Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 5598709, 17 pages
https://doi.org/10.1155/2021/5598709

Research Article

An Optimization Grey Bernoulli Model and Its Application in Forecasting Oil Consumption

Kai Xu,¹ Xinyu Pang,² and Huiming Duan²

¹School of International Business and Management, Sichuan International Studies University, Chongqing 400031, China
²School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Correspondence should be addressed to Kai Xu; 99002142@sisu.edu.cn

Received 7 February 2021; Revised 19 April 2021; Accepted 13 May 2021; Published 2 June 2021

Academic Editor: Xin Ma

Copyright © 2021 Kai Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Energy consumption in the world is mainly dependent on fossil energy, and oil is one of the main energy sources. Accurate prediction of oil consumption can provide an important basis for national energy security, which can provide reference and early warning for the implementation of the environmental strategy developed by the government. According to the nonlinearity of the energy system, this paper uses the principle of the grey nonlinear prediction model NGBM(1,1) to improve the background value of the model, and by the simulated annealing algorithm, we put forward the optimized grey nonlinear model ONGBM(1,1). At the same time, the model is applied to the oil consumption of China, Chile, Mexico, and Japan. Based on the validity analysis of the existing data of the four countries, the model ONGBM(1,1) is basically superior to the other six grey forecast models. Finally, ONGBM(1,1) is used to predict the oil consumption of the four countries in the next five years, which can provide effective information for energy economic policy.

1. Introduction

With the continuous development of the economy, the increasing consumption of fossil energy has caused high carbon emissions, which has led to a series of environmental and ecological problems. In particular, it is the main cause of global warming, greenhouse effect, and extreme weather. Ecologically sustainable development requires us to pay attention to energy consumption. Energy development is currently in a critical period of transformation and change, facing unprecedented opportunities and challenges. In terms of energy consumption results, nowadays, energy consumption in the world is mainly dependent on fossil energy. Oil is one of the major energy sources, and the proportion of oil in energy consumption shows a rising trend, and oil affects the development of the global economy. Thus, the prediction of oil consumption is a necessary basis for the formulation of economic and social plans and can also provide an important basis for national energy security.

In addition, with the rapid development of the world, the global economic structure has undergone major changes, and the energy structure has also changed accordingly. Therefore, some historical data may have little reference value for the current energy situation or the future energy development trend. Therefore, only the data of recent years are reliable for energy consumption forecasts. In a modelling system with small samples and little information, the grey prediction model has more advantage, which is suitable for some systems with unclear information, and its forecast is to generate the original data sequence, weaken the randomness of the original data, to mine the hidden laws in the data sequence, and finally simulate and predict the data through the reduction operation.

The grey model was proposed by Deng [1] based on the grey theory of small sample and poor information system. The grey prediction model is one of the core parts of the grey system, which is characterized by a few data modelling and simple modelling. At present, it is widely used in energy [2–5], transportation [6–9], environment [10, 11], and other industries [12–18].

Grey prediction model is also expanded from GM(1,1) model for predicting exponential growth data to univariate discrete models DGM(1,1) model and NDGM(1,1) model [19, 20], as well as the single variable nonlinear grey Verhulst model and grey Bernoulli NGBM (1,1) model [21, 22]. Since there are many influencing factors in the model, univariate model is extended to multivariate models, such as GM(1,N), NGM(1,N), GMC(1,N), and other multivariate models [23–25]. At the same time, a lot of researchers work on improving the accuracy of the model, such as optimizing the background value of the grey model, using appropriate algorithms to optimize the parameters, and combining with other machine learning methods to improve the accuracy, which promotes the development and improvement of the grey theoretical system.

The grey prediction model has been widely used by domestic and foreign scholars when predicting energy consumption. Ma et al. [26] established a delayed fractional-order grey prediction model to predict coal and natural gas consumption in Chongqing, China, using the grey Wolf optimization algorithm; Jia et al. [27] proposed a modified GM(1,1) model by Markov chain to predict coal consumption in Gansu Province; Wu et al. [28] used the new multivariable grey prediction model to forecast the power consumption in Shandong Province. Wang et al. [29] proposed the combinational optimization model of GM(1,1)-ARIMA based on the IOWGA operator to predict the total coal consumption in the United States. Duan et al. [30] proposed a new model based on the property of data accumulation and fractional-order accumulation and used particle swarm optimization algorithm to seek the optimal fraction and then predicted the crude oil consumption in China from 2015 to 2020 by the model. In addition, the authors in [2–5, 31–33] have effectively predicted various energy consumption methods, and the grey model has achieved good results in the field of energy.

The energy system presents obvious nonlinear characteristics in real life. Based on this characteristic, in this paper, the nonlinear grey Bernoulli NGBM(1,1) prediction model was optimized, and its modelling principle was perfected by optimizing the background value of the model, the model nonlinearity is embodied, the model structure is optimized, and the model accuracy is improved.

The simulated annealing algorithm is combined with the probabilistic jumping to randomly find the global optimal solution of the objective function in the solution space. Compared with other greedy algorithms, which may be trapped in the local optimal solution and cannot find the solution, the simulated annealing algorithm can jump out of the local optimal solution probabilistically and eventually approach the global optimal solution. At present, many scholars of grey systems also apply this algorithm to optimize grey models. For example, Luo et al. [34] put forward a new grey CCRGM(1,1) prediction model by using the simulated annealing algorithm to optimize and apply it to the prediction of clean energy; Pai and Hong [35] proposed a combined method based on support vector machines and the simulated annealing algorithms for annual electricity load forecasting. Guo et al. [36] optimized the grey prediction model based on the simulated annealing algorithm and applied it to landslide prediction. In this paper, the simulated annealing algorithm is used to calculate the optimal undetermined coefficient in consideration of the operation characteristics, time processing cost, and optimization effect of the algorithm. The model is applied to four countries, China, Chile, Mexico, and Japan, and the oil consumption data of the four countries are taken as the effectiveness analysis of the model. The optimized grey Bernoulli model NGBM(1,1) is compared with the other six grey prediction models, which all show better results. So, the model is used to predict oil consumption in four countries over the next five years.

The remainder sections are organized as follows. Section 2 establishes the optimized grey Bernoulli NGBM(1,1). Section 3 analyzes the validity of the model by using the energy data of four countries. Section 4 forecasts the oil consumption of four countries over the next five years. Section 5 provides the conclusion.

2. Grey Bernoulli Extension Model ONGBM(1,1)

In this section, firstly, the grey Bernoulli model NGBM (1,1) is summarized. According to the grey action of the model and the parameters of the model nonlinearity, the model is extended into a new grey Bernoulli extended model ONGBM(1,1,k,c), and the properties of the model are studied.

2.1. Optimized Grey Bernoulli Model ONGBM (1,1)

The original sequence is

The first-order accumulating generation sequence is defined as

where .

is adjacent to the mean generation sequence of :

where .

Definition 1. The sequences , , and are as shown in equations (1)–(3), so

It is called the grey Bernoulli model and it is abbreviated as NGBM(1,1) model [23], and the whitening equation has the following forms:

where is a real number.
The time response of the NGBM(1,1) model is given by

The reducing value of is

The background value in Definition 1 is defined as follows [24]:

So, the model is defined as follows.

Definition 2. The sequences , , and are as shown in equations (1), (2), and (8), so

It is the optimized model of the grey Bernoulli model and is called ONGBM(1,1) model for short.
The parameters and of the ONGBM(1,1) model are as follows:

where the matrices and are

The reduction function is given by equations (6) and (7).

2.2. Optimization of ONGBM(1,1) Model by the Simulated Annealing Algorithm

The idea of the simulated annealing algorithm (hereinafter referred to as SA) was first proposed by Metropolis et al. In 1983, Kirkpatrick et al. used it for combinatorial optimization and VLSI design. SA to solve combinatorial optimization problem is based on the similarity between solid material annealing process in physics and general combinatorial optimization problem. When a solid substance is annealed, it is usually warmed and melted first, so that the particles can move freely. Then, as the temperature drops, the particles also gradually form a lattice of low energy. If the temperature near the condensation point drops slowly enough, the solid must form the lowest energy ground state.

The specific steps of SA to solve the model are as follows.

Step 1. Initialization: for general data (not big fluctuations), set the variation range of as . An initial solution is selected at random within this range, and the corresponding target value is calculated. Set the initial temperature as and the ending temperature as , generate a random number as the threshold, the cooling law:, is the annealing coefficient, and is the th iteration. Since must be reduced slowly, should be a number close to 1. Finally, convergence conditions are set according to actual requirements, such as the magnitude of error, number of iterations, or ending temperature. If the data are large fluctuation data, the variation range of is set as .

Step 2. At temperature , another point in the neighborhood of the current solution is selected as the new solution, and the difference between the objective function value and is calculated, .

Step 3. When , it means that the new solution is accepted; when , it means that the new solution is accepted according to probability ; and if , then the new solution is accepted, and .

Step 4. Following the cooling rule: , the temperature is lowered.

Step 5. Repeat Steps 2–4 until the convergence condition is satisfied.

3. Verification of ONGBM(1,1) Model

This section is mainly divided into three parts. The first part describes the evaluation indicators mainly adopted in this research. In order to measure the effectiveness of the model, two evaluation indicators APE and MAPE are quoted in the research process, and the calculation method is given in Section 3.1. In the second part, the model ONGBM(1,1) is applied to the data from different countries for verification, and the simulation results are compared and analyzed with the existing grey univariate models, including NGBM(1,1), Verhulst, GM(1,1), NGM(1,1), TDGM(1,1), and AR-GM(1,1) models. Meanwhile, the APE trend comparison diagram and MAPE comparison diagram of different cases have been given in Section 3.2. In the third part, based on the cases in the second part, the annual oil consumption in different regions in the next five years is predicted and analyzed. In the following example analysis, the calculation process of the ONGBM(1,1) model is implemented in accordance with the steps described in Section 2.2.

3.1. Model Evaluation Indexes

The model evaluation indexes adopt the APE and MAPE that are commonly used in the grey model, and the specific calculation formulas are shown as equations (12) and (13). In the comparison process, the smaller the calculated values of APE and MAPE, the better the model effect. The calculation of MAPE is divided into two parts in this study, including MAPEFIT of the fitting part and MAPEPRE of the prediction part. These two evaluation indexes should be comprehensively considered in the evaluation process of the model.

3.2. Case Analysis

In order to illustrate the effectiveness of the model, this section applies the optimized NGBM(1,1) model to the oil data of four countries, respectively, and verifies the effectiveness of the model from different regions, which included the annual oil consumption data for China, Chile, Mexico, and Japan, and the data based on the BP Statistical Review of World Energy.

3.2.1. Case 1: Annual Oil Consumption of China

China is the most populous developing country and the second largest economy in the world. With the rapid development of China, the social productivity of China was significantly enhanced, and at the same time, the energy consumption ability also constantly improved, while nonfossil energy sources such as coal and oil are still the main energy sources in China currently. According to the data released by the National Bureau of Statistics of China, during the development process of China, there has been an obvious gap between the value of energy produced and the value of energy consumption and there has been a situation of short supply. Therefore, part of the energy begins to rely on imports; among them, oil accounts for a large proportion of imported energy. Therefore, the security of oil supply and demand is a problem that China needs to consider in the current development process.

In this paper, the oil consumption data from 1999 to 2018 in China were selected for model building, in which the data from 1999 to 2010 were used to establish the model and train the optimal parameter values, and the optimal parameter values and are calculated. Then the data from 2011 to 2018 were used to verify the validity of the model. The simulation results and APE of the different models are shown in Table 1.

Table 1: Metrics of models for fitting in Case 1.

Year
Raw data
ONGBM
APE
NGBM
APE
GM
APE
NGM
APE
TDGM
APE
Verhulst
APE
AR-GM
APE

(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(%)
(1,1)
(%)

1999
213.7
213.7000
0
213.7000
0
213.7000
0
213.7000
0
213.7000
0
213.7000
0
213.7000
0

2000
228.8
215.0852
−5.9942
212.7806
−7.0015
232.9317
−1.8058
208.0514
−9.0684
219.8465
3.9133
72.66238
−68.2420
231.0545
0.9854

2001
234.0
243.8493
4.2091
241.1982
3.0762
248.7854
−6.3186
230.602
−1.4522
242.3371
3.5628
95.60745
−59.1421
249.255
6.5192

2002
252.7
268.6015
6.2926
265.726
5.1547
265.7182
−5.1516
253.0378
0.1337
264.7132
4.7539
124.4687
−50.7445
268.3428
6.1903

2003
281.3
291.4472
3.6073
288.4105
2.5277
283.8034
−0.8900
275.3594
−2.1118
286.9755
2.0176
159.8201
−43.1852
288.3612
2.5102

2004
328.5
313.3144
−4.6227
310.1574
−5.5837
303.1196
7.7262
297.5674
−9.4163
309.1244
5.8982
201.6125
−38.6263
309.3555
−5.8279

2005
334.3
334.7051
0.1212
331.4585
−0.8500
323.7504
3.1557
319.6625
−4.3786
331.1607
0.9391
248.7314
−25.5963
331.3732
−0.8755

2006
359.2
355.928
−0.9109
352.6165
−1.832827
345.7855
3.73456
341.6451
−4.8872
353.0848
1.7024
298.5667
−16.8801
354.4644
−1.3184

2007
377.0
377.1903
0.0505
373.8355
−0.8394
369.3202
2.0371
363.5158
−3.5767
374.8974
0.5577
346.8195
−8.0054
378.6812
0.4459

2008
384.7
398.6414
3.6240
395.2631
2.7458
394.4568
−2.5362
385.2752
0.1495
396.599
3.0931
387.8443
0.8173
404.0787
5.0373

2009
400.6
420.3955
4.9415
417.0120
4.0969
421.3042
−5.1683
406.9239
1.5786
418.1901
4.3909
415.7108
3.7720
430.7143
7.5173

2010
455.5
442.5437
−2.8444
439.1731
−3.5844
449.9789
1.2121
428.4625
−5.9358
439.6714
3.4750
425.8055
−6.5191
458.6484
0.6912

MAPEFIT (%)

3.3835

3.3903

3.6124

3.8808

3.1185

29.2300

3.4471

2011
472.4
465.1620
−1.5322
461.8220
−2.2392
480.6053
−1.7369
449.8914
−4.7647
461.0433
2.4040
416.3413
−11.8668
487.9443
3.2905

2012
495.3
488.3161
−1.4100
485.0246
−2.0746
513.3161
−3.6374
471.2113
−4.8635
482.3065
2.6234
388.9955
−21.4626
518.6684
4.7180

2013
517.3
512.0644
−1.0121
508.8396
−1.6355
548.2533
−5.9836
492.4227
−4.8091
503.4615
2.6751
348.3102
−32.6677
550.8903
6.4934

2014
539.3
536.4604
−0.5265
533.3208
−1.1087
585.5684
−8.5793
513.5262
−4.7791
524.5088
2.7427
300.1995
−44.3354
584.6831
8.4152

2015
573.3
561.5537
−2.0489
558.5185
−2.5783
625.4232
−9.0918
534.5223
−6.7640
545.4491
4.8580
250.3389
−56.3337
620.1233
8.1673

2016
587.0
587.3918
0.0667
584.4808
−0.4292
667.9906
−13.7974
555.4115
−5.3813
566.2827
3.5293
203.0806
−65.4037
657.2912
11.9747

2017
610.7
614.0203
0.5437
611.2542
0.0907
713.4552
−16.8258
576.1944
−5.6502
587.0104
3.8791
161.089
−73.6222
696.2711
14.0120

2018
641.2
641.4836
0.0442
638.8841
−0.3612
762.0142
−18.8419
596.8716
−6.9133
607.6326
5.2351
125.5214
−80.4240
737.1513
14.9643

MAPEPRE (%)

0.8981

1.3147

9.8118

5.4906

3.4933

48.2645

9.0044

As can be seen in Table 1, the Verhulst model shows poor performance in both the simulation and prediction parts. Except for the Verhulst model, the remaining six models show no significant difference in the fitting part, and the APE all fluctuated at about 3.5%. However, in the prediction part, although the prediction results of the GM(1,1), NGM(1,1), and AR-GM(1,1) models are all below 10%, while they are higher than 5%, compared with the other three models, and the deviation is relatively obvious. Then among the ONGBM(1,1), NGBM(1,1), and TDGM(1,1) models, the accuracy of the optimized NGBM(1,1) model is higher, and its MAPEPRE reaches 0.8981%, less than 1%, and that compared with the unoptimized model, the original prediction accuracy is improved by 0.4166%.

Figure 1 shows the APE comparison chart in Case 1. It can be clearly seen that the Verhulst model shows poor performance in the application of these data, so the results of the Verhulst model have a greater impact on the overall comparison effect of all models. Then the comparison chart that removed the calculated data of the Verhulst model is drawn as shown in Figure 2.

Figure 1: Overall APE comparison of seven models in Case 1.

Figure 2: Overall APE comparison after removing the Verhulst model in Case 1.

In Figure 2, it can be seen that the relative error in the prediction part fluctuates within a small range, among which the NGM(1,1) model has a few large deviations at some points. However, the overall relative error of NGM(1,1) of both simulation and prediction parts is in a relatively stable state. In the prediction part shown in Figure 2, it can also be seen that the relative errors of GM(1,1) and AR-RM(1,1) models show a trend of a gradual rise, and the deviation is gradually larger.

Figure 3 shows the MAPE comparison of the seven models in Case 1, which can more intuitively see the performance difference between different models. The overall level of the fitting part is similar except for the Verhulst model while the ONGBM(1,1) model has more advantages in the prediction part.

Figure 3: Comparison of MAPE in Case 1.

3.2.2. Case 2: Annual Oil Consumption of Chile

Chile is the longest and narrowest country in the world, which is rich in mineral resources, but short of energy. The production of coal, oil, and natural gas in Chile is low and mainly depends on imports from other countries. Chile is not a populous country, and thus, the import volume will fluctuate greatly with the demand of the country. Therefore, it can be seen from Table 2 that the annual oil consumption data of Chile fluctuated to a certain extent. In this case, the oil consumption data of Chile from 2004 to 2018 were selected for analysis. The original data and the calculated data of seven models are shown in Table 2. The data from 2004 to 2012 were used to establish the model, and the data from 2013 to 2018 were used to test the validity of the model.

Table 2: Metrics of models for fitting in Case 2.

Year
Raw data
ONGBM
APE
NGBM
APE
GM
APE
NGM
APE
TDGM
APE
Verhulst
APE
AR-GM
APE

(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(%)
(1,1)
(%)

2004
12.4
12.4
0
12.4
0
12.4
0
12.4
0
12.4
0
12.4
0
12.4
0

2005
12.9
13.2328
2.5796
13.9921
8.4659
15.2246
−18.0201
7.8722
−38.9750
12.7664
1.0358
6.4967
−49.6382
14.5834
13.0500

2006
14.4
15.0446
4.4766
15.8268
9.9086
15.7179
−9.1522
13.6896
−4.9336
15.9771
10.9519
9.3351
−35.1732
15.9538
10.7902

2007
18.5
16.2433
−12.1984
16.9431
−8.4157
16.2272
12.2853
16.2658
−12.0767
17.3294
6.3274
12.7894
−30.8684
16.8138
−9.1144

2008
19.2
17.0617
−11.1369
17.6239
−8.2089
16.7530
12.7446
17.4067
−9.3400
17.8991
6.7758
16.4237
−14.4598
17.3536
−9.6166

2009
18.7
17.6170
−5.7917
18.0100
−3.6901
17.2959
7.5087
17.912
−4.2141
18.1390
3.0001
19.4232
3.8671
17.6924
−5.3884

2010
16.6
17.9785
8.3043
18.1844
9.5445
17.8563
−7.5682
18.1357
9.2513
18.2400
9.8797
20.8529
25.6197
17.9050
7.8614

2011
18.1
18.1921
0.5086
18.205
0.5609
18.4349
−1.8504
18.2348
0.7448
18.2826
1.0088
20.1871
11.5308
18.0384
−0.3402

2012
18.1
18.2896
1.0474
18.0992
−0.0042
19.0322
−5.1506
18.2787
0.9873
18.3005
1.1079
17.6770
−2.3373
18.1222
0.1225

MAPEFIT (%)

5.7554

6.0998

9.2850

10.0653

5.0109

21.6868

7.0355

2013
17.5
18.2946
4.5406
17.9052
2.3154
19.6490
−12.2798
18.2981
4.5607
18.3081
4.6176
14.1695
−19.0316
18.1747
3.8556

2014
17.0
18.2251
7.2062
17.6402
3.7660
20.2856
−19.3273
18.3067
7.6867
18.3113
7.7133
10.5746
−37.7964
18.2077
7.1042

2015
17.1
18.0949
5.8180
17.3204
1.2891
20.9430
−22.4734
18.3105
7.0792
18.3126
7.0912
7.4784
−56.2670
18.2284
6.5989

2016
18.2
17.9152
−1.5646
16.9585
−6.8215
21.6216
−18.7998
18.3122
0.6167
18.3132
0.6218
5.0897
−72.0347
18.2414
0.2275

2017
17.7
17.6952
−0.0273
16.5645
−6.4154
22.3222
−26.1140
18.313
3.46318
18.3134
3.4656
3.3742
−80.9367
18.2496
3.1049

2018
18.1
17.4421
−3.6349
16.1466
−10.7924
23.0455
−27.3231
18.3133
1.1785
18.3135
1.1796
2.1982
−87.8555
18.2547
0.8546

MAPEPRE (%)

3.7986

5.2333

21.0529

4.0975

4.1148

58.9870

3.6243

It can be seen from the calculation results in Table 2 that, in the fitting part, the effects of ONGBM(1,1), NGBM(1,1), and TDGM(1,1) are closer, and the effect of TDGM(1,1) is slightly better than that of the optimized NGBM(1,1) model, and the accuracy of ONGBM(1,1) is 0.3444% higher than that of the unoptimized NGBM(1,1) model. In the prediction part, the AR-GM(1,1) model has the best effect, and compared with ONGBM (1,1) model, the MAPEPRE of the AR-GM(1,1) model was 0.1743% higher, but the fitting error of the AR-GM(1,1) model is relatively higher than that of ONGBM (1,1) model. Although the MAPEFIT and MAPEPRE of the ONGBM (1,1) model are not optimal in this case, the comprehensive consideration shows that the ONGBM(1,1) model still has certain desirability, and the and can be calculated in this case.

In the APE comparison shown in Figure 4, it can be seen that the APE of the Verhulst model in the prediction part is obviously larger than that of other models, and from the calculation results of Table 3, it can be known that its MAPEPRE has reached 58.9870%, which led the selection of the error interval too large in the process of mapping and then affected the comparison effect of other models. Therefore, in this case, the relative error value of the Verhulst model is removed and the APE comparison chart is redrawn, as shown in Figure 5.

Figure 4: Overall APE comparison of seven models in Case 2.

Table 3: Metrics of models for fitting in Case 3.

Year
Raw data
ONGBM
APE
NGBM
APE
GM
APE
NGM
APE
TDGM
APE
Verhulst
APE
AR-GM
APE

(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(%)
(1,1)
(%)

1999
87.1
87.1
0
87.1
0
87.1
0
87.1
0
87.1
0
87.1
0
87.1
0

2000
91.2
81.9017
−10.1955
80.1035
−12.1672
89.1865
2.2078
84.6546
−7.1770
88.4599
3.0045
24.3087
−73.3457
89.4691
−1.8980

2001
89.7
87.9096
−1.9960
86.1203
−3.9907
89.7028
−0.0031
85.8011
−4.3466
89.3157
0.4284
30.4944
−66.0040
90.9500
1.3935

2002
85.7
91.5129
6.7828
89.8657
4.8608
90.2221
−5.2766
86.8598
1.3534
90.1060
5.1412
37.8939
−55.7830
91.8757
7.2062

2003
88.0
93.7467
6.5304
92.3067
4.8940
90.7444
−3.1186
87.8376
−0.1845
90.8359
3.2226
46.5390
−47.1148
92.4545
5.0619

2004
91.6
95.0866
3.8063
93.8903
2.5003
91.2697
0.3606
88.7406
−3.1216
91.5099
0.0984
56.3387
−38.4948
92.8162
1.3277

2005
93.8
95.7964
2.1284
94.8653
1.1357
91.7981
2.1342
89.5745
−4.5048
92.1323
1.7779
67.0245
−28.5453
93.0424
−0.8077

2006
92.8
96.0402
3.4916
95.3867
2.7874
92.3295
0.5070
90.3447
−2.6458
92.7071
0.1001
78.1046
−15.8355
93.1837
0.4135

2007
96.4
95.9279
−0.4897
95.5584
−0.8731
92.8640
3.6680
91.0559
−5.5437
93.2379
3.2802
88.8524
−7.8294
93.2721
−3.2447

2008
96.2
95.5373
−0.6888
95.4540
−0.7755
93.4017
2.9089
91.7127
−4.6645
93.7280
2.5696
98.3542
2.2392
93.3273
−2.9861

2009
92.9
94.9256
2.1804
95.1278
2.3981
93.9424
−1.1220
92.3193
−0.6251
94.1807
1.3785
105.6322
13.7053
93.3619
0.4972

2010
93.3
94.1363
0.8964
94.6212
1.4161
94.4862
−1.2714
92.8795
−0.4507
94.5987
1.3919
109.8311
17.7182
93.3834
0.0894

2011
94.9
93.2032
−1.7880
93.9663
−0.9838
95.0332
−0.1404
93.3968
−1.5840
94.9847
0.0892
110.4153
16.3491
93.3969
−1.5838

2012
96.4
92.1533
−4.4053
93.1889
−3.3310
95.5834
0.8471
93.8746
−2.6198
95.3412
1.0984
107.3079
11.3153
93.4054
−3.1065

2013
93.8
91.0081
−2.9764
92.3096
−1.5889
96.1367
−2.4912
94.3158
0.5499
95.6704
1.9940
100.9113
7.5813
93.4107
−0.4151

MAPEFIT (%)

3.4540

3.1216

1.8612

2.8122

1.8268

28.7044

2.1451

2014
89.5
89.7856
0.3191
91.3456
2.0621
96.6933
−8.0372
94.7232
5.8360
95.9744
7.2339
92.0010
2.7944
93.4139
4.3731

2015
88.5
88.5004
0.0004
90.3111
2.0464
97.2530
−9.8904
95.0995
7.4571
96.2551
8.7628
81.5382
−7.8665
93.4160
5.5548

2016
89.1
87.1648
−2.1720
89.2179
0.1323
97.8160
−9.7823
95.4470
7.1235
96.5143
8.3214
70.4734
−20.9052
93.4173
4.8454

2017
85.8
85.7891
−0.0127
88.0761
2.6528
98.3823
−14.6647
95.7680
11.6177
96.7538
12.7666
59.6004
−30.5357
93.4181
8.8789

2018
82.8
84.3822
1.9108
86.8944
4.9449
98.9519
−19.5071
96.0643
16.0197
96.9748
17.1194
49.4852
−40.2353
93.4186
12.8244

MAPEPRE (%)

0.8830

2.3677

12.3763

9.6108

10.8408

20.4674

7.2953

Figure 5: Overall APE comparison after removing the Verhulst model in Case 2.

It can be seen in Figure 5 that the fitting error of the NGM(1,1) model in 2005 was obviously higher, and there was an abnormality at one point, and then, the subsequent nodes tended to be stable. In the fitting part, before 2010, the errors of the seven models were relatively obvious, and the fluctuation range was relatively large. After 2011, except for the GM(1,1) and NGBM(1,1) models, the relative errors of the other models showed no obvious abnormal changes and were relatively stable.

In order to see the MAPE comparison of the different models in Table 2 intuitively, Figure 6 presents the MAPE comparison diagram of the simulation part and prediction part, through that the comparison effect is more obvious.

Figure 6: Comparison of MAPE in Case 2.

3.2.3. Case 3: Annual Oil Consumption of Mexico

Mexico is rich in oil and gas resources and is one of the largest oil producers in the world. EIA data show that the crude oil production of Mexico has shown a downward trend since 2010, and corresponding to the “BP Statistical Review of World Energy” data released by BP, it can be seen that the oil consumption data of Mexico have shown a gradual decrease since 2007. This paper selects the oil consumption data of Mexico from 1999 to 2018. The original data of Mexico’s annual oil consumption and the simulation data of the ONGBM(1,1) model and the other comparison models are given in Table 1. Then, the data from 1999 to 2013 are used to build the model and calculate the model parameters, and the data from 2014 to 2018 are used to test the validity of the model.

The oil consumption data of Mexico are substituted into the model, and the optimal parameters of the ONGBM(1,1) model are calculated by the simulated annealing algorithm, and then and . From the calculation results of the ONGBM(1,1) model and the other six comparison models in Table 1, it can be seen that, except for the Verhulst model, all other models exhibit higher accuracy in the fitting process, among which the MAPEFIT of the GM(1,1), NGM(1,1), TDGM(1,1), and AR-GM(1,1) model are all lower than 3%, and there was no significant difference between them, and it is slightly higher than that of the ONGBM(1,1) and NGBM(1,1) models in the process of fitting. And the best performance in the simulation part is shown by the TDGM(1,1) model, whose MAPEFIT reaches 1.8268%. In the prediction part, the Verhulst model also shows poor results compared with other models, while GM(1,1) and TDGM(1,1) models showed a slightly poor performance in the prediction part, with MAPEPRE higher than 10%, and the prediction errors of the NGM(1,1) and AR-GM(1,1) models are below 10%, while there is a slight deviation compared with the ONGBM(1,1) model. The comparison between ONGBM(1,1) and NGBM(1,1) shows that the accuracy of the optimized model in the fitting effect is slightly increased by 0.3324%, but the prediction error is significantly improved, which reaches less than 1%, reflecting the better performance.

In order to see the comparison calculation results in Table 1 more intuitively, Figure 7 shows the APE comparison diagram of seven models. It can be clearly seen that the deviation of the Verhulst model is very large, and the largest deviation of which reached more than 70%, because the error of the Verhulst model is too large and the value of the vertical axis of the image is increased, making the contrast effect of other models not obvious, and therefore, Figure 8 shows the contrast effect after removing the Verhulst model.

Figure 7: Overall APE comparison of seven models in Case 3.

Figure 8: Overall APE comparison after removing the Verhulst model in Case 3.

It can be seen from Figure 7 that the fitting errors of the first four points of the ONGBM(1,1) model fluctuate greatly in the fitting part and tend to be stable in the subsequent nodes. In the fitting part, the relative errors of most nodes of the seven models fluctuate less than 5% after the fifth time node. In the prediction part, the APE of GM(1,1) was significantly increased by node, while the relative error of ONGM(1,1) and ONGBM(1,1) models fluctuated around 2% in this part, and ONGBM(1,1) showed a better effect.

The comparison of the MAPE values of the seven models in the simulation and prediction part is shown in Figure 9, and the effects of each model can be clearly distinguished from the figure.

Figure 9: Comparison of MAPE in Case 3.

3.2.4. Case 4: Annual Oil Consumption in Japan

Japan is a highly developed capitalist country, which is located on an island in East Asia. Because the domestic resources in Japan are relatively scarce and the proven oil reserves are limited, its energy is mainly dependent on imports. However, as the third largest economy in the world, Japan has become the world’s fifth largest oil consumer after the United States, China, India, and Russia. Japan is also the fourth largest crude oil importer after the United States, China, and India. Therefore, the analysis and forecast of oil consumption in Japan are particularly important. According to the data from “BP Statistical Review of World Energy,” annual oil consumption of Japan has shown a downward trend since 2012, which is closely related to the aging of the country's internal population, the decline of the population, and the improvement of energy efficiency by Japanese domestic technology. This case selects the data from 2008 to 2018 to build the model and evaluates the validity of the model. The data from 2008 to 2012 are used to build the model, and the data from 2013 to 2018 are used to verify the validity of the model. The original data and simulation data of the seven models are shown in Table 4.

Table 4: Metrics of models for fitting in Case 4.

Year
Raw data
ONGBM
APE
NGBM
APE
GM
APE
NGM
APE
TDGM
APE
Verhulst
APE
AR−GM
APE

(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(1,1)
(%)
(%)
(1,1)
(%)

2008
208.2061
208.2
0
208.2
0
208.2061
0
208.2061
0
208.2061
0
208.2
0
208.2061
0

2009
210.516
211.2094
0.3370
207.897
−1.2366
211.3859
−0.4132
157.4182
−25.2227
209.8949
0.2951
159.7675
−24.1010
215.0505
2.1540

2010
211.032
218.3389
3.4781
216.5369
2.6241
213.9545
−1.3848
187.5574
−11.1237
215.0468
1.9024
219.7777
4.1600
215.3288
2.0361

2011
224.8901
218.4848
−2.8525
218.4805
−2.8544
216.5543
3.7066
203.2692
−9.6140
217.6672
3.2118
241.2857
7.2858
215.3401
−4.2465

2012
214.6511
215.0810
0.1775
216.9167
1.0325
219.1857
−2.1125
211.4598
−1.4868
219.0000
2.0260
207.317
−3.4387
215.3406
0.3212

MAPEFIT (%)

1.7113

1.9369

1.9043

11.8618

1.8588

9.7464

2.1894

2014
204.0364
209.6452
2.7672
213.2665
4.5424
221.8490
−8.7301
215.7296
5.7310
219.6779
7.6660
143.8108
−29.5045
215.3406
5.5403

2015
196.5073
202.9962
3.3059
208.3029
6.0065
224.5448
−14.2679
217.9555
10.9147
220.0227
11.9667
85.5894
−56.4431
215.3406
9.5840

2016
190.96
195.6266
2.4223
202.4959
6.0188
227.2733
−19.0162
219.1159
14.7444
220.1980
15.3111
46.3702
−75.7224
215.3406
12.7674

2017
187.8064
187.8537
0.0286
196.1525
4.4475
230.0349
−22.4852
219.7208
16.9933
220.2872
17.2949
23.8493
−87.3007
215.3406
14.6610

2018
182.3648
179.8905
−1.3758
189.4826
3.8830
232.8301
−27.6727
220.0361
20.6571
220.3326
20.8197
11.9385
−93.4548
215.3406
18.0823

MAPEPRE (%)

1.9800

4.9797

18.4344

13.8081

14.6117

68.4851

12.1270

From the results shown in Table 4, it can be observed that the NGM(1,1) model has the largest deviation in the fitting part. The ONGBM(1,1), NGBM(1,1), GM(1,1), and AR-GM(1,1) models have similar performance effects, but the effect of ONGBM(1,1) is slightly better than the other three models. Substituting the data in Case 4 into the Verhulst model, it can be found that the fitting part showed good results. The MAPEFIT value was less than 10%, but its prediction error was about 65%, and the prediction performance was poor. In fact, except for the ONGBM(1,1) model and NGBM(1,1) model, the prediction error of other models exceeds 10%. But on the whole, the prediction effect of the optimized NGBM(1,1) model has been significantly improved, and the MAPEFIT value has increased by 2.997%. The parameters corresponding to the optimized ONGBM(1,1) are and .

It can be seen from Figure 10 that, except for the large deviation of the Verhulst and the NGM(1,1) model in 2010, the relative error of the other years presents a relatively stable state. In fact, it can be known from the calculation results of Table 4 that the MAPEPRE of the Verhulst model reached 68.4851%. It can also be seen in the comparison chart of Figure 10 that in the prediction part, the relative error value of the Verhulst model shows a gradually increasing trend. Because the error calculated by it is too large and affects the comparison effect of other models, the comparison result after removing the relative error data of the Verhulst model is shown in Figure 11.

Figure 10: Overall APE comparison of seven models in Case 4.

Figure 11: Overall APE comparison after removing the Verhulst model in Case 4.

In the APE comparison chart shown in Figure 11, it can be clearly seen that the relative errors calculated by the GM(1,1), NGM(1,1), TDGM(1,1), and AR-GM(1,1) models are gradually increasing in the prediction part, and each model showed the best performance in 2013. Moreover, the ONGBM(1,1) model is indeed better than other models.

The comparison of the MAPE values of the seven models in the simulation and prediction part is shown in Figure 12, and the effects of each model can be clearly distinguished from the figure.

Figure 12: Comparison of MAPE in Case 4.

4. Application of the ONGBM(1,1) Model

Based on the modelling analysis in Section 3.2, this paper predicts the oil consumption of the above four countries in the next five years, and the forecast data are shown in Table 5.

Table 5: Forecast results of oil consumption in the four countries in the next 5 years.

Year
Country

Mexico
Chile
China
Japan

2019
82.9514
17.1622
669.8260
164.3109

2020
81.5033
16.8605
699.0912
155.9098

2021
80.0434
16.5415
729.3230
147.7182

2022
78.5763
16.2089
760.5658
139.7819

2023
77.1063
15.8656
792.8642
132.1313

It can be analyzed from the forecast results in Table 5 that the annual oil consumption of most countries in the next five years will still show a continuous downward trend. However, China is the most populous country in the world, which is in a stage of full-speed development currently. Although it has to take the initiative to adjust its energy structure, due to its large population, vast areas of economic development, and too many related factors, it will often be implemented after the policy is implemented. At the same time, since coal is the first energy source in China, reducing the dependence on coal is a major goal of China. Therefore, the consumption of oil as a substitute for coal may still increase; Chile is an energy-scarce country; in recent years, as the world gradually began to adjust its state of energy structure, Chile also established its own policy because Chile belongs to small population, and thus, the implementation of measures can see the effect relatively quickly. Therefore, the country may have a downward trend in its annual oil consumption under the current environment of energy restructuring. The oil production of Mexico has been declining since 2010, and its annual oil consumption has been declining year by year since 2007; the resources in Japan are relatively scarce, and it mainly depends on imports, and the data show that the oil consumption of Japan has shown a downward trend since 2012. Based on the above reasons and the forecast results of annual oil consumption in the next five years, it can be concluded that oil consumption is on a downward trend in most countries.

5. Conclusion

This paper optimizes the background value of the NGBM(1,1) model and proposes the ONGBM(1,1) model on this basis, and the simulated annealing algorithm was used to optimize undetermined coefficients in the NGBM(1,1) model. Actual oil consumption data from four countries, China, Chile, Mexico, and Japan, were used to test the effectiveness of the model. The results show that the optimized ONGBM(1,1) model is basically better than the other six grey prediction models. At the same time, applying the model to the oil consumption forecasts of the above four countries, it can be concluded that the oil consumption of the other three countries is showing a downward trend, except for China. This phenomenon is related to different factors such as the development measures of the own country, geographic location, resource ownership, and the economic development status of the country.

In this paper, the background values of the NGBM(1,1) model were optimized considering the nonlinear structure of the energy system. The simulated annealing algorithm is used to find the optimal undetermined coefficients, and an optimized ONGBM(1,1) model is established. From the two perspectives of modelling mechanism and data applicability, the actual oil consumption data of China, Chile, Mexico, and Japan were selected to analyze the validity of the model. The selected data contain three types, namely, the data with steady rise, the data with irregular fluctuation, and the data with gradual decline and fluctuation. The comparison results show that the new model has a good effect under the two indexes, and the effect of the optimized ONGBM(1,1) model is basically better than that of the other six grey prediction models. Therefore, on the basis of the modelling analysis in Section 3.2, the model is continued to be applied to the oil consumption forecast of these four countries in 2019–2023. The forecast results show that, except China, the oil consumption of the other three countries shows a downward trend. This phenomenon is related to different factors such as the country's development measures, geographical location, resource availability, and national economic development.

The optimization model proposed in this paper mainly considers the development trend of the variable itself and then research and analysis on it. The nonlinear characteristic of the energy system is considered in the research, but the influence of other factors on energy development is rarely considered. To a certain extent, this problem restricts the application range of the model, so how to solve this problem better considering the influence of other factors on variables will be the focus of our future work.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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