Fractional-Order Accumulative Gray Model for Carbon Emission Prediction: A Case Study of Shandong Province

Abstract

Against the backdrop of global climate change, accurate prediction of carbon emissions is crucial for formulating effective emission reduction policies. Utilizing data from the China Energy Statistical Yearbook and the Shandong Statistical Yearbook between 2010 and 2022, this study estimates carbon emissions in Shandong Province from 2016 to 2022 using the carbon emission factor method and projects future trends through the fractional-order accumulated grey model FAGM(1,1). The forecast results indicate that both total carbon emissions and per capita carbon emissions in Shandong will follow a trajectory characterized by ‘slow increase-peak-steady decline’, while carbon emission intensity is expected to decrease consistently year by year. Based on these projections, this study proposes that Shandong should accelerate the optimization of its energy supply structure to establish a clean and low-carbon energy system, promote green transformation and upgrading of industries to cultivate new economic growth drivers, and enhance policy-market coordination mechanisms to strengthen institutional incentives and constraints. These findings provide a scientific basis for Shandong to achieve its carbon peak and carbon neutrality goals and also offer methodological references for other industrialized provinces facing similar challenges.

1. Introduction

Global warming and the intensification of the greenhouse effect continue to escalate, posing multifaceted threats to ecosystems, climate systems, and socioeconomic development. In response, the 2015 Paris Agreement established a global framework for temperature control. However, the United Nations Environment Program’s 2023 Emissions Gap Report indicates that global greenhouse gas emissions increased by 1.2% from 2021 to 2022, reaching a record high of 57.4 billion tons of carbon dioxide equivalent [1]. This growth underscores a significant misalignment between current mitigation efforts and the Paris Agreement goals. There is now greater urgency than ever to foster economy-wide low-carbon transitions across all nations. According to the International Energy Agency’s CO₂ Emissions in 2023 report, China’s energy-related carbon dioxide emissions reached 12.6 billion tons in 2023, retaining its position as the world’s largest emitter of CO₂ [2]. As a responsible major country, China announced its dual carbon goals in 2020, to peak carbon emissions before 2030 and achieve carbon neutrality before 2060. This commitment not only demonstrates China’s resolve to address climate change but also offers a pivotal Chinese contribution to the evolution of global climate governance.

As a major energy-producing and consuming province in China, Shandong accounts for nearly one-tenth of the nation’s total energy consumption and carbon emissions, making it a representative high-carbon-emitting region [3]. The province has developed an industrial structure dominated by heavy industries, chemical manufacturing, energy production, and manufacturing. While sectors such as petroleum, chemicals, steel, and building materials have substantially contributed to economic growth, they have also imposed considerable energy and environmental pressures. Shandong’s energy consumption is characterized by a distinct “high-carbon, high-coal” profile [4,5], with coal representing 63% of total energy consumption, while clean energy utilization remains relatively low. This coal-centric energy mix, coupled with an industrial composition in which heavy industries constitute over 60% of total output, has consistently placed Shandong among the top three provinces in China for total carbon emissions. These structural constraints pose several developmental challenges: economically, they result in high energy intensity and relatively low resource utilization efficiency; environmentally, they generate significant ecological pressure and limited carbon sink capacity. With the deepening implementation of China’s dual-carbon strategy, Shandong faces mounting pressure to accelerate its green and low-carbon transition. In August 2022, the State Council issued guidelines supporting Shandong in its shift from old to new growth drivers to pursue green, low-carbon, and high-quality development, designating it as China’s first provincial-level pilot zone for green, low-carbon, high-quality development. This elevates Shandong’s low-carbon transition to a national strategic priority. Within this context, scientifically forecasting future carbon emission trends and conducting in-depth analyses of influencing factors and their mechanisms are of strategic importance for formulating province-specific emission reduction pathways and achieving carbon peak and neutrality targets in alignment with national objectives. This research is not only critical for Shandong’s own green transition but also provides valuable insights for the sustainable development of other industrial provinces with similar characteristics.

Although significant research has been conducted on carbon emission forecasting [6,7,8,9,10], there remains a scarcity of high-accuracy models suited to small-sample contexts for predicting short-term emission trends in high-carbon provinces like Shandong. This study employs a fractional-order accumulative grey model to analyze and forecast carbon emissions in Shandong. The findings will offer scientific support for devising differentiated carbon peaking pathways and optimizing the province’s industrial and energy structures. The potential innovations of this study are twofold: First, it introduces an improved fractional-order grey prediction model. By integrating fractional-order calculus and applying particle swarm optimization algorithms, the model effectively reduces prediction errors under conditions of limited observational data, thereby achieving higher accuracy. Second, the selected model is particularly suitable for small-sample data analysis and is well-adapted to short-term trend prediction or datasets without obvious saturation trends. The application of the fractional-order cumulative grey model to provincial-level short-term carbon emission analysis constitutes an important methodological contribution to existing research.

2. Literature Review

Scientific prediction of carbon emissions provides critical decision support for formulating and optimizing emission reduction policies. In recent years, significant advances have been made in carbon accounting models and methodologies, leading to the emergence of a series of innovative prediction approaches. These approaches can be divided into four main types: First is the parametric econometric forecasting models. Conventional econometric models utilize parameterized methods to construct explicit relationships between economic factors and carbon emissions, maintaining their significance in emission prediction studies. For instance, Wang et al. [6] extended the conventional model of stochastic impacts by regression on population, affluence, and technology by incorporating random effects to improve electricity-sector carbon emission predictions in Shanghai. Nibbering and Paap [7] first proposed a heterogeneous forecasting framework for panel data analysis, introducing an asymmetric grouping estimation method specifically tailored for per capita carbon emission projections across national datasets. Second is non-parametric machine learning forecasting methods. These data-driven approaches relax the restrictive assumptions of traditional models by leveraging feature engineering and algorithmic optimization techniques. He et al. [8] developed the FD3 framework, which innovatively combines signal processing techniques with deep neural networks for enhanced carbon emission forecasting. Wu et al. [9] established a scenario-based prediction model integrating systematic scenario analysis with long short-term memory networks. Separately, Cui et al. [10] originally proposed a federated learning system employing seasonal auto-regressive integrated moving average clustering for distributed emission prediction. Third, hybrid ensemble forecasting approaches. The synergistic integration of complementary analytical techniques has gained significant research attention. Sun and Huang [11] proposed a novel predictive framework that integrates factor analysis with extreme learning machine algorithms to forecast carbon emission intensity. Concurrently, Xu et al. [12] developed an innovative tensor-long short term memory-autoregressive integrated moving average model incorporating tensor decomposition methods for multidimensional carbon emission forecasting.

Fourth, extended grey prediction modeling approaches. The grey prediction model (GM) has established itself as a canonical forecasting method due to its exceptional performance with small datasets and uncertain information. This approach demonstrates robust predictive capability under conditions of data scarcity and information incompleteness, with demonstrated applications spanning economic, industrial, agricultural, and environmental systems [13,14,15,16,17,18]. Given the inherently small-sample nature of carbon emission datasets, the grey prediction model has become widely adopted in emission forecasting due to their demonstrated efficacy in limited-data scenarios, contributing substantially to the field. Researchers have systematically improved grey prediction models’ predictive performance and domain adaptability through iterative model refinements and novel methodological developments. In terms of model improvements, Sahin et al. [19] proposed an optimized rolling metabolic grey model incorporating both linear and nonlinear components, which was successfully applied to greenhouse gas emission forecasting in Turkey. Javed et al. [20] innovatively developed a novel grey prediction model specifically designed for systematic forecasting of bio-fuel supply-demand relationships in major CO₂-emitting nations, including China, the United States, and India. Meanwhile, Wu et al. [21] introduced a deformable fractional heterogeneous grey model, providing a new methodological approach for analyzing carbon emission trends in BRICS countries. Regarding model integration approaches, Cheng et al. [22] combined a fractional-order GM(1,1) model with Markov chains to develop a hybrid forecasting framework for evaluating emission reduction potential in China’s transportation sector under the Paris Agreement framework. Separately, Gao et al. [23] enhanced a fractional grey model by incorporating Gompertz law, establishing a specialized prediction system tailored to the emission characteristics of U.S. industrial sectors. In multivariate carbon emission forecasting, Nie et al. [24] developed a multivariate grey differential dynamic forecasting model, providing a novel analytical tool for analyzing carbon emission system. Zhao et al. [25] established a provincial-level emission forecasting framework by integrating heterogeneous grey models with the manta ray foraging optimization algorithm, explicitly incorporating environmental investment factors. Recent advances demonstrate the expanding applicability of grey prediction models: He et al. [26] successfully adapted a fractional-order logistic grey model for small-to-medium-sized nations’ emission forecasting, while Gu and Wu [27] proposed an impulsive fractional grey model offering new methodological insights for global emission trend analysis. These developments not only enrich grey prediction theory but also provide critical scientific foundations for carbon policy formulation.

Gray system theory utilizes addition and subtraction operations in information processing as its fundamental mathematical methodology, with this distinctive algorithmic framework demonstrating essential theoretical significance in the field. As a pivotal theoretical development, the fractional-order accumulation generation operator establishes a rigorous mathematical foundation for fractional-order grey prediction models, with contemporary research in this domain yielding significant theoretical breakthroughs. For instance, Wu [28] demonstrated that the fractional-order grey model, through the incorporation of fractional-order operators and the integration of the particle swarm optimization algorithm, effectively reduces prediction errors in small-sample forecasting compared to traditional grey models. This approach enhances the weighting of new information, thereby improving prediction accuracy. As a cutting-edge research direction in grey prediction modeling, fractional-order grey model has been successfully employed to multiple domains, including environmental studies [29], technology and engineering [30], energy systems [31], economic management [32], agriculture [33], and ecological and climatic research [34].

In summary, significant progress has been made in both methodological innovation and model development for carbon emission forecasting. Among existing approaches, grey prediction models have garnered considerable attention due to their strong performance in handling small-sample datasets and situations with incomplete information. Unlike earlier studies relying on conventional grey models, this study proposes an improved fractional-order grey forecasting model. By integrating fractional-order calculus with a particle swarm optimization algorithm, the proposed approach effectively reduces prediction errors commonly associated with traditional grey models under data-sparse conditions [28]. Furthermore, in contrast to methods employing fractional-order logistic grey models for carbon emission prediction, the present study adopts a grey model specifically tailored for small-sample data analysis. While fractional-order logistic models are more suitable for medium- and long-term projections, the proposed model demonstrates superior efficiency in short-term forecasting and when processing data that do not exhibit saturation trends.

3. Materials

3.1. Data Sources and Carbon Emission Estimation

Due to the limitations in real-time monitoring feasibility and data acquisition completeness, the quantification of carbon dioxide emissions continues to rely predominantly on indirect estimation methods. The carbon emission accounting system from an energy consumption perspective has been widely adopted, owing to its theoretical foundation: fossil fuel combustion represents the primary source of anthropogenic carbon emissions. This study employs a top-down carbon emission accounting framework, selecting eight typical fossil fuels as accounting subjects: coal, crude oil, gasoline, diesel, kerosene, fuel oil, natural gas, and liquefied petroleum gas. With reference to existing research [35], the standard coal conversion coefficient (GB/T 2589-2020) [36] and corresponding carbon emission factors are incorporated to establish a carbon emission calculation model for energy consumption in Shandong Province. The mathematical formulation of the proposed model is expressed as follows:

$$\mathrm{C}={\sum }_{\mathrm{i}=1}^8{\mathrm{E}}_{\mathrm{i}}\times {\mathrm{SCC}}_{\mathrm{i}}\times {\mathrm{CEC}}_{\mathrm{i}}\times {\displaystyle \frac{44}{12}}$$

(1)

where C denotes the estimated carbon emissions. E_i represents the consumption of the i type fossil fuel. SCC_i and CEC_i refer to the standard coal conversion coefficient and carbon emission coefficient of the i type fossil fuel, respectively (in

Table 1. Standard coal coefficients (SCC) and carbon emission coefficients (CEC) for carbon emission calculation.

Energy Type	SCC	CEC
Raw Coal	0.7143	0.7559
Coke	0.9714	0.8550
Crude Oil	1.4286	0.5857
Gasoline	1.4714	0.5538
Kerosene	1.4714	0.5714
Diesel	1.4571	0.5921
Fuel Oil	1.4286	0.6185
Natural Gas	1.3300	0.4483

), and ${\displaystyle \frac{44}{12}}$ indicates the CO₂-to-carbon molecular weight ratio.

In regional carbon emission assessment systems, reliance on a singular aggregate emission metric proves insufficient for comprehensively characterizing emission profiles, thereby necessitating integration with indicators of regional economic development. To address this limitation, the present study employs carbon emission intensity, defined as CO₂ emissions per unit of gross domestic product (GDP), as a core environmental-economic indicator. This measure serves as a proxy for the carbon intensity of economic growth.

Given that CO₂ emissions fundamentally stem from socio-economic activities, examining demographic drivers is essential for a mechanistic understanding of emission patterns. Accordingly, this study introduces a per capita carbon emission indicator, calculated as the ratio of annual estimated CO₂ emissions to permanent resident population, to preliminarily capture scale effects associated with population size.

Empirical data pertaining to fossil energy consumption were obtained from the China Energy Statistical Yearbook. Standard coal conversion coefficients adhere to the General Rules for Comprehensive Energy Consumption Calculation (GB/T 2589-2020), while carbon emission factors are consistent with the IPCC Guidelines for National Greenhouse Gas Inventories. Regional economic and demographic data, including GDP and permanent resident population figures for Shandong Province, were derived from the Shandong Statistical Yearbook.

3.2. Current Status Analysis of Carbon Emissions in Shandong Province

Driven by the continuous advancement of Shandong’s new-old economic momentum conversion strategy and the comprehensive implementation of the green, low-carbon, and high-quality development pilot zone, the province’s economy has demonstrated a positive trajectory characterized by improved quality and enhanced efficiency. Using officially published energy and economic data from the China Energy Statistical Yearbook and the Shandong Statistical Yearbook (2010–2022), key metrics, including carbon emissions, carbon emission intensity, and per capita carbon emissions, were calculated and compared with Shandong’s GDP. The corresponding trends are depicted in Figure 1 and Figure 2.

As a major industrial province in China, Shandong exhibited distinct carbon emission dynamics between 2013 and 2022: high-volume fluctuations in total emissions accompanied by a consistent decline in carbon intensity. As illustrated in Figure 1, the province’s total CO₂ emissions fluctuated from 1.213 billion metric tons in 2013 to 1.498 billion metric tons in 2022, peaking at 1.584 billion metric tons in 2020. Under China’s dual-carbon policy framework, emissions subsequently declined consecutively in 2021 and 2022. Over the same period, Shandong sustained robust economic growth, with its GDP increasing from 4.73 trillion yuan to 8.76 trillion yuan, reflecting an average annual growth rate of 7.4%. Notably, the province achieved absolute decoupling during 2021–2022, marked by simultaneous GDP growth (reaching 8.76 trillion yuan in 2022) and a reduction in carbon emissions. This trend underscores the effectiveness of Shandong’s industrial restructuring and energy efficiency policies in decoupling economic expansion from carbon emissions.

As illustrated in Figure 2, Shandong Province exhibited a marked divergence between carbon emission intensity and per capita carbon emissions during the period 2013–2022. Carbon emission intensity demonstrated a consistent downward trend, declining from 2.56 to 1.71 t CO₂/(10⁴ CNY) over this period, representing a cumulative reduction of 33.2%. This translates to an average annual improvement in carbon efficiency per unit GDP of 3.7%. In contrast, per capita carbon emissions followed a fluctuating trajectory, rising from 12.44 tons per capita in 2013 to a peak of 15.58 tons per capita in 2020, before decreasing to 14.74 tons per capita in 2022. This inverse development highlights Shandong’s successful decoupling of economic growth from carbon-intensive activities through industrial restructuring and enhanced energy efficiency. Notably, following the introduction of China’s dual-carbon policy framework in 2020, the province achieved simultaneous reductions in both carbon intensity and per capita emissions. This dual decline provides compelling evidence of substantive progress in Shandong’s low-carbon transition, demonstrating that economic expansion can be maintained alongside reductions in both production- and consumption-based carbon metrics.

4. Methodology

4.1. Fractional-Order Accumulated Gray Model

Wu [28] originally proposed the fractional-order accumulated grey model (FAGM(1,1)), which extends the conventional first-order accumulated generating operator to a fractional-order framework. This model is constructed using fractional-order accumulated sequences, thereby improving its predictive performance. In contrast to traditional grey models, the FAGM(1,1) reduces the inherent randomness of the original data and suppresses solution-process disturbances through fractional-order accumulation, leading to significantly enhanced prediction accuracy in small-sample settings. Furthermore, by adjusting the fractional-order parameter, the model can assign greater weight to recent observations, improving dynamic adaptability and resulting in superior forecast performance. The detailed modeling procedure of the FAGM(1,1) is described below:

Step 1: Define the original non-negative data sequence as,

$$X^{\left(0\right)}=\left\{\left.X^{\left(0\right)}\left(1\right),X^{\left(0\right)}\left(2\right),\cdots ,X^{\left(0\right)}\left(n\right)\right\}\right.$$

(2)

And using the equation

$$X^{\left(r\right)}\left(k\right)={{\displaystyle {\sum }_{i=1}^{k}C}}_{k-i+r-1}^{k-i}{x}^{\left(0\right)}\left(i\right)$$

(3)

to generate the r-th order accumulated generation sequence

$$X^{\left(r\right)}=\left\{\left.X^{\left(r\right)}\left(1\right),X^{\left(r\right)}\left(2\right),\cdots ,X^{\left(r\right)}\left(n\right)\right\}\right.$$

(4)

We make the following fundamental assumptions:

$$C_{r-1}^0=1,C_k^{k+i}=0,C_{k-i+r-1}^{k-i}={\displaystyle \frac{\left(k-i+r-1\right)\left(k-i+r-2\right)\cdots \left(r+1\right)r}{\left(k-i\right)!}}$$

(5)

Subsequently, construct the background value sequence Z^(r) and solve the grey differential equation.

$$Z^{\left(r\right)}\left(k\right)=0.5\left[X^{\left(r\right)}\left(k\right)+X^{\left(r\right)}\left(k-1\right)\right]$$

(6)

Step 2: For the r-order accumulated generating sequence X^(r), its whitened differential equation is formulated as:

$${\displaystyle \frac{d{X}^{\left(r\right)}\left(k\right)}{dt}}+a{X}^{\left(r\right)}\left(k\right)=b$$

(7)

where a denotes the development coefficient and b represents the grey action quantity. The time response function is obtained by solving this whitened differential equation:

$$X^{\left(r\right)}\left(k+1\right)=\left[X^{\left(0\right)}\left(1\right)-{\displaystyle \frac{b}{a}}\right]e^{-ak}+{\displaystyle \frac{b}{a}}$$

(8)

Step 3: The parameters are estimated using the least squares method, which minimizes the sum of squared errors between the background value sequence and the simulated sequence.

$$\left(\begin{array}{c}\widehat{a}\\ \widehat{b}\end{array}\right)={\left(B^{T}B\right)}^{-1}{B}^{T}Y$$

(9)

where

$$B=\left(\begin{array}{cc}-0.5\left(X^{\left(r\right)}\left(1\right)+X^{\left(r\right)}\left(2\right)\right)& 1\\ -0.5\left(X^{\left(r\right)}\left(2\right)+X^{\left(r\right)}\left(3\right)\right)& 1\\ ⋮& ⋮\\ -0.5\left(X^{\left(r\right)}\left(n-1\right)+X^{\left(r\right)}\left(n\right)\right)& 1\end{array}\right)Y=\left(\begin{array}{c}{X}^{\left(r\right)}\left(2\right)-X^{\left(r\right)}\left(1\right)\\ X^{\left(r\right)}\left(3\right)-X^{\left(r\right)}\left(2\right)\\ ⋮\\ X^{\left(r\right)}\left(n\right)-X^{\left(r\right)}\left(n-1\right)\end{array}\right)$$

(10)

Step 4: The obtained parameter estimates $\widehat{a}$ and $\widehat{b}$ are substituted into the time response function:

$${\widehat{X}}^{\left(r\right)}\left(k+1\right)=\left[X^{\left(0\right)}\left(1\right)-{\displaystyle \frac{\widehat{b}}{\widehat{a}}}\right]e^{-\widehat{a}k}+{\displaystyle \frac{\widehat{b}}{\widehat{a}}}$$

(11)

where ${\widehat{x}}^{\left(r\right)}(k+1)$ represents the simulated output value of the model at time step k + 1, from which the complete fitted sequence can be obtained through recursive calculation:

$${\widehat{X}}^{\left(r\right)}=\left\{{\widehat{X}}^{\left(r\right)}\left(1\right),{\widehat{X}}^{\left(r\right)}\left(2\right),\cdots ,{\widehat{X}}^{\left(r\right)}\left(n\right),\cdots \right\}$$

(12)

Step 5: By implementing the r-order inverse accumulated generating operation on Equation (12), we obtain the restoration equation of the original sequence:

$${\widehat{X}}^{\left(0\right)}={\alpha }^{\left(r\right)}{\widehat{X}}^{\left(r\right)}=\left\{{\alpha }^{\left(1\right)}{\widehat{X}}^{\left(r\right)\left(1-r\right)}\left(1\right),{\alpha }^{\left(1\right)}{\widehat{X}}^{\left(r\right)\left(1-r\right)}\left(2\right),\cdots ,{\alpha }^{\left(1\right)}{\widehat{X}}^{\left(r\right)\left(1-r\right)}\left(n\right),{\alpha }^{\left(1\right)}{\widehat{X}}^{\left(r\right)\left(1-r\right)}\left(n+1\right),\cdots \right\}$$

(13)

where

$${\alpha }^{\left(1\right)}{\widehat{X}}^{\left(r\right)\left(1-r\right)}\left(k\right)={\widehat{X}}^{\left(r\right)\left(1-r\right)}\left(k\right)-{\widehat{X}}^{\left(r\right)\left(1-r\right)}\left(k-1\right)$$

(14)

Consequently, the fitted sequence of the original data is ${\widehat{x}}^{\left(0\right)}\left(1\right),{\widehat{x}}^{\left(0\right)}\left(2\right),\dots ,{\widehat{x}}^{\left(0\right)}\left(n\right)$, and the predicted sequence is ${\widehat{x}}^{\left(0\right)}\left(n+1\right),{\widehat{x}}^{\left(0\right)}\left(n+2\right),\dots$.

To better illustrate the aforementioned methodological procedures, a flowchart is presented below, as shown in Figure 3.

4.2. Principles and Implementation of Particle Swarm Optimization

The choice of the fractional-order accumulation parameter r in the FAGM(1,1) method is a crucial parameter that significantly influences the prediction accuracy. Different values of r yield distinct modeling results. To optimize model performance, the optimal r is determined by minimizing the prediction error, typically measured using the mean absolute percentage error (MAPE).

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{k=1}^n\left|\frac{{\widehat{X}}^{\left(0\right)}\left(k\right)-X^{\left(0\right)}\left(k\right)}{X^{\left(0\right)}\left(k\right)}\right|}\times 100\%$$

(15)

The determination of the optimal fractional order r requires substantial repetitive computations, which proves difficult to achieve through conventional approaches. The Particle Swarm Optimization (PSO) algorithm provides an effective solution for identifying the optimal radius parameter r.

The PSO algorithm is a population-based stochastic optimization method proposed by Kennedy and Eberhart [37], grounded in swarm intelligence principles. Inspired by the collective behaviors of biological swarms including fish schools, bird flocks, and insect colonies [38], this optimization algorithm achieves solution convergence by simulating their swarm intelligence mechanisms. The core principle of PSO lies in achieving optimal solutions through collaboration and information exchange among individuals (termed particles) within a swarm. Each particle is characterized by two primary attributes: velocity (indicating movement speed) and position (representing direction). The PSO algorithm initializes with a population of randomly generated particles and iteratively refines the solution. During each iteration, particles update their states by tracking two key benchmarks: the personal best (p_best), denoting the particle’s own optimal solution discovered thus far, and the global best (g_best), representing the best solution identified by the entire swarm. Specifically, each particle conducts an independent search within the solution space to identify its p_best, which is then shared across the swarm to determine the collective g_best. Subsequently, all particles adjust their velocities and positions based on their current p_best and the swarm’s g_best. Mathematically, the algorithm combines particle-specific information to re-calibrate velocity components across each dimension, thereby computing updated particle positions. This process continues until particles converge to an equilibrium or optimal state within the D-dimensional search space. Extensive empirical studies validate PSO as an efficient optimization tool, which demonstrates applicability across multiple domains where genetic algorithms have been conventionally employed, including function optimization tasks, neural network parameter training, and fuzzy logic control systems.

Consider a particle swarm optimization algorithm operating within a D-dimensional search space with a population size of m, where each particle’s position represents a potential solution. The position vector of the i-th particle is denoted as X_i = (x_i¹,x_i²,…,x_i^D), and its velocity vector as V_i = (v_i¹,v_i²,…,v_i^D). During the optimization process, each particle maintains its historical best position, known as the personal best (p_best), while the best position found by the entire swarm up to the current iteration is termed the global best (g_best). During each iteration, the particle’s velocity is updated based on its p_best and g_best, using the following formula to compute the velocity change.

$$V_{i+1}=w{v}_i^d=c_1{r}_1\left(p_i^d-x_i^d\right)+c_2{r}_2\left(p_g^d-x_i^d\right)$$

(16)

where V_i+1 denotes the updated velocity vector of the particle, w represents the inertia coefficient, controlling momentum retention. r₁ and r₂ are uniformly distributed random variables in [0, 1], c₁ and c₂ are cognitive and social acceleration coefficients (typically set to c₁ = c₂ = 2), and v_max defines the velocity clamping threshold to prevent divergence. During each iteration, the particle’s position is updated via Euler integration:

$$x_{i+1}=x_i+v_i$$

(17)

where x_i+1 denotes the updated position of the i + 1-th particle. v_i represents its corresponding velocity vector. The optimization process terminates if either of the following criteria is satisfied: (1) the best global solution meets a predefined fitness threshold, indicating sufficient convergence, or (2) the maximum iteration count is reached, ensuring computational efficiency.

5. Prediction Results and Discussion

5.1. Prediction and Analysis of Carbon Emissions in Shandong Province

In this study, a FAGM(1,1) was developed using actual carbon emission data from Shandong Province between 2016 and 2022 to predict carbon emissions for the period 2023–2025. First, as presented in

Table 2. PSO algorithm settings and optimized fractional order results when predicting carbon emissions in Shandong province.

Parameter	The Optimal Order r	Population Size	Maximum Iterations	Inertia Weight	Convergence Threshold	Search Range
Value	0.275	50	200	0.8	0.0001	[0, 1]

, the optimal order r = 0.275 as identified via the PSO algorithm. In the parameter configuration of PSO, the choice of each parameter substantially affects algorithmic performance. The PSO parameters in this research are configured as follows: The population size is set to 50, which is appropriate for low-dimensional optimization problems of this kind, thereby maintaining computational efficiency without introducing unnecessary overhead. The maximum number of iterations is set to 200, within the conventional range for medium-complexity optimization tasks, ensuring adequate iterations to converge to a high-quality solution. The inertia weight is set to 0.8, slightly above the lower limit of the typical interval [0.4, 0.9], to strengthen global exploration and avoid premature convergence. The convergence threshold is set to 0.0001 to halt iterations once further improvements become negligible, thus enhancing computational efficiency. The learning factors c₁ and c₂ are both set to the standard value of 2.0 from classical PSO, balancing the influence of individual experience and social collaboration on particle movement. The search range for the fractional-order r is defined as [0, 1], based on its mathematical definition and common practical applications, ensuring the optimization process remains within a feasible domain. Collectively, these parameters ensure that the algorithm achieves a balance between convergence, stability, and computational efficiency in identifying the optimal fractional order for the grey model.

Second, when the fractional order r = 1, the FAGM(1,1) model reduces to the conventional grey GM(1,1) model. To systematically evaluate the predictive performance of the FAGM(1,1) model, a comparative analysis was conducted with the traditional GM(1,1) model. Using actual carbon emission data from Shandong Province between 2016 and 2022, the fitting results and MAPE were calculated for both the FAGM(1,1) model with r = 0.275 and the GM(1,1) model. The detailed comparative outcomes are presented in

Table 3. Prediction results of CO₂ emissions in Shandong province (million tons).

Year	Actual Value	0.275-Order Fitted Value	1-Order Fitted Value
2016	1452.30	1452.30	1452.30
2017	1463.50	1463.50	1500.64
2018	1514.62	1517.66	1518.50
2019	1533.43	1544.57	1520.50
2020	1583.85	1547.99	1532.56
2021	1557.92	1535.38	1540.86
2022	1498.19	1512.67	1552.17
MAPE (%)		0.934	1.6687

. The results demonstrate that the 0.275-order FAGM(1,1) model achieved a mean absolute percentage error of 0.934%, significantly lower than the 1.6687% error of the first-order GM(1,1) model. Since the MAPE evaluation criterion considers values below 10% as excellent, both models attained outstanding predictive performance. However, the FAGM(1,1) model demonstrates higher predictive accuracy, significantly reducing fitting deviations. This finding strongly aligns with the studies [33,34], who also concluded that the fractional grey model outperforms conventional models, further supporting the superior applicability of FAGM(1,1) in the context of this study.

Finally, based on the optimal fractional order r = 0.275, this study utilizes the FAGM(1,1) to forecast the carbon emission trends from 2023 to 2025. The prediction and model validation results are presented in

Table 4. The prediction and model validation results (million tons).

Year	Actual	Predicted	95% CI Lower	95% CI Upper
2016	1452.30	1452.30	1416.42	1488.18
2017	1463.50	1463.50	1427.62	1499.38
2018	1514.62	1517.66	1481.77	1553.54
2019	1533.43	1544.57	1508.69	1580.45
2020	1583.85	1547.99	1512.11	1583.87
2021	1557.92	1535.38	1499.50	1571.26
2022	1498.19	1512.67	1476.79	1548.55
2023	-	1484.07	1448.19	1519.95
2024	-	1452.46	1407.61	1497.31
2025	-	1419.71	1365.89	1473.53
r	0.275
a	0.275
b	869.154
Relative Residual Q	0.801%
Variance Ratio C	0.382
Small Error Probability	1

Note: The model parameters r, a, and b, namely the optimal order, the development coefficient, and the grey action quantity as introduced in Equation (8).

, with the predicted values plotted in Figure 4. The validation results in Table 4 demonstrate that the model exhibits excellent fitting accuracy for CO₂ emission intensity data. Specifically, the relative residual value Q is 0.801%, which is below the first-grade threshold of 1%, indicating an extremely low average relative error in the model’s predictions. The variance ratio value C is 0.382, which is slightly above the first-grade accuracy benchmark but remains well below 0.50. This suggests that the fluctuation of the residuals is considerably smaller than that of the original data, reflecting favorable model stability. Moreover, the small error probability value reaches 1, denoting that all residuals fall strictly within an acceptable error range, which further confirms a highly ideal fit.

Figure 4 illustrates the projected trajectory of carbon emissions in Shandong Province from 2016 to 2025. The overall trend follows a three-phase pattern: ‘slow increase–stabilization–slight decline’. Between 2016 and 2019, carbon emissions continued to rise, a trend strongly associated with Shandong’s industrial structure, which remains heavily reliant on heavy chemical industries. As a major national base for heavy industry, Shandong has a high concentration of energy-intensive sectors such as steel, petrochemicals, and building materials. Its energy mix has long been dominated by coal, resulting in high emission intensity alongside economic growth. Although several energy-saving policies were implemented during this period, ongoing industrialization and urbanization contributed to rigid growth in carbon emissions. The fluctuation observed in 2020 was mainly attributable to exogenous shocks, including the COVID-19 pandemic, which temporarily reduced industrial output and transportation activity, leading to an anomalous deviation in emissions.

During the forecast period (2021–2025), carbon emissions gradually stabilize and exhibit a slight decline, reflecting structural transformations driven by Shandong’s initiative to replace old growth drivers with new ones. Since the launch of the 14th five-year plan, the province has accelerated energy structure adjustments by vigorously promoting renewable energy sources such as wind and solar power, strictly controlling total coal consumption, and phasing out outdated production capacity. As a pilot zone for national industrial transition, Shandong has also made initial progress in upgrading energy-intensive industries, advancing green manufacturing, and establishing a carbon market mechanism. These efforts have facilitated a gradual decoupling of economic growth from carbon emissions. The fractional-order grey model, capable of effectively capturing non-stationary time series, successfully represents this policy-induced convergence in emissions, which aligns broadly with Shandong’s pathway towards achieving peak carbon before 2030.

Nevertheless, it should be noted that Shandong continues to face deep-seated challenges, including an excessively heavy industrial structure and high dependence on conventional energy. Persistent pressures remain regarding future emission control. Achieving sustained reductions will require further energy restructuring, enhanced technological innovation, and stronger policy coordination along with rigorous regulatory enforcement.

5.2. Forecasting and Analysis of Carbon Emission Intensity

This study used actual carbon emission intensity data from Shandong Province from 2016 to 2022 and established FAGM(1,1) to predict the carbon emissions for the years 2023 to 2025. First, the optimal order r = 0.039 of the model was determined using the PSO algorithm, as shown in

Table 5. PSO algorithm settings and optimized fractional order results when predicting carbon emission intensity in Shandong province.

Parameter	The Optimal Order r	Population Size	Maximum Iterations	Inertia Weight	Convergence Threshold	Search Range
Value	0.039	50	200	0.8	0.0001	[0, 1]

. The hyperparameters of the PSO algorithm were maintained consistently with those utilized in the carbon emission prediction model.

Second, a comparative performance analysis was conducted between the FAGM(1,1) and GM(1,1) models, revealing that the fractional-order variant offers substantial advantages in forecasting carbon emission intensity. As summarized in

Table 6. Prediction results of carbon emission intensity in Shandong province.

Year	Actual Value	0.039-Order Fitted Value	1-Order Fitted Value
2016	2.47	2.47	2.47
2017	2.32	2.35	2.40
2018	2.27	2.27	2.27
2019	2.17	2.18	2.25
2020	2.18	2.08	2.04
2021	1.88	1.93	1.94
2022	1.71	1.71	1.81
MAPE (%)		1.489	2.773

, although both models attained an “excellent” accuracy rating based on standard evaluation criteria, the fractional-order FAGM(1,1) model with r = 0.039 achieved a MAPE value of 1.4892%, an improvement of 0.615 percentage points over the conventional GM(1,1) model, which yielded a MAPE of 2.773%. This enhancement is attributed to the incorporation of a fractional accumulation operator, which provides a more precise representation of the nonlinear variation patterns inherent in carbon emission data. Consequently, the FAGM(1,1) model not only reduces prediction bias but also demonstrates stronger adaptability when applied to complex environmental datasets.

Finally, using the optimal order r = 0.039, the FAGM(1,1) model was developed to predict regional carbon intensity from 2023 to 2025. The resulting predictions and validation metrics are summarized in

Table 7. The prediction and model validation results.

Year	Actual	Predicted	95% CI Lower	95% CI Upper
2016	2.47	2.47	2.38	2.56
2017	2.32	2.35	2.26	2.44
2018	2.27	2.27	2.18	2.36
2019	2.17	2.18	2.09	2.27
2020	2.18	2.08	1.98	2.17
2021	1.88	1.93	1.84	2.02
2022	1.71	1.71	1.62	1.80
2023	-	1.39	1.30	1.48
2024	-	0.90	0.78	1.01
2025	-	0.15	0.01	0.29
r	0.038
a	−0.432
b	−1.088
Relative Residual Q	1.278%
Variance Ratio C	0.178
Small Error Probability	1

Note: The model parameters r, a, and b, namely the optimal order, the development coefficient, and the grey action quantity as introduced in Equation (8).

, while the projected values are depicted in Figure 5.

As shown in Table 7, the model exhibits excellent fitting performance. Specifically, the variance ratio C-value is 0.178, considerably lower than the first-grade threshold of 0.35, and the small error probability p-value reaches 1. Although the relative residual Q-value is 1.278%, slightly exceeding the 1% benchmark, it remains well below 2%. Given the model’s strong performance in both C and p values, it comprehensively satisfies the criteria for excellent accuracy. This indicates that the residual fluctuations are substantially smaller than those of the original data, and all predicted values lie within an acceptable error range, reflecting high stability and reliability.

According to the projections illustrated in Figure 5, carbon dioxide emission intensity in Shandong Province has exhibited a consistent decline from 2016 to 2025, suggesting a year-on-year reduction in carbon emissions per unit of gross domestic product (GDP). This trend not only reflects the implementation of the national dual-carbon strategy at the provincial level but also highlights Shandong’s progress in decoupling economic growth from carbon emissions through industrial transformation and the adoption of cleaner energy sources.

The observed decrease in emission intensity is mainly driven by substantial industrial restructuring and a shift towards a cleaner energy mix. Historically, the province’s economy relied heavily on heavy chemical industries, steel manufacturing, and coal-intensive energy consumption, which contributed to a high carbon emission baseline. In recent years, through the rigorous enforcement of the new and old kinetic energy conversion initiative, obsolete production capacity has been progressively phased out. Concurrently, energy-efficient technologies and renewable energy sources have been actively promoted. The capacity of renewable energy installations, particularly wind and solar, has risen markedly, resulting in a continued optimization of the energy structure and a consequent reduction in carbon intensity per unit of economic output. Data indicate that coal consumption in Shandong declined from roughly 420 million tons in 2016 to 400 million tons in 2022 (https://www.gonyn.com/industry/1541808.html (accessed on 27 July 2025)), while the installed capacity of renewable energy (including non-fossil sources) grew at a compound annual rate of 24.7% (http://fgw.shandong.gov.cn/art/2025/4/10/art_91548_10463843.html (accessed on 27 July 2025)), providing essential support for emission mitigation. Additionally, with provincial GDP growth remaining stable at 5–6% (https://m.askci.com/news/data/hongguan/20250207/175228273892194784215875.shtml (accessed on 27 July 2025).), the annual average decline in energy intensity per unit GDP of approximately 3.5% underscores the role of technological advancements and efficiency improvements, such as industrial process optimization and energy-saving technologies, in driving down emission intensity. Furthermore, urban low-carbon initiatives and the electrification of the transport sector, evidenced by the promotion of new energy vehicles and the enhancement of public transport systems, have also played a significant role in reducing emission intensity.

Nevertheless, Shandong continues to face structural challenges, such as its heavy dependence on traditional industries and a continued reliance on coal in its energy mix. To address these issues, future efforts should focus on accelerating green industrial transformation, fostering and deploying technological innovations, and advancing regional collaborative governance. These measures will be critical to achieving enhanced synergy between emission reduction and sustainable economic development.

5.3. Prediction and Analysis of per Capita Carbon Emissions

The FAGM(1,1) model was constructed using per capita carbon emissions data from Shandong Province covering the period 2016 to 2022, with the aim of forecasting carbon emissions from 2023 to 2025. The specific modeling procedure involved determining the optimal order of r = 0.258 via the PSO algorithm, as summarized in

Table 8. PSO algorithm settings and optimized fractional order results when predicting per capita carbon emissions in Shandong province.

Parameter	The Optimal Order r	Population Size	Maximum Iterations	Inertia Weight	Convergence Threshold	Search Range
Value	0.258	50	200	0.8	0.0001	[0, 1]

. The hyperparameters of the PSO algorithm were kept consistent with those employed in the provincial carbon emission prediction model.

Second, this study conducts a systematic comparative analysis of the predictive performance between the 0.258-order FAGM(1,1) model and the conventional GM(1,1) model. Empirical results summarized in

Table 9. Prediction results of per capita carbon emissions in Shandong province (metric tons per capita).

Year	Actual Value	0.258-Order Fitted Value	1-Order Fitted Value
2016	14.56	14.56	14.56
2017	14.59	14.59	15.02
2018	15.03	15.05	15.12
2019	15.17	15.26	15.25
2020	15.58	15.26	15.41
2021	15.32	15.11	15.41
2022	14.742	14.87	15.48
MAPE (%)		0.842	1.543

demonstrate that the 0.258-order FAGM(1,1) model achieves a substantially higher accuracy, registering a MAPE value of 0.8425%, which represents a reduction of 45.4% compared to the traditional GM(1,1) model (1.5434%).

Finally, based on the optimal order r = 0.258, this study developed an FAGM(1,1) model to forecast per capita carbon emission intensity from 2023 to 2025. The prediction and validation outcomes are summarized in

Table 10. The prediction and model validation results (metric tons per capita).

Year	Actual	Predicted	95% CI Lower	95% CI Upper
2016	14.56	14.56	14.24	14.88
2017	14.59	14.59	14.27	14.91
2018	15.03	15.05	14.73	15.37
2019	15.17	15.26	14.94	15.58
2020	15.58	15.26	14.94	15.58
2021	15.32	15.11	14.79	15.43
2022	14.74	14.87	15.55	15.20
2023	-	14.59	14.27	14.91
2024	-	14.28	13.88	14.68
2025	-	13.96	13.48	14.44
r	0.258
a	0.291
b	8.617
Relative Residual Q	0.722%
Variance Ratio C	0.424
Small Error Probability	0.857

Note: The model parameters r, a, and b, namely the optimal order, the development coefficient, and the grey action quantity as introduced in Equation (8).

and illustrated in Figure 6. The validation results presented in Table 10 indicate that the model attains a high level of fitting accuracy. Specifically, the relative residual value Q is 0.722%, surpassing the first-grade benchmark of 1% and reflecting an exceptionally low mean relative error in the predictions. The variance ratio C is 0.424, satisfying the second-grade criterion, which suggests that the residual fluctuation is less than that of the original data and indicates favorable model stability. The small error probability p is 0.857, meeting the second-grade standard and implying that the majority of residuals lie within an acceptable error margin. Although the C- and p-values do not reach the first-grade thresholds, the outstanding performance of the Q-value, coupled with all metrics substantially exceeding the minimum requirements, demonstrates that the model exhibits excellent predictive capability and overall reliability.

As illustrated in Figure 6, per capita carbon emissions in Shandong Province from 2016 to 2025 demonstrate a distinct trajectory characterized by ‘slow increase–peak–steady decline’. This trend reflects the dynamic equilibrium achieved by Shandong, a region traditionally reliant on industry and energy-intensive sectors, in balancing economic restructuring with green low-carbon development.

During the historical period from 2016 to 2020, per capita carbon emissions continued to experience a gradual rise. This was mainly driven by the province’s longstanding industrial framework, centered on heavy chemical sectors, and its continued dependence on coal-based energy consumption. As a major industrial base in China, Shandong’s economy remains heavily influenced by energy-intensive industries such as steel, chemicals, and building materials. The incomplete decoupling of economic growth from energy consumption resulted in persistent upward pressure on carbon emissions during this phase. However, since the mid-to-late stages of the 13th five-year plan, Shandong has actively pursued a strategy of replacing old growth drivers with new ones, enforced strict caps on coal consumption, and accelerated the development of renewable energy sources. These structural changes established a foundation for the subsequent peaking of emissions.

Projections indicate that carbon emissions peaked around 2021, ushering in a phase of consistent decline. This turning point is of considerable significance, demonstrating the initial success of Shandong’s low-carbon transition policies. With the commencement of the 14th five-year plan period, guided by the national dual-carbon targets, Shandong has intensified its emission reduction efforts. These include optimizing the industrial structure, phasing out obsolete production capacity, promoting green manufacturing technologies, and improving energy efficiency. Concurrently, the rapid expansion of clean energy initiatives, such as offshore wind and photovoltaic power generation, has significantly reduced carbon intensity. Additionally, growing public environmental awareness and the adoption of low-carbon lifestyles have further supported these efforts.

6. Conclusions

6.1. Key Findings

Based on data from 2016 to 2022, this study employs the FAGM(1,1) model to forecast carbon emission trends in Shandong Province, yielding the following conclusions: both total and per capita carbon emissions in Shandong follow a three-phase trajectory of ‘slow growth-peak-steady decline’, while carbon emission intensity continues its downward trend. This pattern suggests that Shandong has made initial progress in the green transformation of its industrial structure and the transition to cleaner energy, underpinned by effective policy implementation, particularly through the strategy of replacing old growth drivers with new ones, stringent control of coal consumption, and vigorous development of renewable energy sources. The findings not only elucidate the dynamic trends and underlying drivers of carbon emissions in Shandong Province but also highlight the methodological advantages of fractional-order grey models in carbon emission forecasting with limited data, offering potential applicability to other industrialized regions.

6.2. Policy Implications

As a pivotal industrial base and major energy-consuming region in China, Shandong faces profound structural and energy mix challenges in achieving its dual-carbon goals, characterized by a disproportionately high share of industrial energy use and a persistent heavy reliance on coal. In light of these findings, this study proposes the following policy recommendations:

First, the energy supply system should be optimized to establish a clean and low-carbon development framework. To facilitate low-carbon energy transition, Shandong should implement a coal consumption cap mechanism, thereby accelerating the orderly phase-out of coal-intensive industries through structural adjustments. Key measures include advancing the ‘triple transformation’ strategy, encompassing energy-saving retrofits, heating upgrades, and flexibility modifications, for existing coal-fired power units, while progressively substituting decentralized coal consumption with cleaner alternatives. Concurrently, the province should expand its deployment of renewable energy, significantly scaling up the installed capacity of wind, solar, and other renewable sources, and pursue the development of an integrated energy system that incorporates wind, solar, nuclear, and hydrogen. Furthermore, cross-regional green power trading and dispatching mechanisms should be strengthened to increase the share of clean energy imports, with the long-term objective of establishing a renewable-dominated power system. It is advisable to prioritize the phased retirement of inefficient coal power units and industrial boilers by 2025, complemented by necessary grid enhancements to support renewable integration.

Second, industrial transformation and upgrading should be advanced to foster new drivers of green growth. Strict controls ought to be enforced on new energy-intensive projects, while promoting intelligent and low-carbon upgrades in sectors such as steel and chemicals. Shandong should accelerate the modernization of traditional high-energy-consumption industries, including steel, petrochemicals, and building materials, by phasing out outdated capacity through ’replacing small-scale, inefficient facilities with larger, more efficient alternatives’ and implementing ultra-low emission retrofits. Additionally, digital technologies ought to be harnessed to optimize production processes through smart monitoring, thereby curbing energy intensity per unit of output. Strategic emerging industries, such as renewable energy equipment, energy conservation, and environmental protection, should be actively promoted, with specific emphasis placed on hydrogen energy, energy storage, and other green technologies to cultivate new economic growth engines. Projections indicate that industrial emissions are expected to remain a dominant component of the provincial total in the near term. Therefore, sectoral interventions should prioritize steel, cement, and chemical industries, in which pilot applications of digital smart monitoring and ultra-low emission retrofits have demonstrated the potential to reduce energy intensity by over 15%. Implementing differentiated capacity replacement policies, tailored to regional and technological feasibility, would be crucial to guiding these sub-sectors toward achieving peak emissions by 2025.

Third, policy and market mechanisms should be strengthened to enhance institutional constraints. Specific measures include leveraging Shandong’s coastal advantages to innovate carbon emission trading through the establishment of a distinctive regional carbon market, incorporating mechanisms that convert renewable energy generation into tradable carbon reduction credits. The province should also improve regional collaborative governance via initiatives such as the carbon inclusive mechanism alliance in the Jiaodong region to implement cross-city ecological compensation schemes. Furthermore, financial innovation should be prioritized through instruments like carbon reduction bill financing and transition finance mechanisms, which are pivotal in facilitating the transformation of high-carbon industries while supporting green development. To better align short-term emission forecasts with the long-term 2060 neutrality goal, Shandong could consider introducing a province-specific carbon peak action plan that integrates dynamic sectoral carbon budgets and regular assessments. Additionally, gradually expanding the carbon market’s coverage to include all key industries, particularly power, steel, and chemicals, and introducing innovative incentive mechanisms such as carbon inclusion could accelerate low-carbon transitions. Transition finance mechanisms should also be advanced to support early coal retirement and the demonstration of renewable hydrogen projects in the near term.

6.3. Limitations and Future Research

The FAGM(1,1) model demonstrates significant advantages over the traditional GM(1,1) model in predicting carbon emission trends in Shandong Province, as its integration of a fractional-order differential operator allows for more accurate capture of long-term memory effects and nonlinear growth dynamics inherent in carbon emission data, while adaptively optimizing fitting performance across varying temporal scales. Empirical results indicate that the FAGM(1,1) model substantially enhances prediction accuracy, which aligns with findings from studies [33,34] that also reported the superior performance of fractional grey models over the conventional GM(1,1).

However, several limitations warrant attention. First, model validation is based exclusively on data from Shandong Province. Although the FAGM(1,1) model exhibits strong performance within this context, its generalizability to other regions with distinct economic structures, policy environments, or development stages requires further verification. Future studies should assess the model’s applicability across multiple provinces, especially those with differing industrial profiles, to evaluate its broader utility.

Second, the analysis relies on a relatively short time series (2016–2022), which could heighten the risk of overfitting and diminish the robustness of long-term projections. Although the fractional-order mechanism partially mitigates this issue by improving adaptability, the limited number of observations remains a constraint. Future research could enhance estimation stability by introducing a logistic growth kernel to construct a fractional-order logistic grey model, following the approach proposed by He et al. [26].

Third, as a univariate time-series model, the FAGM(1,1) does not explicitly incorporate influential external factors such as technological progress, industrial restructuring, energy efficiency improvements, or major policy shocks. This simplification may result in mechanical extrapolations that overlook nonlinear disruptions, such as breakthroughs in clean technology or abrupt economic and climatic events. Hence, the projections reflect trend-based pathways rather than scenario-driven dynamics. Future research should explore multivariate extensions, such as integrating FAGM(1,1) with STIRPAT models or machine learning algorithms, to incorporate multi-dimensional drivers like energy intensity, GDP growth, and policy variables.

Finally, the model’s sensitivity to exogenous shocks, such as the COVID-19 pandemic in 2020 or the energy crisis in 2022, is limited. Such events can induce significant deviations from trend-derived forecasts. While grey models excel in small-sample forecasting, they are not designed to account for sudden structural breaks. Further studies could combine grey models with anomaly detection techniques or regime-switching frameworks to improve predictive resilience during turbulent periods. Moreover, the incorporation of variable-weight buffer operators, as investigated in pulse fractional grey models by Gu and Wu [27], represents a promising approach to mitigate the impact of exogenous shocks and enhance forecasting stability.

Despite these limitations, the FAGM(1,1) model provides a valuable tool for short-term carbon emission forecasting in data-scarce contexts. Future efforts should focus on refining its adaptability to external shocks, extending its application to multi-regional datasets, and developing multivariate formulations to better support policy-sensitive emission trajectory analysis.