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Fractional Order Systems with Application to Electrical Power Engineering, 3rd...
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Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition

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

Deadline for manuscript submissions: 30 June 2026 | Viewed by 6253

Special Issue Editors

Dr. Arman Oshnoei

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Guest Editor
Department of Energy, Aalborg University, 9220 Aalborg, Denmark
Interests: power electronics; power systems; smart grid; AC/DC microgrid; intelligent control; fractional order system
Special Issues, Collections and Topics in MDPI journals
Dr. Soroush Oshnoei

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Guest Editor
Department of Electrical and Computer Engineering, Aarhus University, 8000 Aarhus, Denmark
Interests: power systems; microgrid; cyber-physical system; machine learning; optimal control; fractional order system

Special Issue Information

Dear Colleagues,

As the Guest Editors, we invite scientists and professionals  to submit their theoretical and applied contributions, as well as review articles, to this Special Issue of Fractal and Fractional on the subject of “Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition”. This Special Issue aims to advance the modeling, design, analysis, and control of fractional order systems for energy and power engineering applications, such as power electronics and electric motor drives, power systems, distributed generation, and multi-energy systems.

Fractional calculus plays a crucial role in accurately describing practical dynamic behaviors in engineering systems using fractional-order models. As a non-standard operator, fractional-order calculus addresses the limitations of classical differential equations, which cannot accurately describe the dynamic behavior of complex systems. It provides an effective tool for describing practical models with memory properties and historical dependence, offering additional degrees of freedom and enhancing design flexibility. The fractal nature of fractional calculus enables the formulation of more accurate mathematical models compared to those based on integer calculus.

Topics of interest for this Special Issue include, but are not limited to, the following:

  • Development of fractional order modeling of energy systems;
  • Fractional order simulation of energy systems with power electronic topologies;
  • Fractional order modeling and analysis of hybrid energy storage systems;
  • Artificial intelligence application in fractional order energy systems;
  • Robust control of fractional order energy systems;
  • Energy efficiency in fractional order energy systems;
  • Grid integration of fractional order power converters;
  • Power quality issues in fractional order energy systems;
  • Reliability and resilience issues in fractional order energy systems;
  • Intelligent control of fractional order energy systems;
  • Stability issues in fractional order energy systems;
  • Application of fractional order control strategies;
  • Fractional control design of renewable energy systems.

Dr. Arman Oshnoei
Dr. Soroush Oshnoei
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 250 words) can be sent to the Editorial Office for assessment.

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 system
  • distributed energy resources
  • energy storage system
  • multi-energy systems
  • power electronic systems
  • power converters
  • renewable energy systems
  • artificial intelligence
  • stability analysis
  • intelligent control
  • fractional calculus
  • reliability and resiliency

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Related Special Issues

Published Papers (7 papers)

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Research

25 pages, 10186 KB  
Article
Optimization Design Method for Full-Bridge LLC Resonant Converter Based on Fractional-Order Characteristics of Resonant Tank
by Xiaoquan Zhu, Chentao Ma and Haochi He
Fractal Fract. 2026, 10(3), 194; https://doi.org/10.3390/fractalfract10030194 - 16 Mar 2026
Viewed by 572
Abstract
The full-bridge LLC resonant converter is one of the most suitable converters for high-power, high-efficiency applications. Although the design methodologies for full-bridge LLC resonant converters are already well-established, the development of the fractional-order domain has brought new flexibility to converter design. Based on [...] Read more.
The full-bridge LLC resonant converter is one of the most suitable converters for high-power, high-efficiency applications. Although the design methodologies for full-bridge LLC resonant converters are already well-established, the development of the fractional-order domain has brought new flexibility to converter design. Based on the fact that inductors and capacitors have fractional-order characteristics, this paper presents a de-normalized fractional-order FHA gain model, which reveals the impact of fractional-order characteristics of practical inductors and capacitors on the converter gain. By maintaining the convenience of the FHA design method, this work identifies the fractional orders of a resonant tank inductor and capacitor and incorporates them into the parameter design as part of the design requirements, making the design results more accurate than the conventional FHA design method. Specifically, compared with the conventional FHA-based design, the proposed approach improves the DC voltage gain margin of the full-bridge LLC converter by 26% and expands the ZVS operating range margin by 23.3%. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)
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33 pages, 6279 KB  
Article
Maximum Power Extraction from a PMSG-Based Standalone WECS via Neuro-Adaptive Fuzzy Fractional Order Super-Twisting Sliding Mode Control Approach with High Gain Differentiator
by Ameen Ullah, Safeer Ullah, Umair Hussan, Dapeng Zheng, Danyang Bao and Xuewei Pan
Fractal Fract. 2026, 10(3), 158; https://doi.org/10.3390/fractalfract10030158 - 28 Feb 2026
Viewed by 390
Abstract
Maximum Power Point Tracking (MPPT) in permanent-magnet synchronous generator (PMSG)-based wind energy conversion systems (WECS) remains challenging owing to strong nonlinearities, parametric uncertainties, and external disturbances. Conventional sliding mode control (SMC) strategies, while robust, suffer from chattering, dependence on full-state measurements, and degraded [...] Read more.
Maximum Power Point Tracking (MPPT) in permanent-magnet synchronous generator (PMSG)-based wind energy conversion systems (WECS) remains challenging owing to strong nonlinearities, parametric uncertainties, and external disturbances. Conventional sliding mode control (SMC) strategies, while robust, suffer from chattering, dependence on full-state measurements, and degraded performance under model mismatch, limiting their practical deployment. To address these issues, this study proposes a neuroadaptive fuzzy fractional-order super-twisting sliding mode control (Fuzzy-FOSTSMC) integrated with a high-gain observer (HGO) and a radial basis function neural network (RBFNN). The HGO estimates unmeasurable higher-order states (e.g., angular acceleration), enabling output-feedback implementation. In contrast, the RBFNN online approximates unknown nonlinear system dynamics Lf2h(x) and LgLfh(x), rendering the controller model-free. Chattering is eliminated by replacing the discontinuous signum function with an adaptive fuzzy boundary layer that dynamically modulates the slope near the sliding surface. Stability is theoretically confirmed by Lyapunov analysis. Extensive MATLAB/Simulink simulations verify that the proposed approach yields a tracking precision of 99.96%, a steady-state speed error of 0.018 rad/s, and a 58.2% reduction in the integral absolute error (IAE) compared to the traditional FOSTSMC. It achieves the optimal power coefficient (Cp=0.4762) via TSR control at 7.000±0.002, under ±30% parametric uncertainties, demonstrating excellent robustness and MPPT effectiveness. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)
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24 pages, 1474 KB  
Article
A Fractional Hybrid Strategy for Reliable and Cost-Optimal Economic Dispatch in Wind-Integrated Power Systems
by Abdul Wadood, Babar Sattar Khan, Bakht Muhammad Khan, Herie Park and Byung O. Kang
Fractal Fract. 2026, 10(1), 64; https://doi.org/10.3390/fractalfract10010064 - 16 Jan 2026
Cited by 2 | Viewed by 522
Abstract
Economic dispatch in wind-integrated power systems is a critical challenge, yet many recent metaheuristics suffer from premature convergence, heavy parameter tuning, and limited ability to escape local optima in non-smooth valve-point landscapes. This study proposes a new hybrid optimization framework, the Fractional Grasshopper [...] Read more.
Economic dispatch in wind-integrated power systems is a critical challenge, yet many recent metaheuristics suffer from premature convergence, heavy parameter tuning, and limited ability to escape local optima in non-smooth valve-point landscapes. This study proposes a new hybrid optimization framework, the Fractional Grasshopper Optimization algorithm (FGOA), which integrates fractional-order calculus into the standard Grasshopper Optimization algorithm (GOA) to enhance its search efficiency. The FGOA method is applied to the economic load dispatch (ELD) problem, a nonlinear and nonconvex task that aims to minimize fuel and wind-generation costs while satisfying practical constraints such as valve-point loading effects (VPLEs), generator operating limits, and the stochastic behavior of renewable energy sources. Owing to the increasing role of wind energy, stochastic wind power is modeled through the incomplete gamma function (IGF). To further improve computational accuracy, FGOA is hybridized with Sequential Quadratic Programming (SQP), where FGOA provides global exploration and SQP performs local refinement. The proposed FGOA-SQP approach is validated on systems with 3, 13, and 40 generating units, including mixed thermal and wind sources. Comparative evaluations against recent metaheuristic algorithms demonstrate that FGOA-SQP achieves more accurate and reliable dispatch outcomes. Specifically, the proposed approach achieves fuel cost reductions ranging from 0.047% to 0.71% for the 3-unit system, 0.31% to 27.25% for the 13-unit system, and 0.69% to 12.55% for the 40-unit system when compared with state-of-the-art methods. Statistical results, particularly minimum fitness values, further confirm the superior performance of the FGOA-SQP framework in addressing the ELD problem under wind power uncertainty. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)
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16 pages, 7290 KB  
Article
Transfer Learning Fractional-Order Recurrent Neural Network for MPPT Under Weak PV Generation Conditions
by Umair Hussan, Mudasser Hassan, Umar Farooq, Huaizhi Wang and Muhammad Ahsan Ayub
Fractal Fract. 2026, 10(1), 41; https://doi.org/10.3390/fractalfract10010041 - 8 Jan 2026
Viewed by 797
Abstract
Photovoltaic generation systems (PVGSs) face significant efficiency challenges under partial shading conditions and rapidly changing irradiance due to the limitations of conventional maximum power point tracking (MPPT) methods. To address these challenges, this paper proposes a Transfer Learning-based Fractional-Order Recurrent Neural Network (TL-FRNN) [...] Read more.
Photovoltaic generation systems (PVGSs) face significant efficiency challenges under partial shading conditions and rapidly changing irradiance due to the limitations of conventional maximum power point tracking (MPPT) methods. To address these challenges, this paper proposes a Transfer Learning-based Fractional-Order Recurrent Neural Network (TL-FRNN) for robust global maximum power point (GMPP) tracking across diverse operating conditions. The incorporation of fractional-order dynamics introduces long-term memory and non-local behavior, enabling smoother state evolution and improved discrimination between local and global maxima, particularly under weak and partially shaded conditions. The proposed approach leverages Caputo fractional derivatives with Grünwald–Letnikov approximation to capture the history-dependent behavior of PVGSs while implementing a parameter-partitioning strategy that separates shared features from task-specific parameters. The architecture employs a multi-head design with GMPP regression and partial shading classification capabilities, trained through a two-stage process of pretraining on general PV data followed by efficient fine-tuning on target systems with limited site-specific data. The TL-FRNN achieved 99.2% tracking efficiency with 98.7% GMPP detection accuracy, reducing convergence time by 53% compared to state-of-the-art alternatives while requiring 72% less retraining time through transfer learning. This approach represents a significant advancement in adaptive, intelligent MPPT control for real-world photovoltaic energy-harvesting systems. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)
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31 pages, 2482 KB  
Article
Fractional-Order African Vulture Optimization for Optimal Power Flow and Global Engineering Optimization
by Abdul Wadood, Hani Albalawi, Shahbaz Khan, Bakht Muhammad Khan and Aadel Mohammed Alatwi
Fractal Fract. 2025, 9(12), 825; https://doi.org/10.3390/fractalfract9120825 - 17 Dec 2025
Cited by 1 | Viewed by 524
Abstract
This paper proposes a novel fractional-order African vulture optimization algorithm (FO-AVOA) for solving the optimal reactive power dispatch (ORPD) problem. By integrating fractional calculus into the conventional AVOA framework, the proposed method enhances the exploration–exploitation balance, accelerates convergence, and improves solution robustness. The [...] Read more.
This paper proposes a novel fractional-order African vulture optimization algorithm (FO-AVOA) for solving the optimal reactive power dispatch (ORPD) problem. By integrating fractional calculus into the conventional AVOA framework, the proposed method enhances the exploration–exploitation balance, accelerates convergence, and improves solution robustness. The ORPD problem is formulated as a constrained optimization task with the objective of minimizing real power losses while satisfying generator voltage limits, transformer tap ratios, and reactive power compensator constraints. The general optimization capability of the FO-AVOA is verified using the CEC 2017, 2020, and 2022 benchmark functions. In addition, the method is applied to the IEEE 30-bus and IEEE 57-bus test systems. The results demonstrate significant power loss reductions of up to 15.888% and 24.39% for the IEEE 30-bus and IEEE 57-bus systems, respectively, compared with the conventional AVOA and other state-of-the-art optimization algorithms, along with strong robustness and stability across independent runs. These findings confirm the effectiveness of the FO-AVOA as a reliable optimization tool for modern power system applications. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)
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24 pages, 2858 KB  
Article
An Advanced Control Framework for Frequency Regulation in Renewable-Dominated Power Systems: Fractional-Order Virtual Primary–Secondary Control with Synthetic Inertia Control
by Sherif A. Zaid, Gaber Magdy, Hani Albalawi, Omar A. Alatawi, Abderrahim Lakhouit and Ayman E. M. Ahmed
Fractal Fract. 2025, 9(12), 766; https://doi.org/10.3390/fractalfract9120766 - 25 Nov 2025
Cited by 1 | Viewed by 943
Abstract
This paper suggests a developed virtual inertia-damping control (VIDC) scheme to improve the frequency stability of low-inertia, renewable-dominated power systems. The introduced approach integrates fractional-order control theory into both virtual primary and secondary control loops (FO-VPSC), coordinated with a Battery Energy Storage System [...] Read more.
This paper suggests a developed virtual inertia-damping control (VIDC) scheme to improve the frequency stability of low-inertia, renewable-dominated power systems. The introduced approach integrates fractional-order control theory into both virtual primary and secondary control loops (FO-VPSC), coordinated with a Battery Energy Storage System (BESS)-based VIDC, to provide richer phase shaping and greater flexibility in mitigating diverse system disturbances. To ensure robustness and adaptability, the FO-VPSC parameters are optimally tuned using the Dandelion Optimizer (DO) algorithm. The effectiveness of the proposed strategy is validated on a two-area interconnected renewable power system comprising heterogeneous sources, including a thermal power plant, solar and wind units, and BESSs. Simulation results reveal that the proposed FO-VPSC significantly enhances the system’s dynamic response, achieving a 35–45% reduction in frequency overshoot, up to a 42% improvement in settling time, and a 30% reduction in frequency deviation magnitude compared with conventional VIDC. Moreover, the proposed control strategy improves inter-area oscillation damping and ensures stable operation under renewable penetration above 60% reduction in system inertia and damping constants. These results confirm that the FO-VPSC-based VIDC provides superior adaptability and resilience for next-generation low-inertia grids. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)
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23 pages, 5273 KB  
Article
Federated Learning Detection of Cyberattacks on Virtual Synchronous Machines Under Grid-Forming Control Using Physics-Informed LSTM
by Ali Khaleghi, Soroush Oshnoei and Saeed Mirzajani
Fractal Fract. 2025, 9(9), 569; https://doi.org/10.3390/fractalfract9090569 - 29 Aug 2025
Cited by 9 | Viewed by 1793
Abstract
The global shift toward clean production, like using renewable energy, has significantly decreased the use of synchronous machines (SMs), which help maintain stability and control, causing serious frequency stability issues in power systems with low inertia. Fractional order controller-based virtual synchronous machines (FOC-VSMs) [...] Read more.
The global shift toward clean production, like using renewable energy, has significantly decreased the use of synchronous machines (SMs), which help maintain stability and control, causing serious frequency stability issues in power systems with low inertia. Fractional order controller-based virtual synchronous machines (FOC-VSMs) have become a promising option, but they rely on communication networks to work together in real time, causing them to be at risk of cyberattacks, especially from false data injection attacks (FDIAs). This paper suggests a new way to detect FDI attacks using a federated physics-informed long short-term memory (PI-LSTM) network. Each FOC-VSM uses its data to train a PI-LSTM, which keeps the information private but still helps it learn from a common model that understands various operating conditions. The PI-LSTM incorporates physical constraints derived from the FOC-VSM swing equation, facilitating residual-based anomaly detection that is sensitive to minor deviations in control dynamics, such as altered inertia or falsified frequency signals. Unlike traditional LSTMs, the physics-informed architecture minimizes false positives arising from benign disturbances. We assessed the proposed method on an IEEE 9-bus test system featuring two FOC-VSMs. The results show that our method can successfully detect FDI attacks while handling regular changes, proving it could be a strong solution. Full article
(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 3rd Edition)
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