Modeling, simulation, and prediction of global energy indices: a differential approach

doi:10.1007/s11708-021-0723-6

Front. Energy

2022, Vol. 16

Issue (2) : 375-392 https://doi.org/10.1007/s11708-021-0723-6

RESEARCH ARTICLE

Modeling, simulation, and prediction of global energy indices: a differential approach

Stephen Ndubuisi NNAMCHI¹(

), Onyinyechi Adanma NNAMCHI², Janice Desire BUSINGYE³, Maxwell Azubuike IJOMAH⁴, Philip Ikechi OBASI⁵

¹. Department of Mechanical Engineering, Kampala International University, 20000 Kampala, Uganda
². Department of Agricultural Engineering and Bio Resources, Michael Okpara University of Agriculture, Umudike, Umuahia, Nigeria
³. Directorate of Human Resource/Finance, Kampala International University, 20000 Kampala, Uganda
⁴. Department of Mathematics and Statistics, Faculty of Sciences, University of Port Harcourt, PMB 5323 Choba Port Harcourt, Nigeria
⁵. Department of Macroeconomic Analysis (Ministry of Finance), Budget and National Planning, Abuja, Nigeria

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Abstract

Modeling, simulation, and prediction of global energy indices remain veritable tools for econometric, engineering, analysis, and prediction of energy indices. Thus, this paper differentially modeled, simulated, and non-differentially predicated the global energy indices. The state-of-the-art of the research includes normalization of energy indices, generation of differential rate terms, and regression of rate terms against energy indices to generate coefficients and unexplained terms. On imposition of initial conditions, the solution to the system of linear differential equations was realized in a Matlab environment. There was a strong agreement between the simulated and the field data. The exact solutions are ideal for interpolative prediction of historic data. Furthermore, the simulated data were upgraded for extrapolative prediction of energy indices by introducing an innovative model, which is the synergy of deflated and inflated prediction factors. The innovative model yielded a trendy prediction data for energy consumption, gross domestic product, carbon dioxide emission and human development index. However, the oil price was untrendy, which could be attributed to odd circumstances. Moreover, the sensitivity of the differential rate terms was instrumental in discovering the overwhelming effect of independent indices on the dependent index. Clearly, this paper has accomplished interpolative and extrapolative prediction of energy indices and equally recommends for further investigation of the untrendy nature of oil price.

Keywords energy indices differential model normalization simulation inflation/deflation predictive factor and prediction rate

Corresponding Author(s): Stephen Ndubuisi NNAMCHI

About author:

Tongcan Cui and Yizhe Hou contributed equally to this work.

Online First Date: 19 March 2021 Issue Date: 25 May 2022

Cite this article:

Stephen Ndubuisi NNAMCHI,Onyinyechi Adanma NNAMCHI,Janice Desire BUSINGYE, et al. Modeling, simulation, and prediction of global energy indices: a differential approach[J]. Front. Energy, 2022, 16(2): 375-392.

URL:

https://academic.hep.com.cn/fie/EN/10.1007/s11708-021-0723-6
https://academic.hep.com.cn/fie/EN/Y2022/V16/I2/375

Fig.1 Global energy indices.

Index	Function and the differential	Regression coefficient, R²
ec	$ec (-) = 1 .07525039 − 2 .63821125 τ + 1 .74979270 τ 2$ $d ec d τ = − 2 .63821125 + 3 .4995854 τ$	0.95957781
gdp	$gdp (-) = 0 .98114756 − 0 .11750495 τ − 0 .91242700 τ 2$ $d gdp d τ = − 0 .11750495 − 1 .824854 τ$	0.97513034
cde	$gdp (-) = 0 .85504123 − 0 .41672116 τ − 0 .49400846 τ 2$ $d gdp d τ = − 0 .41672116 − 0 .98801692 τ$	0.93317944
hdi	$hdi (-) = 1 .10224884 − 2 .00347452 τ + 0 .92117598 τ 2$ $d hdi d τ = − 2 .00347452 + 1 .84235196 τ$	0.96667570
op	$op (-) = 0 .75398663 + 1 .53674211 τ − 2 .74011768 τ 2$ $d op d τ = 1 .53674211 − 5 .48023536 τ$	0.90916101

Tab.1 Differential of normalized indices with respect to normalized time (τ)

Index	Unit	Coefficients	ec, gdp, cde, hdi, op
Index	Unit	Coefficients	i = 1	i = 2	i = 3	i = 4	i = 5
ec	(-)	$α ′ i$	–0.40058	–2.18962	0.02215	–1.05080	0.31149
gdp	(-)	$β ′ i$	1.14178	0.20888	–0.01155	0.54794	–0.16243
cde	(-)	$δ ′ i$	–0.00625	0.11309	0.61818	0.29666	–0.08794
hdi	(-)	$ε ′ i$	–0.55319	–0.21088	–1.15273	0.01166	0.16399
op	(-)	$φ ′$	–0.48779	0.62730	3.42888	–0.03468	1.64551

Tab.2 Coefficients of normalized system of linear Eqs. (6)–(10)

Constant term (unexplained coefficient)	Indices
	ec	gdp	cde	hdi	op
	(-)	(-)	(-)	(-)	(-)
$g ′ i$	0.80684	–1.91392	–1.38934	–0.18983	–3.85809

Tab.3 Constant terms in the normalized system of linear Eqs. (6)–(10)

Initial condition	Indices
	ec₀	gdp₀	cde₀	hdi₀	op₀
	(-)	(-)	(-)	(-)	(-)
$τ = 0$	1.05000	1.00000	0.85000	1.00000	0.80000

Tab.4 Initial conditions of normalized indices in Eqs. (6)–(10)

Energy index	Unit	Mathematical function, $∃ ? ? ? ? ? ? ? 1982 ≤ t ≤ 2017$	Regression coefficient, R²
EC	Mtoe	$EC = − 4545220 .6308 + 4531 .9472 t − 1 .1291 t 2 ?$	1.0000
GDP	US$ Billion	$GDP = 10807510 .9879 − 10918 .1087 t + 2 .7576 t 2 ?$	1.0000
CDE	Btoe	$EC = 32377 .8112 − 32 .8857 t + 0 .0084 t 2 ?$	1.0000
HDI	%	$HDI = − 444527 .2179 + 44 .0924 t − 0 .0109 t 2 ?$	1.0000
OP	US$/Mtoe	$EC = 6205651 .8570 − 6233 .62602 t + 1 .5655 t 2 ?$	1.0000

Tab.5 Simulated (or interpolative) results of dimensional energy indices

Tab.6 Prediction rates (R) for innovative prediction models

Fig.2 Normalized history.

Fig.3 Normalized rate of change.

Fig.4 Normalized field data and simulated result for energy indices.

Fig.5 Dimensional field data and simulated result for energy indices.

Fig.6 Relative error in simulation or interpolative prediction of energy indices.

Fig.7 Predicted energy index for EC.

Fig.8 Predicted energy index for GDP.

Fig.9 Predicted energy index for CDE.

Fig.10 Predicted energy index of HDI.

Fig.11 Predicted energy index for OP.

Symbol	Index	Unit
EC	Dimensional energy consumption	Mtoe
GDP	Dimensional gross domestic product	US$ Billion
CDE	Dimensional carbon dioxide emission	Btoe
HDI	Dimensional human development index	%
OP	Dimensional oil price	US$/Mtoe
ec	Normalized energy consumption	–
gdp	Normalized gross domestic product	–
cde	Normalized carbon dioxide emission	–
hdi	Normalized human development index	–
op	Normalized oil price	–
t	Dimensional time	year
τ	Normalized time	–
$α i, i = {e c, g d p, c d e, h d i, o p}$	Dimensional EC linear model coefficients	(Mtoe\|year) \|Mtoe, (Mtoe\|year) \|US$ Billion, (Mtoe\|year) \|Btoe, (Mtoe\|year) \|%, (US$ Billion\|year) \|US$ Billion
$β i, i = {g d p, e c, c d e, h d i, o p}$	Dimensional GDP linear model coefficients	(US$ Billion\|year) \|US$ Billion, (US$ Billion\|year) \|Mtoe, (US$ Billion\|year) \|Btoe, (US$ Billion\|year) \|%, (US$ Billion\|year) \|US$ Billion
$δ i, i = {c d e, e c, g d p, h d i, o p}$	Dimensional CDE linear model coefficients	(Btoe\|year) \|Btoe, (Btoe\|year) \|Mtoe, (Btoe\|year) \|US$ Billion, (Btoe\|year) \|%, (Btoe\|year) \|US$ Billion
$ε i, i = {h d i, e c, g d p, c d e, o p}$	Dimensional HDI linear model coefficients	(%\|year) \|%, (%\|year) \|Mtoe, (%\|year) \|US$ Billion, (%\|year) \|Btoe, (%\|year) \|US$ Mtoe
$φ i, i = {o p, e c, g d p, c d e, h d i}$	Dimensional OP linear model coefficients	(US$/Mtoe/year) \|US%/Mtoe, (US$/Mtoe/year) \| Mtoe, (US$/Mtoe/year) \| US$ Billion, (US$/Mtoe/year) \| Btoe, (US$/Mtoe/year) \|%
$α i', i = {e c, g d p, c d e, h d i, o p}$	Normalized ec linear model coefficients	–, –, –, –, –
$β i', ? i = {g d p, e c, c d e, h d i, o p}$	Normalized gdp linear model coefficients	–, –, –, –, –
$δ i', ? i = {c d e, e c, g d p, h d i, o p}$	Normalized cde linear model coefficients	–, –, –, –, –
$ε i', ? i = {h d i, e c, g d p, c d e, o p}$	Normalized hdi linear model coefficients	–, –, –, –, –
$φ i', ? i = {o p, e c, g d p, c d e, h d i}$	Normalized op linear model coefficients	–, –, –, –, –
$g i, ? i = {e c, g d p, c d e, h d i, o p}$	The unexplained coefficients in dimensional linear equations	Mtoe/year, (US$ Billion)/year, Btoe/year, %/year, (US$/Mtoe)/year
$g i', ? i = {e c, g d p, c d e, h d i, o p}$	The unexplained coefficients in normalized linear equations	–, –, –, –, –
${E C 0, G D P 0, C D E 0, H D I 0 O P 0}$	Reference or initial dimensional indices	Mtoe, US$ Billion, Btoe, %, US$/Mtoe
${e c 0, g d p 0, c d e 0, h d i 0, o p 0}$	Reference or initial normalized indices	–, –, –, –, –
rand( )	Random number
$R i, ? i = {e c, g d p, c d e, h d i, o p}$	Rate factors	–, –, –, –, –
R²	Regression coefficient	–
Subscripts
f	Final
0	Initial
Superscript
k	Power of time

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