Forecasting total energy demand based on the DGM(1,N) model Open Access

Achieving carbon peak by 2030 and carbon neutrality by 2060 (Luo et al., 2024) represents China’s dual carbon goals—a crucial commitment to addressing global climate change and a key strategy for driving high-quality economic development and energy structure transformation. In this context, the development trend of China’s energy demand will profoundly affect the country’s emission reduction path, industrial policy adjustment and international climate governance cooperation. Currently, as the world’s largest energy consumer and carbon emitter, changes in China’s energy demand not only concern the success or failure of the domestic green low-carbon transition, but also have great significance for the global carbon emission reduction process. In recent years, China’s total energy consumption has continued to grow, but the energy structure has been gradually optimized, and the proportion of renewable energy has increased significantly. However, the traditional high energy-consuming industries still occupy an important position, coupled with the continuous progress of urbanization and industrialization, energy demand will continue to face growth pressure for a period of time in the future. At the same time, technological progress, policy regulation and market-based instruments (e.g. carbon trading system) are becoming the core driving forces for energy efficiency improvement and carbon emission reduction. Against this complex background, scientific forecasting of the future trend of China’s energy demand and analysis of its driving factors and potential challenges are of great significance to the formulation of reasonable emission reduction policies and optimization of energy resource allocation (Hao et al., 2018).

Based on this, this paper selects the national total energy demand and related macro indicators from 2010 to 2021, screens the core influencing factors through grey correlation analysis, and establishes a DGM(1,4) model for prediction. The study not only improves the theoretical basis for the formulation of energy structure adjustment policies, but also provides methodological support for the accurate prediction of energy demand under the goal of “dual-carbon”, which provides data support for the formulation of differentiated regional energy policies by the governmental departments. The study not only provides a theoretical basis for the formulation of energy restructuring policies, but also provides methodological support for the accurate prediction of energy demand under the goal of “dual-carbon”, which in turn provides data support for the governmental departments to formulate differentiated regional energy policies.

Energy serves as the cornerstone of national development security. Rational analysis and precise forecasting of factors influencing energy demand and their evolving trends provide crucial reference for formulating scientific energy policies and rationally planning energy construction projects. In recent years, scholars both domestically and internationally have conducted extensive research on the determinants of energy demand, focusing primarily on economic factors, industrial structure, population and urbanization, energy prices, and regional disparities. Economic growth is the most fundamental driver of energy demand. Research indicates a significant positive correlation between energy consumption and GDP. Hidayat et al. (2024) conducted a regression analysis on panel data from over ten ASEAN countries, demonstrating that energy consumption exhibits a significant positive coefficient for real GDP growth. However, with economic restructuring and technological advancement, the correlation between energy demand and economic growth may weaken, leading to a phenomenon known as “decoupling” (Gao et al., 2020). Research indicates that industrial structure is a key structural factor influencing energy demand: the higher the proportion of secondary industries—particularly heavy and chemical industries—the greater the energy consumption intensity tends to be (Sun and Gao, 2021). Population size and urbanization processes also directly influence total energy consumption. Population growth expands the baseline demand for residential energy use (Wei and Shen, 2019). Simultaneously, urbanization significantly increases energy demand by altering lifestyles, enhancing infrastructure density, and expanding industrial activities (Zhou et al., 2018). Energy prices serve as a crucial economic lever for regulating energy demand. Theoretically, rising energy prices suppress consumption while promoting energy conservation and substitution. It is noteworthy that the factors influencing energy demand exhibit significant regional heterogeneity. Resource-rich regions, economically developed areas, and underdeveloped regions demonstrate distinct energy consumption patterns and driving mechanisms (Liu and Lu, 2024).

Energy demand forecasting serves as the foundational work for energy system planning, policy formulation, and market operations. With the acceleration of the global energy transition and the introduction of the “dual carbon” goals, energy demand exhibits complex characteristics of nonlinearity, non-stationarity, and high volatility, placing higher demands on forecasting methodologies. In recent years, research on energy demand forecasting has primarily been categorized into three major types: traditional statistical and econometric models, AI prediction model and grey models. Autoregressive Integrated Moving Average (ARIMA) models and their variants, such as SARIMA, have been widely applied in short-term energy demand forecasting. These models extrapolate based on the statistical characteristics of historical data and are suitable for stationary time series. Li et al. (2022) employed time series analysis to conduct short-term forecasts of daily coal consumption at coastal power plants, thereby enhancing the accuracy of electricity coal consumption predictions. Multiple linear regression (MLR) and econometric models predict energy demand by establishing functional relationships between demand and influencing factors such as GDP, population, and prices. Li et al., (2025) developed a stepwise regression-based double-log demand function forecasting method that accurately captures nonlinear inflection points in natural gas demand trends, demonstrating strong predictive performance. However, this model relies on variable selection and the validity of model assumptions, and is sensitive to multicollinearity and heteroscedasticity. With the increase in data volume and the advancement of computational capabilities, machine learning methods represented by artificial neural networks (ANNs) have rapidly developed in the field of energy forecasting due to their powerful nonlinear fitting capabilities. Machado et al. (2021) applied feedforward neural networks (FFNN) to short-term electricity load forecasting in industrial zones. They noted that FFNNs rely solely on current input vectors and lack internal state retention of historical sequences, resulting in insufficient memory capacity when handling load sequences with significant temporal dependencies. Shanmugam and Ramana (2024) developed multiple short-term electricity consumption forecasting models based on radial basis function networks (RBFNs). They validated the advantages of RBFNs in nonlinear approximation and rapid convergence, comparing them with traditional feedforward networks to highlight their model effectiveness. Deep learning models, owing to their unique gating mechanisms, effectively capture long-term dependencies in time series, significantly enhancing prediction accuracy. Congxiang et al. (2025) proposed integrating photovoltaic thermodynamics using LSTM networks. By training bidirectional LSTM (BiLSTM) models with thermal and meteorological features, prediction accuracy can be effectively enhanced. Support vector machines (SVMs) in machine learning are also frequently applied to energy consumption and demand forecasting (Morteza et al., 2023).

Although modern forecasting methods such as AI models demonstrate formidable capabilities in capturing complex nonlinear relationships, and traditional econometric models possess a theoretical foundation for analyzing causal relationships between variables, intelligent AI models involve computationally complex processes, while traditional models are sensitive to data fluctuations. Therefore, this study selects the DGM(1,N) model from the grey forecasting model as its core forecasting tool. Gray prediction models are suitable for systems characterized by “small samples and sparse information.” China’s energy system is undergoing rapid transformation. The introduction of the “dual carbon” goals has introduced new policy variables, disrupting the long-term regularity of certain historical data. Against the backdrop of limited data and potential structural changes, grey system theory specifically addresses uncertain systems characterized by “partially known and partially unknown information.” It does not require large sample sizes or typical probability distributions, enabling more effective modeling and trend extrapolation using limited annual macroeconomic data. This study selected GDP, total population, coal demand, the proportion of secondary industry output value, and the energy demand elasticity coefficient as independent variables for energy demand forecasting. It employed the grey correlation method to quantitatively describe the degree of correlation among variables, screened out indicators with higher correlation, and constructed a DGM(1,4) forecasting model based on the screening results to evaluate forecasting accuracy.

This paper is followed by Section 2, which introduces the theoretical knowledge of grey prediction model and grey correlation; Section 3, which utilizes the total national energy demand and related macro indicators from 2010 to 2021, screens the core influencing factors through grey correlation analysis, and establishes a DGM(1,4) model to make a prediction; and Section 4, which contains conclusions and recommendations.

Grey relational analysis is an important analytical tool in grey systems theory for quantifying the degree of dynamic correlation between factors within a system. The basic idea is to determine the magnitude of the degree of correlation by comparing the degree of geometric similarity between data series. If the trend of two sequences is closer, i.e. the higher the degree of synchronized change, the greater the degree of correlation between them is considered to be. This method does not require the data to obey a specific distribution, and is applicable to the degree of direct correlation of a small sample of data series for which part of the information is known and part of the information is unknown (Liu et al., 2014). The grey relational degree is used to portray the degree of influence of Chinese energy demand influencing factors on energy demand, and its calculation steps are as follows:

1. Perform dimensionless processing on the raw data:

To eliminate dimensional effects, the data was normalized. The calculation steps are as follows:

1. Find the absolute difference between the reference sequence and the comparison sequence:

1. Calculate the grey relational coefficient between the reference sequence and the comparison sequence:

Here, $min$ and $max$ denote the minimum and maximum values of the data; $ρ$ is the resolution coefficient, typically set to 0.5.

1. Calculate the grey relational degree:

Grey relational analysis $R0i$ can be able to portray the extent to which China’s energy demand influencing factors affect energy demand. Grey relation analysis $R0i$ greater, it indicates that the greater the degree of influence, the greater the importance to energy demand forecasting, and vice versa. Grey relational analysis not only quantitatively describes the degree of importance of the influencing factors to the energy demand forecast, but also determines the obvious multicollinearity problem based on the correlation.

In this study, a grey relational degree threshold of 0.8 was set to screen the core influencing factors. This established threshold is widely recognized in grey systems theory as indicative of a strong correlation (Liu et al., 2014). The inter-factor grey relational degrees exceeding 0.8 also signal the presence of significant multicollinearity among the potential influencing factors. Adhering to this threshold allows for the elimination of factors less critical to energy demand forecasting based on their correlation magnitude. This process not only identifies the most impactful drivers but also provides an effective safeguard for the construction of a robust multivariate forecasting model by mitigating the risk of multicollinearity.

The gray multivariate DGM(1,N) model represents a first-order difference time series model with one response variable and N influencing factors. It constructs model expressions through differential and difference relationships between first-order cumulative generation and cumulative reduction generation sequences, thereby enabling model parameter estimation (Ma and Liu, 2015).

In the gray multivariate DGM(1,N) model, let the system’s characteristic behavior sequence be:

The first-order cumulative generating sequence for the system’s characteristic behavior sequence $Y(0)$ is:

The sequence of relevant influencing factors is as follows:

The first-order cumulative generating sequence for the sequence X of relevant influencing factors is:

The differential equation form of the DGM(1,N) model is given by

Among these, parameter $a$ represents the system’s development coefficient, parameter $bixi(1)(t)$ denotes the system’s driving term, and parameter $bi$ signifies the driving coefficient (Ma and Liu, 2015).

The discrete values of the gray derivative of $y(1)(t)$ is

Substituting the discrete values of the gray derivative of $y(1)(t)$ (6) into the differential equation form (5) of the DGM(1,N) model yields

Let $β0=1a+1,βi=bia+1,μ=ua+1,i=1,2,⋯,N$

Then, the difference equation form of the DGM(1,N) model can be obtained as:

The parameter sequence $Pˆ=(β0,β1,β2,⋯,βN,μ)T$ in the differential equation form of the DGM(1,N) model can be estimated using the least squares method:

Among these,

Based on the difference equation form of the DGM(1,1) model and the results of its parameter estimation (9), the time response sequence of the DGM(1,1) model can be obtained:

Reduce ${yˆ(1)(t)},t=1,2,⋯,n$ by successive subtractions to obtain

In light of China’s energy situation, five specific indicators were selected as input variables, including Gross Domestic Product (GDP) and total energy demand, from the macro categories of population, economy, energy demand, and industrial structure. The total energy demand is taken as the reference series, and is denoted as $X1$⁠. The factors influencing this are $X2,X3,X4,X5,X6$⁠, and the total energy demand for 2010–2021 and the five factors are constructed as shown in Table 1 below, which is derived from The China Statistical Yearbook (2022).

1. Pretreatment

Standardize the data according to formula (1). The standardized data are shown in Table 2 below.

1. Grey relational analysis

This study employed grey relational analysis to identify factors with significant influence on China’s total energy consumption. Perform calculations using MATLAB software. The values of the grey correlation coefficients between the solved reference series and the contrasting series are shown in Table 3 below; the correlation coefficients reflect the degree of correlation between each contrasting series and the reference series at each dynamic point in time.

Following the steps for calculating the grey correlation, The grey relational degree for each indicator were calculated, with the results shown in Table 4.

As shown in Table 4, the top three indicators in terms of grey relational degree are GDP overall amount (0.9281), total population (0.8386), and coal demand (0.8061). Therefore, these three indicators are selected to construct a grey multivariate prediction model.

Based on the grey correlation coefficient, GDP total amount, total population, and coal demand were selected as the influence factor sequence. The DGM(1,4) model, along with two comparative models—GM(1,4) and MLR—were constructed to forecast total energy demand from 2010 to 2021. The prediction results and model performance evaluations are presented in Table 5.

By comparing the forecast results of the three models with actual values (Table 5), it is evident that the DGM(1,4) model demonstrates the most outstanding predictive performance. Its Mean Absolute Percentage Error (MAPE) is only 0.06%, significantly lower than the 0.32% of the MLR model and the 0.77% of the GM(1,4) model. Examining the forecast values for each year reveals that the DGM(1,4) model’s prediction curve closely aligns with the actual value curve, particularly after 2015 where they nearly coincide completely, demonstrating exceptional fitting accuracy and stability. In contrast, the MLR model exhibited significant deviations in 2011 and 2020, indicating limited adaptability of linear models to abnormal fluctuations. Meanwhile, the GM(1,4) model showed substantial prediction errors during the early phase of the series (2011–2012), reflecting potential issues such as initial value sensitivity in standard grey models. This comparative analysis fully validates the superiority of the DGM(1,4) model in forecasting China’s energy demand. By incorporating differential operations, it effectively enhances model stability and predictive accuracy, providing more reliable quantitative support for energy policy formulation.

This figure visually demonstrates the fitting performance of three forecasting models for China’s total energy demand between 2010 and 2021 (Figure 1). As shown, the forecast curve of the DGM(1,4) model nearly perfectly aligns with the actual observed curve, indicating its ability to accurately capture the dynamic trends in energy demand. The GM(1,4) model exhibits noticeable fluctuations in the early stages of the sequence, with forecast values significantly deviating from actual values, revealing initial instability. While the MLR method maintains a consistent overall trend, visible deviations occur at multiple points (e.g. 2011, 2020), reflecting the limitations of linear models in fitting complex nonlinear relationships. Overall, this illustration further validates the superior performance and reliability of the DGM(1,4) model in energy demand forecasting.

1. Through grey correlation analysis, this study clarifies the core factors affecting China’s total energy demand, which are, in order of correlation, GDP overall amount (correlation of 0.928), total population (correlation of 0.839), and coal demand (correlation of 0.806), suggesting that economic growth (GDP) is the strongest driver of energy demand growth, with population size and coal dependence coming next, and the proportion of secondary industry output, and the elasticity coefficient of energy demand have relatively weak effects.

2. The DGM(1,4) model shows high accuracy (MAPE = 0.06%) in forecasting total energy demand, and fits well especially in medium- and long-term forecasts. The grey correlation coefficient indicates that coal demand exerts the greatest influence on overall energy demand, reflecting China’s continued reliance on coal as the dominant energy source. Population size contributes secondarily, demonstrating that population growth remains a persistent driver of energy demand. GDP overall amount exhibits relatively low influence, likely attributable to economic restructuring—such as the increased share of the service sector—which has reduced energy intensity per unit of GDP.

1. Optimize the energy mix. Reduce dependence on coal, replace coal through clean energy (wind energy, photovoltaic) and reduce negative impacts. Promote industrial upgrading, increase the proportion of tertiary industry, and weaken the correlation between secondary industry (high energy consumption) and energy demand.

2. Improve the forecasting system. Dynamically adjust the model by introducing time-lag variables (e.g. policy lag effect) or segmented modeling (distinguishing between high/low economic growth stages). Integrate multi-methods and combine machine learning (LSTM, Random Forest) to deal with non-linear features and improve forecast robustness.

3. Strengthen data support. Expand the dimensions of the indicators to include new variables such as the proportion of renewable energy and carbon emission constraints, reflecting the trend of energy transition under the “dual-carbon” goal. Increase the frequency of data and adopt monthly or quarterly data to enhance the ability to capture short-term fluctuations.

From the above conclusions, we can see that the DGM(1, N) model has good accuracy in forecasting, but the model can still be optimized to further improve its accuracy, such as increasing the time lag and adding an intelligent optimization algorithm to search for the optimal parameters. This will be the focus of my next research, and I will strive to learn more about this topic so that I can improve the model as soon as possible to increase the accuracy of the model and make more accurate forecasts of China’s energy demand.