.024356537
0.029827898
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
252.8476943
141.1491367
260.5294979
275.1057498
170.7912875
189.0706106
236.7945303
83.450789
MAE
190.0408036
102.9062759
198.9204499
274.0837795
124.4124534
137.7478703
176.7256754
82.8804665
NRMSE
0.179344392
0.08089408
0.187724249
0.258657024
0.106690266
0.120734047
0.166778703
0.07821556
MAPE
9.954008505
5.411556534
10.4350859
14.92924272
6.519477311
7.211856848
9.251358159
4.50171212
RMSPE
13.13711368
7.326292761
13.54158261
14.96064621
8.861865967
9.811145772
12.30065852
4.5142693
MSE
63931.95652
19923.07878
67875.61925
75683.17359
29169.6639
35747.69578
56071.64958
6964.03418
IA
0.593522143
0.786941885
0.580405823
0.462920793
0.731260314
0.699839537
0.616307649
0.83617379
U1
0.06517862
0.037446493
0.067003457
0.080883402
0.044965721
0.049568808
0.061267078
0.02322958
U2
0.137579314
0.076801972
0.141759132
0.149690352
0.092930838
0.102876971
0.128844478
0.04540719Figure 3
Comparison of the true value and the produced value by different grey models in Example A.


### 5.3. Example B: Predicting the Total Energy Consumption of China

In this example, China’s total energy consumption from 2008 to 2018 is employed to examine the accuracy of the grey prediction models. The raw sequence is listed in Table7, which was collected from China’s National Bureau of Statistics. To construct models and validate the prediction performance, the raw dataset is broken into two groups, including training set and test set. The training set, including the consumption from 2008 to 2015, is used to build models of the proposed model and other contrast grey models. The test set containing the rest digits is utilized to test the prediction accuracy of all models.Table 7
The raw sequence of China’s total energy consumption (TEC) (2008–2018).
Year
TEC
2008
320611
2009
336126
2010
360648
2011
387043
2012
402138
2013
416913
2014
425806
2015
429905
2016
435819
2017
448529.1
2018
464000During the stage of constructing the new proposed model, the WOA optimizer is utilized to seek the optimum value of power-driven grey model’s nonlinear parameter. From Figure4, which shows convergence curve of WOA, it can be noticed that the fitness function converges to a constant rapidly after dozens of iterations. Then, the linear parameters a,b,andc and nonlinear parameter α are obtained and are equal to −0.003666, −224002.105686, 458089.69852, and −0.235980, respectively. The order of FGM(1,1) is equal to 0.209972 obtained by the whale optimization algorithm. All established models are employed to forecast the energy consumption from 2016 to 2018 in China. For each grey model, the raw data, fitted data, predicted data, and their absolute percentage error (APE) are tabulated in Table 8. The evaluation results of the eight models are given in Table 9. Based on Lewis’ criteria, the GM(1,1,eαt) model shows an excellent forecasting ability. Although other grey models also exhibit excellent prediction performances, the proposed model has the lowest value of MAPE. It can be noticed that the forecasted values of the GM(1,1,eαt) model are more approximate to the actual values than those of the other seven comparative grey models in Figure 5. Overall, the power-driven grey model has the highest prediction performance of the total energy consumption though it is not the best one for fitted performance compared with the other contrast grey models.Figure 4
Convergence curve of WOA in Example B.Table 8
Fitted and predicted results of various grey models in Example B.
Year
Real data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2008
320611
320611
0
320611
0
320611
0
320611
0
311679.0155
−2.785925775
320611
0
320611
0
320611
0
2009
336126
349189.7029
3.886549367
344430.113
2.47E + 00
349311.5014
3.922785319
195251.0697
−41.91134582
343554.0921
2.209912975
336126
−6.09E − 09
334656.6702
−0.437136603
302216.2049
−10.08841777
2010
360648
363180.4997
0.70220817
364871.1079
1.17097776
363285.296
0.731265934
324553.7721
−10.00815973
368260.8022
2.110867706
362542.7915
0.525385288
363846.1266
0.886772319
336515.6299
−6.691391627
2011
387043
377731.8583
−2.405712472
382413.0823
−1.196228253
377818.0957
−2.383431369
378605.0893
−2.18009644
387411.2353
0.095140671
384858.2814
−0.564464039
385905.0726
−0.294005419
363964.4755
−5.962780493
2012
402138
392866.2383
−2.305616897
397467.1865
−1.161495191
392932.263
−2.289198485
401199.7052
−0.233326558
402254.9392
0.029079382
402143.1792
0.001287911
402575.3761
0.108762691
386004.0797
−4.012035752
2013
416913
408606.9995
−1.992262303
410386.2597
−1.565492142
408651.0547
−1.981695294
410644.7418
−1.503493099
413760.4517
−0.756164554
414991.5431
−0.46087718
415173.395
−0.417258508
403772.8105
−3.15E + 00
2014
425806
424978.4373
−0.194352063
421473.1004
−1.017575986
424998.6581
−0.189603218
414592.9707
−2.633365728
422678.4964
−0.734490251
424176.7152
−0.382635476
424693.9224
−0.261170013
418169.7514
−1.793363307
2015
429905
442005.8207
2.814766209
430987.5627
0.251814401
442000.2281
2.813465333
416243.4155
−3.177814742
429590.9676
−0.073046936
430405.4018
0.116398218
431888.7395
0.461436707
429905.0001
1.97E − 08
2016
435819
459715.4311
5.483109076
439152.6474
0.764915571
459681.9259
5.475421199
416933.337
−4.333373018
434948.8973
−0.199647723
434274.3904
−0.354415389
437325.9795
0.345781049
439539.3936
0.853655668
2017
448529.1
478134.6031
6.600575768
446159.7284
−0.528253708
478070.9591
6.58638629
417221.7391
−6.980006624
439101.8855
−2.101806668
436274.3346
−2.732211883
441434.99
−1.581638728
447515.8839
−0.225897516
2018
464000
497291.7661
7.174949597
452173.0382
−2.548914172
497195.624
7.154229303
417342.2974
−10.05553935
442320.9107
−4.672217527
436804.2529
−5.861152385
444540.236
−4.193914654
454184.3177
−2.115448779Table 9
Evaluation result of different grey models in Example B.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
8410.121727
4906.499126
8410.286218
51928.17151
5158.480682
1367.489689
1660.996872
18561.19627
MAE
6926.686251
4221.269336
6927.244243
27186.02944
3881.517201
1016.979092
1384.772743
14753.88106
NRMSE
1.79E −05
0.006017121
0.000384048
0.199776443
5.88E −16
0.003063485
1.47E −04
0.108418844
MAPE
1.787683435
1.104264936
1.788930619
7.705950265
1.099328531
0.256381015
0.358317783
3.9624714
RMSPE
2.197483638
1.313602202
2.200080172
15.3330731
1.509443367
0.34767392
0.437458618
5.139941157
MSE
70730147.47
24073733.67
70732914.28
2696534997
26609922.95
1870028.049
2758910.609
344518006.9
IA
0.987856201
0.995717785
0.987850143
0.775004734
0.995552338
0.999687628
0.99954467
0.95091572
U1
0.010871273
0.00635032
0.010870745
0.069089208
0.006667366
0.001768468
0.002146699
0.024435002
U2
0.021739295
0.012682793
0.02173972
0.134228952
0.013334139
0.003534819
0.004293505
0.047978772
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
29188.27919
7225.052506
29121.06984
34223.58019
13657.87156
17245.03553
11989.99302
6088.66592
MAE
28931.23345
5843.326918
28866.80301
32283.57548
10658.80219
13665.04068
9353.617817
4849.76408
NRMSE
0.111492795
0.013953907
0.111244499
0.124411774
0.041076011
0.052661204
0.03217455
0.00913139
MAPE
6.419544814
1.280694484
6.405345597
7.122972999
2.324557306
2.982593219
2.04044481
1.06500067
RMSPE
6.457863097
1.566430259
6.443180338
7.496943508
2.960127649
3.739148714
2.595513162
1.32349051
MSE
851955642.2
52201383.72
848036708.6
1171253441
186537455.5
297391250.5
143759932.6
37071852.7
IA
0.521410498
0.823769072
0.52205901
0.349931321
0.596904989
0.495386509
0.615535006
0.88095424
U1
0.031445311
0.008068562
0.031375108
0.039484363
0.015373596
0.019477492
0.013461203
0.00678995
U2
0.064920974
0.016070062
0.064771486
0.07612056
0.030378027
0.038356646
0.026668308
0.0135425Figure 5
Comparison of the true value and the produced value by different grey models in Example B.


## 5.1. Contrast Grey Models and Performance Criteria

To illustrate the advantages of the power-driven grey model with WOA in the aspect of prediction performance, the numerical validation study is conducted on two real-world data sequences. A series of classic existing grey models listed in Table1 are used to compare with the power-driven grey model. The power-driven grey model and the contrastive grey models are all realized by MATLAB. Then, all validation experiments and case studies are performed on the MATLAB platform 2019a.Table 1
The definition of the comparative grey models used.
No.
Name
Model
Reference
1
GM(1,1)
d
x
1
t
/
d
t
+
a
x
1
t
=
b
[11]
2
ARGM
x
0
k
=
a
x
0
k
−
1
+
b
[43]
3
DGM
x
1
k
=
a
x
1
k
−
1
+
b
[44]
4
NGM
x
0
k
+
a
z
1
k
=
b
k
[32]
5
NIGM
∑
i
=
k
n
x
0
i
=
a
∑
i
=
k
n
x
0
i
−
1
+
n
−
k
+
1
b
[43]
6
FGM(1,1)
d
x
r
t
/
d
t
+
a
x
r
t
=
b
[45]
7
SAIGM
x
0
k
+
a
z
1
k
=
b
k
+
c
[20]Nine evaluation criteria tabulated in Table2 are adopted to evaluate the prediction ability of the aforementioned grey prediction models. Meanwhile, Lewis’ criteria [47]shown in Table 3 are also adopted to illustrate the prediction power of grey models.Table 2
The definition of performance metrics used.
No.
Name
Formula
Reference
1
RMSE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
0.5
[24]
2
MAE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
[24]
3
NRMSE
1
/
x
¯
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
×
100
[24]
4
MAPE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
x
0
m
−
1
×
100
[24]
5
RMSPE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
x
0
m
−
1
2
0.5
×
100
[24]
6
MSE
1
/
v
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
[24]
7
IA
1
−
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
∑
m
=
1
v
x
0
m
−
x
¯
+
x
^
0
m
−
x
¯
2
−
1
[5]
8
U1
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
0.5
∑
m
=
1
v
x
0
m
2
0.5
+
∑
m
=
1
v
x
^
0
m
2
0.5
−
1
[46]
9
U2
∑
m
=
1
v
x
0
m
−
x
^
0
m
2
0.5
∑
m
=
1
v
x
0
m
2
−
0.5
[46]Table 3
Lewis’ criterion for model evaluation.
MAPE (%)
Prediction performance
<
10
Excellent
10
−
20
Good
20
−
50
Reasonable
>
50
IncorrectFor seeking out the optimal nonlinear parameter of the proposed model, the necessary parameters of WOA are set to the same values in validation experiments and applications as follows. The population size of the humpbacks is set to 30. Maximum iteration is set to 200. The minimum value and maximum value of the nonlinear parameter are set to −10 and 10, respectively.


## 5.2. Example A: Predicting the Natural Gas Consumption of China

In this section, the validation experiment is to study and analyze China’s natural gas consumption (NGC). The original sequence of the natural gas consumption during 2005–2015 is listed in Table4, which was collected from China’s National Bureau of Statistics. To construct a power-driven grey model and validate its superiority, the dataset is partitioned into two sub-datasets, including training set and test set. The test set, including the natural gas consumption in the last 3 years, is employed to check the prediction performance of grey models. The training set, including the natural gas consumption from 2005 to 2012, is utilized to build models of the eight grey models.Table 4
The original data of China’s natural gas consumption (NGC) (2005–2015).
Year
NGC
2005
467.63
2006
561.41
2007
705.23
2008
812.94
2009
895.2
2010
1069.41
2011
1305.3
2012
1463
2013
1705.37
2014
1868.94
2015
1931.75For achieving better prediction accuracy, the WOA method is used to seek the optimum value of the proposed model’s nonlinear parameter. Figure2 shows the convergence curve. It can be noticed that the fitness function converges to a constant after dozens of iterations. Then, the linear parameters a,b,andc and nonlinear parameter α are obtained and are equal to −0.175843, −4.869699, 463.613598, and 0.469429, respectively. The order of FGM(1,1) is equal to 0.862399 obtained by the whale optimization algorithm. All established models are used to predict the consumption from 2013 to 2015. For each grey model, the produced results and their absolute percentage error (APE) [12] are tabulated in Table 5. The evaluation values of the eight models are given in Table 6. Based on Lewis’ criteria, GM(1,1,eαt), GM(1,1), ARGM, SAIGM, NIGM, and FGM models exhibit excellent prediction performance while the other grey models only show good prediction performance. Furthermore, the GM(1,1,eαt) model obtains the lowest MAPE of prediction. From Figure 3, it can be noticed that the results produced by GM(1,1,eαt) are more approximate to real values than those of the other seven contrast models. Overall, the proposed model can more accurately predict natural gas consumption, though its fitted performance is not better than the other comparative models.Figure 2
Convergence curve of WOA in Example A.Table 5
Fitted and predicted results of various grey models in Example A.
Year
Raw data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2005
467.63
467.63
0
467.63
0
467.63
0
467.63
0
469.7263546
0.448293434
467.63
2.43E − 14
467.63
0
467.63
0
2006
561.41
581.0296492
3.494709602
570.8747479
1.69E + 00
582.5467989
3.764948767
326.4241266
−41.85637473
566.9866424
0.993327942
561.41
−1.44E − 09
578.1643099
2.984326951
585.984921
4.377357191
2007
705.23
678.504639
−3.789595023
685.1861578
−2.842170945
680.4246572
−3.517340834
541.2409827
−23.25326735
676.4109333
−4.086477709
675.7856868
−4.175136232
678.9353041
−3.728527697
692.1766157
−1.850940014
2008
812.94
792.3322774
−2.534962309
811.7504501
−0.146326902
794.7476753
−2.237843466
735.3614988
−9.542955343
799.5205363
−1.650732361
798.9734068
−1.718034933
795.5276016
−2.141904496
814.9110141
0.242455057
2009
895.2
925.2559256
3.357453712
951.8809942
6.331657078
928.2789222
3.695143228
910.7796456
1.740353617
938.0270257
4.784073466
935.7082618
4.525051583
930.4251873
3.934895813
955.0483743
6.685475232
2010
1069.41
1080.479178
1.035073373
1107.031938
3.518008797
1084.24571
1.387279884
1069.297286
−0.010539815
1093.856036
2.28593671
1089.675641
1.895030039
1086.502038
1.598268044
1112.224417
4.00E + 00
2011
1305.3
1261.743072
−3.336928487
1278.813297
−2.029165926
1266.417594
−2.978809925
1212.542685
−7.106206627
1269.174034
−2.767637051
1264.347874
−3.137372698
1267.083353
−2.927805624
1283.782232
−1.648492178
2012
1463
1473.416252
0.711978939
1469.007663
0.410640007
1479.197481
1.107141546
1341.987232
−8.271549391
1466.418438
0.233659495
1463.349784
0.023908691
1476.016395
0.88970577
1463
2.62E − 08
2013
1705.37
1720.600254
0.89307624
1679.588701
−1.511771576
1727.72804
1.311037472
1458.96056
−14.44903103
1688.331509
−0.999108165
1690.650448
−0.863129502
1717.752459
0.726086333
1636.180316
−4.057165562
2014
1868.94
2009.252464
7.507595978
1912.74163
2.343661633
2018.015997
7.976499897
1564.664195
−16.28066204
1937.998469
3.695060789
1950.695202
4.374415544
1997.4417
6.875646112
1777.904354
−4.870977476
2015
1931.75
2346.329692
21.4613533
2170.885899
12.37923639
2357.077313
22.01772034
1660.183907
−14.05803509
2218.8904
14.86426298
2248.518858
16.3980255
2321.042867
20.15234203
1843.333931
−4.576993327Table 6
Evaluation result of different grey models in Example A.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
23.86274134
27.06135354
23.66839461
118.1206638
24.48123827
24.4715338
23.55366157
28.84466814
MAE
20.25637699
19.68692977
20.89112313
88.25197924
19.59112421
18.18583986
20.50145901
20.47248493
NRMSE
0.007664991
0.024109321
0.00130884
0.262191084
3.09E − 16
0.009028807
6.38E − 05
0.036768004
MAPE
2.28258768
2.120482298
2.336063456
11.47265586
2.156267271
1.934316772
2.275679299
2.351034331
RMSPE
2.668940276
2.90890819
2.674700195
17.69775476
2.656105278
2.605146741
2.617187635
3.280388261
MSE
569.4304242
732.3168555
560.1929035
13952.49123
599.3310274
598.8559665
554.7749735
832.01488
IA
0.998657193
0.998291523
0.998685903
0.969110255
0.998594898
0.998594002
0.998699542
0.998042174
U1
0.01235891
0.013937773
0.012237389
0.063448569
0.012659895
0.012673831
0.012179981
0.014834273
U2
0.024676152
0.027983796
0.024475181
0.122147051
0.025315732
0.025305697
0.024356537
0.029827898
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
252.8476943
141.1491367
260.5294979
275.1057498
170.7912875
189.0706106
236.7945303
83.450789
MAE
190.0408036
102.9062759
198.9204499
274.0837795
124.4124534
137.7478703
176.7256754
82.8804665
NRMSE
0.179344392
0.08089408
0.187724249
0.258657024
0.106690266
0.120734047
0.166778703
0.07821556
MAPE
9.954008505
5.411556534
10.4350859
14.92924272
6.519477311
7.211856848
9.251358159
4.50171212
RMSPE
13.13711368
7.326292761
13.54158261
14.96064621
8.861865967
9.811145772
12.30065852
4.5142693
MSE
63931.95652
19923.07878
67875.61925
75683.17359
29169.6639
35747.69578
56071.64958
6964.03418
IA
0.593522143
0.786941885
0.580405823
0.462920793
0.731260314
0.699839537
0.616307649
0.83617379
U1
0.06517862
0.037446493
0.067003457
0.080883402
0.044965721
0.049568808
0.061267078
0.02322958
U2
0.137579314
0.076801972
0.141759132
0.149690352
0.092930838
0.102876971
0.128844478
0.04540719Figure 3
Comparison of the true value and the produced value by different grey models in Example A.


## 5.3. Example B: Predicting the Total Energy Consumption of China

In this example, China’s total energy consumption from 2008 to 2018 is employed to examine the accuracy of the grey prediction models. The raw sequence is listed in Table7, which was collected from China’s National Bureau of Statistics. To construct models and validate the prediction performance, the raw dataset is broken into two groups, including training set and test set. The training set, including the consumption from 2008 to 2015, is used to build models of the proposed model and other contrast grey models. The test set containing the rest digits is utilized to test the prediction accuracy of all models.Table 7
The raw sequence of China’s total energy consumption (TEC) (2008–2018).
Year
TEC
2008
320611
2009
336126
2010
360648
2011
387043
2012
402138
2013
416913
2014
425806
2015
429905
2016
435819
2017
448529.1
2018
464000During the stage of constructing the new proposed model, the WOA optimizer is utilized to seek the optimum value of power-driven grey model’s nonlinear parameter. From Figure4, which shows convergence curve of WOA, it can be noticed that the fitness function converges to a constant rapidly after dozens of iterations. Then, the linear parameters a,b,andc and nonlinear parameter α are obtained and are equal to −0.003666, −224002.105686, 458089.69852, and −0.235980, respectively. The order of FGM(1,1) is equal to 0.209972 obtained by the whale optimization algorithm. All established models are employed to forecast the energy consumption from 2016 to 2018 in China. For each grey model, the raw data, fitted data, predicted data, and their absolute percentage error (APE) are tabulated in Table 8. The evaluation results of the eight models are given in Table 9. Based on Lewis’ criteria, the GM(1,1,eαt) model shows an excellent forecasting ability. Although other grey models also exhibit excellent prediction performances, the proposed model has the lowest value of MAPE. It can be noticed that the forecasted values of the GM(1,1,eαt) model are more approximate to the actual values than those of the other seven comparative grey models in Figure 5. Overall, the power-driven grey model has the highest prediction performance of the total energy consumption though it is not the best one for fitted performance compared with the other contrast grey models.Figure 4
Convergence curve of WOA in Example B.Table 8
Fitted and predicted results of various grey models in Example B.
Year
Real data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2008
320611
320611
0
320611
0
320611
0
320611
0
311679.0155
−2.785925775
320611
0
320611
0
320611
0
2009
336126
349189.7029
3.886549367
344430.113
2.47E + 00
349311.5014
3.922785319
195251.0697
−41.91134582
343554.0921
2.209912975
336126
−6.09E − 09
334656.6702
−0.437136603
302216.2049
−10.08841777
2010
360648
363180.4997
0.70220817
364871.1079
1.17097776
363285.296
0.731265934
324553.7721
−10.00815973
368260.8022
2.110867706
362542.7915
0.525385288
363846.1266
0.886772319
336515.6299
−6.691391627
2011
387043
377731.8583
−2.405712472
382413.0823
−1.196228253
377818.0957
−2.383431369
378605.0893
−2.18009644
387411.2353
0.095140671
384858.2814
−0.564464039
385905.0726
−0.294005419
363964.4755
−5.962780493
2012
402138
392866.2383
−2.305616897
397467.1865
−1.161495191
392932.263
−2.289198485
401199.7052
−0.233326558
402254.9392
0.029079382
402143.1792
0.001287911
402575.3761
0.108762691
386004.0797
−4.012035752
2013
416913
408606.9995
−1.992262303
410386.2597
−1.565492142
408651.0547
−1.981695294
410644.7418
−1.503493099
413760.4517
−0.756164554
414991.5431
−0.46087718
415173.395
−0.417258508
403772.8105
−3.15E + 00
2014
425806
424978.4373
−0.194352063
421473.1004
−1.017575986
424998.6581
−0.189603218
414592.9707
−2.633365728
422678.4964
−0.734490251
424176.7152
−0.382635476
424693.9224
−0.261170013
418169.7514
−1.793363307
2015
429905
442005.8207
2.814766209
430987.5627
0.251814401
442000.2281
2.813465333
416243.4155
−3.177814742
429590.9676
−0.073046936
430405.4018
0.116398218
431888.7395
0.461436707
429905.0001
1.97E − 08
2016
435819
459715.4311
5.483109076
439152.6474
0.764915571
459681.9259
5.475421199
416933.337
−4.333373018
434948.8973
−0.199647723
434274.3904
−0.354415389
437325.9795
0.345781049
439539.3936
0.853655668
2017
448529.1
478134.6031
6.600575768
446159.7284
−0.528253708
478070.9591
6.58638629
417221.7391
−6.980006624
439101.8855
−2.101806668
436274.3346
−2.732211883
441434.99
−1.581638728
447515.8839
−0.225897516
2018
464000
497291.7661
7.174949597
452173.0382
−2.548914172
497195.624
7.154229303
417342.2974
−10.05553935
442320.9107
−4.672217527
436804.2529
−5.861152385
444540.236
−4.193914654
454184.3177
−2.115448779Table 9
Evaluation result of different grey models in Example B.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
8410.121727
4906.499126
8410.286218
51928.17151
5158.480682
1367.489689
1660.996872
18561.19627
MAE
6926.686251
4221.269336
6927.244243
27186.02944
3881.517201
1016.979092
1384.772743
14753.88106
NRMSE
1.79E −05
0.006017121
0.000384048
0.199776443
5.88E −16
0.003063485
1.47E −04
0.108418844
MAPE
1.787683435
1.104264936
1.788930619
7.705950265
1.099328531
0.256381015
0.358317783
3.9624714
RMSPE
2.197483638
1.313602202
2.200080172
15.3330731
1.509443367
0.34767392
0.437458618
5.139941157
MSE
70730147.47
24073733.67
70732914.28
2696534997
26609922.95
1870028.049
2758910.609
344518006.9
IA
0.987856201
0.995717785
0.987850143
0.775004734
0.995552338
0.999687628
0.99954467
0.95091572
U1
0.010871273
0.00635032
0.010870745
0.069089208
0.006667366
0.001768468
0.002146699
0.024435002
U2
0.021739295
0.012682793
0.02173972
0.134228952
0.013334139
0.003534819
0.004293505
0.047978772
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
29188.27919
7225.052506
29121.06984
34223.58019
13657.87156
17245.03553
11989.99302
6088.66592
MAE
28931.23345
5843.326918
28866.80301
32283.57548
10658.80219
13665.04068
9353.617817
4849.76408
NRMSE
0.111492795
0.013953907
0.111244499
0.124411774
0.041076011
0.052661204
0.03217455
0.00913139
MAPE
6.419544814
1.280694484
6.405345597
7.122972999
2.324557306
2.982593219
2.04044481
1.06500067
RMSPE
6.457863097
1.566430259
6.443180338
7.496943508
2.960127649
3.739148714
2.595513162
1.32349051
MSE
851955642.2
52201383.72
848036708.6
1171253441
186537455.5
297391250.5
143759932.6
37071852.7
IA
0.521410498
0.823769072
0.52205901
0.349931321
0.596904989
0.495386509
0.615535006
0.88095424
U1
0.031445311
0.008068562
0.031375108
0.039484363
0.015373596
0.019477492
0.013461203
0.00678995
U2
0.064920974
0.016070062
0.064771486
0.07612056
0.030378027
0.038356646
0.026668308
0.0135425Figure 5
Comparison of the true value and the produced value by different grey models in Example B.


## 6. Applications

Residential energy consumption refers to the energy consumption of urban residents, rural residents, and public facilities. The residential energy consumption has been the second largest part of total energy consumption in China [48]. Effective forecasting of the residential energy consumption plays an indispensable role in programming and planning of energy for governments and companies. So, the research studies of forecasting the total residential energy and the residential thermal energy consumption are, respectively, conducted in this section.


### 6.1. Case 1: Predicting the Total Residential Energy Consumption of China

Original data on total residential energy consumption (2005–2015) were collected from China’s National Bureau of Statistics and are tabulated in Table10. The original data are partitioned into two groups, including training set and test set. The training set, including annual total residential energy consumption between 2005 and 2012, is employed to establish models of the proposed and the other seven contrast grey models. The test set, including the total residential energy consumption in the last three years, is utilized to verify the prediction performance of these grey models.Table 10
Raw sequence of China’s total residential energy consumption (TREC) (2005–2015).
Year
TREC
2005
27573
2006
27765
2007
30814
2008
31898
2009
33843
2010
36470
2011
39584
2012
42306
2013
45531
2014
47212
2015
50099To get a better prediction performance of the power-driven grey model, it is essential to seek out the optimal parameters of the model. WOA algorithm is employed to seek for the optimal parameter of the proposed model. Furthermore, the linear parameter is estimated by using the least-squares method once the nonlinear parameter is determined. The convergence curve of WOA is shown in Figure6. Obviously, it can be noticed that the fitness function converges to a constant after dozens of iterations from the convergence curve. According to the computational steps of the power-driven grey model, the linear parameters a,b,andc and nonlinear parameter α are equal to −0.085448, −1868.619677, 27446.346885, and 0.171344, respectively. The order of FGM(1,1) obtained by the WOA optimizer is equal to 1.047609. By using these established grey models, the fitted and predicted results are produced and are given in Table 11. The evaluation results of the 8 grey models are tabulated in Table 12. Obviously, the prediction performance of the power-driven grey method is much better than those of other seven contrast grey prediction methods; though the fit ability is not the best, it is not the worst. The results produced by each model are also shown in Figure 7 in which the horizontal axis denotes the raw values and the vertical axis denotes the simulated or predicted values. It can be found that the values produced by the GM(1,1,eαt) model are almost equal to the real consumption. Meanwhile, the correlation coefficient between the raw values and the produced values of the proposed model is also the highest among the eight prediction methods. Above all, the performance of the power-driven grey model is the best for forecasting the total residential consumption compared with the other seven grey prediction methods.Figure 6
Convergence curve of WOA in Case 1.Table 11
Fitted and predicted results of various grey models in Case 1.
Year
Real data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2005
27573
27573
0
27573
0
27573
0
27573
0
27058.56265
−1.865728606
27573
0
27573
0
27573
0
2006
27765
27978.32853
0.768336125
29005.12306
4.47E + 00
27991.46972
0.815666186
15193.77532
−45.27723636
28538.6003
2.786242738
28291.15519
1.895030383
28213.87712
1.616701298
28590.2434
2.972243475
2007
30814
29952.57086
−2.795577125
30630.24572
−0.596333737
29966.86358
−2.749193289
26179.55591
−15.04006001
30206.15237
−1.972634606
29924.47954
−2.886741277
29999.16318
−2.644372115
30669.18209
−0.4699744
2008
31898
32066.1222
0.527061881
32474.3775
1.806939309
32081.66352
0.575783803
32092.74897
0.610536628
32084.97616
0.5861689
31901.97945
0.012475549
31981.14675
0.260664469
32845.45897
2.970277048
2009
33843
34328.81263
1.435489247
34567.03309
2.139388033
34345.7076
1.485410853
35275.57717
4.233008816
34201.83887
1.060304543
34142.0597
0.883667805
34181.49939
1.000205036
35111.28887
3.747566316
2010
36470
36751.16589
0.770951158
36941.70473
1.293404786
36769.52816
0.821300144
36988.76246
1.422436136
36586.89902
0.320534735
36622.63513
0.418522422
36624.28035
0.423033577
37455.11423
2.70E + 00
2011
39584
39344.44832
−0.605173011
39636.3982
0.132372173
39364.40085
−0.554767454
37910.89951
−4.22670899
39274.13608
−0.782800938
39341.62499
−0.612305507
39336.19965
−0.626011396
39860.69337
0.699003063
2012
42306
42120.72123
−0.437949155
42694.24116
0.917697623
42142.39703
−0.386713401
38407.2479
−9.215600867
42301.83456
−0.009845972
42306
1.14E − 08
42346.91016
0.096700621
42305.99989
−2.61E − 07
2013
45531
45092.89704
−0.962208079
46164.17331
1.390642227
45116.43995
−0.910500641
38674.41178
−15.05916456
45713.12948
0.400012041
45527.84857
−0.006921508
45689.33182
0.347745099
44761.89622
−1.689187102
2014
47212
48274.79929
2.251121097
50101.72973
6.120752634
48300.3649
2.305271744
38818.21509
−17.77892253
49556.62086
4.966154493
49022.73478
3.835327421
49400.01157
4.634439477
47190.53749
−0.045459867
2015
50099
51681.22697
3.158200705
54569.92965
8.924189408
51708.9835
3.213604062
38895.61849
−22.36248529
53887.06613
7.56116117
52808.95862
5.409207009
53519.52298
6.827527459
49543.45316
−1.108898065Table 12
Evaluation result of different grey models for fitting and forecasting China’s total residential energy consumption in Case 1.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
390.9308897
592.0396847
390.5257193
4998.407423
433.8786898
393.6339653
366.3717398
729.3183711
MAE
304.3361034
454.5790022
305.3384416
3115.451246
359.0785841
264.2156166
266.0438679
555.9520645
NRMSE
0.001442512
0.034214153
0.000188067
0.215925463
3.43E − 16
0.001570568
3.22E − 05
0.04351679
MAPE
0.917567213
1.41907916
0.923604391
10.00319848
1.17303263
0.838592869
0.833461064
1.695028338
RMSPE
1.219447364
1.958309635
1.219282349
17.31812039
1.471344436
1.287221758
1.186116654
2.227857299
MSE
152826.9605
350510.9882
152510.3374
24984076.77
188250.7174
154947.6986
134228.2517
531905.2864
IA
0.998469768
0.996464675
0.998474462
0.832432082
0.998113113
0.99844636
0.998656649
0.994674042
U1
0.005725206
0.008618064
0.005717969
0.07547321
0.006352639
0.005765005
0.005364149
0.010598626
U2
0.011447175
0.01733601
0.011435311
0.146362557
0.012704766
0.011526326
0.010728038
0.021355782
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
1129.146983
3095.192957
1147.235089
8999.707111
2574.222417
1881.723156
2346.09186
547.909444
MAE
1027.709741
2664.610899
1037.636147
8817.91821
2104.938825
1507.948278
1922.28879
448.704651
NRMSE
0.02676036
0.096930345
0.027692398
0.320768731
0.076571197
0.054778092
0.069926951
0.0163225
MAPE
2.123843293
5.478528089
2.143125482
18.4001908
4.309109234
3.083818646
3.936570679
0.94784892
RMSPE
2.307063232
6.299170718
2.343113335
18.64536826
5.227933254
3.828372361
4.768440944
1.16691584
MSE
1274972.909
9580219.441
1316148.349
80994728.08
6626621.053
3540882.034
5504147.015
300204.759
IA
0.941388626
0.738632316
0.939683635
0.267264603
0.797293581
0.866620632
0.819041097
0.9797724
U1
0.011752671
0.031568432
0.011937749
0.104105989
0.026407079
0.019427025
0.024114228
0.00577616
U2
0.02369601
0.064954984
0.024075603
0.188865715
0.05402202
0.039489395
0.049234526
0.0114983Figure 7
Comparison of the true value and the produced value by different grey models in Case 1.


### 6.2. Case 2: Predicting the Residential Thermal Energy Consumption

In this case study, the main aim is to forecast China’s residential thermal consumption, which is considered as a raw sequence. The raw data listed in Table13 were collected from China’s National Bureau of Statistics from 2005 to 2016. The first seven digits are used to build models of the power-driven grey model and the other seven comparative grey models, respectively. The rest of the digits of raw data are utilized to validate the predicted values.Table 13
Raw sequence of China’s residential thermal energy consumption (RThEC) (2005–2016).
Year
RThEC
2005
52044
2006
56948
2007
57689
2008
62765
2009
67000
2010
67410
2011
70044
2012
77608
2013
81472
2014
86482
2015
93841
2016
98623The results produced by these established grey prediction methods are tabulated in Table14. The comparisons of various grey prediction methods are plotted in Figure 8, in which the horizontal axis denotes the original value and the vertical axis represents the generated values. From the regression lines in Figure 8, it can be found that the simulated and forecasted consumptions of the power-driven grey prediction method are very close to the actual consumption while the other seven grey prediction methods are worse than the proposed method. Meanwhile, the correlation coefficient between the raw consumptions and the generated consumptions of the proposed model is the highest among the eight grey prediction methods. The fitted and predicted metrics of these models are listed in Table 15. It can be clearly noticed that the prediction accuracy of the proposed prediction model is the best; though the fitting accuracy is not the best, it is not the worst. The convergence curve of WOA during the stage of building power-driven grey model is plotted in Figure 9. The fitness value rapidly stabilizes to a constant after a few iterations. It shows that the WOA method is effective to search for the optimum value of nonlinear parameter. According to the calculation procedures of the power-driven grey model, the linear parameters a,b,andc and nonlinear parameter α are equal to −0.015782, 35273.868015, 13840.161440, and 0.057975, respectively, in this case. The order of FGM(1,1), which is equal to −0.071090, is also calculated by the WOA optimizer. To sum up, the power-driven grey prediction method has better accuracy for forecasting the residential thermal energy consumption than the other contrast grey prediction models.Table 14
Fitted and predicted results of various grey models in Case 2.
Year
Real data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2005
52044
52044
0
52044
0
52044
0
52044
0
51888.19898
−0.299364038
52044
0
52044
0
52044
0
2006
56948
56831.2207
−0.205063046
56445.26181
−8.83E − 01
56850.51634
−0.171180124
32790.06199
−42.42104728
56047.04046
−1.582074074
54662.18926
−4.013856046
56093.70088
−1.500138938
53564.22871
−5.941861514
2007
57689
59393.10731
2.953955369
60129.28218
4.230064973
59409.97388
2.983192421
53714.75243
−6.88909077
59664.98819
3.425242585
59743.78058
3.561823886
59515.12026
3.165456597
56731.39771
−1.659939135
2008
62765
62070.48086
−1.106538906
63212.9447
0.713685488
62084.66032
−1.083947558
61922.32276
−1.342590999
62812.39016
0.075504119
63668.89356
1.440123579
62613.27866
−0.241729217
60087.13419
−4.266495356
2009
67000
64868.54734
−3.18127262
65794.08529
−1.799872708
64879.76336
−3.164532294
65141.68751
−2.773600732
65550.44495
−2.163515
66447.8307
−0.82413329
65418.71828
−2.360121975
63642.68594
−5.01091651
2010
67410
67792.74748
0.567790358
67954.59627
0.807886474
67800.70427
0.579593939
66404.46187
−1.491675015
67932.39175
0.774946961
68301.59137
1.32263963
67959.09564
0.814561098
67409.97201
−4.15E − 05
2011
70044
70848.76722
1.148945267
69763.02459
−0.401141304
70853.14838
1.155200125
66899.7767
−4.488925964
70004.54551
−0.056328148
69432.19338
−0.873460429
70259.45423
0.307598404
71401.62276
1.938242767
2012
77608
74042.54885
−4.594179917
71276.74656
−8.157990718
74043.01605
−4.593577918
67094.06064
−13.54749428
71807.19754
−7.474490333
70001.66656
−9.80096568
72342.47139
−6.784775551
75631.02234
−2.547389011
2013
81472
77380.30251
−5.02221314
72543.78834
−10.95862586
77376.49422
−5.026887497
67170.26722
−17.55416927
73375.3989
−9.937894128
70137.08261
−13.91265391
74228.68188
−8.890561319
80112.35362
−1.668851119
2014
86482
80868.5183
−6.490924933
73604.34959
−14.89055573
80860.04835
−6.500718823
67200.15874
−22.29578555
74739.64194
−13.57780586
69937.99929
−19.12999319
75936.6803
−12.19365845
84860.64593
−1.8747879
2015
93841
84513.97888
−9.939174904
74492.07893
−20.61883513
84500.43498
−9.953607718
67211.88349
−28.37684649
75926.45323
−19.09031955
69482.71953
−25.95697027
77483.30439
−17.43128868
89891.82555
−4.208367825
2016
98623
88323.77266
−10.44302783
75235.14151
−23.71440586
88304.71485
−10.46235173
67216.48244
−31.84502353
76958.90932
−21.96657035
68833.09601
−30.20583839
78883.80106
−20.01480278
95222.76925
−3.447705659Figure 8
Comparison of the true value and the produced value by different grey models in Case 2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Table 15
Evaluation result of different grey models for fitting and forecasting China’s residential thermal energy consumption in Case 2.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
1117.216715
1084.839675
1117.196143
9369.107976
1008.424292
1112.393086
995.4137672
2159.813561
MAE
833.481874
774.6359237
831.2695007
4997.562392
727.3628862
644.5599287
739.7103295
1676.346441
NRMSE
0.000311765
0.008800034
0.000138821
0.21331217
4.44E − 16
0.005034
2.05E − 05
0.054996175
MAPE
1.309080795
1.26220759
1.305378066
8.486704395
1.196710704
1.046859636
1.198515175
2.688266118
RMSPE
1.762923227
1.821901401
1.764262819
16.38324835
1.673963545
1.83762531
1.632581987
3.487383104
MSE
1248173.188
1176877.121
1248127.222
87780184.26
1016919.552
1237418.377
990848.5679
4664794.619
IA
0.991453209
0.991823554
0.991452721
0.70140632
0.99308751
0.991283835
0.993269893
0.971417611
U1
0.008969537
0.008695398
0.00896862
0.077822239
0.008095409
0.008923061
0.00799091
0.017509782
U2
0.017936366
0.017416568
0.017936035
0.150416426
0.016189757
0.017858925
0.015980879
0.034674746
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
7127.875052
15537.08007
7138.65228
21828.1977
14334.23205
10884.47247
13027.7043
2666.27728
MAE
6579.37576
14174.77901
6588.258311
20426.62949
13043.67981
10068.06587
11830.2122
2461.54694
NRMSE
0.167934454
0.361802375
0.168161176
0.521376951
0.332931775
0.256981089
0.301958773
0.06282945
MAPE
7.297904145
15.66808266
7.307428738
22.72386382
14.40941604
11.1683294
13.06301736
2.74950079
RMSPE
7.69811988
16.70857475
7.709280273
23.69867698
15.40101991
11.76115179
13.9871564
2.91164244
MSE
50806602.76
241400857
50960356.37
476470215
205470208.5
118471740.9
169721079.4
7109034.54
IA
0.770969645
0.473466789
0.770285171
0.372333584
0.502732671
0.618658116
0.538683921
0.96778228
U1
0.042144608
0.096270537
0.042210649
0.140714021
0.088194495
0.065740224
0.079555449
0.01537902
U2
0.081048357
0.176666229
0.0811709
0.248200135
0.162989102
0.123763197
0.148133072
0.03031723Figure 9
Convergence curve of WOA in Case 2.


## 6.1. Case 1: Predicting the Total Residential Energy Consumption of China

Original data on total residential energy consumption (2005–2015) were collected from China’s National Bureau of Statistics and are tabulated in Table10. The original data are partitioned into two groups, including training set and test set. The training set, including annual total residential energy consumption between 2005 and 2012, is employed to establish models of the proposed and the other seven contrast grey models. The test set, including the total residential energy consumption in the last three years, is utilized to verify the prediction performance of these grey models.Table 10
Raw sequence of China’s total residential energy consumption (TREC) (2005–2015).
Year
TREC
2005
27573
2006
27765
2007
30814
2008
31898
2009
33843
2010
36470
2011
39584
2012
42306
2013
45531
2014
47212
2015
50099To get a better prediction performance of the power-driven grey model, it is essential to seek out the optimal parameters of the model. WOA algorithm is employed to seek for the optimal parameter of the proposed model. Furthermore, the linear parameter is estimated by using the least-squares method once the nonlinear parameter is determined. The convergence curve of WOA is shown in Figure6. Obviously, it can be noticed that the fitness function converges to a constant after dozens of iterations from the convergence curve. According to the computational steps of the power-driven grey model, the linear parameters a,b,andc and nonlinear parameter α are equal to −0.085448, −1868.619677, 27446.346885, and 0.171344, respectively. The order of FGM(1,1) obtained by the WOA optimizer is equal to 1.047609. By using these established grey models, the fitted and predicted results are produced and are given in Table 11. The evaluation results of the 8 grey models are tabulated in Table 12. Obviously, the prediction performance of the power-driven grey method is much better than those of other seven contrast grey prediction methods; though the fit ability is not the best, it is not the worst. The results produced by each model are also shown in Figure 7 in which the horizontal axis denotes the raw values and the vertical axis denotes the simulated or predicted values. It can be found that the values produced by the GM(1,1,eαt) model are almost equal to the real consumption. Meanwhile, the correlation coefficient between the raw values and the produced values of the proposed model is also the highest among the eight prediction methods. Above all, the performance of the power-driven grey model is the best for forecasting the total residential consumption compared with the other seven grey prediction methods.Figure 6
Convergence curve of WOA in Case 1.Table 11
Fitted and predicted results of various grey models in Case 1.
Year
Real data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2005
27573
27573
0
27573
0
27573
0
27573
0
27058.56265
−1.865728606
27573
0
27573
0
27573
0
2006
27765
27978.32853
0.768336125
29005.12306
4.47E + 00
27991.46972
0.815666186
15193.77532
−45.27723636
28538.6003
2.786242738
28291.15519
1.895030383
28213.87712
1.616701298
28590.2434
2.972243475
2007
30814
29952.57086
−2.795577125
30630.24572
−0.596333737
29966.86358
−2.749193289
26179.55591
−15.04006001
30206.15237
−1.972634606
29924.47954
−2.886741277
29999.16318
−2.644372115
30669.18209
−0.4699744
2008
31898
32066.1222
0.527061881
32474.3775
1.806939309
32081.66352
0.575783803
32092.74897
0.610536628
32084.97616
0.5861689
31901.97945
0.012475549
31981.14675
0.260664469
32845.45897
2.970277048
2009
33843
34328.81263
1.435489247
34567.03309
2.139388033
34345.7076
1.485410853
35275.57717
4.233008816
34201.83887
1.060304543
34142.0597
0.883667805
34181.49939
1.000205036
35111.28887
3.747566316
2010
36470
36751.16589
0.770951158
36941.70473
1.293404786
36769.52816
0.821300144
36988.76246
1.422436136
36586.89902
0.320534735
36622.63513
0.418522422
36624.28035
0.423033577
37455.11423
2.70E + 00
2011
39584
39344.44832
−0.605173011
39636.3982
0.132372173
39364.40085
−0.554767454
37910.89951
−4.22670899
39274.13608
−0.782800938
39341.62499
−0.612305507
39336.19965
−0.626011396
39860.69337
0.699003063
2012
42306
42120.72123
−0.437949155
42694.24116
0.917697623
42142.39703
−0.386713401
38407.2479
−9.215600867
42301.83456
−0.009845972
42306
1.14E − 08
42346.91016
0.096700621
42305.99989
−2.61E − 07
2013
45531
45092.89704
−0.962208079
46164.17331
1.390642227
45116.43995
−0.910500641
38674.41178
−15.05916456
45713.12948
0.400012041
45527.84857
−0.006921508
45689.33182
0.347745099
44761.89622
−1.689187102
2014
47212
48274.79929
2.251121097
50101.72973
6.120752634
48300.3649
2.305271744
38818.21509
−17.77892253
49556.62086
4.966154493
49022.73478
3.835327421
49400.01157
4.634439477
47190.53749
−0.045459867
2015
50099
51681.22697
3.158200705
54569.92965
8.924189408
51708.9835
3.213604062
38895.61849
−22.36248529
53887.06613
7.56116117
52808.95862
5.409207009
53519.52298
6.827527459
49543.45316
−1.108898065Table 12
Evaluation result of different grey models for fitting and forecasting China’s total residential energy consumption in Case 1.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
390.9308897
592.0396847
390.5257193
4998.407423
433.8786898
393.6339653
366.3717398
729.3183711
MAE
304.3361034
454.5790022
305.3384416
3115.451246
359.0785841
264.2156166
266.0438679
555.9520645
NRMSE
0.001442512
0.034214153
0.000188067
0.215925463
3.43E − 16
0.001570568
3.22E − 05
0.04351679
MAPE
0.917567213
1.41907916
0.923604391
10.00319848
1.17303263
0.838592869
0.833461064
1.695028338
RMSPE
1.219447364
1.958309635
1.219282349
17.31812039
1.471344436
1.287221758
1.186116654
2.227857299
MSE
152826.9605
350510.9882
152510.3374
24984076.77
188250.7174
154947.6986
134228.2517
531905.2864
IA
0.998469768
0.996464675
0.998474462
0.832432082
0.998113113
0.99844636
0.998656649
0.994674042
U1
0.005725206
0.008618064
0.005717969
0.07547321
0.006352639
0.005765005
0.005364149
0.010598626
U2
0.011447175
0.01733601
0.011435311
0.146362557
0.012704766
0.011526326
0.010728038
0.021355782
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
1129.146983
3095.192957
1147.235089
8999.707111
2574.222417
1881.723156
2346.09186
547.909444
MAE
1027.709741
2664.610899
1037.636147
8817.91821
2104.938825
1507.948278
1922.28879
448.704651
NRMSE
0.02676036
0.096930345
0.027692398
0.320768731
0.076571197
0.054778092
0.069926951
0.0163225
MAPE
2.123843293
5.478528089
2.143125482
18.4001908
4.309109234
3.083818646
3.936570679
0.94784892
RMSPE
2.307063232
6.299170718
2.343113335
18.64536826
5.227933254
3.828372361
4.768440944
1.16691584
MSE
1274972.909
9580219.441
1316148.349
80994728.08
6626621.053
3540882.034
5504147.015
300204.759
IA
0.941388626
0.738632316
0.939683635
0.267264603
0.797293581
0.866620632
0.819041097
0.9797724
U1
0.011752671
0.031568432
0.011937749
0.104105989
0.026407079
0.019427025
0.024114228
0.00577616
U2
0.02369601
0.064954984
0.024075603
0.188865715
0.05402202
0.039489395
0.049234526
0.0114983Figure 7
Comparison of the true value and the produced value by different grey models in Case 1.


## 6.2. Case 2: Predicting the Residential Thermal Energy Consumption

In this case study, the main aim is to forecast China’s residential thermal consumption, which is considered as a raw sequence. The raw data listed in Table13 were collected from China’s National Bureau of Statistics from 2005 to 2016. The first seven digits are used to build models of the power-driven grey model and the other seven comparative grey models, respectively. The rest of the digits of raw data are utilized to validate the predicted values.Table 13
Raw sequence of China’s residential thermal energy consumption (RThEC) (2005–2016).
Year
RThEC
2005
52044
2006
56948
2007
57689
2008
62765
2009
67000
2010
67410
2011
70044
2012
77608
2013
81472
2014
86482
2015
93841
2016
98623The results produced by these established grey prediction methods are tabulated in Table14. The comparisons of various grey prediction methods are plotted in Figure 8, in which the horizontal axis denotes the original value and the vertical axis represents the generated values. From the regression lines in Figure 8, it can be found that the simulated and forecasted consumptions of the power-driven grey prediction method are very close to the actual consumption while the other seven grey prediction methods are worse than the proposed method. Meanwhile, the correlation coefficient between the raw consumptions and the generated consumptions of the proposed model is the highest among the eight grey prediction methods. The fitted and predicted metrics of these models are listed in Table 15. It can be clearly noticed that the prediction accuracy of the proposed prediction model is the best; though the fitting accuracy is not the best, it is not the worst. The convergence curve of WOA during the stage of building power-driven grey model is plotted in Figure 9. The fitness value rapidly stabilizes to a constant after a few iterations. It shows that the WOA method is effective to search for the optimum value of nonlinear parameter. According to the calculation procedures of the power-driven grey model, the linear parameters a,b,andc and nonlinear parameter α are equal to −0.015782, 35273.868015, 13840.161440, and 0.057975, respectively, in this case. The order of FGM(1,1), which is equal to −0.071090, is also calculated by the WOA optimizer. To sum up, the power-driven grey prediction method has better accuracy for forecasting the residential thermal energy consumption than the other contrast grey prediction models.Table 14
Fitted and predicted results of various grey models in Case 2.
Year
Real data
GM(1,1)
Error (%)
ARGM
Error (%)
DGM
Error (%)
NGM
Error (%)
NIGM
Error (%)
FGM(1,1)
Error (%)
SAIGM
Error (%)
GM(1,1,eαt)
Error (%)
2005
52044
52044
0
52044
0
52044
0
52044
0
51888.19898
−0.299364038
52044
0
52044
0
52044
0
2006
56948
56831.2207
−0.205063046
56445.26181
−8.83E − 01
56850.51634
−0.171180124
32790.06199
−42.42104728
56047.04046
−1.582074074
54662.18926
−4.013856046
56093.70088
−1.500138938
53564.22871
−5.941861514
2007
57689
59393.10731
2.953955369
60129.28218
4.230064973
59409.97388
2.983192421
53714.75243
−6.88909077
59664.98819
3.425242585
59743.78058
3.561823886
59515.12026
3.165456597
56731.39771
−1.659939135
2008
62765
62070.48086
−1.106538906
63212.9447
0.713685488
62084.66032
−1.083947558
61922.32276
−1.342590999
62812.39016
0.075504119
63668.89356
1.440123579
62613.27866
−0.241729217
60087.13419
−4.266495356
2009
67000
64868.54734
−3.18127262
65794.08529
−1.799872708
64879.76336
−3.164532294
65141.68751
−2.773600732
65550.44495
−2.163515
66447.8307
−0.82413329
65418.71828
−2.360121975
63642.68594
−5.01091651
2010
67410
67792.74748
0.567790358
67954.59627
0.807886474
67800.70427
0.579593939
66404.46187
−1.491675015
67932.39175
0.774946961
68301.59137
1.32263963
67959.09564
0.814561098
67409.97201
−4.15E − 05
2011
70044
70848.76722
1.148945267
69763.02459
−0.401141304
70853.14838
1.155200125
66899.7767
−4.488925964
70004.54551
−0.056328148
69432.19338
−0.873460429
70259.45423
0.307598404
71401.62276
1.938242767
2012
77608
74042.54885
−4.594179917
71276.74656
−8.157990718
74043.01605
−4.593577918
67094.06064
−13.54749428
71807.19754
−7.474490333
70001.66656
−9.80096568
72342.47139
−6.784775551
75631.02234
−2.547389011
2013
81472
77380.30251
−5.02221314
72543.78834
−10.95862586
77376.49422
−5.026887497
67170.26722
−17.55416927
73375.3989
−9.937894128
70137.08261
−13.91265391
74228.68188
−8.890561319
80112.35362
−1.668851119
2014
86482
80868.5183
−6.490924933
73604.34959
−14.89055573
80860.04835
−6.500718823
67200.15874
−22.29578555
74739.64194
−13.57780586
69937.99929
−19.12999319
75936.6803
−12.19365845
84860.64593
−1.8747879
2015
93841
84513.97888
−9.939174904
74492.07893
−20.61883513
84500.43498
−9.953607718
67211.88349
−28.37684649
75926.45323
−19.09031955
69482.71953
−25.95697027
77483.30439
−17.43128868
89891.82555
−4.208367825
2016
98623
88323.77266
−10.44302783
75235.14151
−23.71440586
88304.71485
−10.46235173
67216.48244
−31.84502353
76958.90932
−21.96657035
68833.09601
−30.20583839
78883.80106
−20.01480278
95222.76925
−3.447705659Figure 8
Comparison of the true value and the produced value by different grey models in Case 2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Table 15
Evaluation result of different grey models for fitting and forecasting China’s residential thermal energy consumption in Case 2.
Fitting
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
1117.216715
1084.839675
1117.196143
9369.107976
1008.424292
1112.393086
995.4137672
2159.813561
MAE
833.481874
774.6359237
831.2695007
4997.562392
727.3628862
644.5599287
739.7103295
1676.346441
NRMSE
0.000311765
0.008800034
0.000138821
0.21331217
4.44E − 16
0.005034
2.05E − 05
0.054996175
MAPE
1.309080795
1.26220759
1.305378066
8.486704395
1.196710704
1.046859636
1.198515175
2.688266118
RMSPE
1.762923227
1.821901401
1.764262819
16.38324835
1.673963545
1.83762531
1.632581987
3.487383104
MSE
1248173.188
1176877.121
1248127.222
87780184.26
1016919.552
1237418.377
990848.5679
4664794.619
IA
0.991453209
0.991823554
0.991452721
0.70140632
0.99308751
0.991283835
0.993269893
0.971417611
U1
0.008969537
0.008695398
0.00896862
0.077822239
0.008095409
0.008923061
0.00799091
0.017509782
U2
0.017936366
0.017416568
0.017936035
0.150416426
0.016189757
0.017858925
0.015980879
0.034674746
Prediction
GM(1,1)
ARGM
DGM
NGM
NIGM
FGM(1,1)
SAIGM
GM(1,1,eαt)
RMSE
7127.875052
15537.08007
7138.65228
21828.1977
14334.23205
10884.47247
13027.7043
2666.27728
MAE
6579.37576
14174.77901
6588.258311
20426.62949
13043.67981
10068.06587
11830.2122
2461.54694
NRMSE
0.167934454
0.361802375
0.168161176
0.521376951
0.332931775
0.256981089
0.301958773
0.06282945
MAPE
7.297904145
15.66808266
7.307428738
22.72386382
14.40941604
11.1683294
13.06301736
2.74950079
RMSPE
7.69811988
16.70857475
7.709280273
23.69867698
15.40101991
11.76115179
13.9871564
2.91164244
MSE
50806602.76
241400857
50960356.37
476470215
205470208.5
118471740.9
169721079.4
7109034.54
IA
0.770969645
0.473466789
0.770285171
0.372333584
0.502732671
0.618658116
0.538683921
0.96778228
U1
0.042144608
0.096270537
0.042210649
0.140714021
0.088194495
0.065740224
0.079555449
0.01537902
U2
0.081048357
0.176666229
0.0811709
0.248200135
0.162989102
0.123763197
0.148133072
0.03031723Figure 9
Convergence curve of WOA in Case 2.


## 7. Conclusions

A novel power-driven grey prediction method called GM(1,1,eαt) is proposed to forecast the total residential energy consumption and the residential thermal energy consumption of China in this paper. The grey input of GM(1,1,eαt) is an exponential term which is different from the grey action quantity of the traditional grey model. It plays an imperative role in raising the prediction performance of GM(1,1,eαt). The optimal value of nonlinear parameter α is sought out by using the WOA algorithm. Compared with seven classical grey prediction methods such as ARGM, DGM, NGM, NIGM, FGM(1,1), GM(1,1), and SAIGM, the proposed grey model obtains a more superior prediction accuracy in validation experiments and case studies. From their fitted and predicted results, it can be clearly noticed that the GM(1,1,eαt) model reaches the lowest forecast errors though the fit errors are not the lowest. The main reason for taking higher fitting error is that the strategy similar to cross validation is chosen to build the GM(1,1,eαt) model. In fact, the strategy can overcome the overfitting phenomenon of the prediction problem. According to Lewis’ criterion of accuracy evaluation, the fit abilities of the proposed grey model are still excellent in all validations and case studies because the MAPEs of fitting are less than 10%. In summary, three conclusions are drawn as follows. Firstly, the improvement of grey action quantity with an exponential term of time is one of the effective methods to improve the prediction accuracy of the grey prediction method. Secondly, the nonlinear parameter of exponential grey action quantity plays a significant role in forecasting future data accurately. Moreover, the heuristic optimization algorithm WOA can be used to seek out the optimal value of the nonlinear parameter effectively. Thirdly, the strategy similar to cross validation can be used to conquer the overfitting problem in prediction task.As future work, more studies and applications about the strategy of determining the nonlinear parameters in the grey model should be carried out to overcome the overfitting problem in prediction task. Besides, the power-driven grey model can also be employed to solve more problems such as solving the industrial, rural, and urban energy supply and demand prediction problems. At the same time, it is worth studying to extend the power-driven grey model to other more prediction applications.



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*Source: 1510257-2019-11-06.xml*