A Novel Riccati Equation Grey Model And Its Application In Forecasting Clean Energy.

and accurate prediction of clean energy can supply an important reference for governments to formulate social and economic development policies. This paper begins with the logistic equation which is the whitening equation of the Verhulst model, introduces the Riccati equation with constant coefficients to optimize the whitening equation, and establishes a grey prediction model (CCRGM(1,1)) based on the Riccati equation. This model organically combines the characteristics of the grey model, and flexibly improves the modelling precision. Furthermore, the nonlinear term is optimized by the simulated annealing algorithm. To illustrate the validation of the new model, two kinds of clean energy consumption in the actual area are selected as the research objects. Compared with six other grey prediction models, CCRGM(1,1) model has the highest accuracy in simulation and prediction. Finally, this model is used to predict the nuclear and hydroelectricity energy consumption in North America from 2019 to 2028. The results predict that nuclear energy consumption will keep rising in the next decade, while hydroelectricity energy consumption will rise to a peak and subsequently fall back, which offers important information for the governments of North America to formulate energy measures.

Introduction

British Petroleum (BP) noted that in 2018, coal consumption increased by 1.4%, oil consumption increased by 1.5%, and natural gas consumption increased by 5.3%. Thus, the energy consumption structure dominated by fossil energy increased carbon emissions by 2.0%, the fastest growth rate in nearly seven years, which also complicates the global environment and climate governance situation. However, the good news is that the consumption of clean energy represented by nuclear energy (up 2.4%) and hydroelectricity (up 3.1%) continues to increase. Clean energy has played a huge role in global environmental governance by virtue of its health, safety and pollution-free characteristics [1], thus reducing the proportion of fossil energy in total energy consumption. Therefore, in recent years, with the global promotion of energy conservation and emission reduction, countries have increased their research into clean energy [2]. Clean energy has become an important reference factor for governments to formulate policies.

The International Energy Agency (IEA) states that with vigorous promotion of clean energy, the global energy structure is undergoing significant changes, and this topic has spurred researchers to focus on building accurate energy prediction models to supply effective information for global energy policy-making. The most common models include commonly used mathematical statistical models (such as the autoregressive distributed lag model, ARIMA, Markov model [[3], [4], [5]]) and intelligent computer technology model [6,7]. These models predict the energy data accurately. However, the main disadvantage of statistical models is that sufficient input data are generally needed for parameter estimation to achieve accurate prediction [8]. The main disadvantage of intelligent computer models is that they usually require a large amount of data for training, and too little data might lead to an inaccurate model. However, significant changes have taken place in the global energy structure, and historical data are no longer reliable for future energy prediction, which has led to a significant reduction in the amount of useful energy data. In addition, no matter how much data are used to build the model, the computational results are not the real data. If the model can be built with less data to obtain relatively effective results, then the model is considered attractive [9]. In fact, grey models can meet these requirements. Therefore, in recent years, researchers have turned to the grey prediction models with small samples (at least four independent data points are required [10]).

To address “small-sample and poor information” systems, Professor Deng proposed the grey system [11], which is based on information coverage, via sequence generation and the grey model to explore the real law of movement of things, with the feature of “less data modelling”. Now, the grey model is widely used in predictions related to energy, finance, transportation, environment, manufacturing and other industries [[12], [13], [14], [15], [16]]. In this process, the grey model has also been extended from the original GM(1,1) to Verhulst model, DGM(1,1), NGM (1,1, K, c), FDGM (1, n), NIPGM (1,1,t,α) [[17], [18], [19], [20], [21], [22]], and combined models, such as GM-ARIMA,GM-MARKOV [23,24] and the combined models of grey model and intelligent computer method [[25], [26], [27]]. To improve the performance of the grey model, researchers have conducted much in-depth and systematic research on the grey prediction model from the perspective of the data accumulation mode, optimization of the background value, model property, modelling mechanism, etc. [[28], [29], [30], [31], [32]], which has advanced the development and refinement of the grey prediction models.

Many grey energy prediction models consider only a single variable and can be divided into three main types: the first is the grey basic model. Zeng et al. [33] proposed the UGM(1,1) model based on the unbiased grey model and a weakening buffer operator to predict the shale gas constant in China. Wang et al. [34] proposed the DGGM(1,1) model to predict China’s hydroelectricity based on a small sample and data grouping. Ding [35] proposed a new grey model to predict China’s total and industrial electricity consumption from 2015 to 2020. The second type is the grey combination model, Wang [36] proposed MNGM-ARIMA model by combining linear and nonlinear models to predict shale gas production in the United States every month. Li [37] proposed a grey model with the regression method, and proposed the GM-ARIMA model to predict the annual average new installed capacity of China’s coal-fired growth in 2017–2026, which will reach 740 GW. Wang et al. [38] used the MVO-MNGBM model to predict that natural gas consumption will reach 354.1 billion cubic meters by 2020 in different regions of China. The third type is the grey model optimized by an optimization algorithm. Ding et al. [39] proposed the adaptive grey system SIGM(1,1) to predict natural gas demand in China via the ant colony optimization algorithm. Wu et al. [10] proposed the FANGBM(1,1) model to predict the renewable energy in a short period by PSO algorithm. Ma et al. [40] proposed a fractional time-delay grey model to predict coal and natural gas consumption in Chongqing with the latest grey wolf optimizer.

The above models have achieved good results, but they ignore the characteristics of energy data. The trend of consumption for fossil energy such as coal and oil presents a disordered and unsaturated S-shaped, and clean energy is no exception. In grey theory, the grey Verhulst model is better for ordered S-shaped data or single-peak data, and the classical GM(1,1) or DGM (1,1) model is better for exponential growth data. For the disordered S-shaped trend of energy data, the performance of the Verhulst model is affected. Therefore, based on the data characteristics and the above literature analysis, the grey Verhulst model is optimized and extended by the Riccati equation, and the mathematical properties of the new model are analysed. The nonlinear term of the new model is optimized by the simulated annealing algorithm, and modelling is performed. The good performance of the new model is verified by the validation cases. Therefore, this model is applied to predict clean energy consumption in North America. The results of the validation and application cases show that the new model is superior to the GM(1,1), DGM(1,1), NGM(1,1), ENGM(1,1) [41], ARGM(1,1) [42] and Verhulst models. Finally, the clean energy consumption over the next 10 years is predicted, providing important information to the government for policy-making.

In the full text, the different abbreviations are for different grey prediction models. Abbreviations and their meanings are listed in Table 1 .

Table 1

Abbreviations of the models.

Number	Abbreviation	Definition
1	GM(1, 1)	Grey model with one variable and one first order equation [17]
2	NGM(1,1, k, c)	Non-homogeneous grey model [20]
3	DGM(1,1)	Discrete grey model with one variable and one first-order equation [19]
4	ARGM(1,1)	Autoregressive grey model [42]
5	ENGM(1,1)	Exact nonhomogeneous grey model [41]
6	Verhulst	Verhulst grey model [18]
7	CCRGM(1,1)	Constant coefficients Riccati grey model

The remainder of this paper is arranged as follows. The CCRGM(1,1) model is discussed in detail in Section 2. Section 3 studies the accuracy of the CCRGM(1,1) model in three cases. Application is presented in Section 4. Conclusions and future work are summarized in Section 5.

CCRGM(1,1) model

The basis of the verhulst model

Definition 2.1 Assume that is a real data sequence, is a 1-accumulating generation operator(AGO) sequence of , where , and is the mean sequence generated by consecutive neighbors of , where

The Verhulst model is

Definition 2.2 The whitening equation of grey Verhulst model is

Definition 2.3 (1) The solution of whitening equation is

The time response equation is

In definitions 2.1–2.3, the least square estimate of the series parameter is , where

The basis of riccati equation

The general Riccati equation iswhere and are continuous on the interval , and is not always 0. Eq. (5) generally has no elementary product decomposition. When and Eq. (5) is the classical logistic model, and it is expressed as follows:

Eq. (6) is the whitening equation of the Verhulst model.

The Verhulst model has certain limitations in dealing with unordered S-shaped or single-peak data. To make the model more applicable, optimization is necessary. Because the logistic model is a special case of the Riccati equation, and can be considered from Riccati equation, and the following equation can be achieved:

Grey prediction model based on riccati equation

In this section, according to the whitening equation of the Verhulst model and the relationship between the logistic equation and the Riccati equation, a new grey prediction model is established. The properties of this model are studied, and the time response function of this model is obtained.

Definition 2.4 Set and as in definition 2.1. Thus,is CCRGM(1,1). The following whitening equation can be obtained from Eq. (7):

CCRGM(1,1) is the grey Verhulst model and its extended model, and are nonlinear correction and constant correction terms respectively. In Eq. (9), when and , CCRGM(1,1) is the grey Verhulst model.

Definition 2.5 Set parameter list is , and

Then parameters and satisfy

Proof. Substituting non-negative raw data into Eq. (8), sothat is, .

Then, replacing with is obtained.

Setand the that minimize should satisfy

According to the above equations, the following equation can be obtained:

Therefore, definition 2.5 is proven.

Definition 2.6 Set and are as definition 2.5, the time response sequence of CCRGM(1,1) iswhere

When,the first equation takes on a minus sign; otherwise, it takes on a positive sign.

Proof. Set ,Eq. (9) is changed as

Then, integrating the two sides of Eq. (14):

Setand , so that

Then, integrating the two sides of Eq. (16) to get

The right end of Eq. (17) can be discussed in three cases:

When:

Set the initial value condition as, and substitute these into Eq. (18). The constant can be calculated as follows:

Move the left and right sides of Eq. (18) and solve to yield

Set , thus,

When , the above equation takes on a minus sign; otherwise, it takes on a positive sign.

When :

Set the initial value condition as , and substitute these into Eq. (22). Then,

The time response equation can be obtained from Eq. (23):

When :

Take the initial value condition into Eq. (25), the constant isand the time response equation is

Definition 2.6 is thus proven, and the reduced value can be obtained:

Optimization of nonlinear correction term

This section derives the optimal order of nonlinear terms. First, to evaluate the accuracy of the model and the effect of order selection, the absolute percentage error (APE) and the mean absolute percentage error (MAPE) are introduced as evaluation metrics. The APE and MAPE are defined as follows:

The nonlinear term optimization is primarily aimed at the orderin Eq. (8). CCRGM(1,1) uses the simulated annealing (SA) algorithm proposed by Kirkpatrick [43] to search. The SA is used to simulate the annealing and cooling process of disordered thermal power system. The entire process is composed of a discrete-time nonhomogeneous Markov chains. The SA is not a predefined mechanical calculation sequence, but a strategy to solve combinatorial optimization problems [44]. Therefore, it is an efficient random global optimization method [45].

When using the SA algorithm, first, set the convergence conditions according to the actual needs, such as the size of the error, number of iterations or termination temperature. The cooling law is as follows:is the annealing coefficient, andis theiteration. Becausehas to decrease slowly, should be close to 1. The SA algorithm is given, as shown in Table 2 .

Table 2

The steps of the SA algorithm.

Algorithm: The SA algorithm to find the optimalr

Set the objective function and the maximum iteration numberT(t+1)=V·T(t)Input: The original seriesrand the number of modelling dataOutput: The best orderrforr∈[0.1,n]do SubstitutertoPˆ=(B^TB)^−1B^TYand obtain parametersP=(a,b,c,d)^T Substitute parameters to discrete equation Eq. (13) and compute the simulation value to obtainXˆ^(1)(k) ComputeXˆ^(1)(k)in Eq. (28) Compute MAPE in Eq. (29), (30)End Update the minimum MAPE valueReturn the bestrby the SA algorithm.

Modelling steps

Based on the definition of CCRGM(1,1) and SA algorithm, this paper proposes a prediction process using CCRGM(1,1):

Step 1: Input the original series .

Step 2: Compute the 1-AGO series of , and the mean sequence of the 1-AGO series .

Step 3: Substitute the data of step 1 into Eq. (10) and initial nonlinear order to obtain parameters and.

Step 4: According to the relationship betweenand 0, choose the appropriate time response equation. When , choose Eq. (21); When , choose Eq. (24); When , choose Eq. (27). Then, compute the restored value by Eq. (28).

Step 5: Substitute the data of above three steps into Eq. (8) to construct the CCRGM(1,1) and compute the MAPE.

Step 6: Use the SA algorithm to optimize nonlinear term and compute the lowest MAPE value.

Step 7: Substitute the optimalto reconstruct CCRGM(1,1) model and compute the simulated data and MAPE.

Combined with the above modelling steps of the model, the modelling flow chart is obtained, as shown in Fig. 1 .

Fig. 1

Flowchart of the CCRGM(1,1) model.

Validation of CCRGM (1,1) model

To illustrate the validity of the CCRGM(1,1) model, three numerical cases are selected. The first is the share of renewable energy consumption, and the last two cases are nuclear energy and hydroelectricity, which are the most consumed forms of clean energy. In three numerical experiments, the results of CCRGM(1,1) are compared with those of GM(1,1) [20], DGM(1,1) [22], NGM(1,1, K, c) (referred to as NGM(1,1)) [23], ARGM(1,1) [46], ENGM(1,1) [45], and Verhulst models [21]. To comprehensively evaluate the prediction performance of the selected models, this paper evaluates the models from two different aspects: the first is to measure the performance of the evaluation metrics generated by these competition models. In addition to the APE and MAPE commonly used in grey model, RMSPE, MAE, MSE, IA, U1, U2, and R are also introduced. The definitions of these metrics are listed in Table 3 . MAPE value is calculated in two stages: MAPEFIT in model building stage and MAPEPRE in prediction stage. Other metrics are calculated by the value of the entire process. Second, the APE comparison chart and curve trend chart are used to show the model performance. The curve trend chart is used to evaluate the fitting and approximation degree of the model simulation trend line and the actual data trend line. The higher the degree of fitting and approximation, the better the data fitting ability of the model is. This standard is reflected mainly by the data trend chart. In the data trend chart of three cases, because the error of the NGM(1,1) is very large, it is omitted.

Table 3

Metrics for evaluating the effectiveness of the models [40].

Name	Abbreviation	Formulation
Mean absolute percentage error	MAPE	1n−1∑k=1n|X^(0)(k)−Xˆ^(0)(k)X^(0)(k)|×100%
Root mean squares percentage error	RMSPE	1n∑i=1n(X^(0)(k)−Xˆ^(0)(k)X^(0)(k))^2×100%
Mean absolute percentage error	MAE	1n∑i=1n|X^(0)(k)−Xˆ^(0)(k)|
Mean squares error	MSE	1n∑i=1n(X^(0)(k)−Xˆ^(0)(k))^2
Index of agreement	IA	1-∑k=1nX^(0)k)−Xˆ^(0)(k))^2∑k=1n(|Xˆ^(0)(k)−x¯|+|X^(0)(k)−x¯|)^2
Theil U statistic 1	U1	1n∑i=1nX^(0)k)−Xˆ^(0)(k))^21n∑i=1n[X^(0)(k)]^2+1n∑i=1n[Xˆ^(0)(k)]^2
Theil U statistic 2	U2	[∑i=1nX^(0)k)−Xˆ^(0)(k))^2]^1/2[∑i=1n[X^(0)(k)]^2]^1/2
Correlation coefficient	R	Cov(Xˆ^(0),X^(0))Var(Xˆ^(0))Var(X^(0))

In the following experiments, the CCRGM(1,1) model is calculated according to the steps in Fig. 1, for which the steps of the SA algorithm are calculated using MATLAB according to the steps in Table 2.

Numerical simulation experiments

Validation case 1 Simulation experiment on the percentage of renewable energy consumption:

In the first case, data of the S-shaped renewable energy consumption proportion are taken from CHINA ENERGY STATISTICAL YEARBOOK 2018. The data from 1999 to 2013 are used to establish seven models in Table 1, and the other data are used to test the models. This numerical experiment primarily shows the advantages of the models under disordered S-shaped data. Table 4 shows the original data and calculation results of seven models. Table 5 lists the evaluation metrics of the seven models. The optimalof the CCRGM(1,1) is.

Table 4

Forecasting results of grey models in Validation Case 1.

Year	Raw data	GM	APE	DGM	APE	NGM	APE	ARGM	APE
		(1,1)	(%)	(1,1)	(%)	(1,1)	(%)	(1,1)	(%)
1999	5.9	5.9	0	5.9	0	5.9	0	5.9	0
2000	7.3	7.1894	−1.5152	7.1983	−1.3933	8.8069	20.6429	6.8761	−5.8063
2001	8.4	7.3417	−12.5992	7.3495	−12.5055	9.2502	10.1216	7.5359	−10.2865
2002	8.2	7.4972	−8.5711	7.504	−8.4881	9.8463	20.0763	7.9819	−2.6598
2003	7.4	7.656	3.459	7.6617	3.5358	10.6477	43.8878	8.2833	11.9369
2004	7.6	7.8181	2.8701	7.8226	2.9295	11.7253	54.2804	8.4871	11.6721
2005	7.4	7.9837	7.8881	7.987	7.9327	13.1743	78.0308	8.6248	16.5513
2006	7.4	8.1528	10.1732	8.1548	10.2006	15.1226	104.3588	8.7179	17.8092
2007	7.5	8.3255	11.0067	8.3262	11.016	17.7422	136.5628	8.7808	17.0773
2008	8.4	8.5018	1.2124	8.5012	1.2042	21.2646	153.15	8.8233	5.0396
2009	8.5	8.6819	2.1402	8.6798	2.1151	26.0008	205.8919	8.8521	4.142
2010	9.4	8.8658	−5.6829	8.8622	−5.7216	32.3691	244.3524	8.8715	−5.6224
2011	8.4	9.0536	7.7809	9.0484	7.7189	40.932	387.2854	8.8846	5.7694
2012	9.7	9.2454	−4.6871	9.2385	−4.7576	52.4456	440.6761	8.8935	−8.3144
2013	10.2	9.4412	−7.4394	9.4326	−7.5231	67.9268	565.9489	8.8995	−12.7499
MAPE(%)			6.2161		6.2173		176.0945		9.6741
2014	11.3	9.6412	−14.6801	9.6308	−14.7713	88.7428	685.3349	8.9036	−21.2074
2015	12.1	9.8454	−18.6334	9.8332	−18.7338	116.7321	864.7283	8.9063	−26.3942
2016	13.3	10.0539	−24.4068	10.0398	−24.5125	154.3665	1060.6507	8.9082	−33.0214
2017	13.8	10.2668	−25.6026	10.2508	−25.7189	204.9698	1385.2886	8.9094	−35.4391
MAPE(%)			20.8307		20.9341		999.0006		29.0155
	Year	Raw data	ENGM (1,1)	APE (%)	Verhulst	APE (%)	CCRGM (1,1)	APE (%)
	1999	5.9	5.9	0	5.9	0	5.9	0
	2000	7.3	7.8284	7.238	1.525	−79.1098	7.8868	8.0386
	2001	8.4	7.9565	−5.2794	1.896	−77.429	7.7762	−7.4261
	2002	8.2	8.1094	−1.1047	2.3441	−71.4134	7.7171	−5.8885
	2003	7.4	8.2918	12.0514	2.8783	−61.1039	7.7048	4.1196
	2004	7.6	8.5094	11.9655	3.5046	−53.8873	7.7379	1.814
	2005	7.4	8.7689	18.4993	4.2235	−42.9254	7.8163	5.6251
	2006	7.4	9.0786	22.6837	5.0275	−32.0614	7.9413	7.3153
	2007	7.5	9.448	25.9733	5.8972	−21.3713	8.1156	8.2076
	2008	8.4	9.8887	17.7224	6.7993	−19.0563	8.3427	−0.6823
	2009	8.5	10.4144	22.5223	7.6856	−9.5814	8.6278	1.5036
	2010	9.4	11.0415	17.4633	8.4952	−9.6252	8.9776	−4.4932
	2011	8.4	11.7897	40.3539	9.1611	9.0607	9.4009	11.9155
	2012	9.7	12.6823	30.745	9.6198	−0.8273	9.9087	2.1517
	2013	10.2	13.747	34.7748	9.8232	−3.6939	10.5153	3.091
	MAPE(%)			19.1697		35.0818		5.1623
	2014	11.3	15.0173	32.8961	9.7491	−13.7248	11.2388	−0.5418
	2015	12.1	16.5326	36.633	9.4056	−22.2676	12.1026	0.0213
	2016	13.3	18.3403	37.8971	8.8298	−33.6106	13.137	−1.2253
	2017	13.8	20.4969	48.528	8.0792	−41.4549	14.382	4.2171
	MAPE(%)			38.9886		27.7645		1.5014

Table 5

Metrics of models in Validation Case 1.

Metrics	GM	DGM	NGM	ARGM	ENGM	Verhulst	CCRGM	CCRGM rank
	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)
RMSPE	11.5431	11.5821	517.7336	16.4476	26.1834	40.7137	5.261	1
MAE	0.9413	0.9443	38.7245	1.3615	2.2479	2.7831	0.35	1
MSE	1.8801	1.8957	4254.7012	3.6698	8.2028	12.242	0.1914	1
IA	0.8098	0.8074	−0.3316	0.506	0.7666	0.4096	0.9892	1
U1	0.0767	0.077	0.7975	0.1078	0.1351	0.215	0.0233	1
U2	0.1473	0.1479	7.0091	0.2058	0.3078	0.376	0.047	1
R	0.8849	0.8842	0.9531	0.5453	0.9748	0.6328	0.9814	1
MAPEFIT	6.2161	6.2173	176.0945	9.6741	19.1697	35.0818	5.1623	1
MAPEPRE	20.8307	20.9341	999.0006	29.0155	38.9886	27.7645	1.5014	1

In comparison with other grey models, Table 4 shows that the MAPEFIT values of CCRGM(1,1), GM(1,1), DGM(1,1), and ARGM(1,1) are all lower than 10%, indicating that the fitting effect of these models is good, but the lowest is that of CCRGM(1,1), which is 5.1623%. The fitting effect of NGM(1,1) is the worst, and the MAPEFIT is as high as 176.0954%. In the prediction stage, the MAPEPRE of CCRGM(1,1) is the lowest, only 1.5014%, and that of other models exceeds 20%. Table 5 shows that all evaluation metrics of the CCRGM(1,1) are better than other models. For RMSPE, MSE, IA, U1, and U2 evaluation indexes, CCRGM(1,1) values are far better than those of the other models. To intuitively show the fitting error of all models, Table 4 is transformed into APE comparison chart and curve trend chart, as shown in Fig. 2, Fig. 3 respectively. Because the error of NGM(1,1) model is too large to affect Fig. 2, Fig. 3, it is omitted in the charts.

Fig. 2

The APE of the seven models in Validation Case 1.

Fig. 3

Overall trend of simulation results of models in Validation Case 1.

Fig. 2 shows that the APE values of CCRGM(1,1) are not the lowest in 7 years, but the those of other 12 years are much lower than other models, and those of six years are close to the zero line. Fig. 3 shows that the actual trend line exhibits a disordered S-shaped, the fitting lines of GM(1,1), DGM(1,1) ARGM(1,1), ENGM(1,1) are increasing, and far from the actual data trend line, while Verhulst model presents a complete S-shaped, and underestimates the actual value. Only the fitting line of CCRGM(1,1) is closest to the actual data line. In conclusion, two aspects show that CCRGM(1,1) has the best effect in dealing with disordered S-shaped data and alleviates the dependence of Verhulst model on saturated S-shaped data.

Validation case 2 Predicting nuclear clean energy consumption：

This case selects China’s nuclear energy consumption as the research object, with data taken from the literature [[46]]. One of the largest countries in the world, China began to develop clean energy later than other countries, but its development proceeded rapidly. Thus, the data in recent years are not S-shaped data, and are used to illustrate the ability of CCRGM(1,1) model for non-S-shaped data. The data from 2006 to 2012 are used to establish the seven models, while the data from 2013 to 2017 are used to test the models. The original data and calculation results are shown in Table 6, Table 7 . The optimalof CCRGM(1,1) is.

Table 6

Forecasting results of grey models in Validation Case 2.

Year	Raw data	GM(1,1)	APE(%)	DGM(1,1)	APE(%)	NGM(1,1)	APE(%)	ARGM(1,1)	APE(%)
2006	12.4	12.4	0	12.4	0	12.4	0	12.4	0
2007	14.1	13.6523	3.1752	13.6726	−3.0313	17.5093	24.1795	13.4684	4.4793
2008	15.5	14.9241	3.7156	14.9433	−3.5914	19.5799	26.3217	14.7098	5.0979
2009	15.9	16.3143	−2.6059	16.3322	2.7181	22.6345	42.3556	16.1522	−1.5863
2010	16.7	17.8341	−6.7911	17.8501	6.8868	27.1411	62.5216	17.8281	−6.7553
2011	19.5	19.4955	0.0233	19.5091	0.0466	33.7896	73.2802	19.7754	−1.4122
2012	22	21.3116	3.1293	21.3223	−3.0806	43.5982	98.1736	22.0379	−0.1722
MAPE(%)			3.2401		3.2258		54.472		3.2505
2013	25.3	23.2969	7.9176	23.304	−7.8894	58.0688	129.5208	24.6667	−2.5032
2014	30	25.4671	15.1097	25.4699	15.1004	79.4171	164.7238	27.7211	−7.5963
2015	38.6	27.8395	27.877	27.8371	27.8833	110.9124	187.3377	31.27	18.9896
2016	48.2	30.4329	36.8613	30.4242	36.8792	157.3772	226.5087	35.3935	26.5694
2017	56.2	33.2679	40.8045	33.2519	40.8329	225.9267	302.0048	40.1846	28.4971
MAPE(%)			25.714		25.717		202.0191		16.8311

Table 7

Metrics of models for fitting in Validation Case 2.

Metrics	GM	DGM	NGM	ARGM	ENGM	Verhulst	CCRGM	CCRGM rank
	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)
RMSPE	18.6602	18.6671	142.4507	13.0347	55.6844	21.4582	5.6917	1
MAE	5.1051	5.1055	41.1629	3.5150	15.2066	3.8693	1.2187	1
MSE	82.0302	82.1168	4193.7546	40.1883	786.4197	22.0579	9.1250	1
IA	0.7970	0.7966	0.2960	0.9193	0.6897	0.9769	0.9894	1
U1	0.1745	0.1746	0.5300	0.1171	0.3299	0.0768	0.0498	1
U2	0.3062	0.3063	2.1892	0.2143	0.9480	0.1588	0.1021	1
R	0.9641	0.9640	0.9903	0.9833	0.9784	0.9807	0.9889	2
MAPEFIT	3.2401	3.2258	54.472	3.2505	9.6595	18.8782	2.2628	1
MAPEPRE	25.714	25.717	202.0191	16.8311	75.6177	16.1082	4.7724	1

According to Table 6, the model with the best fitting effect is CCRGM(1,1), whose MAPEFIT value is only 2.2628%, which is far better than that of NGM(1,1), ENGM(1,1) and Verhulst models, but approximately only a percentage point lower than that of GM(1,1), DGM(1,1) and ARGM(1,1). However, the MAPEPRE of CCRGM(1,1) model is much lower than that of other models, only 4.7724%, and that of other models exceeds 16%. In the comparison of other evaluation indexes in Table 7, the R metric of CCRGM(1,1) ranks second of the seven models, and other metrics are slightly better than those of the other models. However, NGM(1,1), which ranks first in R metric, has the worst results in the other metrics. To visually highlight the error of all models, APE comparison chart and curve trend chart are shown in Fig. 4, Fig. 5 respectively.

Fig. 4

The APE of the models in Validation Case 2.

Fig. 5

Overall trend of simulation results of models in Validation Case 2.

Fig. 4 shows that the APE values of CCRGM(1,1) are the lowest in 8 out of 12 years, most of which are close to the zero line, while the APE values of other four years are also not the largest. In Fig. 5, the actual data show an upward trend, not an S-shaped trend. GM(1,1), DGM(1,1) and ARGM(1,1) underestimate the actual value; ENGM(1,1) overestimates the actual value. The fitting line of Verhulst model shows an S-shaped trend, only that of CCRGM(1,1) is the closest to the actual data line. In conclusion, CCRGM(1,1) can not only effectively predict nuclear energy consumption, but also eliminate the dependence of Verhulst model on the saturated S-shaped.

Validation case 3 Predicting the hydroelectricity clean energy consumption：

These data are collected from BP Statistical Review of World Energy 2019. Countries in Commonwealth of Independent States (CIS) region, which is influenced by European and North American countries, have vigorously developed clean energy. Therefore, the data in recent years show a disordered S-shaped. The data of the first ten years are used to build the models, and the data of the last years are used to test the model. The optimalof CCRGM(1,1) model is. The original data and fitting results are shown in Table 8 , and the results of evaluation metrics are shown in Table 9 .

Table 8

Forecasting results of grey models in Validation Case 3.

Year	Raw data	GM (1,1)	APE (%)	DGM (1,1)	APE (%)	NGM (1,1)	APE (%)	ARGM (1,1)	APE (%)
2008	47.0	47.0000	0.0000	47.0000	0.0000	47.0000	0.0000	47.0000	0.0000
2009	49.4	47.9984	−2.8372	48.0122	−2.8093	64.6302	30.8304	49.04	−0.7287
2010	49.1	48.5632	−1.0932	48.574	−1.0713	74.9167	52.5798	50.0817	1.9993
2011	48.1	49.1347	2.1512	49.1423	2.167	91.8805	91.0197	50.6136	5.2257
2012	48.2	49.7129	3.1389	49.7174	3.148	119.8561	148.6641	50.8851	5.5708
2013	51.9	50.298	−3.0868	50.2991	−3.0846	165.9918	219.83	51.0238	−1.6882
2014	50	50.8899	1.7797	50.8876	1.7753	242.0759	384.1518	51.0946	2.1893
2015	48.8	51.4887	5.5097	51.4831	5.4981	367.5492	653.1745	51.1308	4.7762
2016	53.1	52.0946	−1.8933	52.0855	−1.9106	574.4719	981.868	51.1492	−3.6737
2017	54.3	52.7077	−2.9324	52.6949	−2.9559	915.7161	1586.4017	51.1587	−5.7851
MAPE(%)			2.7136		2.7134		460.9467		3.5152
2018	55.4	53.3279	−3.7402	53.3115	−3.7699	1478.4751	2568.7276	51.1635	−7.6471
MAPE(%)			3.7402		3.7699		2568.7276		7.6471

Table 9

Metrics of models for fitting in Validation Case 3.

Metrics	GM	DGM	NGM	ARGM	ENGM	Verhulst	CCRGM	CCRGM rank
	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)	(1,1)
RMSPE	2.9069	2.9099	987.9148	4.2509	2.9559	27.1680	2.2172	1
MAE	1.3033	1.3048	326.1149	1.8337	1.2923	9.8783	0.7675	1
MSE	2.1854	2.1917	290,770	4.8765	2.2197	180.2411	1.2763	1
IA	0.8911	0.8904	−0.5425	0.6343	0.9026	0.1082	0.9469	1
U1	0.0147	0.0147	0.8649	0.0219	0.0148	0.1390	0.0112	1
U2	0.0292	0.0293	10.6671	0.0437	0.0295	0.2656	0.0223	1
R	0.8391	0.8389	0.8722	0.5568	0.8353	0.3467	0.9060	1
MAPEFIT	2.7136	2.7134	460.9467	3.5152	2.7854	22.2604	1.8476	1
MAPEPRE	3.7402	3.7699	2568.7276	7.6471	2.9731	17.8566	0.0018	1

As seen in Table 8, the difference between the MAPEFIT of GM(1,1), DGM(1,1), ARGM(1,1), ENGM(1,1) and CCRGM(1,1) models is not large. The MAPEFIT of CCRGM(1,1) is approximately 1% lower than the other values, only 1.8476%, while NGM(1,1) has the worst fitting effect, with MAPEFIT reaching 460.9467%. The MAPEFIT of Verhulst model is slightly better than that of NGM(1,1), at 22.2604%. The MAPEPRE of CCRGM(1,1) model is close to zero, only 0.0018%, far lower than that of the other models. Table 9 clearly shows that the results of evaluation metrics of CCRGM(1,1) are better than those of other models. In the same way as the first two cases, Table 8 is transformed into APE comparison chart and curve trend chart, as shown in Fig. 6, Fig. 7 respectively.

Fig. 6

The APE of the six models in Validation Case 3.

Fig. 7

Overall trend of simulation results of models in Validation Case 3.

In Fig. 6, except for 2016, the APE of CCRGM(1,1) is the lowest every year, and all of the values are close to zero. In Fig. 7, the original data show a continuous multiple S-shaped trend, Verhulst model shows single saturated S-shaped; and the fitting curves of GM(1,1), DGM(1,1), ARGM(1,1) and ENGM(1,1) show a straight upward trend. Only the fitting trend of CCRGM(1,1) is close to the actual data trend. The above analysis shows that CCRGM(1,1) is effective and accurate for the prediction of hydroelectricity energy.

Analysis of results

In this section, the results of three cases are analysed in combination with figures, and the following conclusions can be obtained:

The Riccati equation is introduced into the classical Verhulst model to achieve a new grey model, and the accuracy of Verhulst model is substantially improved after the expansion. In the validation cases, the regularity of clean energy data is not obvious, but the effect of CCRGM(1,1) model is far better than that of the Verhulst model, which shows that the CCRGM(1,1) model has no obvious data requirements and alleviates the dependence of Verhulst model on S-shaped data, which makes CCRGM(1,1) model more suitable for clean energy prediction.

In the comparison of CCRGM(1,1) with the GM(1,1), DGM(1,1), NGM(1,1), ARGM(1,1) and ENGM(1,1) models, the evaluation metrics of CCRGM(1,1) model are the best, and the fitting and approximation degree between the trend lines of CCRGM(1,1) model and original data is the highest, far superior to those of the other grey models. This result shows that the CCRGM(1,1) model is effective and accurate in short-term and metaphase prediction of clean energy consumption.

Applications

North America usually refers to the United States, Canada and other regions. It is the most developed continent in the world and the first region to vigorously develop clean energy. According to BP Statistical Review of World Energy 2018, the annual growth rate of new energy consumption in North America from 2007 to 2017 was 13.9%, which effectively reduced the consumption of coal, oil and other resources. In 2016, North America reached a consensus that clean energy should account for 50% of energy consumption, thus, its predicted clean energy consumption trend plays a crucial role in formulating energy consumption policies.

Due to the good performance of the CCRGM(1,1) model, this model is applied to predict nuclear and hydroelectricity energy consumption in North America. The data are taken from BP Statistical Review of World Energy 2019, and divided into two parts: one part is used to build the model, and the other part is used to test and compare the prediction results of the models.

Case 1: prediction of nuclear clean energy consumption

The actual data are shown in Table 10 , in which the data from 2002 to 2007 are used to build the model, and the other data are used to test the models. The results of the seven models are shown in Table 10. The optimalultimately identified is.

Table 10

Forecasting results of grey models in Case 1.

Year	Raw data		GM (1,1)	APE (%)	DGM (1,1)	APE (%)	NGM (1,1)	APE (%)	ARGM (1,1)	APE (%)
2002	205		205.0000	0.0000	205.0000	0.0000	205.0000	0.0000	205.0000	0.0000
2003	201.1		203.6043	1.2453	203.6174	1.2518	120.1210	−40.268	208.0793	3.4706
2004	210.2		206.5670	−1.7284	206.5757	−1.7242	190.7175	−9.2685	209.9471	−0.1203
2005	209.4		209.5728	0.0825	209.5769	0.0845	206.4427	−1.4123	211.0801	0.8023
2006	212		212.6224	0.2936	212.6217	0.2933	209.9454	−0.9691	211.7673	−0.1098
2007	215.4		215.7164	0.1469	215.7108	0.1443	210.7256	−2.1701	212.1842	−1.493
MAPE (%)				0.6993		0.6996		13.522		1.1992
2008	215.8		218.8553	1.4158	218.8448	1.4109	210.8994	−2.1701	212.437	−1.5584
2009	212.9		222.0400	4.2931	222.0243	4.2857	210.9381	−2.2709	212.5904	−0.1454
2010	213.9		225.2710	5.3160	225.2499	5.3062	210.9468	−0.9215	212.6834	−0.5688
2011	211.5		228.5490	8.0610	228.5225	8.0485	210.9487	−1.3807	212.7398	0.5862
2012	206.5		231.8747	12.2880	231.8426	12.2724	210.9491	−0.2607	212.7741	3.0383
2013	213.8		235.2488	10.0322	235.2109	10.0145	210.9492	2.1545	212.7948	−0.4701
2014	216.2		238.6720	10.3941	238.6282	10.3738	210.9492	−1.3334	212.8074	−1.5692
2015	215.4		242.1450	12.4164	242.0951	12.3933	210.9492	−2.4287	212.8151	−1.2001
2016	217		245.6686	13.2113	245.6123	13.1854	210.9492	−2.0663	212.8197	−1.9264
2017	216.9		249.2434	14.9117	249.1807	14.8828	210.9492	−2.7884	212.8225	−1.8799
2018	217.9		252.8702	16.0488	252.801	16.0170	210.9492	−2.7436	212.8242	−2.3294
MAPE(%)				9.8535		9.8355		1.9757		1.3884
		Year	Raw data	ENGM (1,1)	APE (%)	Verhulst	APE (%)	CCRGM (1,1)	APE (%)
		2002	205	205.0000	0.0000	205.0000	0.0000	205.0000	0.0000
		2003	201.1	211.9158	5.3783	134.4224	33.1564	200.5235	−0.2867
		2004	210.2	212.897	1.2831	188.8725	10.1463	208.2622	−0.9219
		2005	209.4	213.9062	2.1520	230.0859	9.8786	210.9926	0.7606
		2006	212	214.9442	1.3888	236.1629	11.3976	212.6156	0.2904
		2007	215.4	216.0118	0.2840	203.3911	−5.5752	213.7055	−0.7867
		MAPE (%)			2.0972		14.0308		0.6092
		2008	215.8	217.1098	0.6070	150.3347	30.3361	214.4788	−0.6122
		2009	212.9	218.2392	2.5078	249.3627	17.1267	215.0417	1.0059
		2010	213.9	219.4007	2.5716	309.7385	44.8053	215.4545	0.7267
		2011	211.5	220.5954	4.3004	344.8282	63.0393	215.7549	2.0118
		2012	206.5	221.8241	7.4209	364.6568	76.5892	215.9682	4.5851
		2013	213.8	223.0879	4.3442	375.684	75.7175	216.1119	1.0813
		2014	216.2	224.3878	3.7871	381.7621	76.5782	216.1989	−0.0005
		2015	215.4	225.7247	4.7932	385.0958	78.7817	216.239	0.3895
		2016	217	227.0997	4.6542	386.9194	78.3039	216.2395	−0.3505
		2017	216.9	228.5139	5.3545	387.9154	78.8453	216.2064	−0.3198
		2018	217.9	229.9684	5.5385	388.459	78.2740	216.1444	−0.8057
		MAPE(%)			4.1709		63.4907		1.0808

In Table 10, the errors of CCRGM(1,1) are the smallest, the MAPEFIT is only 0.6092%, and the MAPEPRE is only 1.0808%. The model with the largest MAPE value is Verhulst model. To visually compare the differences among the models, Table 10 is transformed into charts of the MAPE comparison and curve trend, as shown in Fig. 8, Fig. 9 respectively. Table 10 shows that the MAPEPRE of Verhulst model is very large, so it is omitted in Fig. 8, Fig. 9.

Fig. 8

The MAPE of the models in Case 1.

Fig. 9

Overall trend of simulation results of models in Case 1.

In Fig. 8, the MAPEFIT values of GM(1,1) and DGM(1,1) are slightly 0.1% higher than that of CCRGM(1,1), but the MAPEPRE values are far higher than that of CCRGM(1,1). The MAPEPRE values of ARGM(1,1) and NGM(1,1) are similar to that of CCRGM(1,1), but the MAPEFIT values are higher. In Fig. 9, except for 2012, the CCRGM(1,1) predictions generally coincide with the original data, ARGM(1,1) and NGM(1,1) underestimate the nuclear energy consumption, and the fitting lines of GM(1,1) and DGM(1,1) essentially coincide, but similar to ENGM(1,1), they overestimate the nuclear energy consumption. In addition, Verhulst model presents a saturated S-shaped, which is contrary to the actual situation. Therefore, the CCRGM (1,1) model is the best for prediction of nuclear energy.

Case 2: prediction of hydroelectricity clean energy consumption

In this case, the CCRGM(1,1) model is established based on the hydroelectricity energy consumption in 2000–2009, and the consumption in subsequent nine years is used for prediction. The optimalof CCRGM(1,1) is. The actual data and the results of seven grey models are shown in Table 11 .

Table 11

Forecasting results of grey models in Case 2.

Year	Raw data		GM (1,1)	APE (%)	DGM (1,1)	APE (%)	NGM (1,1)	APE (%)	ARGM (1,1)	APE (%)
2000	149.9		149.9000	0.0000	149.9000	0.0000	149.9000	0.0000	149.9000	0.0000
2001	129		136.4277	5.7579	136.4622	5.7846	103.7961	−19.5379	144.7347	12.1975
2002	143.1		138.4104	−3.2771	138.4377	−3.2581	120.8235	−15.5671	144.6373	1.0743
2003	141.6		140.422	−0.8319	140.4418	−0.8179	131.6053	−7.0584	144.6355	2.1437
2004	141.7		142.4628	0.5383	142.475	0.5469	138.4322	−2.3061	144.6355	2.0716
2005	148.5		144.5333	−2.6712	144.5375	−2.6683	142.7551	−3.8686	144.6355	−2.6024
2006	151.4		146.6338	−3.1481	146.63	−3.1506	145.4923	−3.902	144.6355	−4.468
2007	144.4		148.7649	3.0228	148.7527	3.0143	147.2255	1.9567	144.6355	0.1631
2008	151.1		150.927	−0.1145	150.9062	−0.1283	148.323	−1.8379	144.6355	−4.2783
2009	150.9		153.1205	1.4715	153.0908	1.4518	149.0179	−1.2473	144.6355	−4.1514
MAPE(%)				2.3148		2.3134			6.3647	3.6834
2010	146.1		155.3458	6.3284	155.3071	6.3019	149.4579	2.2983	144.6355	−1.0024
2011	164.7		157.6035	−4.3087	157.5554	−4.3379	149.7365	−9.0853	144.6355	−12.1825
2012	155.3		159.894	2.9582	159.8363	2.921	149.9129	−3.4688	144.6355	−6.8671
2013	155.3		162.2178	4.4545	162.1502	4.4109	150.0246	−3.3969	144.6355	−6.8671
2014	153.2		164.5754	7.4252	164.4976	7.3744	150.0954	−2.0265	144.6355	−5.5904
2015	149.2		166.9673	11.9083	166.879	11.8492	150.1401	0.6301	144.6355	−3.0593
2016	154.2		169.3939	9.8533	169.2949	9.7892	150.1685	−2.6145	144.6355	−6.2027
2017	164.1		171.8557	4.7262	171.7457	4.6592	150.1865	−8.4787	144.6355	−11.8614
2018	160.3		174.3534	8.7669	174.232	8.6912	150.1978	−6.302	144.6355	−9.772
MAPE(%)				6.7477		6.7039		4.2557		7.0450
	Year		Raw data	ENGM (1,1)	APE (%)	Verhulst	APE (%)	CCRGM (1,1)	APE (%)
	2000		149.9	149.9000	0.0000	149.9000	0.0000	149.9000	0.0000
	2001		129	145.9006	13.1012	60.7186	−52.9313	136.2155	5.5934
	2002		143.1	147.2977	2.9334	81.0047	−43.3929	142.3433	−0.5288
	2003		141.6	148.7978	5.0832	104.466	−26.2246	144.7516	2.2257
	2004		141.7	150.4084	6.1457	128.986	−8.9725	146.338	3.2731
	2005		148.5	152.1377	2.4496	150.9712	1.6641	147.529	−0.6539
	2006		151.4	153.9944	1.7136	166.0327	9.6650	148.485	−1.9253
	2007		144.4	155.988	8.0249	170.5425	18.1042	149.2848	3.3828
	2008		151.1	158.1284	4.6515	163.3174	8.0857	149.9728	−0.7460
	2009		150.9	160.4265	6.3131	146.2315	−3.0938	150.5769	−0.2141
	MAPE(%)				5.6018		19.1260		2.0603
	2010		146.1	162.894	11.4948	123.2598	−15.6332	151.1157	3.4330
	2011		164.7	165.5433	0.5120	221.9766	34.7763	151.6019	−7.9527
	2012		155.3	168.3877	8.4274	297.8487	91.7892	152.0452	−2.0958
	2013		155.3	171.4418	10.394	354.3381	128.1636	152.4526	−1.8335
	2014		153.2	174.7209	14.0476	395.4088	158.0997	152.8295	−0.2418
	2015		149.2	178.2417	19.4649	424.7562	184.6891	153.1802	2.6677
	2016		154.2	182.0218	18.0427	445.4678	188.8896	153.5083	−0.4486
	2017		164.1	186.0805	13.3946	459.9570	180.2907	153.8164	−6.2667
	2018		160.3	190.4383	18.8012	470.0312	193.2197	154.107	−3.8634
	MAPE(%)				12.731		130.6168		3.2004

According to Table 11, the errors of CCRGM(1,1) are the smallest: the MAPEFIT is only 2.0603%, and the MAPEPRE is only 3.2004%. Similar to the previous case, the model with the largest error is the Verhulst model, whose MAPEFIT is 19.1260% and MAPEPRE is 130.6168%. This result also shows that in the prediction of clean energy, CCRGM(1,1) improves the prediction accuracy of Verhulst model. To directly compare the differences between models, the MAPE comparison chart and curve trend chart are constructed according to Table 11, as detailed in Fig. 10, Fig. 11 , respectively. Because the error of Verhulst model is too large to affect these Fig. 10, Fig. 11, it is omitted.

Fig. 10

The MAPE of the models in Case 2.

Fig. 11

Overall trend of simulation results of models in Case 2.

In Fig. 10, the MAPEFIT of six models does not exceed 6.5%. In the prediction stage, only the MAPEPRE of ENGM(1,1) model is more than 10%, and that of the other models is less than 7.1%. These models can be used to predict hydroelectricity energy consumption, but the CCRGM(1,1) model has the best fitting effect. In Fig. 11, the trend of original data shows a disordered S-shaped, whereas GM(1,1), DGM(1,1) and ENGM(1,1) overestimate the hydroelectricity consumption, and ARGM(1,1) and NGM(1,1) slightly underestimate it. The Verhulst model shows a saturated S-shaped. Each point of CCRGM(1,1) predictions generally coincides with the original data. The above results show that CCRGM(1,1) is the most accurate model for prediction of hydroelectricity energy consumption in North America.

Future discussions

According to the results of the five cases, regardless of the shape of the data presented, the CCRGM(1,1) model is the most effective and accurate in predicting the clean energy consumption represented by nuclear energy and hydroelectricity energy. Therefore, this section uses this new model to predict the future energy consumption and supply information for the region to formulate energy consumption strategies.

Prediction of nuclear consumption in North America in the next 10 years

According to the prediction method of nuclear energy consumption in the previous section, CCRGM(1,1) model is established by data from 2013 to 2018. At this time, and the MAPEFIT is only 0.3900%. Therefore, this model is used to predict nuclear consumption in the next 10 years, as shown in Table 12 .

Table 12

Prediction of nuclear energy consumption in the next decade.

Year	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028
Data	217.8680	218.1555	218.4013	218.6161	218.8067	218.9779	219.1333	219.2756	219.4066	219.5282

To more intuitively show the future consumption trend of nuclear energy, the data in Table 12 are converted into Fig. 12 , showing clearly that the consumption of nuclear energy in North America is predicted to continue to increase, reaching 219.5282 million tons of oil equivalent in 2028, 0.9222% higher than that in 2018, but it does not reach a peak, indicating that the consumption of nuclear energy in North America will maintain an increasing trend. Therefore, North America is expected to increase the use of nuclear power and reduce the use of fossil fuels.

Fig. 12

The predicted trend of nuclear energy consumption in the next decade.

Prediction of hydroelectricity consumption in North America in the next 10 years

According to the hydroelectricity energy consumption prediction method in Section 4.2, CCRGM(1,1) is established by data from 2009 to 2018. In this case, , and the MAPEFIT is only 3.0028%. Then, the hydroelectricity consumption in the next 10 years is predicted, as shown in Table 13 .

Table 13

Prediction of hydroelectricity energy consumption in the next decade.

Year	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028
Data	160.9689	161.5271	161.9217	162.0885	161.9502	161.4136	160.3676	158.681	156.1997	152.7455

By presenting the data in Table 13 as Fig. 13 , it can be clearly observed that the hydroelectricity consumption in North America shows a trend of continuous rise and fall and is predicted to reach a peak in 2022, reaching 161.9502 million tons of oil equivalent, followed by a fall to 152.7455 million tons of oil equivalent in 2029, down 4.71232% compared with 2018.

Fig. 13

The predicted trend of hydroelectricity energy consumption in the next decade.

Uncertainty analysis

Two application cases show that CCRGM(1,1) model can effectively predict the consumption trend of nuclear energy and hydroelectricity in North America, but the energy situation is complex and changeable, especially with respect to the occurrence of unforeseen events, which leads to the results of prediction model might seem counterintuitive. According to literature [47], many large dams were built in North America before 1975. In recent years, environmental problems involving geological disasters and river ecology have become increasingly severe, and social problems have arisen. These problems were not foreseen, resulting in more dams being demolished by the government than are being built. The United States Energy Information Administration noted that due to the development of wind power technology, wind power in the United States is expected to surpass hydropower for the first time in 2019, accounting for a large proportion of domestic power structure. Therefore, hydropower might show a downward trend after 2022 and the prediction results in Section 4.3.2 conform to the above analysis. To avoid the impact of such unforeseen events, CCRGM(1,1) model can be used for short-term prediction to avoid uncertainty due to medium and long-term prediction.

Conclusion

In this paper, based on the properties of Riccati equation with constant coefficients, the whitening equation of Verhulst model is proposed, and CCRGM(1,1) model is established. In practical cases, the eight evolution metrics of CCRGM(1,1) model are much better than those of GM (1,1), DGM(1,1), NGM(1,1), ARGM(1,1), ENGM(1,1), and Verhulst models, that is, the accuracy of CCRGM(1,1) model is the highest. In short, this paper makes two major contributions:

CCRGM(1,1) model optimizes the whitening equation of Verhulst grey model by mathematical equation, and optimizes the nonlinear term by SA algorithm, which reduces or even eliminates the dependence of the traditional Verhulst model on saturated S-shaped data, thus improving the accuracy and applicability of the traditional grey model. This is a generalization of traditional grey Verhulst model.

CCRGM(1,1) model can effectively predict the consumption of nuclear and hydroelectricity energy consumption in North America in the next decade, which will help local governments make policy decisions.

The CCRGM(1,1) model uses the classical Riccati equation to increase the nonlinearity of the Verhulst model, while the Verhulst model has a better effect on saturated S-shaped data or single-peak data, which has its own limitations [48]. The CCRGM(1,1), as an extension model, cannot completely eliminate these limitations. For clean energy data, the CCRGM (1,1) model alleviates the dependence of the Verhulst model on saturated S-shaped data to a certain extent, but this dependence has not been completely eliminated. Therefore, for countries that develop clean energy later, such as China, India and other countries, the various clean energy data of these countries may show insignificant data characteristics, and the prediction results may show large deviations. In addition, in the process of energy prediction, unforeseen events may occur, such as the global COVID-19 pandemic, which may lead to significant changes in energy consumption in the short term, resulting in a large deviation between the prediction results and the actual situation.

As a single variable grey model, CCRGM(1,1) model can effectively predict energy consumption. However, with the vigorous development of clean energy in China, the United States and other major countries, the world energy system is expected to become increasingly complex, and the factors affecting clean energy consumption are predicted to gradually increase, such as economic, population, and environmental factors. Furthermore, the influencing factors of different energy sources are different. which results in uncertainty in the prediction process. Therefore, fully elucidating the characteristics of these factors affecting different energy consumption and introducing them into CCRGM(1,1) model is a topic of interest. Via subsequent expansion of the CCRGM (1,1) to a multivariate model, the prediction effect might be further improved. How to further weaken the dependence of the Verhulst model on the trend of data change and how to build a multivariable CCRGM model for a variety of clean energy sources is our anticipated next main research direction.

CRediT authorship contribution statement

Xilin Luo: Software, Visualization, Writing - original draft, Writing - review & editing, Validation, Data curation. Huiming Duan: Conceptualization, Methodology, Funding acquisition, Project administration, Supervision. Leiyuhang He: Investigation, Formal analysis, Validation, Data curation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.