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Fractal Fract
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Applications of Fractional-Order Grey Models, 2nd Edition
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Applications of Fractional-Order Grey Models, 2nd Edition

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: 15 January 2026 | Viewed by 1834

Special Issue Editors

Prof. Dr. Lifeng Wu

grade E-Mail Website
Guest Editor
College of Management Engineering and Business, Hebei University of Engineering, Handan 056038, China
Interests: fractional models; grey system model
Special Issues, Collections and Topics in MDPI journals
Dr. Shuli Yan

E-Mail Website
Guest Editor
1. School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
2. Department of Computer Science and Artificial Intelligence, University of Granada, 18071 Granada, Spain
Interests: fractional grey models; prediction; decision analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional-order grey systems refers to a class of emerging theories that incorporate concepts from fractional-order calculus into the establishment of grey models. Most of the systems in real life are fractional-order, and the fractional-order grey model can essentially reflect the characteristics and behavior of objects. Compared with traditional grey models, the fractional-order grey models have stronger stability, higher flexibility and wider application prospects. They can be further divided into fractional-order accumulating grey models and fractional-order derivative grey models. The main difference is whether the modeling is implemented directly using the raw data or accumulating the raw data firstly. In addition, fractional grey models have been widely used in various fields such as agriculture, industry, society, economy, management, transportation, and energy, and relevant achievements are constantly emerging.

The focus of this Special Issue is to advance research on topics relating to the application of fractional-order grey models. The submitted papers must demonstrate sufficient novelty in the solution of practical problems by using fractional-order grey models. Topics that are invited for submission include (but are not limited to):

  • Applications of fractional-order grey models for energy consumption;
  • Fractional-order grey modelling algorithm;
  • Fractional-order grey model theory improvements;
  • Applications of fractional-order grey models for air quality;
  • Applications of fractional-order grey models for traffic flow;
  • Applications of fractional-order grey models in agriculture;
  • Applications of fractional-order grey models in settlement prediction;
  • Applications of fractional-order grey models in supply chains;
  • Applications of fractional-order grey models for natural disasters.

Prof. Dr. Lifeng Wu
Dr. Shuli Yan
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional-order
  • grey model
  • prediction
  • algorithm
  • application

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Related Special Issue

Published Papers (4 papers)

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Research

21 pages, 2627 KB  
Article
Fractional-Order Accumulative Gray Model for Carbon Emission Prediction: A Case Study of Shandong Province
by Lei Wu, Wei-Feng Gong, Wei-Jie Zhang and Xue-Yan Liu
Fractal Fract. 2025, 9(9), 595; https://doi.org/10.3390/fractalfract9090595 - 12 Sep 2025
Viewed by 27
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models, 2nd Edition)
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30 pages, 4012 KB  
Article
A Novel Nonlinear Different Fractional Discrete Grey Multivariate Model and Its Application in Energy Consumption
by Jun Zhang and Jiayi Liu
Fractal Fract. 2025, 9(9), 555; https://doi.org/10.3390/fractalfract9090555 - 22 Aug 2025
Viewed by 335
Abstract
With global energy demand escalating and climate change posing unprecedented challenges, accurate forecasting of regional energy consumption has emerged as a cornerstone for national energy planning and sustainable development strategies. This study develops a novel nonlinear different fractional discrete grey multivariate model (NDFDGM( [...] Read more.
With global energy demand escalating and climate change posing unprecedented challenges, accurate forecasting of regional energy consumption has emerged as a cornerstone for national energy planning and sustainable development strategies. This study develops a novel nonlinear different fractional discrete grey multivariate model (NDFDGM(ri,N)). This model improves the shortcomings of the conventional GM(1,N) in handling nonlinear relationships and variable differences by introducing different fractional order accumulation and nonlinear logarithmic conditioning terms. In addition, the Firefly Algorithm (FA) was utilized to optimize the model’s hyperparameters, significantly enhancing the prediction accuracy. Through empirical analysis of energy consumption data in China’s eastern, central and western regions and across the country, it has been confirmed that the NDFDGM model outperforms others during both the simulation and forecasting phases, and its predicted MAPE values are, respectively, 1.4585%, 1.4496%, 2.0673% and significantly lower than that of compared models. The findings indicate that this model can effectively capture the complex characteristics of energy consumption, and its prediction results offer a solid scientific foundation for guiding energy strategies and shaping policy decisions. Finally, this paper conducts extrapolation and predictive analysis using the NDFDGM(ri,N) to explore the development trends of energy consumption in the whole country in the coming three years and puts forward energy policy suggestions for different regions to promote the optimization and sustainable development of the energy structure. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models, 2nd Edition)
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27 pages, 2185 KB  
Article
A Novel Fractional Order Multivariate Partial Grey Model and Its Application in Natural Gas Production
by Hui Li, Huiming Duan and Hongli Chen
Fractal Fract. 2025, 9(7), 422; https://doi.org/10.3390/fractalfract9070422 - 27 Jun 2025
Viewed by 565
Abstract
Accurate prediction of natural gas production is of great significance for optimizing development strategies, simplifying production management, and promoting decision-making. This paper utilizes partial differentiation to effectively capture the spatiotemporal characteristics of natural gas data and the advantages of grey prediction models. By [...] Read more.
Accurate prediction of natural gas production is of great significance for optimizing development strategies, simplifying production management, and promoting decision-making. This paper utilizes partial differentiation to effectively capture the spatiotemporal characteristics of natural gas data and the advantages of grey prediction models. By introducing the fractional damping accumulation operator, a new fractional order partial grey prediction model is established. The new model utilizes partial capture of details and features in the data, improves model accuracy through fractional order accumulation, and extends the metadata of the classic grey prediction model from time series to matrix series, effectively compensating for the phenomenon of inaccurate results caused by data fluctuations in the model. Meanwhile, the principle of data accumulation is effectively expressed in matrix form, and the least squares method is used to estimate the parameters of the model. The time response equation of the model is obtained through multiplication transformation, and the modelling steps are elaborated in detail. Finally, the new model is applied to the prediction of natural gas production in Qinghai Province, China, selecting energy production related to natural gas production, including raw coal production, oil production, and electricity generation, as relevant variables. To verify the effectiveness of the new model, we started by selecting the number of relevant variables, divided them into three categories for analysis based on the number of relevant variables, and compared them with five other grey prediction models. The results showed that in the seven simulation experiments of the three types of experiments, the average relative error of the new model was less than 2%, indicating that the new model has strong stability. When selecting the other three types of energy production as related variables, the best effect was achieved with an average relative error of 0.3821%, and the natural gas production for the next nine months was successfully predicted. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models, 2nd Edition)
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16 pages, 2958 KB  
Article
Fractional Uncertain Forecasting of the Impact of Groundwater Over-Exploitation on Temperature in the Largest Groundwater Depression Cone
by Xiangyue Ren, Liyuan Ren and Lifeng Wu
Fractal Fract. 2025, 9(5), 299; https://doi.org/10.3390/fractalfract9050299 - 5 May 2025
Viewed by 540
Abstract
China currently faces critical climatic conditions, with persistent global warming trends and extreme heat waves across the northern hemisphere. To explore the predictive trajectory of regional extreme high temperature influenced by groundwater over-exploitation, the SGMC(1,N) was established. Additionally, the SGMC(1,N) model was validated [...] Read more.
China currently faces critical climatic conditions, with persistent global warming trends and extreme heat waves across the northern hemisphere. To explore the predictive trajectory of regional extreme high temperature influenced by groundwater over-exploitation, the SGMC(1,N) was established. Additionally, the SGMC(1,N) model was validated using 2019–2023 observational data from the world’s largest groundwater depression cone. The results demonstrate superior performance, with the model achieving a MAPE of 1.97% compared to benchmark models. Scenario simulations with annual groundwater reduction rates (−15%, −20%, −25%) successfully project extreme heat evolution for 2024–2028. When the decline rate of annual groundwater over-exploitation is set at −20%, a 6.66 °C temperature reduction from baseline by 2028 is projected. Stable decline trends emerge when GOE reduction exceeds 20%. To mitigate regional extreme heat, implementing phased groundwater extraction quotas and total extraction cap regulations is recommended. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models, 2nd Edition)
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