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In order to accurately predict the uncertain and nonlinear characteristics of China's three clean energy generation, this paper presents a novel time-varying grey Riccati model (TGRM(1,1)) based on interval grey number sequences.

By combining grey Verhulst model and a special kind of Riccati equation and introducing a time-varying parameter and random disturbance term the authors advance a TGRM(1,1) based on interval grey number sequences. Additionally, interval grey number sequences are converted into middle value sequences and trapezoid area sequences by using geometric characteristics. Then the predicted formula is obtained by using differential equation principle. Finally, the proposed model's predictive effect is evaluated by three numerical examples of China's clean energy generation.

Based on the interval grey number sequences, the TGRM(1,1) is applied to predict the development trend of China's wind power generation, China's hydropower generation and China's nuclear power generation, respectively, to verify the effectiveness of the novel model. The results show that the proposed model has better simulated and predicted performance than compared models.

Due to the uncertain information and continuous changing of clean energy generation in the past decade, interval grey number sequences are introduced to characterize full information of the annual clean energy generation data. And the novel TGRM(1,1) is applied to predict upper and lower bound values of China's clean energy generation, which is significant to give directions for energy policy improvements and modifications.

The main contribution of this paper is to propose a novel TGRM(1,1) based on interval grey number sequences, which considers the changes of parameters over time by introducing a time-varying parameter and random disturbance term. In addition, the model introduces the Riccati equation into classic Verhulst, which has higher practicability and prediction accuracy.

To develop the theory and application of the grey prediction model, this investigation constructs a novel discrete grey Riccati model termed DGRM(1,1).

By examining a special kind of Riccati difference equation and the structure of the conventional discrete grey model (DGM), we advance a novel DGRM, and the model's prediction effect is evaluated by two numerical examples and an application case and compared with that of other conventional grey models.

The average relative simulation error of DGRM(1,1) does not change if the model is built after the original sequence has been transformed by a multiplier, and the new model is suitable to predict monotonically increasing, monotonically decreasing and unimodal sequences.

DGRM(1,1) is utilized to forecast the development cost of a small plane owned by the Aviation Industry Corporation of China (AVIC) with an original data sequence from 2006 to 2013. The outcomes indicate that DGRM(1,1) exhibits high precision and potential in development cost prediction.

Combining the Riccati difference equation with the conventional DGM, the author advances a new grey model that is suitable to predict three kinds of data series with different changing trends.

The purpose of this paper is to summarize progress of grey forecasting modelling, explain mechanism of grey forecasting modelling and classify exist grey forecasting models.

General modelling process and mechanism of grey forecasting modelling is summarized and classification of grey forecasting models is done according to their differential equation structure. Grey forecasting models with linear structure are divided into continuous single variable grey forecasting models, discrete single variable grey forecasting models, continuous multiple variable grey forecasting models and discrete multiple variable grey forecasting models. The mechanism and traceability of these models are discussed. In addition, grey forecasting models with nonlinear structure, grey forecasting models with grey number sequences and grey forecasting models with multi-input and multi-output variables are further discussed.

It is clearly to explain differences between grey forecasting models with other forecasting models. Accumulation generation operation is the main difference between grey forecasting models and other models, and it is helpful to mining system developing law with limited data. A great majority of grey forecasting models are linear structure while grey forecasting models with nonlinear structure should be further studied.

Mechanism and classification of grey forecasting models are very helpful to combine with suitable real applications.

The main contributions of this paper are to classify models according to models' structure are linear or nonlinear, to analyse relationships and differences of models in same class and to deconstruct mechanism of grey forecasting models.

The purpose of this study is to make a prediction of China's energy consumption structure from the perspective of compositional data and construct a novel grey model for forecasting compositional data.

Due to the existing grey prediction model based on compositional data cannot effectively excavate the evolution law of correlation dimension sequence of compositional data. Thus, the adaptive discrete grey prediction model with innovation term based on compositional data is proposed to forecast the integral structure of China's energy consumption. The prediction results from the new model are then compared with three existing approaches and the comparison results indicate that the proposed model generally outperforms existing methods. A further prediction of China's energy consumption structure is conducted into a future horizon from 2021 to 2035 by using the model.

China's energy structure will change significantly in the medium and long term and China's energy consumption structure can reach the long-term goal. Besides, the proposed model can better mine and predict the development trend of single time series after the transformation of compositional data.

The paper considers the dynamic change of grey action quantity, the characteristics of compositional data and the impact of new information about the system itself on the current system development trend and proposes a novel adaptive discrete grey prediction model with innovation term based on compositional data, which fills the gap in previous studies.

A time-varying grey Fourier model (TVGFM(1,1,N)) is proposed for the simulation of variable amplitude seasonal fluctuation time series, as the performance of traditional grey models can't catch the time-varying trend well.

The proposed model couples Fourier series and linear time-varying terms as the grey action, to describe the characteristics of variable amplitude and seasonality. The truncated Fourier order N is preselected from the alternative order set by Nyquist-Shannon sampling theorem and the principle of simplicity, then the optimal Fourier order is determined by hold-out method to improve the robustness of the proposed model. Initial value correction and the multiple transformation are also studied to improve the precision.

The new model has a broader applicability range as a result of the new grey action, attaining higher fitting and forecasting accuracy. The numerical experiment of a generated monthly time series indicates the proposed model can accurately fit the variable amplitude seasonal sequence, in which the mean absolute percentage error (MAPE) is only 0.01%, and the complex simulations based on Monte-Carlo method testify the validity of the proposed model. The results of monthly electricity consumption in China's primary industry, demonstrate the proposed model catches the time-varying trend and has good performances, where MAPE_F and MAPE_T are below 5%. Moreover, the proposed TVGFM(1,1,N) model is superior to the benchmark models, grey polynomial model (GMP(1,1,N)), grey Fourier model (GFM(1,1,N)), seasonal grey model (SGM(1,1)), seasonal ARIMA model seasonal autoregressive integrated moving average model (SARIMA) and support vector regression (SVR).

The parameter estimates and forecasting of the new proposed TVGFM are studied, and the good fitting and forecasting accuracy of time-varying amplitude seasonal fluctuation series are testified by numerical simulations and a case study.

Traditional multivariable grey prediction models define the background-value coefficients of the dependent and independent variables uniformly, ignoring the differences between their physical properties, which in turn affects the stability and reliability of the model performance.

A novel multivariable grey prediction model is constructed with different background-value coefficients of the dependent and independent variables, and a one-to-one correspondence between the variables and the background-value coefficients to improve the smoothing effect of the background-value coefficients on the sequences. Furthermore, the fractional order accumulating operator is introduced to the new model weaken the randomness of the raw sequence. The particle swarm optimization (PSO) algorithm is used to optimize the background-value coefficients and the order of the model to improve model performance.

The new model structure has good variability and compatibility, which can achieve compatibility with current mainstream grey prediction models. The performance of the new model is compared and analyzed with three typical cases, and the results show that the new model outperforms the other two similar grey prediction models.

This study has positive implications for enriching the method system of multivariable grey prediction model.

This paper aims to develop a novel grey Bernoulli model with memory characteristics, which is designed to dynamically choose the optimal memory kernel function and the length of memory dependence period, ultimately enhancing the model's predictive accuracy.

This paper enhances the traditional grey Bernoulli model by introducing memory-dependent derivatives, resulting in a novel memory-dependent derivative grey model. Additionally, fractional-order accumulation is employed for preprocessing the original data. The length of the memory dependence period for memory-dependent derivatives is determined through grey correlation analysis. Furthermore, the whale optimization algorithm is utilized to optimize the cumulative order, power index and memory kernel function index of the model, enabling adaptability to diverse scenarios.

The selection of appropriate memory kernel functions and memory dependency lengths will improve model prediction performance. The model can adaptively select the memory kernel function and memory dependence length, and the performance of the model is better than other comparison models.

The model presented in this article has some limitations. The grey model is itself suitable for small sample data, and memory-dependent derivatives mainly consider the memory effect on a fixed length. Therefore, this model is mainly applicable to data prediction with short-term memory effect and has certain limitations on time series of long-term memory.

In practical systems, memory effects typically exhibit a decaying pattern, which is effectively characterized by the memory kernel function. The model in this study skillfully determines the appropriate kernel functions and memory dependency lengths to capture these memory effects, enhancing its alignment with real-world scenarios.

Based on the memory-dependent derivative method, a memory-dependent derivative grey Bernoulli model that more accurately reflects the actual memory effect is constructed and applied to power generation forecasting in China, South Korea and India.

This study aims to predict China's carbon emission intensity and put forward a set of policy recommendations for further development of a low-carbon economy in China.

In this paper, the Interaction Effect Grey Power Model of N Variables (IEGPM(1,N)) is developed, and the Dragonfly algorithm (DA) is used to select the best power index for the model. Specific model construction methods and rigorous mathematical proofs are given. In order to verify the applicability and validity, this paper compares the model with the traditional grey model and simulates the carbon emission intensity of China from 2014 to 2021. In addition, the new model is used to predict the carbon emission intensity of China from 2022 to 2025, which can provide a reference for the 14th Five-Year Plan to develop a scientific emission reduction path.

The results show that if the Chinese government does not take effective policy measures in the future, carbon emission intensity will not achieve the set goals. The IEGPM(1,N) model also provides reliable results and works well in simulation and prediction.

The paper considers the nonlinear and interactive effect of input variables in the system's behavior and proposes an improved grey multivariable model, which fills the gap in previous studies.

With the massive use of fossil energy polluting the natural environment, clean energy has gradually become the focus of future energy development. The purpose of this article is to propose a new hybrid forecasting model to forecast the production and consumption of clean energy.

Firstly, the memory characteristics of the production and consumption of clean energy were analyzed by the rescaled range analysis (R/S) method. Secondly, the original series was decomposed into several components and residuals with different characteristics by the ensemble empirical mode decomposition (EEMD) algorithm, and the residuals were predicted by the fractional derivative grey Bernoulli model [FDGBM (p, 1)]. The other components were predicted using artificial intelligence (AI) models (least square support vector regression [LSSVR] and artificial neural network [ANN]). Finally, the fitting values of each part were added to get the predicted value of the original series.

This study found that clean energy had memory characteristics. The hybrid models EEMD–FDGBM (p, 1)–LSSVR and EEMD–FDGBM (p, 1)–ANN were significantly higher than other models in the prediction of clean energy production and consumption.

Consider that clean energy has complex nonlinear and memory characteristics. In this paper, the EEMD method combined the FDGBM (P, 1) and AI models to establish hybrid models to predict the consumption and output of clean energy.

H₂O, CO₂ and CO are three main species in combustion systems which have high volume fractions. In addition, soot has strong absorption in the infrared band. Thus, H₂O, CO₂, CO and soot may take important roles in radiative heat transfer. To provide calculations with high accuracy, all of the participating media should be considered non-gray media. Thus, the purpose of this paper is to study the effect of non-gray participating gases and soot on radiative heat transfer in an inhomogeneous and non-isothermal system.

To solve the radiative heat transfer, the fluid flow as well as the pressure, temperature and species distributions were first computed by FLUENT. The radiative properties of the participating media are calculated by the Statistical Narrow Band correlated K-distribution (SNBCK), which is based on the database of EM2C. The calculation of soot properties is based on the Mie scattering theory and Rayleigh theory. The radiative heat transfer is calculated by the discrete ordinate method (DOM).

Using SNBCK to calculate the radiative properties and DOM to calculate the radiative heat transfer, the influence of H₂O, CO₂, CO and soot on radiation heat flux to the wall in combustion system was studied. The results show that the global contribution of CO to the radiation heat flux on the wall in the kerosene furnace was about 2 per cent, but that it can reach up to 15 per cent in a solid fuel gasifier. The global contribution of soot to the radiation heat flux on the wall was 32 per cent. However, the scattering of soot has a tiny influence on radiation heat flux to the wall.

This is the first time H₂O, CO₂, CO and the scattering of soot were all considered simultaneously to study the radiation heat flux in combustion systems.