Elsevier

Renewable Energy

Volume 181, January 2022, Pages 803-819
Renewable Energy

A novel method for carbon emission forecasting based on Gompertz's law and fractional grey model: Evidence from American industrial sector

https://doi.org/10.1016/j.renene.2021.09.072Get rights and content

Highlights

  • This paper discussed the Gompertz's law in carbon emissions.

  • A FAGGM(1,1) model is created for predicting carbon emissions.

  • The case study reveals the rising of peak in the industrial carbon emission in U.S.

  • The reduction percentage of carbon emission on 2005 levels by 2025 will be 17.01% (in total), 17.89% (in industry) and 26.23% (in commerce) in U.S.

Abstract

With the manufacturing reshoring to the US, increasing attention are focus on its energy consumption and environmental effects and accurate prediction of carbon emissions is vital to controlling growth from the source. Considering the slowing growth in carbon emissions with the Gompertz's law, this paper establishes a Gompertz differential equation. According to the differential information principle and fractional accumulation operator, this differential equation is transformed into a fractional accumulation grey Gompertz model. Furthermore, the chaotic whale optimization algorithm is used to optimize the order of accumulation generation and the grey background value in the proposed model. Then the Gompertz's datasets and six validation cases about carbon emissions are used to show that the proposed model demonstrates better accuracy in all cases and efficiency in the carbon emissions forecasting with several existing models. Three case studies indicate that the proposed model can fit the trend of American industrial carbon emissions better. The model results also reveal the recent policy changes have promoted the uptrend of the industrial and the total carbon emissions in the U.S. The future forecasting suggests that U.S. carbon emission is estimated to be 17.01% (in total emissions) or 17.89% (in industrial emission) percent below 2005 levels by 2025 under current policies, falling short of its commitment submitted to the United Nations Framework Convention on Climate Change.

Introduction

With the continuous progress of global industrialization, world economic development has made tremendous progress. At the same time, the demand for energy is increasing. Energy consumption promotes the development of the global economy, and it is also accompanied by a large number of carbon emissions. Excessive carbon dioxide emissions have caused a series of social and environmental problems, which seriously threatened human survival and social development. According to the Environment Protection Authority data, CO2 was responsible for 82% of all greenhouse gas emissions in 2017. Transportation accounted for 37% of total carbon emissions, and the largest sources of transportation-related carbon emissions were passenger cars (41%), medium- and heavy-duty trucks (24%), and light-duty trucks (17%). Commercial air travel accounted for a relatively small share, at 7% of transportation-related emissions. Increased carbon emissions have become an important factor restricting the sustainable development of the global economy [1]. In order to solve this problem, carbon dioxide emissions forecasting has attracted increasing attention. Accurate prediction of carbon emissions can not only provide a basis for policymakers, especially in the rapid development of block chain technology today, but it also can improve the management of carbon emissions [2].

At present, many forecasting methodologies, which can be divided into three categories, namely, statistical analysis models, nonlinear intelligent models, and grey forecasting methods, have been proposed to settle carbon emissions prediction problems. From the mathematical perspective, most models were obtained according to the carbon emissions time series. In practice, the slowing growth in carbon emissions is usually considered for governments striving to keep a balance between environmental issues and economic development. It is therefore necessary to construct a model that represents the growth law in carbon emissions. In another aspect to gain a high level of prediction accuracy and the model stability, the designing of accumulation generating, the grey background value optimization, and the intelligent optimization algorithm are also our focus.

In aspects of statistical analysis models, Fang et al.[3] proposed an improved Gaussian processes regression method for carbon dioxide emissions forecasting based on a modified PSO algorithm. Xia et al. [4] combined multivariate regression analysis with input-output technology and a structural decomposition analysis to examine their impact and predict carbon emissions in the future. A disadvantage of statistical models is that the accuracy of their predictions is highly dependent on the availability of sufficient input data for parameter estimation. At the same time, some scholars used nonlinear intelligent models to predict carbon dioxide emissions, the nonlinear intelligent models included neural networks, SVM, and ensemble learning method. Zhou et al. [5] proposed a GA-SVM to analyze and predict carbon emissions in China. Heydari et al.[6] combined a GRNN method for optimizing training hyper parameters with the grey wolf optimization algorithm and proposed a GRNN-based prediction method for predicting the total carbon emissions of Iran, Italy, and Canada. However, like much statistical models, nonlinear intelligent models usually require large sample sizes.

Due to the significant changes that have occurred and are occurring in the global energy structure, historical data are not reliable for future energy prediction [7,8]. Moreover, the length of the collected data is short, and carbon emissions are affected by many factors, such as GDP, energy use, and population, which may result in an abundance of uncertainty. As an alternative, grey prediction models represent another class of modelling techniques that can achieve accurate predictions even with sparse samples [46,47]. Grey prediction models have also been successfully applied for the prediction of carbon emissions. Lin et al. [11] applied the GM(1,1) model to estimate carbon emissions in Taiwan from 2010 until 2012 [12]. used a GM and ARIMA model to predict carbon emissions in Brazil from 2008 to 2013. In recent years, grey models, especially the nonlinear grey models, have been used to predict carbon emissions. The nonlinear grey model has been shown to be more effective than ARIMA and other existing linear grey models [13]. [15] introduced the relevant variables of the power exponent term, and analyzed the nonlinearity and uncertainty of carbon emissions and economic growth, and established a multivariate grey model for predicting China's carbon emissions due to fossil energy consumption. Ding et al.[16] employed grey incidence analysis to identify the influential factors that may generate strong and nonlinear effects on emissions, then designed a new discrete grey power model called DGPM(1,N) to estimate Chinese energy-related carbon emissions from 2016 to 2020. Duan et al.[17] established a new multi-kernel GMC(1,N) model based on a Gaussian vector basis kernel function and a global polynomial kernel function combined with the characteristics of grey prediction models to predict the carbon dioxide emissions in Chongqing from 2016 to 2020. Xiao (2020) established a novel NGBM(1,1) optimization model with constraints using Box-Cox transformation to forecast biomass energy consumption in China, the United States, Brazil, and Germany [10].

In addition, the FGMs were introduced in 2013, they are easy to use and quite efficient in improving the accuracy of existing grey models. With its effectiveness in time series forecasting with small samples, FGMs have been applied to many fields, especially in energy forecasting. Ma et al.[14] used a novel fractional grey model called the fractional time delayed grey model to forecast the natural gas and coal consumption in Chongqing China. Wu et al.[18] used the novel FAGMO(1,1,k) to predict China's nuclear energy consumption. Gao et al.[19] presented a new discrete fractional accumulation GM(1,1) model known as FAGM(1,1,D) and applied it to China's carbon emissions. The current FGMs use fractional order accumulation generating operator, which have been proven to be efficient in reducing the errors of existing grey models [20]. With its effectiveness, this operator has been used to rebuild numerous existing grey models [21]. proposed a nonhomogeneous discrete grey model with the first accumulated generating operator. Moreover, Wu et al.[22] presented a novel fractional nonlinear grey Bernoulli model to forecast short-term renewable energy consumption of China.

Gompertz's law (or Gompertz–Makeham law of mortality) states the rate of absolute mortality falls exponentially with current size in bio-demography. Then researchers finds this law also has a good performance on depicting the biology [23], growth of tumors [24] and diffusion of new product or technical innovation [25]. Furthermore, the energy environmentalists also find that the growth of energy consumption follows Gompertz's law [26]. Gutiérrez et al.[26] and Gutiérrez et al. [27] have proposed the stochastic Gompertz innovation diffusion process to respectively model the natural-gas consumption in Spain and the electricity consumption in Morocco. Adam et al. [28] provided the non-homogeneous Gompertz diffusion process to model the monthly peak electricity demand in Mauritius in order to predict the future values on the basis of a GA approach. However, these existing researches of Gompertz model are focused on the enough training data and presetting statistical assumptions, and few researches discuss the application with the limited information data. On the other hands, the four main advantages of FGMs can be summarized through reviewing these exiting researchers. Namely, (1) The fractional accumulation can characterize more information from the initial data. As referred in Wu's researches [29], the initial point can be characterized in the fractional accumulation, which used to be ignored in the classic first order accumulation generating operator. (2) The fractional accumulation can help to weaken the disturbance effect of initial data. Wu has proved that the fractional accumulation can weaken the disturbance error [20] when r(0,1). (3) The fractional accumulation is usually more flexible than the classical integer order accumulation generation because the fractional (or rational) set is denser than the integer set. Thus, fractional accumulation could improve the quality of the modeling sequence more flexibly, which directly affects the performance of the grey prediction model [30]. This accumulation operator also has been adopted successfully in carbon emissions prediction [19]. (4) Fractional accumulation can improve the memory of system. As proven in Ref. [31], the close relation between memory and reality has led to the wide application of fractional accumulation to various field. Thus, the novel grey model combing the FGMs and Gompertz models may be the potential meth to for carbons emission forecasting during limited training data. And designing the parameters optimization for the background value and the order of accumulation generation may further improve the forecasting performance of FAGGM(1,1) model.

The contributions of this paper include the following four aspects:

  • (1)

    The slowing growth in carbon emissions with the Gompertz's law is considered to establish a forecasting model.

  • (2)

    The FAGGM(1,1) for carbon emissions prediction is obtained through transforming the differential equation into a grey model using the differential information principle.

  • (3)

    The CWOA is used to optimize the order of accumulation generation and the grey background value in the FAGGM(1,1) model.

  • (4)

    The novel model is verified by 100 Gompertz's datasets and regional carbon emissions datasets, and is also applied to forecast American industrial carbon emissions.

The rest of this paper is organized as follows. A novel FAGGM(1,1) model for forecasting the carbon emissions is proposed in Section 2. Section 3 presents three kinds of validation analyses, including the ablation research using Gompertz's dataset, the model comparisons with other grey models, and the algorithm comparison in the proposed model. The applications from the U.S. are enumerated in Section 4. And Section 5 summarizes the main conclusions of this paper.

Section snippets

Methodology

As shown in Fig. 1, this section proposed a novel FAGGM(1,1) model for forecasting the carbon emissions. Firstly, the Gompertz differential equation model is introduced for forecasting the carbon emissions which follows the Gompertz's law in subsection 2.1. Considering the difference information principle and the fractional order accumulation, the fractional accumulation grey Gompertz model is derived from the Gompertz differential equation in subsection 2.2. Then a parameter optimization model

Validation

In this section, the advantages of the FAGGM(1,1) model over the existing grey model are demonstrated using two kinds of datasets. In subsection 3.2, the Gompertz's datasets with are used for an ablation validation, and then the annual carbon emissions of countries from the Statistical Review of World Energy are used to validate the prediction performance of FAGGM(1,1) in subsection 3.3. The algorithm comparison has been done in subsection 3.4.

Data collection

This section studies the carbon emissions of the US, which has the top GDP in the world. In the U.S., the major carbon emissions are from its industry and commerce, and the detailed datasets are uploaded as supplementary materials, whose outlines are displayed in Fig. 6. The US has pledged to cut greenhouse gas emissions by 26–28% from 2005 levels by 2025 in the Paris Agreement. However, Donald Trump announced that the US would withdraw from the Paris climate accord in June 2017, and then

Conclusions

With the return of manufacturing, the increasing attentions focus on American industrial carbon emissions forecasting. On the basis of the limited information characteristics, Gompertz's law, and uncertainty of the annual carbon emissions sequences, this research develops a Gompertz differential equation. Then combining with the differential information principle and fractional accumulation operator, this research proposes a novel fractional accumulation grey Gompertz model. Estimating the

CRediT authorship contribution statement

Mingyun Gao: Software, Validation, Visualization, Formal analysis. Honglin Yang: Conceptualization. Qinzi Xiao: Data curation, Writing – original draft. Mark Goh: Writing – review & editing, Supervision.

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.

Acknowledgement

The authors are grateful to the editors and referees for their valuable comments. This work is supported by the China Scholarship Council (CSC) (No. 201906130025, 201906130049), and the National Natural Science Foundation of China (Grant No. 71790593, 72071072).

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