Forecasting Japan’s Solar Energy Consumption Using a Novel Incomplete Gamma Grey Model

Abstract

Energy consumption is an essential basis for formulating energy policy and programming, especially in the transition of energy consumption structure in a country. Correct prediction of energy consumption can provide effective reference data for decision-makers and planners to achieve sustainable energy development. Grey prediction method is one of the most effective approaches to handle the problem with a small amount of historical data. However, there is still room to improve the prediction performance and enlarge the application fields of the traditional grey model. Nonlinear grey action quantity can effectively improve the performance of the grey prediction model. Therefore, this paper proposes a novel incomplete gamma grey model (IGGM) with a nonlinear grey input over time. The grey input of the IGGM model is a revised incomplete gamma function of time in which the nonlinear coefficient determines the performance of the IGGM model. The WOA algorithm is employed to seek for the optimal incomplete coefficient of the IGGM model. Then, the validations of IGGM are performed on four real-world datasets, and the results exhibit that the IGGM model has more advantages than the other state-of-the-art grey models. Finally, the IGGM model is applied to forecast Japan’s solar energy consumption in the next three years.

1. Introduction

Under the pressure of environmental pollution and the energy crisis, numerous countries have been devoted to the development of renewable energies such as hydro energy, wind energy, solar energy, geothermal energy, and bio-energy. In order to meet the people’s needs for energy in a sustainable way, the sustainable energy policy plays an important role in developing energies and adjusting the energy mix in a country. Accurately forecasting energy consumption contributes to formulating the sustained development strategy of energy. Nowadays, Japan’s energy structure has significantly changed. Before 2011, Japan mainly focused on developing nuclear energy. However, after the Fukushima nuclear accident, the nuclear energy generation declined rapidly because all the nuclear power plants were closed due to safety in 2012 [1]. Until 2015, several plants were gradually restarted for power generation. Nuclear energy consumption was also decreased due to the low production of nuclear energy. In order to alleviate the pressure of energy supply and reduce the dependence on gas and oil abroad at the same time, Japan has vigorously developed renewable energy sources after the nuclear accident. Meanwhile, the consumption of renewable energy is increasing rapidly due to its production. Renewable energy consumption has accounted for a larger proportion of total energy consumption compared with before the nuclear accident. In particular, Japan’s solar energy consumption is the fastest growing one in its renewable energy consumption, according to the BP Statistical Review of World Energy 2019. The consumption of solar energy in 2018 is 23 times that of 10 years ago. It deserves the attention of energy authorities or energy companies for making energy policy. Nevertheless, correctly predicting the trend of solar energy consumption is a challenging task. The current main prediction models include machine learning models and time series models. However, machine learning models usually need a lot of historical data as a training set used to train a model. The time series models with low demand for historical data are widely used to solve the forecasting problems. One of the most effective time-series prediction methods is the grey prediction model, which can be used to handle the uncertain problem with small samples effectively.

The classic grey model (GM(1,1)), initially designed by Professor Deng in 1982, is used to deal with univariate time series prediction problems. Due to the simplicity of modeling process and the lower demand about raw sequence, GM(1,1) and its extended models have been applied in various fields, such as natural gas consumption [2,3,4], electricity consumption [5,6,7], renewable energy consumption [8,9], tertiary industry [10], oil field production [11,12], carbon emissions [13,14,15], electricity and natural gas demand [16,17], and so on. Though grey prediction models have obtained much success in different fields, the prediction accuracy of grey models still requires to be increased by optimization of existing models. There are many optimization approaches about grey model presented as follows. Firstly, the optimization of background value is an effective approach to promote the prediction accuracy of the classical grey models. Li et al. [18] presented an optimized grey prediction model (TBGM(1,1)) based on the data transformation technique and optimization of grey input to predict the short-term energy consumption of Shanghai China. Zhang et al. [19] improved the classical GM(1,1) model by optimizing the coefficients and the time correlation factor of background values, and applied it to forecast the slope deformation field. Zeng et al. [20] optimized the existing multivariate grey model by dynamic background values and applied it to predict the number of China’s wireless communication users. Ye et al. [21] improved the classical Grey–Markov prediction model based on background-value optimization. Li et al. [22] proposed an optimized GM(1,1) model based on hybridization of background-value optimization and weighted weakening buffer operator. Atalay et al. [23] applied multiple linear regression, time series and grey models to estimate the electricity consumption of the heating, cooling, air conditioning system in a commercial building located in France. Rahman et al. [24] employed an improved GM(1,1) model to analyze and forecast the key performance indicators of a medical institution. Ma et al. [25] modified the classical GM(1,1) model based on the Simpson formula and employed it to forecast the gross domestic product (GDP) of Lanzhou, China.

On the other hand, optimization of initial conditions is also a practical approach. Wang et al. [26] optimized the initial condition of the time response sequence with the weighted sum of the first term and the last term of an accumulation generated sequence to improve the prediction precision of the classical GM(1,1) model. Wang et al. [27] proposed the nash nonlinear grey Bernoulli model (Nash NGBM(1,1)) model by optimization of the initial value in its time response sequence. Zeng et al. [28] proposed an unbiased grey model and optimized the initial value by employing the least-square method.

In addition, various accumulation generators are utilized to improve the traditional grey model. Wu et al. [29] initially introduced a fractional accumulation operator into grey prediction theory. Combining the fractional order differential theory and the fractional accumulating operator, Mao et al. [30] put forward a new fractional grey model that could conquer the disadvantages of the classical grey prediction model. Zeng et al. [28] proposed the NHGM(1,1,k) to deal with the non-homogeneous exponential time series based on a wakening buffer operator. With a weighted accumulation operator, Wu et al. [31] proposed the WDGM model that obtained better prediction precision than other models. Ma et al. [32] defined a comfortable fractional accumulation operator and proposed the comfortable fractional grey model (CFGM) that gained more advantages in fluctuating time series.

Moreover, the classical grey model is generalized to deal with multivariate time series. Tien et al. [33] proposed the GMC(1,n) model which corrected the traditional GM(1,n) model designed by Professor Deng et al. Ding et al. [14] designed a new GM(1,n) model based on the trends of system inputs and background value optimization to predict the carbon emissions from fossil fuel of China. Wu et al. [34] proposed a multivariate grey prediction model with a weakening buffer operator. Pei et al. [35] put forward the TNGM(1,n) model based on a nonlinear least-square approach, which had better performance than the traditional GM(1,n) model. Wu et al. [36] designed the GMCN(1,n) model with the new information priority accumulating operator.

Based on the new information priority operator, Ding et al. [12] modified the classical GM(1,1) to forecast the industrial electricity consumption of China. Xia et al. [37] proposed the new information priority accumulated grey model with time power to forecast the wind turbine capacity. The hybrids of the grey model and other classical approaches are also important methods. Al-shanini et al. [38] proposed the hybrid grey model with the Bayesian network to assist the risk management in the chemical process industries. Another optimization measure is the optimization of grey action quantity which is also an important measure to boost the prediction precision of the classical grey models. Zeng et al. [39] proposed an optimized GM(1, n) model by adding a constant term and a linear term over time in the grey input of a traditional GM(1, n) model. The results showed that the proposed grey model obtained the best precision in predicting a material’s tensile strength compared with other classical models. Yin et al. [40] presented a grey model with an exponential grey quantity of time (EOGM(1,1)) model and predicted the GDP of China’s tertiary industry. Wu et al. [41] improved the classical fractional grey model with a linear grey input over time and forecasted the nuclear energy consumption in China. Ma et al. [42] presented an optimized fractional grey prediction method with a nonlinear time-delayed grey input and forecasted the coal and gas consumption. Zeng et al. [43] proposed a new multivariable grey prediction model (NMGM(1,n)) with structure compatibility by optimizing the grey input of the multivariate grey model. Wang et al. [44] proposed an optimized multivariate grey model based on the background values of all variables in the system and employed it to forecast China’s industrial energy consumption. In addition, many scholars optimized classical grey prediction model with kernel technique [45,46,47], rolling mechanism [48,49], intelligent heuristic algorithm [42,50,51,52,53], data preprocess method [10,54,55], regularization method [56], hybridization of the first principles and the data driven models [57], parameter optimization [58], and so on.

In this study, a novel nonlinear grey input is employed to optimize the traditional grey prediction model. Compared with the nonlinear grey action quantities of the existing grey models [40,42], the proposed grey incomplete gamma input has more complicated curves and nonlinear properties. The incomplete gamma grey model (IGGM) can be utilized to manifest the non-exponential properties of time series. Therefore, one of the highlights is that a novel incomplete gamma grey model is proposed based on the WOA algorithm. The validation results prove that IGGM model obtains better performance than the other existing grey model. The primary function of WOA is to seek for the optimal value of the nonlinear coefficient of grey action quantity. Another highlight is that IGGM model is applied to forecast Japan’s solar energy consumption in the next three years.

The remainders of this paper are organized as follows. The classical GM(1,1) model and its improved grey models based on optimization of grey input are presented in Section 2. Then, a novel incomplete gamma grey prediction model named the IGGM model with a nonlinear grey input is minutely illustrated in Section 3. The method of determining the optimal nonlinear parameter of grey input is described in Section 4. Then, the validation experiments performed on the real-life data sequences and analysis are demonstrated in Section 5. In Section 6, the application in forecasting Japan’s solar energy consumption is stated, and the conclusions are drawn in the last section.

2. Basic GM(1,1) and Its Extended Models with Optimization of Grey Action Quantity

Grey prediction theory is one of the most effective approaches to deal with prediction problems with small samples. For a given data sequence $\chi =\left(x\left(1\right),x\left(2\right),\cdots ,x(\ell )\right)$, the classical grey prediction model with one variable and first order accumulated generating operator (GM(1,1)) can be represented as

$$x^{\left(1\right)}\left(k\right)-x^{\left(1\right)}(k-1)+a{z}^{\left(1\right)}\left(k\right)=b,$$

(1)

where $x^{\left(1\right)}\left(k\right)={\displaystyle \sum _{i=1}^k}x\left(i\right)$ denotes the first-order accumulation generating value at kth point, $z^{\left(1\right)}\left(k\right)=\left(x^{\left(1\right)}\left(k\right)+x^{\left(1\right)}(k-1)\right)\times 0.5$ denotes the background value.

The corresponding whitening equation of the classical grey prediction model is defined as

$$\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}+a{x}^{\left(1\right)}\left(t\right)=b,$$

(2)

where a and b are development coefficient and grey action quantity, respectively. The core issue is to estimate the parameters a and b of the grey model. By employing the least square (LS) method, the parameters a and b can be calculated as follows:

$${[a,b]}^T={\left(A{A}^T\right)}^{-1}{A}^{T}Y,$$

(3)

where

$$A={\left[\begin{array}{cccc}0.5\left(x^{\left(1\right)}\left(2\right)+x^{\left(1\right)}\left(1\right)\right)& 0.5\left(x^{\left(1\right)}\left(3\right)+x^{\left(1\right)}\left(2\right)\right)& \cdots & 0.5\left(x^{\left(1\right)}(\ell )+x^{\left(1\right)}(\ell -1)\right)\\ 1& 1& \cdots & 1\end{array}\right]}^T$$

(4)

and

$$Y={\left[\begin{array}{cccc}{x}^{\left(1\right)}\left(2\right)-x^{\left(1\right)}\left(1\right)& x^{\left(1\right)}\left(3\right)-x^{\left(1\right)}\left(2\right)& \cdots & x^{\left(1\right)}(\ell )-x^{\left(1\right)}(\ell -1)\end{array}\right]}^T.$$

(5)

After the two parameters are determined, by solving the whitening Equation (2) and substituting the initial condition into the solution of Equation (2), the time response function of GM(1,1) can be represented as follows:

$$x^{\left(1\right)}(m+1)=\left(x^{\left(0\right)}\left(1\right)-\frac{b}{a}\right)e^{-am}+\frac{b}{a}$$

(6)

By using inverse accumulation operator, the stored value of GM(1,1) can be calculated as:

$$\widehat{x}(m+1)=\left(x^{\left(0\right)}\left(1\right)-\frac{b}{a}\right)(1-e^a)e^{-am}.$$

(7)

The stored value $\widehat{x}\left(m\right)$ denotes the predicted value.

Though the classical GM(1,1) model has been applied in many fields and obtained numerous achievements in solving prediction problems with poor information, GM(1,1) still requires improvement to promote the prediction accuracy and enlarge the application range. Optimization of grey input is one of the most important measures to increase the accuracy of classical grey model. Many studies have been performed by improving the grey input of grey model. When the term $bk$ takes the place of the grey input of GM(1,1), NGM(1,1,k) model [59] can be obtained. When the term $bk+c$ takes the place of the grey input of GM(1,1), the SAIGM model [60] can be obtained. When the term $b{t}^{\alpha }+c$ takes the place of the grey input of GM(1,1), the whitening GM(1,1, $t^{\alpha }$) model [61] can be obtained. When the term $b{e}^{at}$ replaces the grey input of GM(1,1), the whitening equation of EOGM model [40] can be obtained. When the term ${\displaystyle \sum _{i=0}^h}{b}_i{k}^i,(h\ge 1)$ replaces the grey input of GM(1,1), the FOTP-GM model [62] can be obtained. When the term $b{\left(x^{\left(1\right)}\left(t\right)\right)}^n$ replaces the grey input of GM(1,1), the NGBM model [63] can be obtained. When the term $b{\left(x^{\left(1\right)}\left(t\right)\right)}^2$ replaces the grey input of GM(1,1), the GVM model [64] can be obtained.

3. Incomplete Gamma Grey Model

In this section, a novel incomplete gamma grey model called an IGGM model is proposed, in which the grey action quantity of traditional GM(1,1) is taken to replace a new grey input driven by incomplete gamma function.

3.1. IGGM Model

In classical grey prediction theory, the grey action quantity of GM(1,1) model is a constant value. This denotes that the grey input of grey system is stable as time changes. However, the input of the grey system usually changes over time in the real world. Some scholars believe that grey action quantity of the grey system varies linearly with time. The typical grey models such as NGM and SAIGM are proposed to promote the prediction accuracy of classical GM(1,1) model. The grey input of NGM is the term $bk$. The grey input of SAIGM is the term $bk+c$. Moreover, some other scholars deem that the grey input of grey model varies nonlinearly over time. The nonlinear term $t^{\alpha }$ is considered the grey action quantity of GM(1,1, $t^{\alpha }$) model. The nonlinear term $e^{\alpha t}$ is considered as the grey input of EOGM(1,1) model. Figure 1 shows the properties of these two grey action quantities. These two existing grey models with nonlinear grey action quantity exhibit better performance and applicability compared with the other models. In this section, the incomplete gamma function as the grey input is brought into a grey prediction model. The incomplete gamma function has superior nonlinear characteristics which can enrich the state of grey input at different times and cater more data sequences with different features.

From Figure 2, it can be noticed that the properties of incomplete gamma function are significantly different from those of the above-mentioned linear and nonlinear grey input. The classic grey model can be improved with grey input based on an incomplete gamma function. The definition of the improved grey model is described as follows:

The differential equation

$$\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}+a{x}^{\left(1\right)}\left(t\right)=b\gamma (\mu ,t)+c$$

(8)

is defined as a whitening equation of the incomplete gamma grey prediction model called IGGM. The parameter a is the development coefficient of the IGGM model. The term $b\gamma (\mu ,t)+c$ is a grey input of the IGGM model. The function $\gamma (\mu ,t)$ is called an incomplete gamma function and is defined as:

$$\gamma (\mu ,t)={\int }_0^t{t}^{\mu -1}{e}^{-\tau }d\tau ,$$

(9)

where $\mu$ is a tunable parameter called an incomplete coefficient, and $\mu >0$. The parameter $\mu$ determines the property of the incomplete gamma function and profoundly affects the performance of the proposed IGGM model. Through integrating both sides of Equation (9) within interval $[k-1,k]$, the discrete differential form of the IGGM model can be represented as:

$$x^{\left(0\right)}\left(k\right)+a{z}^{\left(1\right)}\left(k\right)=0.5b\left(\gamma (\mu ,k)+\gamma (\mu ,k-1)\right)+c,$$

(10)

where $\gamma (\mu ,k)=k^{\mu }{e}^{-k}{\displaystyle \sum _{m=0}^{\infty }}\frac{\mathsf{\Gamma }\left(\mu \right)k^m}{\mathsf{\Gamma }(\mu +m+1)}=k^{\mu }{e}^{-k}\left(\frac{1}{\mu }+\frac{k}{\mu (\mu +1)}+\frac{k^2}{\mu (\mu +1)(\mu +2)}+\cdots \right)$.

When the parameter b is set to 0, the IGGM model can be degenerated into the classical GM(1,1) model. When the incomplete coefficient $\mu$ is set to 1, the incomplete gamma function can be represented as $\gamma (1,k)=1-e^{-k}$. In addition, then the IGGM model can be reduced into a kind of special EOGM(1,1) model with the form $\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}+a{x}^{\left(1\right)}\left(t\right)=b{e}^{-at}$ in which the development coefficient a is equal to 1.

3.2. Estimating Linear Parameter of IGGM Model

In the IGGM model, there are four parameters $a,b,c$, and $\mu$ which should be estimated. The parameters $a,b$, and c are linear parameters which can be estimated as similar to the traditional GM(1,1) model by using the LS method. However, the coefficient $\mu$ is a nonlinear parameter which can be optimized by an intelligent algorithm introduced in Section 4. Assuming that the parameter $\mu$ is given, the parameters ${[a,b,c]}^T$ can be estimated by the LS method and computed as:

$${[a,b,c]}^T={\left({A_{\ell }}^T{A}_{\ell }\right)}^{-1}{A_{\ell }}^T{Y}_{\ell },$$

(11)

where

$$A_{\ell }=\left[\begin{array}{ccc}-z^{\left(1\right)}\left(2\right)& 0.5\left(\gamma (\mu ,2)+\gamma (\mu ,1)\right)& 1\\ -z^{\left(1\right)}\left(3\right)& 0.5\left(\gamma (\mu ,3)+\gamma (\mu ,2)\right)& 1\\ ⋮& ⋮& ⋮\\ -z^{\left(1\right)}(\ell )& 0.5\left(\gamma (\mu ,\ell )+\gamma (\mu ,\ell -1)\right)& 1\end{array}\right],Y_{\ell }=\left[\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\\ ⋮\\ x^{\left(0\right)}(\ell )\end{array}\right],$$

(12)

in which ℓ is the amount of modeling samples.

3.3. Time Response Function and Restored Values

After estimating the optimal parameters of the IGGM model, we can solve Equation (8) and obtain its solution as the time response function of the grey model. Multiplying both sides of Equation (8) by $e^{at}$, we have

$$e^{at}\left(\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}+a{x}^{\left(1\right)}\left(t\right)\right)=e^{at}\left(b\gamma (\mu ,t)+c\right).$$

(13)

Rearrange Equation (13) and obtain

$$e^{at}\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}+x^{\left(1\right)}\left(t\right)\frac{d{e}^{at}}{dt}=e^{at}(b\gamma (\mu ,t)+c).$$

(14)

Then, we have

$$\frac{d\left(e^{at}{x}^{\left(1\right)}\left(t\right)\right)}{dt}=e^{at}\left(b\gamma (\mu ,t)+c\right).$$

(15)

Integrating both sides of Equation (15) within interval $[1,t]$, the equation can be obtained as

$${\left.\left(e^{at}{x}^{\left(1\right)}\left(t\right)\right)\right|}_{t=1}^{t=t}={\int }_1^t{e}^{ax}\left(b\gamma (\mu ,x)+c\right)dx.$$

(16)

Substituting the initial condition ${\widehat{x}}^{\left(1\right)}\left(1\right)=x^{\left(0\right)}\left(1\right)$, we have

$$e^{at}{x}^{\left(1\right)}\left(t\right)-e^a{x}^{\left(0\right)}\left(1\right)={\int }_1^t{e}^{ax}\left(b\gamma (\mu ,x)+c\right)dx.$$

(17)

Rearrange Equation (17) and obtain

$$x^{\left(1\right)}\left(t\right)=e^{-at}\left(e^a{x}^{\left(0\right)}\left(1\right)+{\int }_1^t{e}^{ax}\left(b\gamma (\mu ,x)+c\right)dx\right).$$

(18)

Then, we have

$$x^{\left(1\right)}\left(t\right)=x^{\left(1\right)}\left(1\right)e^{-a(t-1)}+{\int }_1^t{e}^{-a(t-x)}\left(b\gamma (\mu ,x)+c\right)dx.$$

(19)

Using the Gaussian formula, the convolution integral term of Equation (19) can be discrete as:

$${\int }_1^t{e}^{-a(t-x)}\left(b\gamma (\mu ,x)+c\right)dx={\displaystyle \sum _{m=2}^k}\frac{1}{2}\left(e^{-a(k-m+0.5)}\left(f\left(m\right)+f(m-1)\right)\right),$$

(20)

where $f\left(t\right)=b\gamma (\mu ,t)+c$. Substituting Equation (20) into Equation (19), the time response function of the IGGM model can be represented as:

$${\widehat{x}}_C^{\left(1\right)}\left(k\right)=x^{\left(0\right)}\left(1\right)e^{-a(k-1)}+{\displaystyle \sum _{m=2}^k}\frac{1}{2}\left(e^{-a(k-m+0.5)}\left(f\left(m\right)+f(m-1)\right)\right).$$

(21)

By using inverse first order accumulated operation, the restored values of the IGGM model can be calculated as follows:

$${\widehat{x}}_C^{\left(0\right)}\left(k\right)={\widehat{x}}_C^{\left(1\right)}\left(k\right)-{\widehat{x}}_C^{\left(1\right)}(k-1),$$

(22)

where $k=2,3,\cdots ,\ell ,\cdots r+p$, in which ℓ is the total amount of fitted values while p is the number of predicted values.

4. Determining the Nonlinear Parameter of an Incomplete Gamma Grey Model

4.1. Constructing Optimization Problem for Searching the Optimal Nonlinear Parameter

In Section 3, the methodology of linear parameter estimation of the IGGM model has been introduced based on the hypothesis of given nonlinear parameter $\mu$. In fact, the parameter $\mu$ profoundly determines the accuracy of the two proposed grey models. In order to determine the optimal value of $\mu$, a constrained optimization problem based on the above-mentioned process of building the IGGM model is constructed. The objective function of this optimization problem is to minimize the fit error during the simulation stage. Usually, the fit error can be calculated with mean absolute percentage error. The constraint of the optimization problem is a serial of equality constraint derived from the process of modeling in Section 3. Mathematically, the optimization problem is represented as follows:

$$\begin{array}{c}minError\left(\mu \right)=\frac{1}{\ell -1}{\displaystyle \sum _{m=2}^{\ell }}\left|\frac{{\widehat{x}}_C^{\left(0\right)}\left(m\right)-x^{\left(0\right)}\left(m\right)}{x^{\left(0\right)}\left(m\right)}\right|\hfill \\ s.t.\left\{\begin{array}{c}{[a,b,c]}^T={\left({A_{\ell }}^T{A}_{\ell }\right)}^{-1}{A_{\ell }}^T{Y}_{\ell }\hfill \\ A_u=\left[\begin{array}{ccc}-z^{\left(1\right)}\left(2\right)& 0.5\left(\gamma (\mu ,2)+\gamma (\mu ,1)\right)& 1\\ -z^{\left(1\right)}\left(3\right)& 0.5\left(\gamma (\mu ,3)+\gamma (\mu ,2)\right)& 1\\ ⋮& ⋮& ⋮\\ -z^{\left(1\right)}(\ell )& 0.5\left(\gamma (\mu ,\ell )+\gamma (\mu ,\ell -1)\right)& 1\end{array}\right],Y_{\ell }=\left[\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\\ ⋮\\ x^{\left(0\right)}(\ell )\end{array}\right]\hfill \\ {\widehat{x}}_C^{\left(1\right)}\left(m\right)=x^{\left(0\right)}\left(1\right)e^{-a(m-1)}+{\displaystyle \sum _{i=2}^m}\frac{1}{2}\left(e^{-a(m-i+0.5)}\left(f\left(i\right)+f(i-1)\right)\right)\hfill \\ {\widehat{x}}_C^{\left(0\right)}\left(m\right)={\widehat{x}}_C^{\left(1\right)}\left(m\right)-{\widehat{x}}_C^{\left(1\right)}(m-1),m=2,3,\dots ,\ell \hfill \\ \gamma (\mu ,m)=m^{\mu }{e}^{-k}{\displaystyle \sum _{i=0}^{\infty }}\frac{\mathsf{\Gamma }\left(\mu \right)m^i}{\mathsf{\Gamma }(\mu +i+1)}\hfill \\ f\left(m\right)=\gamma (\mu ,m)+c,\hfill \end{array}\right.\hfill \end{array}$$

(23)

where ℓ denotes the number of modeling samples. Obviously, it can be found that problem (23) is a nonlinear programming problem with an equality constraint. It is difficult to solve such problem (23) exactly because of the complexity of its objective function and constraint. Some scholars have constructed similar optimization problems to search for the optimal parameters. For example, Wang et al. [44] established an optimization problem with constraint to search for the generation coefficient ${\lambda }_i$ of background values. Mao et al. [30] employed the similar measure to choose the optimal order r of the accumulated operator and the order q of differential equations. These studies prove that it is an effective measure to search for optimal parameters to enhance the accuracy of their proposed model.

4.2. Searching Optimal Nonlinear Parameter by the Whale Optimization Algorithm

From the expression of problem (23), it is a nonlinear programming problem with constraints. Obviously, it can be solved by employing a heuristic search algorithm. The Whale Optimization Algorithm (WOA) firstly proposed by Mirjalili et al. in 2016 [65] is employed to search for the optimal value of the incomplete coefficient. The WOA algorithm has been widely employed to settle the optimization problems such as classification, feature extraction, image segment, and so on [66,67]. The inspiration of WOA is to mimic the predation behavior of humpback group. When humpback whales prey on the fish, they encircle the prey and spirally move toward the prey synchronously. Mathematically, the predation behavior of humpback group can be formulated as:

$$L(\kappa +1)\left\{\begin{array}{}\mathrm{(24a)}& L^{*}\left(\kappa \right)-(2rf\left(\kappa \right)-f\left(\kappa \right))\left|2r{L}^{*}\left(\kappa \right)-L\left(\kappa \right)\right|,prob<0.5,\mathrm{(24b)}& cos\left(2\pi \tau \right)e^{\beta \tau }\left|L^{*}\left(\kappa \right)-L\left(\kappa \right)\right|+L^{*}\left(\kappa \right),prob\ge 0.5,\end{array}\right.$$

where $L\left(\kappa \right)$ denotes the location of the humpback at time $\kappa$, $L^{*}\left(\kappa \right)$ denotes the optimal location of humpback at time $\kappa$, the coefficient r is randomly generated in interval $[0,1]$, the coefficient $\tau$ is a random value in $[-1,1]$, the coefficient $\beta$ is a constant, $prob$ is a probability of choosing behavior, and $f\left(\kappa \right)=2-2\kappa /T$ at time $\kappa$. When $prob<0.5$, WOA mimics the encircling behavior formulated by Equation (24a). When $prob\ge 0.5$, WOA mimics the spiral movement behavior represented by Equation (24b). In order to enhance the global search capability of WOA, the location of a randomly chosen humpback replaces the best location of whale when the norm of $\left(2rf\right(\kappa )-f(\kappa \left)\right)$ is greater than 1. Mathematically, it can be represented as:

$$L(\kappa +1)=L_{rand}\left(\kappa \right)-(2rf\left(\kappa \right)-f\left(\kappa \right))\left|2r{L}_{rand}\left(\kappa \right)-L\left(\kappa \right)\right|,$$

(25)

where $L_{rand}\left(\kappa \right)$ denotes the location of a randomly chosen humpback at time $\kappa$. In order to employ WOA to seek for the optimal incomplete coefficient, the necessary parameters are set as follows: the maximum number of iteration T is 300, the size of humpback group is set to 30, the lower bound of the designed parameter is 0, and the upper bound of the designed parameter is set to 10. WOA was initially used to solve the unconstraint optimization problem. Thus, WOA should be revised to solve problem (23) with complex equality constraints. The main computational steps of revised WOA algorithm are summarized as follows:

Step 1: Initialize each humpback’s location $L\left(\kappa \right)$ randomly at $\kappa =1$ in search space $[0,10]$.

Step 2: Calculate the fitness values of humpback group. According to the objective function of Equation (23), the fitness function can be defined as:

$$fitnes{s}_{\kappa }=\frac{1}{\ell -1}{\displaystyle \sum _{m=2}^{\ell }}\left|\frac{{\widehat{x}}_C^{\left(0\right)}\left(m\right)-x^{\left(0\right)}\left(m\right)}{x^{\left(0\right)}\left(m\right)}\right|.$$

(26)

Then, the location of a humpback that has the lowest fitness value is considered as the optimal location $L^{*}\left(\kappa \right)$ at $\kappa$st iteration.

Step 3: Update the locations of all humpbacks. When $prob\ge 0.5$, each humpback’s location is updated by using Equation (24b). When $prob<0.5$ and $\left|2rf\left(\kappa \right)-f\left(\kappa \right)\right|<1$, the location of each humpback is updated by using Equation (25).The location of each humpback is updated by using Equation (24a) in the other condition. Then, each humpback’s fitness is recalculated according to the new locations of humpbacks.

Step 4: If there is a better location that generated a lower fitness than the fitness of last iteration, the best location $L^{*}(\kappa +1)$ is updated.

Step 5: The procedure iterates and continues from Step 3 until the stop criterion is reached. The stop criterion is that the maximum number of iterations has been reached.

4.3. Computational Procedure of the Grey Model

In order to utilize the IGGM model to forecast the future, the key issue is to build an IGGM model with optimal parameters. Firstly, the nonlinear parameter $\mu$ can be sought out by solving the optimization problem (23). Then, the linear parameters $a,b$, and c can be estimated by the LS method after determining the coefficient $\mu$. Finally, the future values can be predicted by using the time response function of IGGM with optimal parameters. The overall procedure of the IGGM model can be summarized in Figure 3.

5. Validation of an Incomplete Gamma Grey Model

5.1. Evaluation Metrics and Benchmark Grey Models

In order to evaluate the performance of the proposed grey models, two common performance metrics Absolute Percentage Error (APE) [68] and Mean Absolute Percentage Error (MAPE) [69] are adopted. The formula of APE is represented as:

$$APE\left(m\right)=\frac{\left|x^{\left(0\right)}\left(m\right)-x_C^{\left(0\right)}\left(m\right)\right|}{x^{\left(0\right)}\left(m\right)}\times 100\%,m=1,2,\dots ,\ell ,\dots ,\ell +p,$$

(27)

where ℓ refers to the number of samples for building the model while the other p samples are used to test the model; $x^{\left(0\right)}\left(m\right)$ and $x_p^{\left(0\right)}\left(m\right)$ respectively denote the actual value and the produced value by grey models at the period of m. Lower APE indicates that the predicted value by the grey model is more approximate to the actual value. The evaluation criterion MAPE is usually used to evaluate the performance of prediction model. Mathematically, the MAPE of fitting is defined as:

$$MAPE{}_{fitting}=\frac{1}{\ell }{\displaystyle \sum _{i=1}^{\ell }}APE\left(i\right).$$

(28)

Meanwhile, MAPE can also be used to evaluate prediction performance of the prediction method. MAPE for prediction is defined as:

$$MAPE{}_{prediction}=\frac{1}{p}{\displaystyle \sum _{i=\ell +1}^{\ell +p}}APE\left(i\right).$$

(29)

In addition, overall MAPE of fitting and prediction is defined as:

$$MAPE{}_{overall}=\frac{1}{\ell +p}{\displaystyle \sum _{i=1}^{\ell +p}}APE\left(i\right).$$

(30)

Lower MAPE indicates that the prediction model has better performance. To reveal the prediction advantages of the IGGM model, six state-of-the-art grey models including GM(1,1) [70], ARGM [2], NGM [59], DGM [71], SAIGM [60], and the GVM [64] model are used as baseline prediction models that are applied in validation cases and application compared with the two proposed grey models.

5.2. Example A: Forecasting the Average Daily Electricity Consumption of China

In this validation study, the raw sequence is China’s average daily electricity consumption (ADEC) from 2006 to 2016 tabulated in

Table 1. China’s average daily electricity consumption (ADEC) from 2006 to 2016 (108 kilowatts hours).

Year	ADEC	Year	ADEC	Year	ADEC	Year	ADEC	Year	ADEC	Year	ADEC
2006	78.3	2008	94.4	2010	114.9	2012	136	2014	154.5	2016	167.5
2007	89.6	2009	101.5	2011	128.8	2013	148.5	2015	159

. The raw consumptions are collected from the National Bureau of Statistics of China. The ADEC from 2006 to 2013 are used to build the model while the rest of the raw data are used to examine the prediction performance of the grey model. The proposed IGGM model and the other classic grey models are applied to predict ADEC of China. The linear parameters of grey models are directly estimated by using the LS method. For the IGGM model, the optimal value of incomplete coefficient $\mu$ is firstly sought out by the WOA algorithm. The optimal incomplete coefficient of the IGGM model is $5.7157$. Then, the linear parameters are calculated based on the optimal coefficient $\mu$. The results of all grey models are listed in

Table 2. Fitted and predicted values of different grey models in Example A.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2006	78.3	78.3	78.3	78.3	78.3	78.3	78.3	78.3
2007	89.6	87.6039	87.1513	87.6789	48.5982	87.5448	39.8951	89.6
2008	94.4	95.7349	96.376	95.8182	81.0928	95.7855	57.3079	94.0626
2009	101.5	104.6205	105.9896	104.713	102.5308	104.7532	79.1194	102.6355
2010	114.9	114.3308	116.0087	114.4336	116.6741	114.5121	103.4536	114.3785
2011	128.8	124.9424	126.4503	125.0565	126.005	125.1321	126.0643	126.9591
2012	136	136.539	137.3323	136.6656	132.1609	136.689	141.0617	138.3768
2013	148.5	149.2118	148.6733	149.3524	136.2222	149.2656	143.547	147.6738
2014	154.5	163.0609	160.4925	163.2168	138.9016	162.9518	132.634	154.777
2015	159	178.1954	172.8101	178.3684	140.6693	177.8456	112.0574	160.0748
2016	167.5	194.7345	185.6473	194.9264	141.8355	194.0534	87.732	164.0766

. The APE and MAPE of all grey models are listed in

Table 3. APEs and MAPEs of different grey models in Example A.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2006	78.3	0	0	0	0	0	0	0
2007	89.6	2.2278	2.7329	2.1441	45.761	2.2937	55.4742	0
2008	94.4	1.4141	2.0932	1.5023	14.0966	1.4677	39.2925	0.3575
2009	101.5	3.0744	4.4233	3.1655	1.0155	3.2051	22.0498	1.1188
2010	114.9	0.4954	0.9649	0.4059	1.5441	0.3376	9.962	0.4539
2011	128.8	2.995	1.8243	2.9064	2.17	2.8478	2.124	1.4293
2012	136	0.3963	0.9797	0.4894	2.8229	0.5066	3.7218	1.7476
2013	148.5	0.4793	0.1167	0.574	8.2679	0.5156	3.3354	0.5564
2014	154.5	5.541	3.8786	5.642	10.0961	5.4704	14.1527	0.1793
2015	159	12.0726	8.6856	12.1814	11.5288	11.8526	29.5237	0.676
2016	167.5	16.2594	10.8342	16.374	15.3221	15.8528	47.6227	2.0438
$MAPE{}_{fitting}$		1.3853	1.6419	1.3985	9.4597	1.3968	16.995	0.7079
$MAPE{}_{prediction}$		11.291	7.7995	11.3991	12.3157	11.0586	30.433	0.9664
$MAPE{}_{overall}$		4.0868	3.3212	4.1259	10.2386	4.0318	20.6599	0.7784

. Fitting MAPE of IGGM is $0.7079$. Prediction MAPE of the IGGM model is $0.9664$. From Figure 4, it can be obviously noticed that the IGGM model achieves the best prediction performance and obtains the best fit compared with the other six classic grey models. Above all, our proposed IGGM model is superior to the other six popular grey models in Example A.

5.3. Example B: Forecasting the Nuclear Energy Consumption of China

In this subsection, the nuclear energy consumption (NEC) of China collected from the BP Statistical Review of World Energy (BPSRWE) 2019 are considered as a raw sequence to validate the performance of our proposed grey models. The raw data listed in

Table 4. China’s nuclear energy consumption (NEC) from 2008 to 2018 (million tonnes of oil equivalent).

Year	NEC	Year	NEC	Year	NEC	Year	NEC	Year	NEC	Year	NEC
2007	14.1	2009	15.9	2011	19.5	2013	25.3	2015	38.6	2017	56.1
2008	15.5	2010	16.7	2012	22	2014	30	2016	48.3	2018	66.6

are divided into two groups. The first one, including the first nine digits, is regarded as a training set to build the model for the proposed grey models and the other six state-of-the-art existing grey models. The second group, including the last three digits, is used to examine the prediction performance of all grey models. By using an intelligent algorithm WOA, the optimal incomplete coefficient of IGGM is sought out. Its optimal value is $3.0101$. Then, all linear parameters of grey models are estimated by the LS method. The results of all grey models are placed in

Table 5. Fitted and predicted values of different grey models in Example B.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2007	14.1	14.1	14.1	14.1	14.1	14.1	14.1	14.1
2008	15.5	12.8213	14.5636	12.8864	6.4378	15.2665	5.3463	15.5266
2009	15.9	14.8391	15.2359	14.9102	10.8562	16.0966	7.2459	15.9023
2010	16.7	17.1745	16.2108	17.2518	15.0671	17.3011	9.7112	16.7
2011	19.5	19.8774	17.6243	19.9612	19.0803	19.0487	12.8221	18.5163
2012	22	23.0057	19.674	23.096	22.9052	21.5846	16.599	21.4709
2013	25.3	26.6263	22.646	26.7232	26.5504	25.264	20.9485	25.5627
2014	30	30.8166	26.9556	30.9201	30.0245	30.6028	25.605	30.8306
2015	38.6	35.6665	33.2044	35.7761	33.3356	38.3493	30.1	37.3969
2016	48.3	41.2796	42.2654	41.3947	36.4911	49.5892	33.8043	45.4683
2017	56.1	47.7762	55.404	47.8957	39.4986	65.8982	36.0712	55.3329
2018	66.6	55.2951	74.4552	55.4176	42.3648	89.5622	36.4488	67.3614

To evaluate the fit and prediction performance, the APE of each period and MAPEs of fitting and predicting for all models are computed, which are tabulated in

Table 6. APEs and MAPEs of different grey models in Example B.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2007	14.1	0	0	0	0	0	0	0
2008	15.5	17.2816	6.041	16.8622	58.4655	1.5064	65.508	0.1716
2009	15.9	6.672	4.1764	6.2255	31.7221	1.2365	54.4284	0.0147
2010	16.7	2.8413	2.9294	3.3041	9.7778	3.5993	41.8489	0
2011	19.5	1.9354	9.6188	2.3649	2.1521	2.3141	34.2458	5.0445
2012	22	4.5712	10.5727	4.982	4.1143	1.8883	24.5501	2.4048
2013	25.3	5.2421	10.4899	5.6255	4.9423	0.1423	17.1994	1.0382
2014	30	2.7221	10.1481	3.067	0.0818	2.0093	14.6498	2.7688
2015	38.6	7.5997	13.9781	7.3159	13.6385	0.6496	22.0207	3.1169
2016	48.3	14.5349	12.4939	14.2967	24.449	2.6692	30.0119	5.8628
2017	56.1	14.8375	1.2406	14.6245	29.5926	17.4656	35.702	1.3674
2018	66.6	16.9744	11.7946	16.7903	36.3892	34.4778	45.272	1.1432
$MAPE{}_{fitting}$		5.4295	7.5505	5.5274	13.8772	1.4829	30.4946	1.6177
$MAPE{}_{prediction}$		15.4489	8.5097	15.2372	30.1436	18.2042	36.9953	2.7911
$MAPE{}_{overall}$		7.9344	7.7903	7.9549	17.9438	5.6632	32.1198	1.9111

. The fit MAPE of SAIGM model is the lowest among the seven grey models while its MAPE of prediction reaches 18.2042%. It indicates that the overfitting phenomenon appears in the modeling process of SIAGM. Though IGGM model’s MAPE of fitting are not best, its overall MAPE and MAPE of prediction are smaller than the six other contrastive grey models. From Figure 5 and Table 6, it can be apparently found that the IGGM model obtains the lowest MAPE of prediction. In conclusion, our proposed IGGM prediction model significantly outperforms the other six grey models in this validation.

5.4. Example C: Forecasting the Hydro Electricity Consumption of China

The raw data are collected form BPSRWE 2019, which is tabulated in

Table 7. China’s hydro electricity consumption (HEC) from 2008 to 2018 (million tonnes of oil equivalent).

Year	HEC	Year	HEC	Year	HEC	Year	HEC	Year	HEC	Year	HEC
2007	109.8	2009	139.3	2011	155.7	2013	205.8	2015	252.2	2017	263.6
2008	144.1	2010	161	2012	195.2	2014	237.8	2016	261	2018	272.1

. In order to evaluate the grey models, the raw data are partitioned into two sets as training set and test set. The training set, including the first ten samples, is used to build a model for different grey models, respectively. The test set, including the other two samples, is used to validate the prediction performance of these grey models. The nonlinear parameter $\mu$ of IGGM is searched by WOA. Its optimum value is $7.6901$. In addition, the linear parameters of all models are easily determined by the LS method. The fitted and predicted values of different grey models are tabulated in

Table 8. Fitted and predicted values of different grey models in Example C.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2007	109.8	109.8	109.8	109.8	109.8	109.8	109.8	109.8
2008	144.1	133.5679	131.0898	133.7852	67.7323	133.8498	44.7947	144.1
2009	139.3	145.7886	150.9755	146.0083	113.2576	145.9752	61.6218	144.1285
2010	161	159.1274	169.5498	159.3481	148.6234	159.2408	83.4428	150.585
2011	155.7	173.6867	186.899	173.9066	176.097	173.7537	110.6087	165.6826
2012	195.2	189.578	203.1041	189.7953	197.4396	189.6313	142.549	187.6051
2013	205.8	206.9233	218.2404	207.1356	214.0194	207.0018	177.193	211.8547
2014	237.8	225.8557	232.3784	226.0601	226.8993	226.0057	210.6237	233.98
2015	252.2	246.5202	245.584	246.7137	236.9049	246.7964	237.4526	251.2433
2016	261	269.0754	257.9187	269.2543	244.6776	269.5422	252.2393	262.8209
2017	263.6	293.6943	269.4399	293.8542	250.7158	294.4267	251.5681	269.2077
2018	272.1	320.5656	280.2012	320.7016	255.4065	321.651	235.6004	271.4988

. The errors of fitting and forecasting during the modeling stage and forecasting stage are placed in

Table 9. APEs and MAPEs of different grey models in Example C.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2007	109.8	0	0	0	0	0	0	0
2008	144.1	7.3089	9.0286	7.1581	52.9963	7.1132	68.9142	0
2009	139.3	4.658	8.3816	4.8157	18.6952	4.792	55.7632	3.4662
2010	161	1.1631	5.3104	1.0261	7.6873	1.0927	48.1722	6.4689
2011	155.7	11.5521	20.0379	11.6934	13.1002	11.5952	28.9604	6.4114
2012	195.2	2.8801	4.0492	2.7688	1.1474	2.8528	26.9729	3.8908
2013	205.8	0.5458	6.0449	0.649	3.9939	0.584	13.9004	2.942
2014	237.8	5.0229	2.2799	4.9369	4.584	4.9598	11.4282	1.6064
2015	252.2	2.2521	2.6233	2.1754	6.0647	2.1426	5.8475	0.3794
2016	261	3.094	1.1806	3.1626	6.2538	3.2729	3.3566	0.6977
2017	263.6	11.4166	2.2154	11.4773	4.8878	11.6945	4.5645	2.1274
2018	272.1	17.8117	2.9773	17.8617	6.1351	18.2106	13.414	0.2209
$MAPE{}_{fitting}$		3.8477	5.8936	3.8386	11.4523	3.8405	26.3316	2.5863
$MAPE{}_{prediction}$		14.6142	2.5964	14.6695	5.5114	14.9525	8.9892	1.1741
$MAPE{}_{overall}$		5.6421	5.3441	5.6437	10.4621	5.6925	23.4412	2.3509

. The IGGM model obtains the best fitting performance and the best prediction performance which are $2.5863$ and $1.1741$, respectively. From Figure 6 and Table 9, it can be apparently found that the proposed model outperforms the other six grey models.

5.5. Example D: Forecasting the Oil Production of Oil Fields

In this section, the raw data are obtained from the paper [11] and listed in

Table 10. The raw data of the cumulative oil production (COP) in Example D.

Year	COP	Year	COP	Year	COP	Year	COP
2001	195.059	2004	342.6394	2007	454.043	2010	552.6569
2002	247.8547	2005	382.4312	2008	485.1171	2011	581.6092
2003	297.0902	2006	420.0399	2009	519.8508	2012	608.1863

. The first eight samples are used to build model while the other four samples are used to validate the prediction performance of the proposed model. Similar to the previous three validation experiments, the incomplete coefficient of IGGM is sought out by the WOA algorithm. In addition, then the linear parameters of all grey models are estimated by the LS method. The optimal value of the incomplete coefficient is $2.8614$. The fitted and predicted results of different grey models are shown in

Table 11. Fitted and predicted values of different grey models in Example D.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2001	195.059	195.059	195.059	195.059	195.059	195.059	195.059	195.059
2002	247.8547	270.5874	248.4202	271.0127	155.6094	247.9491	110.0805	247.8547
2003	297.0902	299.8087	297.2072	300.2579	250.2939	297.165	164.6198	296.3056
2004	342.6394	332.1856	341.8121	332.659	319.9912	342.0147	236.6466	342.8489
2005	382.4312	368.059	382.5935	368.5565	371.2953	382.8853	321.5642	383.328
2006	420.0399	407.8065	419.8791	408.3277	409.0602	420.1301	405.075	419.0918
2007	454.043	451.8463	453.9685	452.3907	436.8589	454.0705	464.2502	452.5551
2008	485.1171	500.6422	485.1358	501.2085	457.3216	484.9998	477.7854	485.6453
2009	519.8508	554.7075	513.6314	555.2943	472.3841	513.1851	440.2383	519.6086
2010	552.6569	614.6116	539.6843	615.2165	483.4716	538.8699	366.1885	555.1964
2011	581.6092	680.9847	563.504	681.605	491.6331	562.276	279.542	592.8695
2012	608.1863	754.5257	585.2818	755.1576	497.6408	583.6055	199.7332	632.9383

. The APE and MAPE of grey models are tabulated in

Table 12. APEs and MAPEs of different grey models in Example D.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2001	195.059	0	0	0	0	0	0	0
2002	247.8547	9.1718	0.2281	9.3434	37.2175	0.0381	55.5867	0
2003	297.0902	0.915	0.0394	1.0662	15.7515	0.0252	44.5893	0.2641
2004	342.6394	3.051	0.2414	2.9128	6.6099	0.1823	30.9342	0.0611
2005	382.4312	3.7581	0.0424	3.628	2.9119	0.1188	15.9158	0.2345
2006	420.0399	2.9124	0.0383	2.7884	2.614	0.0215	3.5627	0.2257
2007	454.043	0.4838	0.0164	0.3639	3.7847	0.0061	2.2481	0.3277
2008	485.1171	3.2003	0.0039	3.317	5.7297	0.0242	1.5113	0.1089
2009	519.8508	6.7051	1.1964	6.818	9.1308	1.2822	15.3145	0.0466
2010	552.6569	11.2103	2.3473	11.3198	12.5187	2.4947	33.7404	0.4595
2011	581.6092	17.0863	3.113	17.193	15.4702	3.3241	51.9365	1.9361
2012	608.1863	24.0616	3.766	24.1655	18.1763	4.0416	67.1592	4.0698
$MAPE{}_{fitting}$		2.9365	0.0762	2.9275	9.3274	0.052	19.2935	0.1528
$MAPE{}_{prediction}$		14.7658	2.6057	14.8741	13.824	2.7857	42.0376	1.628
$MAPE{}_{overall}$		6.8796	0.9194	6.9097	10.8263	0.9632	26.8749	0.6445

. Though the fitting MAPE of IGGM is larger than that of the ARGM and SAIGM model, the IGGM model’s MAPE of prediction is lower than that of the two models. From Table 12 and Figure 7, it can be noticed that the prediction performance of the IGGM model is superior to the other six grey models in this validation experiment.

5.6. Analysis and Discussion

From the results of four validation experiments, it can be noticed that the proposed IGGM model has advantages over the other six classical grey models. In Example A, the MAPEs of GM, ARGM, DGM, SAIGM, and IGGM are less than 2% in terms of data fit while the MAPEs of GM, DGM, and SAIGM are larger than 10% in terms of prediction. It indicates that the over-fitting problem exists in modeling processes of GM, DGM, and SAIGM. The MAPE of fitting and prediction of GVM model are much larger than 10%. Though the prediction abilities ARGM and IGGM are excellent according to Lewis’ criterion, the MAPE of IGGM is significantly lower than that of the ARGM model. In Example B, only fit errors of SAIGM and IGGM model are apparently lower than those of other models and close to each other. However, the prediction’s MAPE of SAIGM is 18.2042%. It denotes that the over-fitting phenomenon occurs in the SAIGM model. In Example C, the MAPEs of fitting in GM, ARGM, DGM, SAIGM, and IGGM are relatively low and very close to each other. However, the prediction’s MAPE of GM, DGM, and SAIGM is markedly higher than 10%. Similarly, there are over-fitting problems in such three models. Only the IGGM model achieves excellent fitting and prediction results. In Example D, ARGM, SAIGM, and IGGM models obtain good fitting and prediction performance simultaneously. Though the MAPEs of ARGM and SAIGM for fitting are slightly lower than the MAPE of the IGGM model, the MAPE of the IGGM model for prediction is distinctly lower than those of ARGM and SAIGM. Above all, it can be drawn that IGGM has more advantages in prediction than the other six grey models. Notably, the various kinds of energy consumption prediction are involved in Example A, B, and C. To some extent, it indicates that the IGGM model has the potential of energy prediction.

6. Applications

In the past decade, Japan’s energy consumption mix has undergone significant changes. In addition, the total primary energy consumption of Japan decreased from 504.7 TKW in 2010 to 454.1 TKW in 2018, according to BP Statistical Review of World Energy (BPSRWE) 2019. Especially after the Fukushima Daiichi nuclear crisis, nuclear energy consumption declined rapidly. Japan’s nuclear power consumption was previously the third-highest level in the world but is now the 14th highest of the 30 nuclear energy-producing countries [72]. The energy consumption transition toward renewable energy is another of the most important phenomena in Japan. From BPSRWE 2019, it can be noticed that the solar consumption of Japan has increased rapidly from 0.7 million tonnes oil equivalent in 2009 to 16.2 million tonnes oil equivalent in 2018, while other renewable energy consumption of Japan changed only slightly in the past few decades. Thus, special attention should be paid to solar energy consumption because it is currently the only rapidly growing one of the renewable energies in Japan. Meanwhile, accurate prediction of energy consumption plays an essential role in making energy policy and supply plans by energy authorities and energy companies. Therefore, the IGGM model is employed to predict Japan’s solar energy consumption based on historical consumption data in the last ten years.

6.1. Raw Data of Japan’s Solar Energy Consumption

The raw data used to build Japan’s solar energy consumption prediction model are collected from the BP Statistical Review of World Energy 2019. The raw data are partitioned into two subsets, including modeling dataset and test dataset. The modeling dataset used to build model includes the first eight years’ solar energy consumption from 2009 to 2016. The test dataset used to validate the prediction performance of the built model includes the two years’ solar energy consumption from 2017 to 2018. The raw data are listed in

Table 13. Solar energy consumption (SEC) of Japan from 2009 to 2018 (million tonnes oil equivalent).

Year	SEC	Year	SEC	Year	SEC	Year	SEC	Year	SEC
2009	0.7	2011	1.2	2013	2.9	2015	7.8	2017	14
2010	0.9	2012	1.7	2014	5.3	2016	11	2018	16.2

.

6.2. Construct and Verify Grey Models

In the grey prediction model, the key issue is to determine the optimal parameters of the model. The incomplete coefficient of the IGGM model can be sought out by using WOA. Then, the other linear parameters of IGGM can be estimated by the LS method. The formulation of the IGGM model used to predict Japan’s solar energy consumption can be represented as $\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}-0.0905x^{\left(1\right)}\left(t\right)=0.0056\gamma (7.6901,t)+0.7836$. The parameters of the other classical grey models can be directly estimated by the LS method. The whitening equation of the GM model can be formulated as $\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}-0.4206x^{\left(1\right)}\left(t\right)=0.4581$. The ARGM model can be represented as $x^{\left(0\right)}\left(k\right)=1.4205x^{\left(0\right)}(k-1)+0.2398$. The DGM model can be formulated as $x^{\left(1\right)}\left(k\right)=1.5310x^{\left(1\right)}(k-1)+0.5917$. The NGM model can be formulated as $x^{\left(0\right)}\left(k\right)-0.3806z^{\left(1\right)}\left(k\right)=0.1698k$. The SAIGM model can be formulated as $x^{\left(0\right)}\left(k\right)-0.3620z^{\left(1\right)}\left(k\right)=0.2633k-0.3089$. The whitening equation of the GVM model can be formulated as $\frac{d{x}^{\left(1\right)}\left(t\right)}{dt}-0.5529x^{\left(1\right)}\left(t\right)=-0.0050{\left(x^{\left(1\right)}\left(t\right)\right)}^2$. The produced results of these grey models are tabulated in

Table 14. Fitted and predicted values of different grey models for forecasting solar energy consumption of Japan.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2009	0.7	0.7	0.7	0.7	0.7	0.7	0.7	0.7
2010	0.9	0.9355	1.2342	0.9634	0.6277	0.5753	0.5111	0.9
2011	1.2	1.4247	1.9931	1.475	1.1252	1.1511	0.8773	1.1164
2012	1.7	2.1697	3.0711	2.2583	1.8531	1.9813	1.4922	1.7278
2013	2.9	3.3043	4.6025	3.4576	2.9182	3.1785	2.499	3.0346
2014	5.3	5.0322	6.7779	5.2937	4.4766	4.9048	4.0786	5.1371
2015	7.8	7.6637	9.8681	8.1048	6.7569	7.3941	6.3848	7.8769
2016	11	11.6713	14.2579	12.4088	10.0934	10.9836	9.3699	10.9619
2017	14	17.7745	20.4938	18.9984	14.9755	16.1595	12.5247	14.1224
2018	16.2	27.0692	29.3522	29.0874	22.119	23.623	14.8047	17.1942

. The valuation results of all grey models are placed in

Table 15. APEs and MAPEs of different grey models for forecasting solar energy consumption of Japan.

Year	Raw Data	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2009	0.7	0	0	0	0	0	0	0
2010	0.9	3.944	37.1354	7.0456	30.2505	36.078	43.2067	0
2011	1.2	18.7244	66.0916	22.9184	6.2321	4.0772	26.8894	6.9703
2012	1.7	27.6297	80.6542	32.8421	9.0069	16.5484	12.2247	1.6324
2013	2.9	13.9415	58.7068	19.2266	0.6272	9.6041	13.8281	4.6403
2014	5.3	5.0526	27.8846	0.1192	15.5357	7.456	23.0447	3.0727
2015	7.8	1.7475	26.5143	3.9082	13.373	5.2035	18.1441	0.9859
2016	11	6.1023	29.6174	12.8076	8.2414	0.149	14.8194	0.346
2017	14	26.9605	46.3845	35.7031	6.9679	15.4252	10.5379	0.8739
2018	16.2	67.094	81.1864	79.5516	36.5368	45.8212	8.6127	6.1373
$MAPE{}_{fitting}$		9.6427	40.8255	12.3585	10.4083	9.8895	19.0196	2.206
$MAPE{}_{prediction}$		47.0273	63.7854	57.6274	21.7524	30.6232	9.5753	3.5056
$MAPE{}_{overall}$		17.1196	45.4175	21.4122	12.6771	14.0363	17.1308	2.4659

. The MAPEs of different grey models for fitting and prediction are plotted in Figure 8. From Table 15 and Figure 8, it can be noticed that the MAPEs of the IGGM model for fitting and prediction are the lowest while the fitting and prediction performance of the other six grey models are significantly worse than those of the IGGM model. According to Lewis’ criterion of prediction valuation [73], the IGGM model has excellent prediction ability because its MAPE of prediction is far less than 10%. Therefore, the IGGM model can be used to forecast the short-term solar energy consumption of Japan.

6.3. Forecasting Japan’s Solar Energy Consumption with Grey Models

This study forecasts Japan’s short-term solar energy consumption during the period 2019–2021. The predicted results of different grey models are filled in

Table 16. The predicted solar energy consumption of Japan.

Year	GM	ARGM	DGM	NGM	SAIGM	GVM	IGGM
2019	41.22448	41.9359	44.53393	32.57134	34.38517	15.15592	20.12687
2020	62.78189	59.81159	68.18324	47.86533	49.90382	13.39473	22.94959
2021	95.61227	85.20477	104.3913	70.24361	72.28124	10.38506	25.73219

. The predicted consumptions of GM, ARGM, DGM, and the NGM model in 2021 rapidly grow more than 70 million tonnes oil equivalent, which is more than four times the consumption in 2018. From the empirical point of view, these results are not reasonable in the real world. The predicted consumptions of GVM are declined from 15.15592 to 10.38506. These results of GVM are also not reasonable based on existing photovoltaic capacity and future ongoing investment. The predicted consumptions from 2019 to 2021 by the IGGM model are 20.12687, 22.94959, and 25.73219, respectively. Based on the trend of solar energy consumption and the renewable energy mix of Japan, the results of the IGGM model are much more reasonable than those of the other six classical grey models.

7. Conclusions

Forecasting energy consumption is one of the critical issues to formulate sustainable energy policy. The grey model is an effective method to solve the prediction problem with small samples. However, there is still room to promote its prediction performance. Optimization of grey action quantity is one of the most effective approaches to enhance the prediction performance of the grey model. In this study, a novel incomplete gamma grey prediction model named the IGGM model is put forward by improving the grey input of the classical grey prediction model. The incomplete gamma grey input of the IGGM model is an incomplete gamma function of time, which is formulated as $b\gamma (\mu ,t)+c$. It is apparently a nonlinear grey input of which the nonlinear coefficient plays a significant role in boosting the prediction performance of the IGGM model. In addition, the optimal value of the incomplete coefficient is searched by employing WOA. On the other hand, the IGGM model can be degenerated into some classical grey models, such as GM(1,1) and the FOTP-GM model, when the coefficient $\mu$ is set to different values. In other words, the IGGM model has the characteristics of the above-mentioned grey models. Finally, the IGGM model is applied to forecast the short-term solar energy consumption of Japan. The MAPE of IGGM for building the solar energy consumption prediction model is 2.206%, which is far superior to those of the other six competitive grey models. The MAPE of IGGM for verifying the prediction performance is 3.5056%, which is also far better than the other six competitive grey models. Japan’s solar energy consumptions forecasted by the IGGM model from 2019 to 2021 are respectively 20.12687, 22.94959, and 25.73219 million tonnes oil equivalent. They can serve as reference information to make energy policy and energy plan. As future work, the IGGM model can be applied in more application fields such as gas and oil prediction [74,75]. Meanwhile, the nonlinear grey action quantity can be adopted to optimize the fractional grey prediction model in the future.