# A Novel Fractional Hausdorff Discrete Grey Model for Forecasting the Renewable Energy Consumption

Authors: Yuzhen Chen; Suzhen Li; Shuangbing Guo
Journal: Journal of Mathematics
 (2022)

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## Abstract

Reducing carbon dioxide emissions and using renewable energy to replace fossil fuels have become an essential trend in future energy development. Renewable energy consumption has a significant impact on energy security so accurate prediction of renewable energy consumption can help the energy department formulate relevant policies and adjust the energy structure. Based on this, a novel Fractional Hausdorff Discrete Grey Model, abbreviated FHDGM (1,1), is developed in this study. The paper investigates the model’s characteristics. The fractional-orderr of the FHDGM (1,1) model is optimized using particle swarm optimization. Subsequently, through two empirical analyses, the prediction accuracy of the FHDGM (1,1) model is proven to be higher than that of other models. Finally, the proposed model is applied with a view to forecasting the consumption of renewable energy for the years 2021 to 2023 in three different areas: the Asia Pacific region, Europe, and the world. The study’s findings will offer crucial forecasting data for worldwide energy conservation and emission reduction initiatives.


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## Body

## 1. Introduction

The world economy is expanding, and so is energy consumption. Renewable energy consumption, as a part of global energy consumption, has continuously increased, reaching 2.9 exajoules in 2020. Indeed, it has become an essential part of energy consumption. The International Energy Agency (IEA) states although oil and other liquid fuels will remain the world’s leading energy source in 2050, renewable energy consumption, including solar and wind energy, will increase by 50% [1]. Spencerdale, the chief economist of the BP group, said renewable energy, as part of total 2020 global power generation, had exhibited the fastest growth in history. This increase was mainly due to the decline in the proportion of coal power generation. The strong growth of renewable energy and the gradual replacement of coal align with the world’s need to transition to net-zero emissions [2]. The accurate prediction of renewable energy consumption is therefore of great significance for both the government and the energy sector in their quest to adjust current energy structures and deploy resources.In recent years, more and more scholars have focused on the prediction method of renewable energy consumption. There are three commonly used renewable energy prediction models: the first is the statistical model [3, 4]; the second is the intelligent predict model [5–7]; the third is the grey prediction model [8, 9]. The establishment of statistical models and intelligent prediction models requires a lot of data, but renewable energy, as a newly emerging energy in recent years, has a limited amount of data. As the grey prediction model is famous for its characteristics of “less data and higher prediction accuracy,” this paper chooses the grey prediction model to forecast renewable energy consumption.In the 1980s, in order to solve the problem of data with the characteristics of “ limited sample and poor information,” Prof. Deng first proposed the grey system theory. Subsequently, numerous fields, such as agriculture, the economy, the environment, and others, have extensively used this theory. As it was the first model, the GM (1,1) model is the basic model of the theory and, since it was put forward, many scholars have used it. For instance, Zeng et al. employed GM (1,1) to analyze the functional status of the human kidney. They found that, even though the renal function index (serum creatinine) was within the normal range, human renal function could nevertheless be abnormal [10]. For practical applications, however, it was later determined that the GM (1,1) model’s prediction accuracy had to be increased. Hence, an increasing number of researchers have optimized the GM (1,1) model to improve its predictive accuracy [11–19].In the early stages of the development of the grey system theory, first-order accumulation generation was used for all raw data preprocessing. However, the first-order accumulation generation operator cannot reflect the importance of new information in the original data. In order to solve this problem, scholars have proposed different accumulation generation operators. Wu et al. used the least-square method perturbation theory to demonstrate how the new information priority principle of the grey system theory was broken in this processing technique. Then, he proposed a new fractional-order accumulation and built a new grey prediction model FGM (1,1) [20]. Ma et al. introduced the conformable fractional accumulation and difference based on the definition of conformable fractional derivative. Then, they came up with a new conformable fractional grey model [21]. In 2020, Chen et al. created a brand-new Fractional Hausdorff Grey Model (FHGM (1,1)) by incorporating the fractional Hausdorff derivative into the grey model [22]. Shen et al. proposed a grey prediction model based on weighted fractional order accumulation generation operator and applied it to the prediction and analysis of the total output value of China’s construction industry [23]. With the introduction of a new accumulation generation operator, a series of new grey prediction models have been established and applied to practice, further enriching grey system theory [24–28].In the study of the above grey prediction model, the estimated value of the model parameters is calculated based on the basic form (difference equation), while the time response function of the model is derived based on the whitening equation (differential equation). The jump from the difference equation to the differential equation will inevitably lead to errors in the model, which reduces the accuracy of the model in some applications. To solve this problem, in 2005, Xie and Liu proposed the discrete grey prediction model (DGM (1,1) model), which formally unified the model parameter estimation and the time response function, fundamentally eliminating the jump between the difference equation and the differential equation of the grey prediction model [29]. Inspired by their work, many scholars have further studied the model and put forward a series of new discrete grey prediction models on this basis, all of which have greatly complemented the research content and application scope of the discrete grey prediction model [30–32].Based on the research results of the literature mentioned above, this paper proposes a new Fractional Hausdorff Discrete Grey Model and utilizes the model to study the consumption of renewable energy. The main contributions of this paper can be exacted below:(1)
The FHGM (1,1) model and the discrete model were combined to reduce errors in the FHGM (1,1) model discretization process(2)
We investigate the properties of the FHDGM (1,1) model(3)
To find the final optimal fractional-orderr, we create an optimization problem and utilize the particle swarm optimization algorithm to determine it(4)
Two examples demonstrate the strong predictive qualities of the FHDGM (1,1) model, with the suggested model used to forecast renewable energy consumption over the following three yearsThe rest of this paper is structured as follows: Section2 discusses the modeling mechanism of the proposed model. Section 3 describes the properties of the FHDGM (1,1) model in detail. Section 4 introduces the process of using PSO to find the optimal order r of the FHDGM (1,1) model. Two numerical examples are used to verify the performance of the FHDGM (1,1) model in Section 5. Section 6 uses the newly proposed model to simulate and forecast renewable energy consumption. The conclusions are given in Section 7.


## 2. Methodology

### 2.1. The Definitions of the Hausdorff Fractional Accumulation and Difference

Before introducing the definitions of the Hausdorff fractional accumulation and difference, the Hausdorff fractional derivative is given as shown in Definition1.Definition 1 (see [33]).
The Hausdorff fractional derivative is defined as(1)dfxdxr=limx0⟶xfx−fx0xr−x0r.
Based on Definition1, the definitions of the Hausdorff fractional accumulation and difference are given in Definition 2 and Definition 3, respectively.Definition 2 (see [34]).
The Hausdorff fractional accumulation (HFA) withr order is
(2)∇rfk=∇tr−t−1rfk=∑j=1kjr−j−1rfk.Definition 3 (see [34]).
The Hausdorff fractional difference (HFD) withr order is
(3)Δrfk=limh⟶1ft−ft−htr−t−hrt=k=fk−fk−1kr−k−1r=Δfkkr−k−1r.


### 2.2. Traditional FHGM (1,1) Model

Definition 4.
AssumeX0=x01,x02,⋯,x0n is a nonnegative sequence; its r-order accumulative sequence is then given by Xr=xr1,xr2,⋯,xrn, so the differential equation of the FHGM (1,1) model is
(4)dxrtdt+axrt=b,where xrk=∑i=1kx0iir−i−1rk=1,2,⋯,n.Definition 5.
The basic form of the FHGM (1,1) model is(5)xrk−xrk−1+azrk=b,where zrk=xrk+xrk−1/2, and k=2,3,⋯,n..Theorem 6.
LetX0,Xr be defined in Definition 1. The least-square estimation of the parameter list μ^=a,bT in Eq. (5) is
(6)μ^=a,bT=BTB−1BTY,where
(7)B=−0.5xr1+xr21−0.5xr2+xr31⋮⋮−0.5xrn−1+xrn1,Y=xr2−xr1xr3−xr2⋮xrn−xrn−1.Theorem 7.
Let the initial value bex∧r1=xr1=x01; then, the time response function of the FHGM (1,1) model is
(8)x∧rk=x01−bae−ak−1+ba,and the restored values are x∧0k=x∧rk−x∧rk−1/kr−k−1r, and k=2,3,⋯,n..Obviously, Eq. (4) is a continuous equation, and Eq. (5) is a discrete equation. The time response function of the model is obtained by directly substituting the parameters obtained from Eq. (5) into Eq. (4). The model’s performance in terms of prediction will be reduced since the prediction of the time response formula derived using this method will result in inaccuracies in the model’s predictive results. To solve the problem, this paper proves the main reasons for the unstable simulation and accuracy of the FHGM (1,1) model through model derivation. In doing so, this paper constructs the FHDGM (1,1) model, which solves the error caused by the FHGM (1,1) model changing from a difference equation to a differential equation.


### 2.3. Fractional Hausdorff Discrete Grey Model

Definition 8.
LetX0 and Xr be the same as Definition 4; the equation
(9)xrk+1=β1xrk+β2,is called the Fractional Hausdorff Discrete Grey Model (FHDGM (1,1) for short).Theorem 9.
Using the least-square method, the parametersβ1,β2 are estimated by
(10)ν^=β1,β2T=BTB−1BTY,where
(11)B=xr1xr2⋮xrn−111⋮1,Y=xr2xr3⋮xrn.Proof.
When we enterk=2,3,⋯,n into Eq. (9), we get Y=Bν^. After that, the error sequence is obtained with ε=Y−Bν^ by changing xrk+1 to β1xrk+β2. Let
(12)S=εTε=Y−Bv∧TY−Bv^=∑k=1n−1xrk+1−β1xrk−β22.
The parametersβ1,β2 that make the S minimum should satisfy
(13)∂S∂β1=−2∑k=1n−1xrk+1−β1xrk−β2xrk=0,∂S∂β2=−2∑k=1n−1xrk+1−β1xrk−β2=0.
Recalculating the equation system, we have(14)γ=δβ1,β2T,
where(15)δ=∑k=1n−1xrk2∑k=1n−1xrk∑k=1n−1xrkn−1,γ=∑k=1n−1xrk+1xrk∑k=1n−1xrk+1
Owing to(16)BTB=∑k=1n−1xrk2∑k=1n−1xrk∑k=1n−1xrkn−1=δ,BTY=∑k=1n−1xrk+1xrk∑k=1n−1xrk+1=γ.
Substituting Eq. (16) into Eq. (14), we get
(17)ν^=β1,β2T=BTB−1BTY.
Quod erat demonstrandum (Q.E.D).Theorem 10.
By assumingB and Y as shown in Theorem 9 and letting the initial condition be x∧r1=xr1=x01. Then, the time response sequence of the FHDGM (1,1) model is given as
(18)x∧rk=β1k−1x01+1−β1k−11−β1β2,β1≠1x01+k−1β2,β1=1,k=2,3,⋯,n.The restored value ofX0 is
(19)x∧0k=β1k−1−β1k−2x01−β2/1−β1kr−k−1r,β1≠1,β2kr−k−1r,β1=1.Proof.
(20)xrk=β1xrk−1+β2=β1β1xrk−2+β2+β2=β12β1xrk−3+β2+β1β2+β2=⋯=β1k−1xr1+β2β1k−1+⋯+β12+β11+β10.
Ifβ1≠1, x∧rk=β1k−1xr1+β21−β1k−1/1−β1=β1k−1x01+β21−β1k−1/1−β1.
Ifβ1=1, x∧rk=β1k−1xr1+k−1β2=x01+k−1β2.
Therefore,(21)x∧0k=β1k−1−β1k−2x01−β2/1−β1kr−k−1r,β1≠1,β2kr−k−1r,β1=1.
Q.E.D.


## 2.1. The Definitions of the Hausdorff Fractional Accumulation and Difference

Before introducing the definitions of the Hausdorff fractional accumulation and difference, the Hausdorff fractional derivative is given as shown in Definition1.Definition 1 (see [33]).
The Hausdorff fractional derivative is defined as(1)dfxdxr=limx0⟶xfx−fx0xr−x0r.
Based on Definition1, the definitions of the Hausdorff fractional accumulation and difference are given in Definition 2 and Definition 3, respectively.Definition 2 (see [34]).
The Hausdorff fractional accumulation (HFA) withr order is
(2)∇rfk=∇tr−t−1rfk=∑j=1kjr−j−1rfk.Definition 3 (see [34]).
The Hausdorff fractional difference (HFD) withr order is
(3)Δrfk=limh⟶1ft−ft−htr−t−hrt=k=fk−fk−1kr−k−1r=Δfkkr−k−1r.


## 2.2. Traditional FHGM (1,1) Model

Definition 4.
AssumeX0=x01,x02,⋯,x0n is a nonnegative sequence; its r-order accumulative sequence is then given by Xr=xr1,xr2,⋯,xrn, so the differential equation of the FHGM (1,1) model is
(4)dxrtdt+axrt=b,where xrk=∑i=1kx0iir−i−1rk=1,2,⋯,n.Definition 5.
The basic form of the FHGM (1,1) model is(5)xrk−xrk−1+azrk=b,where zrk=xrk+xrk−1/2, and k=2,3,⋯,n..Theorem 6.
LetX0,Xr be defined in Definition 1. The least-square estimation of the parameter list μ^=a,bT in Eq. (5) is
(6)μ^=a,bT=BTB−1BTY,where
(7)B=−0.5xr1+xr21−0.5xr2+xr31⋮⋮−0.5xrn−1+xrn1,Y=xr2−xr1xr3−xr2⋮xrn−xrn−1.Theorem 7.
Let the initial value bex∧r1=xr1=x01; then, the time response function of the FHGM (1,1) model is
(8)x∧rk=x01−bae−ak−1+ba,and the restored values are x∧0k=x∧rk−x∧rk−1/kr−k−1r, and k=2,3,⋯,n..Obviously, Eq. (4) is a continuous equation, and Eq. (5) is a discrete equation. The time response function of the model is obtained by directly substituting the parameters obtained from Eq. (5) into Eq. (4). The model’s performance in terms of prediction will be reduced since the prediction of the time response formula derived using this method will result in inaccuracies in the model’s predictive results. To solve the problem, this paper proves the main reasons for the unstable simulation and accuracy of the FHGM (1,1) model through model derivation. In doing so, this paper constructs the FHDGM (1,1) model, which solves the error caused by the FHGM (1,1) model changing from a difference equation to a differential equation.


## 2.3. Fractional Hausdorff Discrete Grey Model

Definition 8.
LetX0 and Xr be the same as Definition 4; the equation
(9)xrk+1=β1xrk+β2,is called the Fractional Hausdorff Discrete Grey Model (FHDGM (1,1) for short).Theorem 9.
Using the least-square method, the parametersβ1,β2 are estimated by
(10)ν^=β1,β2T=BTB−1BTY,where
(11)B=xr1xr2⋮xrn−111⋮1,Y=xr2xr3⋮xrn.Proof.
When we enterk=2,3,⋯,n into Eq. (9), we get Y=Bν^. After that, the error sequence is obtained with ε=Y−Bν^ by changing xrk+1 to β1xrk+β2. Let
(12)S=εTε=Y−Bv∧TY−Bv^=∑k=1n−1xrk+1−β1xrk−β22.
The parametersβ1,β2 that make the S minimum should satisfy
(13)∂S∂β1=−2∑k=1n−1xrk+1−β1xrk−β2xrk=0,∂S∂β2=−2∑k=1n−1xrk+1−β1xrk−β2=0.
Recalculating the equation system, we have(14)γ=δβ1,β2T,
where(15)δ=∑k=1n−1xrk2∑k=1n−1xrk∑k=1n−1xrkn−1,γ=∑k=1n−1xrk+1xrk∑k=1n−1xrk+1
Owing to(16)BTB=∑k=1n−1xrk2∑k=1n−1xrk∑k=1n−1xrkn−1=δ,BTY=∑k=1n−1xrk+1xrk∑k=1n−1xrk+1=γ.
Substituting Eq. (16) into Eq. (14), we get
(17)ν^=β1,β2T=BTB−1BTY.
Quod erat demonstrandum (Q.E.D).Theorem 10.
By assumingB and Y as shown in Theorem 9 and letting the initial condition be x∧r1=xr1=x01. Then, the time response sequence of the FHDGM (1,1) model is given as
(18)x∧rk=β1k−1x01+1−β1k−11−β1β2,β1≠1x01+k−1β2,β1=1,k=2,3,⋯,n.The restored value ofX0 is
(19)x∧0k=β1k−1−β1k−2x01−β2/1−β1kr−k−1r,β1≠1,β2kr−k−1r,β1=1.Proof.
(20)xrk=β1xrk−1+β2=β1β1xrk−2+β2+β2=β12β1xrk−3+β2+β1β2+β2=⋯=β1k−1xr1+β2β1k−1+⋯+β12+β11+β10.
Ifβ1≠1, x∧rk=β1k−1xr1+β21−β1k−1/1−β1=β1k−1x01+β21−β1k−1/1−β1.
Ifβ1=1, x∧rk=β1k−1xr1+k−1β2=x01+k−1β2.
Therefore,(21)x∧0k=β1k−1−β1k−2x01−β2/1−β1kr−k−1r,β1≠1,β2kr−k−1r,β1=1.
Q.E.D.


## 3. Properties of the FHDGM (1,1) Model

Property 11.
AssumeX0=x01,x02,⋯,x0n is the original nonnegative sequence, Y0=y01,y02,⋯,y0n=ρX0=ρx01,ρx02,⋯,ρx0n is the multiple transformation sequence of X0, then, yrk=ρxrk,k=1,2,⋯,n, where ρρ≠0 is a multiplied amount.Proof.
(22)yrk=∑i=1ky0iir−i−1r=∑i=1kρx0iir−i−1r=ρ∑i=1kx0iir−i−1r=ρxrk.
Q.E.D.Property 12.
LetX0 and Y0 be the same as Property 11. Assuming the parameters in FHDGM (1,1) modeled by X0 are β1,β2 and that β1′,β2′ are parameters in FHDGM (1,1) modeled by Y0, then, β1′=β1,β2′=ρβ2.Proof.
Parametersβ1,β2 satisfy β1,β2T=BTB−1BTY, where
(23)B=xr1xr2⋮xrn−111⋮1,Y=xr2xr3⋮xrn.
Parametersβ1′,β2′ satisfy β1′,β2′T=B′TB′−1B′TY′, where
(24)B′=yr1yr2⋮yrn−111⋮1=ρxr1ρxr2⋮ρxrn−111⋮1=xr1xr2⋮xrn−111⋮1×ρ001=BM,(25)Y′=yr2yr3⋮yrn=ρxr2ρxr3⋮ρxrn=ρxr2xr3⋮xrn=ρY.
Therefore,(26)β1′,β2′T=B′TB′−1B′TY′=BMTBM−1BMTρY=MTBTBM−1MTBTYρ=M−1BTB−1MT−1MTBTYρ=ρM−1BTB−1BTY=ρ1ρ001BTB−1BTY=100ρβ1β2=β1ρβ2.
Q.E.D.Property 13.
IfX0, Y0, respectively, formed the FHDGM (1,1) model, and if x∧0k,y∧0k are the restored values of those models, then, y∧0k=ρx∧0k.Proof.
Whenβ1≠1, we have
(27)y∧0k=β1′k−1−β1′k−2y01−β2′/1−β1′kr−k−1r=β1k−1−β1k−2ρx01−ρβ2/1−β1kr−k−1r=ρβ1k−1−β1k−2x01−β2/1−β1kr−k−1r=ρx∧0k.
Whenβ1=1, we have
(28)y∧0k=β2′kr−k−1r=ρβ2kr−k−1r=ρx∧0k.
Q.E.D.Property 14.
Consider thatσXk,σYk are, respectively, the relative errors of the FHDGM (1,1) model built by X0, Y0, then, σXk=σYk.Proof.
(29)σYk=y0k−y∧0ky0k=ρx0k−ρx∧0kρx0k=x0k−x∧0kx0k=σXk.
Q.E.D.
From Properties11 to 12, we see that the original data sequence can affect both the structural parameter β2 and the restored value x∧0k by multiplying it by a certain quantity, but it cannot affect the structural parameter β1 and the relative error σXk. As a result, when modeling with data of a large order of magnitude, we can perform data multiplication transformation on the original data to reduce the order of magnitude of the data for modeling without affecting the model’s ability to predict outcomes accurately and to streamline the modeling procedure.Property 15.
The initial value of the FHDGM (1,1) model is invalid.Proof.
Ifω is defined as the disturbance of the initial value x01, then, x~01=x01+ω, X~0=x01+ω,x02,⋯,x0n. If set λ1,λ2 are the parameters of the FHDGM (1,1) model constructed by X~0, then, λ1,λ2T=B~TB~−1B~TY~, where
(30)B~=xr1+ωxr2+ω⋮xrn−1+ω11⋮1=xr1xr2⋮xrn−111⋮1×10ω1=BP,Y~=xr2+ωxr3+ω⋮xrn−1+ω=xr2xr3⋮xrn−1+ωω⋮ω=Y+ωω⋮ω,(31)λ1,λ2T=B~TB~−1B~TY~=BPTBP−1BPTY~=PTBTBP−1PTBTY~=P−1BTB−1PT−1PTBTY~=P−1BTB−1BTY~=P−1BTB−1BTY+P−1BTB−1BTωω⋮ω=10−ω1β1β2+0ω=β1−ωβ1+β2+ω.
Therefore, ifβ1≠1, then
(32)x~∧0k=λ1k−1−λ1k−2x~01−λ21−λ1×1kr−k−1r=β1k−1−β1k−2x01+ω−−ωβ1+β2+ω1−β1×1kr−k−1r=β1k−1−β1k−2x01−β21−β1×1kr−k−1r=x∧0k.
Ifβ1=1, then
(33)x~∧0k=λ2kr−k−1r=−ωβ1+β2+ωkr−k−1r=x∧0k.
Q.E.D.
The fact that the restored value of the FHDGM (1,1) model is unaffected by the initial value change, as shown by Property15, suggests that the initial value of the FHDGM (1,1) model is invalid.Lemma 16 (see [35]).
SupposeA∈Cm×n, b∈Cm, and A+ are the generalized inverse of matrix A, the column vectors of A are linearly independent, B=A+E and c=b+k∈Cm. Solutions to equations Bx−c2= min and Ax−b2= min are defined as x+h and x, respectively. If rankA=rankB=n, then
(34)h≤κ+γ+E2Ax+kA+κ+γ+E2ArxA.Property 17.
AssumingLx0kk=1,2,⋯,n to be perturbation bound when αkk=1,2,⋯,n is the disturbance of x0k. If αk≠0, B†2ΔB≠0, and B†2ΔB2<1, in which B† represents the generalized inverse of matrix B, then, we have
(35)Lx0k=α1κ+γ+xn−1B+n−1B+κ+γ+n−1BrxB,k=1,kr−k−1rαkκ+γ+xn−kB+n−k+1B+κ+γ+n−kBrxB,k=2,3,⋯,n.Proof.
Whenα1 and Lx01 are defined as, respectively, the disturbance of x01 and its perturbation bound of the solution, then
(36)B^=B+ΔB=xr1xr2⋮xrn−111⋮1+α1α1⋮α100⋮0,Y^=Y+ΔY=xr2xr3⋮xrn+α1α1⋮α1.
Therefore,(37)ΔB2=α12+α12+⋯+α12=α1n−1,ΔY2=α12+α12+⋯+α12=α1n−1.Based on Lemma16, we have
(38)Δx≤α1κ+γ+xn−1B+n−1B+κ+γ+n−1BrxB.That is(39)Lx01=α1κ+γ+xn−1B+n−1B+κ+γ+n−1BrxB.Whenα2 and Lx02 are defined as, respectively, the disturbance of x02 and its perturbation bound of the solution, then
(40)B^=B+ΔB=xr1xr2⋮xrn−111⋮1+02r−1α2⋮2r−1α200⋮0,Y^=Y+ΔY=xr2xr3⋮xrn+2r−1α22r−1α2⋮2r−1α2.Therefore,(41)ΔB2=2r−1α22+2r−1α22+⋯+2r−1α22=2r−1α2n−2,ΔY2=2r−1α22+2r−1α22+⋯+2r−1α22=2r−1α2n−1.Then,(42)Δx≤2r−1α2κ+γ+xn−2B+n−1B+κ+γ+n−2BrxB.Therefore,(43)Lx02=2r−1α2κ+γ+xn−2B+n−1B+κ+γ+n−2BrxB.Whenαkk=3,4,⋯,n and Lx0k are defined as, respectively, the disturbance of x0k and its perturbation bound of the solution, then
(44)Lx0k=αkκ+γ+kr−k−1rxn−kB+n−k+1B+κ+γ+n−kBrxB.Q.E.D.According to Property17, the disturbance bound Lx0k is an increasing function depending on the sample n. As n increases, Lx0k gradually increases, which leads to system instability. In other words, similar to the GM (1, 1) model and the FGM (1,1) model, the newly proposed FHDGM (1,1) model is also suitable for predicting small sample problems.


## 4. Seeking the Optimal Fractional-Order for FHDGM (1,1)

For a new model, it is very significant not only to explore its modeling mechanism but to also seek its optimal fractional-order. Referring to previous papers in the literature, searching for the optimal fractional-order is commonly carried out by constructing a simple constrained optimization problem; its formula is(45)argminfr=1n−1∑k=2nx∧0k−x0kx0k×100%,(46)s.t.β1β2T=BTB−1BTY,B=xr1xr2⋮xrn−111⋮1,Y=xr2xr2⋮xrn,x∧rk=β1k−1x01+1−β1k−11−β1β2,β1≠1,x01+k−1β2,β1=1,x∧0k=x∧rk−x∧rk−1kr−k−1r.It should be pointed out that since Eq. (46) has nonlinear characteristics, it is difficult to solve directly by ordinary methods. PSO is used to determine the optimal fractional-order r. The concrete flowchart of PSO to seek the optimal fractional-order r of the FHDGM (1,1) model is shown in Figure 1. The specific values of PSO are listed in Table 1.Figure 1
The flowchart of the FHDGM (1,1) model.Table 1
Configuration of PSO in this paper.
c1, c22w0.6MaxDT1000Pmin0Pmax1N100


## 5. Validation of the FHDGM (1,1) Model

### 5.1. Model Evaluation Metrics

Several judgment methods are used to evaluate the suitability of the novel model; the absolute percentage error (APE) and the mean absolute percentage error (MAPE) are chosen.By defining as Eq. (47), APE is used to illustrate the absolute percentage error between the real value and the predicted value. The smaller the APE values, the better the predicted accuracy.
(47)APEk=x∧0k−x0kx0k×100%,k=2,3,⋯.MAPE is another widely used metric to measure the overall difference between the real value and the predicted value. By defining Eq. (48), the smaller the MAPE values, the better the predicted accuracy.
(48)MAPE=1n−1∑k=2nx∧0k−x0kx0k×100%.


### 5.2. Case 1: Prediction of Coal Consumption in the United States

Identical information can be found in this example from the literature [36]. Different grey models are constructed using the data from 2010 to 2015 (in-sample data), and these are used to forecast the values from 2016 to 2018 (out-of-sample). Four models were examined, with the actual and forecast values shown in Table 2.Table 2
Fitted and predicted values in coal consumption of the United States.
YearDataDGM (1,1)FHDGM (1,1)GM (1,1,β)FHGM (1,1)ValueAPE (%)ValueAPE (%)ValueAPE (%)ValueAPE (%)2010498.8498.80.000498.80.000498.80.000498.80.0002011470.6461.61.912453.43.651452.03.948452.23.9102012416.0442.26.298450.78.350433.44.172450.68.3272013431.8423.51.922431.80.008415.53.784431.90.0312014430.9405.65.871406.45.683398.37.564406.55.6712015372.2388.54.379378.61.729381.92.595378.41.661MAPE (2010-2015) (%)4.0773.8844.4133.9202016340.6372.29.278350.42.868366.17.484349.72.6832017331.3356.57.606322.62.611351.05.938321.62.9192018317.0341.47.697296.16.608336.56.145294.77.047MAPE (2016-2018) (%)8.1944.0296.5224.216MAPE (2010-2018) (%)5.6213.9385.2044.031Table2 shows that FHDGM (1,1) produced the lowest MAPE in-sample data and out-of-sample, which are 3.884% and 4.029%, respectively. The MAPE of the total data from 2010 to 2018 is 3.938%, which is also the smallest MAPE of the four models. This indicates that the FHDGM (1,1) model is better at predicting coal consumption in the United States.


### 5.3. Case 2: Prediction of the Natural Gas Consumption in North America

We look at a case in the literature [37] that offers sample data. Different grey models are constructed using the data from 2010 to 2015 (in-sample data), and the values from 2016 to 2018 (out-of-sample) were forecast. The actual and forecast values of the four models were then compared.From Table3, FHDGM (1,1) has the lowest MAPE in-sample data, out-of-sample, and total data, 0.220%, 2.732%, and 0.974%, respectively. This indicates that FHDGM (1,1) is better at predicting natural gas consumption in North America.Table 3
Fitted and predicted values in natural gas consumption of North America.
YearDataDGM (1,1)FHDGM (1,1)CFNHGBMFHGM (1,1)ValueAPE (%)ValueAPE (%)ValueAPE (%)ValueAPE (%)2008778.2778.20.000778.20.000778.20.000778.20.0002009769.4775.70.819768.40.127770.20.100770.10.0962010802.5801.00.187802.20.041794.21.029801.80.0872011826.6827.10.060831.30.565830.40.459830.20.4322012859.0854.20.559858.40.067862.60.414857.30.1982013889.1882.10.787884.60.502889.20.015884.00.5792014910.7910.90.022910.40.030910.80.009910.50.0212015934.1940.60.696936.10.210927.90.666937.20.331MAPE (2008-2015) (%)0.4470.2200.3850.2492016938.0971.33.550961.72.529941.10.335964.12.7852017935.51003.17.226980.54.814951.11.669991.45.97820181022.31035.81.3211013.60.852958.26.2681019.10.308MAPE (2016-2018) (%)4.0322.7322.7573.024MAPE (2008-2018) (%)1.5230.9741.0961.081


### 5.4. Case 3: Prediction of Natural Gas Consumption in Thailand

We look at another case in the literature [26] that offers sample data. Different grey models are constructed using the data from 2008 to 2017 (in-sample data), and the value of 2018 (out-of-sample) was forecast. Table 4 displays the actual and forecast values of the four models that were compared.Table 4
Fitted and predicted values in natural gas consumption of Thailand.
YearDataDGM (1,1)FHDGM (1,1)Model 1 [26]FHGM (1,1)ValueAPE (%)ValueAPE (%)ValueAPE (%)ValueAPE (%)200831.8031.800.00031.800.00031.800.00031.800.000%200932.8036.0910.03032.760.10933.151.06733.040.729%201037.2037.140.16136.850.94136.152.82336.831.006%201138.1038.220.31539.343.26738.932.17839.182.843%201241.8039.335.90940.991.93441.121.62740.792.421%201342.0040.483.61942.100.23142.471.11941.920.201%201442.9041.652.91442.820.17942.990.21042.710.442%201543.8042.872.12343.271.21243.001.82643.261.237%201643.5044.111.40243.500.00043.001.14943.620.266%201743.1045.405.33643.561.07043.480.88243.821.676%MAPE (2008-2017) (%)3.5350.9941.4311.202201842.9046.728.90443.491.36744.794.40643.912.349%MAPE (2008-2018) (%)4.0721.0311.7291.317From Table4, we see that FHDGM (1,1) has the lowest MAPE in-sample data, out-of-sample, and total data, 0.994%, 1.367%, and 1.031%, respectively. This indicates that FHDGM (1,1) is better at predicting natural gas consumption in Thailand.


## 5.1. Model Evaluation Metrics

Several judgment methods are used to evaluate the suitability of the novel model; the absolute percentage error (APE) and the mean absolute percentage error (MAPE) are chosen.By defining as Eq. (47), APE is used to illustrate the absolute percentage error between the real value and the predicted value. The smaller the APE values, the better the predicted accuracy.
(47)APEk=x∧0k−x0kx0k×100%,k=2,3,⋯.MAPE is another widely used metric to measure the overall difference between the real value and the predicted value. By defining Eq. (48), the smaller the MAPE values, the better the predicted accuracy.
(48)MAPE=1n−1∑k=2nx∧0k−x0kx0k×100%.


## 5.2. Case 1: Prediction of Coal Consumption in the United States

Identical information can be found in this example from the literature [36]. Different grey models are constructed using the data from 2010 to 2015 (in-sample data), and these are used to forecast the values from 2016 to 2018 (out-of-sample). Four models were examined, with the actual and forecast values shown in Table 2.Table 2
Fitted and predicted values in coal consumption of the United States.
YearDataDGM (1,1)FHDGM (1,1)GM (1,1,β)FHGM (1,1)ValueAPE (%)ValueAPE (%)ValueAPE (%)ValueAPE (%)2010498.8498.80.000498.80.000498.80.000498.80.0002011470.6461.61.912453.43.651452.03.948452.23.9102012416.0442.26.298450.78.350433.44.172450.68.3272013431.8423.51.922431.80.008415.53.784431.90.0312014430.9405.65.871406.45.683398.37.564406.55.6712015372.2388.54.379378.61.729381.92.595378.41.661MAPE (2010-2015) (%)4.0773.8844.4133.9202016340.6372.29.278350.42.868366.17.484349.72.6832017331.3356.57.606322.62.611351.05.938321.62.9192018317.0341.47.697296.16.608336.56.145294.77.047MAPE (2016-2018) (%)8.1944.0296.5224.216MAPE (2010-2018) (%)5.6213.9385.2044.031Table2 shows that FHDGM (1,1) produced the lowest MAPE in-sample data and out-of-sample, which are 3.884% and 4.029%, respectively. The MAPE of the total data from 2010 to 2018 is 3.938%, which is also the smallest MAPE of the four models. This indicates that the FHDGM (1,1) model is better at predicting coal consumption in the United States.


## 5.3. Case 2: Prediction of the Natural Gas Consumption in North America

We look at a case in the literature [37] that offers sample data. Different grey models are constructed using the data from 2010 to 2015 (in-sample data), and the values from 2016 to 2018 (out-of-sample) were forecast. The actual and forecast values of the four models were then compared.From Table3, FHDGM (1,1) has the lowest MAPE in-sample data, out-of-sample, and total data, 0.220%, 2.732%, and 0.974%, respectively. This indicates that FHDGM (1,1) is better at predicting natural gas consumption in North America.Table 3
Fitted and predicted values in natural gas consumption of North America.
YearDataDGM (1,1)FHDGM (1,1)CFNHGBMFHGM (1,1)ValueAPE (%)ValueAPE (%)ValueAPE (%)ValueAPE (%)2008778.2778.20.000778.20.000778.20.000778.20.0002009769.4775.70.819768.40.127770.20.100770.10.0962010802.5801.00.187802.20.041794.21.029801.80.0872011826.6827.10.060831.30.565830.40.459830.20.4322012859.0854.20.559858.40.067862.60.414857.30.1982013889.1882.10.787884.60.502889.20.015884.00.5792014910.7910.90.022910.40.030910.80.009910.50.0212015934.1940.60.696936.10.210927.90.666937.20.331MAPE (2008-2015) (%)0.4470.2200.3850.2492016938.0971.33.550961.72.529941.10.335964.12.7852017935.51003.17.226980.54.814951.11.669991.45.97820181022.31035.81.3211013.60.852958.26.2681019.10.308MAPE (2016-2018) (%)4.0322.7322.7573.024MAPE (2008-2018) (%)1.5230.9741.0961.081


## 5.4. Case 3: Prediction of Natural Gas Consumption in Thailand

We look at another case in the literature [26] that offers sample data. Different grey models are constructed using the data from 2008 to 2017 (in-sample data), and the value of 2018 (out-of-sample) was forecast. Table 4 displays the actual and forecast values of the four models that were compared.Table 4
Fitted and predicted values in natural gas consumption of Thailand.
YearDataDGM (1,1)FHDGM (1,1)Model 1 [26]FHGM (1,1)ValueAPE (%)ValueAPE (%)ValueAPE (%)ValueAPE (%)200831.8031.800.00031.800.00031.800.00031.800.000%200932.8036.0910.03032.760.10933.151.06733.040.729%201037.2037.140.16136.850.94136.152.82336.831.006%201138.1038.220.31539.343.26738.932.17839.182.843%201241.8039.335.90940.991.93441.121.62740.792.421%201342.0040.483.61942.100.23142.471.11941.920.201%201442.9041.652.91442.820.17942.990.21042.710.442%201543.8042.872.12343.271.21243.001.82643.261.237%201643.5044.111.40243.500.00043.001.14943.620.266%201743.1045.405.33643.561.07043.480.88243.821.676%MAPE (2008-2017) (%)3.5350.9941.4311.202201842.9046.728.90443.491.36744.794.40643.912.349%MAPE (2008-2018) (%)4.0721.0311.7291.317From Table4, we see that FHDGM (1,1) has the lowest MAPE in-sample data, out-of-sample, and total data, 0.994%, 1.367%, and 1.031%, respectively. This indicates that FHDGM (1,1) is better at predicting natural gas consumption in Thailand.


## 6. Application of the FHDGM (1,1) Model

In this section, the new model that we propose is utilized to forecast renewable energy consumption in the Asia Pacific region, in all of Europe and globally to evaluate its applicability to other competing models such as the FDGM (1,1) model [30], the CFGM (1,1) model [21], and the DGM (1,1) model [29]. The data came from the BP Statistical Review of World Energy 2020, which can be downloaded at https://www.bp.com/en/global/corporate/energy-economics/statistical-review-of-world-energy/renewable-energy.html. Different grey models are constructed using the data from 2008 to 2017, with the accuracy of the models tested using data from 2018 to 2020. Based on this, the structural chart of forecasting renewable energy consumption can be drawn in Figure 2.Figure 2
The structural chart for forecasting renewable energy consumption.


### 6.1. Renewable Energy Consumption of the Asia Pacific Region

In order to fit and forecast the consumption of renewable energy in the Asia Pacific region, the suggested FHDGM (1,1) model is compared with three conventional grey models, DGM (1,1), FDGM (1,1), and CFGM (1,1). The whole data set is split into two sections: data from 2008 to 2017 (in-sample periods), which are used to evaluate grey model accuracy at fitting data, and data from 2018 to 2020 (out-of-sample periods), which are used to assess their accuracy at forecasting data. Second, PSO determines the parameters for FHDGM (1,1), FDGM (1,1), and CFGM (1,1). The minimum MAPE and corresponding parameters are listed in Table5, with both the track searching for the minimum MAPE, as well as the optimal parameters, graphed in Figure 3.Table 5
Optimal parameters of the three models showing renewable energy consumption in the Asia Pacific region.
FHDGM (1,1)FDGM (1,1)CFGM (1,1)r0.9363r0.9235α0.9230MAPE (%)1.68MAPE (%)1.78MAPE (%)1.87Figure 3
Track of searching for the minimum MAPE and the optimal parameter.
(a)(b)(c)Based on Table5, the structural parameters and the expression of the four grey models can be calculated. These are listed in Table 6.Table 6
The expression of four grey models.
ModelsExpressionsFHDGM (1,1)xrk+1=1.1988xrk+1.1636FDGM (1,1)xrk+1=1.2072xrk+1.1897CFGM (1,1)xαk−xαk−1−0.1799zαk=1.1128DGM (1,1)x1k+1=1.2126x1k+1.3331Based on Table6, the prediction results of the models can be obtained with results and error metrics listed in Tables 7 and 8 and Figures 4 and 5.Table 7
Fitted and predicted values of the four models relating to renewable energy consumption in the Asia Pacific region.
YearDataFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sample20081.271.271.271.271.2720091.551.551.551.551.6020101.951.921.921.911.9420112.402.352.352.342.3620122.882.872.862.852.8620133.553.483.473.463.4720144.284.224.214.204.2020154.925.105.105.085.1020166.066.176.176.136.1820177.637.457.467.407.49Out-of-sample20189.388.999.028.929.09201910.8710.8310.8910.7511.02202012.3613.0613.1612.9513.36Table 8
The values of APE and MAPE are generated by the above four models.
YearFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sampleAPE (%)APE (%)APE (%)APE (%)20080.000.000.000.0020090.000.000.003.4320101.471.681.830.3120111.932.192.381.7820120.430.710.930.7420131.912.172.432.3620141.421.631.961.7920153.753.623.153.6020161.791.791.161.9920172.392.243.031.77MAPE (%)1.681.781.871.97Out-of-sample20184.213.884.883.1120190.320.221.081.3920205.656.484.788.12MAPE (%)3.393.534.213.58Figure 4
APE of renewable energy consumption prediction in Asia Pacific region.Figure 5
Fitted and predicted values of renewable energy consumption in the Asia Pacific region.From Table8 and Figure 5, the four grey model MAPE values in the in-sample period are 1.68%, 1.78%, 1.87%, and 1.97%, respectively, while the competing MAPE values in the out-of-sample period are 3.39%, 3.53%, 4.21%, and 3.58%. From a comparison of these MAPE values, it is apparent that FHDGM (1,1) has the smallest MAPE value in fitting and predicting the renewable energy consumption of the Asia Pacific region. Therefore, the new model performs well with respect to fitting and forecasting the Asia Pacific region’s renewable energy consumption.


### 6.2. Renewable Energy Consumption for All of Europe

The total of European renewable energy data, as it pertains to consumption, is broken down into two categories: in-sample period and out-of-sample period. The raw data from 2008 to 2017 is utilized to test how well the FHDGM (1,1) model fits; out-of-sample period: the raw data from 2018 to 2020 is also applied to verify the prediction accuracy of the FHDGM (1,1) model. A comparison is made between the model and the three other models, FHDGM (1,1), FDGM (1,1), and CFGM (1,1). Similarly, the parameters of FHDGM (1,1), FDGM (1,1), and CFGM (1,1) are decided by PSO, with the minimum MAPE and corresponding parameters listed in Table9. The tracks of both searching for the minimum MAPE and the optimal parameters are given in Figure 6.Table 9
Optimal parameters of the three models with respect to renewable energy consumption in all of Europe.
FHDGM (1,1)FDGM (1,1)CFGM (1,1)r0.7230r0.3660α0.6288MAPE (%)1.90MAPE (%)1.74MAPE (%)1.79Figure 6
Track related to searching for the minimum MAPE and the optimal parameter.
(a)(b)(c)According to Table9, the structural parameters and the expression of the four grey models can be calculated, as shown in Table 10.Table 10
The expression of four grey models.
ModelsExpressionsFHDGM (1,1)xrk+1=1.0450xrk+1.8294FDGM (1,1)xrk+1=1.0100xrk+1.3216CFGM (1,1)xαk−xαk−1−0.0338zαk=2.1892DGM (1,1)x1k+1=1.1016x1k+3.1212We obtain the predictive results of these models based on Table10, with results and error metrics listed in Tables 11 and 12 and Figures 7 and 8.Table 11
Fitted and predicted values of the four models related to renewable energy consumption in all of Europe.
YearDataFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sample20082.652.652.652.652.6520093.013.003.033.003.3920103.483.623.593.603.7320114.114.164.144.154.1120124.794.664.674.664.5320135.265.155.195.164.9920145.645.645.685.655.5020156.326.136.176.146.0620166.416.646.646.646.6720177.147.157.107.147.35Out-of-sample20187.527.687.557.658.1020198.248.248.008.178.9220208.948.818.448.719.83Table 12
Values of APE and MAPE are generated by the four models.
YearFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sampleAPE (%)APE (%)APE (%)APE (%)20080.000.000.000.0020090.490.610.4012.6420104.073.133.587.3220111.210.780.940.1020122.662.412.685.3920132.051.381.915.0920140.000.770.202.5020152.962.442.804.1520163.523.533.564.1020170.160.590.002.95MAPE (%)1.901.741.794.92Out-of-sample20182.180.411.757.6720190.062.940.808.2520201.495.612.579.90MAPE (%)1.242.991.708.61Figure 7
Fitted and predicted values of renewable energy consumption in all of Europe.Figure 8
APE of prediction of total European renewable energy consumption.Table12 and Figure 8 show that the MAPE values of the four grey models are 1.90%, 1.74%, 1.79%, and 4.92% in describing total European renewable energy consumption; the MAPE values of the competitors are 1.24%, 2.99%, 1.70%, and 8.61%, respectively, for predicting renewable energy consumption in Europe. It can be seen that, although the MAPE of the FHDGM (1,1) model is not the smallest in the in-sample period, it is very close to the minimum MAPE 1.74%. The MAPE of the FHDGM (1,1) model is the smallest in the out-of-sample period, which is lower than that of the other three grey models. In general, the FHDGM (1,1) model is better able to predict when compared to the other three grey models.


### 6.3. Renewable Energy Consumption of the World

Four prediction models are developed using data on global renewable energy consumption from 2008 to 2017; the performance of the models is tested using data from 2018 to 2020. Similar comparisons are made between the model and the FHDGM (1,1), the FDGM (1,1), and the CFGM (1,1). PSO then determines the parameters of FHDGM (1,1), FDGM (1,1), and CFGM (1,1). The minimum MAPE and corresponding parameters are listed in Table13; the track of searching for the minimum MAPE and the optimal parameters are shown in Figure 9.Table 13
Optimal parameters of the three models in world renewable energy consumption.
FHDGM (1,1)FDGM (1,1)CFGM (1,1)r0.8831r0.8504α0.8444MAPE (%)0.46MAPE (%)0.51MAPE (%)0.52Figure 9
Track of searching for the minimum MAPE and the optimal parameter.
(a)(b)(c)According to Table13, the structural parameters and the expression of the five grey models can be calculated, as shown in Table 14.Table 14
The expression of four grey models.
ModelsExpressionsFHDGM (1,1)xrk+1=1.1066xrk+6.1316FDGM (1,1)xrk+1=1.1141xrk+6.2853CFGM (1,1)xαk−xαk−1−0.0970zαk=6.2862DGM (1,1)x1k+1=1.1315x1k+7.6652We obtain the predictive results of these models based on Table14, with results and error metrics listed in Tables 15 and 16 and Figures 10 and 11.Table 15
Fitted and predicted values of four models related to global renewable energy consumption.
YearDataFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sample20087.117.117.117.117.1120098.168.168.168.168.6020109.639.609.589.589.73201111.0811.0611.0411.0411.01201212.6012.6012.5912.5912.46201314.3814.2814.2714.2714.10201416.0416.1116.1116.1115.95201518.1018.1318.1418.1218.05201620.1120.3620.3820.3420.42201723.0622.8322.8522.7823.10Out-of-sample201825.8825.5625.6025.4826.14201928.8228.5928.6428.4529.58202031.7131.9532.0531.7533.46Table 16
The values of APE and MAPE are generated by the four models.
YearFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sampleAPE (%)APE (%)APE (%)APE (%)20080.000.000.000.0020090.000.000.005.3920100.300.530.551.0420110.210.380.400.6320120.020.070.081.1320130.700.740.751.9820140.460.460.420.5720150.180.220.120.3020161.261.321.131.5320171.000.901.220.18MAPE (%)0.460.510.521.42Out-of-sample20181.231.091.561.0020190.800.631.272.6220200.761.090.115.53MAPE (%)0.930.940.983.05Figure 10
Fitted and predicted values of world renewable energy consumption.Figure 11
APE of renewable energy consumption prediction in the world.From Table16 and Figure 11, we see that the MAPE values of the four grey models are 0.46%, 0.51%, 0.52%, and 1.42% in the “in-sample” period. The competitor MAPEs are 0.93%, 0.94%, 0.98%, and 3.05% in the “out-of-sample” period. The MAPE value of FHDGM (1,1) is the lowest among these grey models, which also verifies that the predictive ability of the FHDGM (1,1) model is higher than that of the other models. Moreover, the maximum APE of the FHDGM (1,1) model is 1.26%, which is less than the 1.32% of the FDGM model, the 1.56% of the CFGM (1,1) model, and the 5.53% of the DGM model. That is to say, the proposed new model, FHDGM (1,1), outperforms all the other competitors because of its lowest fitting and the predictive ability of the MAPE values.


### 6.4. Forecasting Renewable Energy Consumption from 2021 to 2023

Since the novel FHDGM (1,1) model proposed in this paper performs well in predicting renewable energy consumption, we employ the FHDGM (1,1) model to predict renewable energy consumption from 2021 to 2023. Considering the principle of new information priority, we use the data on renewable energy consumption from 2008 to 2020 to establish the FHDGM (1,1) model and make a three-step prediction to obtain the prediction data from 2021 to 2023. The values of the optimal parameters are shown in Table17, and the prediction results are shown in Table 18 and Figure 12.Table 17
Optimal parameter values of renewable energy consumption.
Asia Pacific regionTotal EuropeThe worldr0.87260.73670.8808Table 18
Predictions for the renewable energy consumption over the next 3 years.
Asia Pacific regionTotal EuropeThe world202115.299.4535.72202218.2410.0839.86202321.7510.7544.45Figure 12
Predictions for the renewable energy consumption over the next 3 years.
(a)(b)(c)As can be seen from the above analysis, the Asia Pacific, European, and global consumption of renewable energy will continue to grow over the next three years, with annual increase rates shown in Table19. Apparently, the annual growth rate of renewable energy consumption in the Asia Pacific region and the world in the next three years will exceed 10%, while the annual growth rate in Europe will exceed 5%. This demonstrates that, as global energy growth moves forward, the coal-based energy system will continue to change and renewable energy sources will take on a more significant role. This is because both the government and the energy industry have pointed towards boosting the use of clean and renewable energy to decrease carbon dioxide emissions, lessen the burden on the environment, and hasten the transition to a low-carbon and green economy.Table 19
The annual increase rates of renewable energy consumption (%).
Asia Pacific regionTotal EuropeThe world202119.396.9111.65202219.316.7611.58202319.236.6211.51Mean value19.316.7611.58


## 6.1. Renewable Energy Consumption of the Asia Pacific Region

In order to fit and forecast the consumption of renewable energy in the Asia Pacific region, the suggested FHDGM (1,1) model is compared with three conventional grey models, DGM (1,1), FDGM (1,1), and CFGM (1,1). The whole data set is split into two sections: data from 2008 to 2017 (in-sample periods), which are used to evaluate grey model accuracy at fitting data, and data from 2018 to 2020 (out-of-sample periods), which are used to assess their accuracy at forecasting data. Second, PSO determines the parameters for FHDGM (1,1), FDGM (1,1), and CFGM (1,1). The minimum MAPE and corresponding parameters are listed in Table5, with both the track searching for the minimum MAPE, as well as the optimal parameters, graphed in Figure 3.Table 5
Optimal parameters of the three models showing renewable energy consumption in the Asia Pacific region.
FHDGM (1,1)FDGM (1,1)CFGM (1,1)r0.9363r0.9235α0.9230MAPE (%)1.68MAPE (%)1.78MAPE (%)1.87Figure 3
Track of searching for the minimum MAPE and the optimal parameter.
(a)(b)(c)Based on Table5, the structural parameters and the expression of the four grey models can be calculated. These are listed in Table 6.Table 6
The expression of four grey models.
ModelsExpressionsFHDGM (1,1)xrk+1=1.1988xrk+1.1636FDGM (1,1)xrk+1=1.2072xrk+1.1897CFGM (1,1)xαk−xαk−1−0.1799zαk=1.1128DGM (1,1)x1k+1=1.2126x1k+1.3331Based on Table6, the prediction results of the models can be obtained with results and error metrics listed in Tables 7 and 8 and Figures 4 and 5.Table 7
Fitted and predicted values of the four models relating to renewable energy consumption in the Asia Pacific region.
YearDataFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sample20081.271.271.271.271.2720091.551.551.551.551.6020101.951.921.921.911.9420112.402.352.352.342.3620122.882.872.862.852.8620133.553.483.473.463.4720144.284.224.214.204.2020154.925.105.105.085.1020166.066.176.176.136.1820177.637.457.467.407.49Out-of-sample20189.388.999.028.929.09201910.8710.8310.8910.7511.02202012.3613.0613.1612.9513.36Table 8
The values of APE and MAPE are generated by the above four models.
YearFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sampleAPE (%)APE (%)APE (%)APE (%)20080.000.000.000.0020090.000.000.003.4320101.471.681.830.3120111.932.192.381.7820120.430.710.930.7420131.912.172.432.3620141.421.631.961.7920153.753.623.153.6020161.791.791.161.9920172.392.243.031.77MAPE (%)1.681.781.871.97Out-of-sample20184.213.884.883.1120190.320.221.081.3920205.656.484.788.12MAPE (%)3.393.534.213.58Figure 4
APE of renewable energy consumption prediction in Asia Pacific region.Figure 5
Fitted and predicted values of renewable energy consumption in the Asia Pacific region.From Table8 and Figure 5, the four grey model MAPE values in the in-sample period are 1.68%, 1.78%, 1.87%, and 1.97%, respectively, while the competing MAPE values in the out-of-sample period are 3.39%, 3.53%, 4.21%, and 3.58%. From a comparison of these MAPE values, it is apparent that FHDGM (1,1) has the smallest MAPE value in fitting and predicting the renewable energy consumption of the Asia Pacific region. Therefore, the new model performs well with respect to fitting and forecasting the Asia Pacific region’s renewable energy consumption.


## 6.2. Renewable Energy Consumption for All of Europe

The total of European renewable energy data, as it pertains to consumption, is broken down into two categories: in-sample period and out-of-sample period. The raw data from 2008 to 2017 is utilized to test how well the FHDGM (1,1) model fits; out-of-sample period: the raw data from 2018 to 2020 is also applied to verify the prediction accuracy of the FHDGM (1,1) model. A comparison is made between the model and the three other models, FHDGM (1,1), FDGM (1,1), and CFGM (1,1). Similarly, the parameters of FHDGM (1,1), FDGM (1,1), and CFGM (1,1) are decided by PSO, with the minimum MAPE and corresponding parameters listed in Table9. The tracks of both searching for the minimum MAPE and the optimal parameters are given in Figure 6.Table 9
Optimal parameters of the three models with respect to renewable energy consumption in all of Europe.
FHDGM (1,1)FDGM (1,1)CFGM (1,1)r0.7230r0.3660α0.6288MAPE (%)1.90MAPE (%)1.74MAPE (%)1.79Figure 6
Track related to searching for the minimum MAPE and the optimal parameter.
(a)(b)(c)According to Table9, the structural parameters and the expression of the four grey models can be calculated, as shown in Table 10.Table 10
The expression of four grey models.
ModelsExpressionsFHDGM (1,1)xrk+1=1.0450xrk+1.8294FDGM (1,1)xrk+1=1.0100xrk+1.3216CFGM (1,1)xαk−xαk−1−0.0338zαk=2.1892DGM (1,1)x1k+1=1.1016x1k+3.1212We obtain the predictive results of these models based on Table10, with results and error metrics listed in Tables 11 and 12 and Figures 7 and 8.Table 11
Fitted and predicted values of the four models related to renewable energy consumption in all of Europe.
YearDataFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sample20082.652.652.652.652.6520093.013.003.033.003.3920103.483.623.593.603.7320114.114.164.144.154.1120124.794.664.674.664.5320135.265.155.195.164.9920145.645.645.685.655.5020156.326.136.176.146.0620166.416.646.646.646.6720177.147.157.107.147.35Out-of-sample20187.527.687.557.658.1020198.248.248.008.178.9220208.948.818.448.719.83Table 12
Values of APE and MAPE are generated by the four models.
YearFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sampleAPE (%)APE (%)APE (%)APE (%)20080.000.000.000.0020090.490.610.4012.6420104.073.133.587.3220111.210.780.940.1020122.662.412.685.3920132.051.381.915.0920140.000.770.202.5020152.962.442.804.1520163.523.533.564.1020170.160.590.002.95MAPE (%)1.901.741.794.92Out-of-sample20182.180.411.757.6720190.062.940.808.2520201.495.612.579.90MAPE (%)1.242.991.708.61Figure 7
Fitted and predicted values of renewable energy consumption in all of Europe.Figure 8
APE of prediction of total European renewable energy consumption.Table12 and Figure 8 show that the MAPE values of the four grey models are 1.90%, 1.74%, 1.79%, and 4.92% in describing total European renewable energy consumption; the MAPE values of the competitors are 1.24%, 2.99%, 1.70%, and 8.61%, respectively, for predicting renewable energy consumption in Europe. It can be seen that, although the MAPE of the FHDGM (1,1) model is not the smallest in the in-sample period, it is very close to the minimum MAPE 1.74%. The MAPE of the FHDGM (1,1) model is the smallest in the out-of-sample period, which is lower than that of the other three grey models. In general, the FHDGM (1,1) model is better able to predict when compared to the other three grey models.


## 6.3. Renewable Energy Consumption of the World

Four prediction models are developed using data on global renewable energy consumption from 2008 to 2017; the performance of the models is tested using data from 2018 to 2020. Similar comparisons are made between the model and the FHDGM (1,1), the FDGM (1,1), and the CFGM (1,1). PSO then determines the parameters of FHDGM (1,1), FDGM (1,1), and CFGM (1,1). The minimum MAPE and corresponding parameters are listed in Table13; the track of searching for the minimum MAPE and the optimal parameters are shown in Figure 9.Table 13
Optimal parameters of the three models in world renewable energy consumption.
FHDGM (1,1)FDGM (1,1)CFGM (1,1)r0.8831r0.8504α0.8444MAPE (%)0.46MAPE (%)0.51MAPE (%)0.52Figure 9
Track of searching for the minimum MAPE and the optimal parameter.
(a)(b)(c)According to Table13, the structural parameters and the expression of the five grey models can be calculated, as shown in Table 14.Table 14
The expression of four grey models.
ModelsExpressionsFHDGM (1,1)xrk+1=1.1066xrk+6.1316FDGM (1,1)xrk+1=1.1141xrk+6.2853CFGM (1,1)xαk−xαk−1−0.0970zαk=6.2862DGM (1,1)x1k+1=1.1315x1k+7.6652We obtain the predictive results of these models based on Table14, with results and error metrics listed in Tables 15 and 16 and Figures 10 and 11.Table 15
Fitted and predicted values of four models related to global renewable energy consumption.
YearDataFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sample20087.117.117.117.117.1120098.168.168.168.168.6020109.639.609.589.589.73201111.0811.0611.0411.0411.01201212.6012.6012.5912.5912.46201314.3814.2814.2714.2714.10201416.0416.1116.1116.1115.95201518.1018.1318.1418.1218.05201620.1120.3620.3820.3420.42201723.0622.8322.8522.7823.10Out-of-sample201825.8825.5625.6025.4826.14201928.8228.5928.6428.4529.58202031.7131.9532.0531.7533.46Table 16
The values of APE and MAPE are generated by the four models.
YearFHDGM (1,1)FDGM (1,1)CFGM (1,1)DGM (1,1)In-sampleAPE (%)APE (%)APE (%)APE (%)20080.000.000.000.0020090.000.000.005.3920100.300.530.551.0420110.210.380.400.6320120.020.070.081.1320130.700.740.751.9820140.460.460.420.5720150.180.220.120.3020161.261.321.131.5320171.000.901.220.18MAPE (%)0.460.510.521.42Out-of-sample20181.231.091.561.0020190.800.631.272.6220200.761.090.115.53MAPE (%)0.930.940.983.05Figure 10
Fitted and predicted values of world renewable energy consumption.Figure 11
APE of renewable energy consumption prediction in the world.From Table16 and Figure 11, we see that the MAPE values of the four grey models are 0.46%, 0.51%, 0.52%, and 1.42% in the “in-sample” period. The competitor MAPEs are 0.93%, 0.94%, 0.98%, and 3.05% in the “out-of-sample” period. The MAPE value of FHDGM (1,1) is the lowest among these grey models, which also verifies that the predictive ability of the FHDGM (1,1) model is higher than that of the other models. Moreover, the maximum APE of the FHDGM (1,1) model is 1.26%, which is less than the 1.32% of the FDGM model, the 1.56% of the CFGM (1,1) model, and the 5.53% of the DGM model. That is to say, the proposed new model, FHDGM (1,1), outperforms all the other competitors because of its lowest fitting and the predictive ability of the MAPE values.


## 6.4. Forecasting Renewable Energy Consumption from 2021 to 2023

Since the novel FHDGM (1,1) model proposed in this paper performs well in predicting renewable energy consumption, we employ the FHDGM (1,1) model to predict renewable energy consumption from 2021 to 2023. Considering the principle of new information priority, we use the data on renewable energy consumption from 2008 to 2020 to establish the FHDGM (1,1) model and make a three-step prediction to obtain the prediction data from 2021 to 2023. The values of the optimal parameters are shown in Table17, and the prediction results are shown in Table 18 and Figure 12.Table 17
Optimal parameter values of renewable energy consumption.
Asia Pacific regionTotal EuropeThe worldr0.87260.73670.8808Table 18
Predictions for the renewable energy consumption over the next 3 years.
Asia Pacific regionTotal EuropeThe world202115.299.4535.72202218.2410.0839.86202321.7510.7544.45Figure 12
Predictions for the renewable energy consumption over the next 3 years.
(a)(b)(c)As can be seen from the above analysis, the Asia Pacific, European, and global consumption of renewable energy will continue to grow over the next three years, with annual increase rates shown in Table19. Apparently, the annual growth rate of renewable energy consumption in the Asia Pacific region and the world in the next three years will exceed 10%, while the annual growth rate in Europe will exceed 5%. This demonstrates that, as global energy growth moves forward, the coal-based energy system will continue to change and renewable energy sources will take on a more significant role. This is because both the government and the energy industry have pointed towards boosting the use of clean and renewable energy to decrease carbon dioxide emissions, lessen the burden on the environment, and hasten the transition to a low-carbon and green economy.Table 19
The annual increase rates of renewable energy consumption (%).
Asia Pacific regionTotal EuropeThe world202119.396.9111.65202219.316.7611.58202319.236.6211.51Mean value19.316.7611.58


## 7. Conclusion

Accurate estimates of renewable energy consumption can be used to inform stakeholders and decision-makers about the worldwide renewable energy supply. This has significant practical implications and will assist in adjusting and enhancing the energy structure. Based on this, Hausdorff fractional-order accumulation is introduced into the discrete model, and a new model FHDGM (1,1) is proposed to predict renewable energy consumption. We then not only examine the features of the FHDGM (1,1) model parameters but also assess their impact on modeling precision using a multiplication transformation. We discover that the prediction performance of the FHDGM (1,1) model is unaffected by the multiplication transformation of modeling data. Subsequently, the optimal orderr of the FHDGM (1,1) model is sought by PSO. Finally, two numerical examples show that the FHDGM (1,1) model has good predictive performance. Using the FHDGM (1,1) model to forecast renewable energy consumption shows that renewable energy consumption will continue to increase over the next three years and that the annual growth rate will be more than 5%. With the increasing use of renewable energy, carbon emissions pressure will be reduced.Although the FHDGM (1,1) model has decent prediction performance, there is still some opportunity for improvement. This paper does not consider some important factors that affect renewable energy consumption, which may reduce the predictive accuracy. In future research, we will add influencing factors to construct a multivariable grey prediction model. We will also consider combining the model with different intelligent optimization methods, including the ant lion algorithm [38], the whale optimizer [39], the moth-flame optimization algorithm [40], and other intelligent optimization algorithms, to improve the prediction accuracy of the FHDGM (1,1) model and enrich grey system theory.



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