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Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition

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

Deadline for manuscript submissions: closed (1 January 2025) | Viewed by 14416

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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
Special Issues, Collections and Topics in MDPI journals

Dr. Mahdieh S. Sadabadi

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Guest Editor

Department of Electrical and Electronic Engineering, The University of Manchester, Manchester, UK
Interests: robust control; networked control systems; power-electronics-based power systems

Special Issue Information

Dear Colleagues,

As Guest Editors, we encourage scientists and colleagues to submit their theoretical and applied contributions, as well as review articles, to this Special Issue of Fractal and Fractional on the subject “Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition”. This Special Issue aims to explore 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 can describe many practical dynamic behaviors in the engineering field as fractional-order systems. As a non-standard operator, fractional-order calculus can describe the dynamic behavior of complex systems that cannot be described by the constitutive model of classical differential equations. It provides an effective tool for defining practical models with memory properties and historical reliance, provides additional degrees of freedom, and increases design flexibility. A more accurate mathematical model of the system can be established using fractional calculus due to the nature of a fractal dimension compared to 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. Mahdieh S. Sadabadi
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 systems
distributed energy resources
energy storage systems
multi-energy systems
power electronic systems
power converters
renewable energy systems
artificial intelligence
stability analysis
intelligent control
fractional calculus
reliability and resilience

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Fractional Order Systems with Application to Electrical Power Engineering in Fractal and Fractional (12 articles)

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Research

34 pages, 12268 KiB

Open AccessArticle

Novel Fractional Order Differential and Integral Models for Wind Turbine Power–Velocity Characteristics

by Ahmed G. Mahmoud, Mohamed A. El-Beltagy and Ahmed M. Zobaa

Fractal Fract. 2024, 8(11), 656; https://doi.org/10.3390/fractalfract8110656 - 11 Nov 2024

Viewed by 1267

Abstract

This work presents an improved modelling approach for wind turbine power curves (WTPCs) using fractional differential equations (FDE). Nine novel FDE-based models are presented for mathematically modelling commercial wind turbine modules’ power–velocity (P-V) characteristics. These models utilize Weibull and Gamma probability density functions to estimate the capacity factor (CF), where accuracy is measured using relative error (RE). Comparative analysis is performed for the WTPC mathematical models with a varying order of differentiation (

α

) from 0.5 to 1.5, utilizing the manufacturer data for 36 wind turbines with capacities ranging from 150 to 3400 kW. The shortcomings of conventional mathematical models in various meteorological scenarios can be overcome by applying the Riemann–Liouville fractional integral instead of the classical integer-order integrals. By altering the sequence of differentiation and comparing accuracy, the suggested model uses fractional derivatives to increase flexibility. By contrasting the model output with actual data obtained from the wind turbine datasheet and the historical data of a specific location, the models are validated. Their accuracy is assessed using the correlation coefficient (R) and the Mean Absolute Percentage Error (MAPE). The results demonstrate that the exponential model at

α = 0.9

gives the best accuracy of WTPCs, while the original linear model was the least accurate. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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24 pages, 6890 KiB

Open AccessArticle

Application of an Optimal Fractional-Order Controller for a Standalone (Wind/Photovoltaic) Microgrid Utilizing Hybrid Storage (Battery/Ultracapacitor) System

by Hani Albalawi, Sherif A. Zaid, Aadel M. Alatwi and Mohamed Ahmed Moustafa

Fractal Fract. 2024, 8(11), 629; https://doi.org/10.3390/fractalfract8110629 - 25 Oct 2024

Viewed by 1029

Abstract

Nowadays, standalone microgrids that make use of renewable energy sources have gained great interest. They provide a viable solution for rural electrification and decrease the burden on the utility grid. However, because standalone microgrids are nonlinear and time-varying, controlling and managing their energy can be difficult. A fractional-order proportional integral (FOPI) controller was proposed in this study to enhance a standalone microgrid’s energy management and performance. An ultra-capacitor (UC) and a battery, called a hybrid energy storage scheme, were employed as the microgrid’s energy storage system. The microgrid was primarily powered by solar and wind power. To achieve optimal performance, the FOPI’s parameters were ideally generated using the gorilla troop optimization (GTO) technique. The FOPI controller’s performance was contrasted with a conventional PI controller in terms of variations in load power, wind speed, and solar insolation. The microgrid was modeled and simulated using MATLAB/Simulink software R2023a 23.1. The results indicate that, in comparison to the traditional PI controller, the proposed FOPI controller significantly improved the microgrid’s transient performance. The load voltage and frequency were maintained constant against the least amount of disturbance despite variations in wind speed, photovoltaic intensity, and load power. In contrast, the storage battery precisely stores and releases energy to counteract variations in wind and photovoltaic power. The outcomes validate that in the presence of the UC, the microgrid performance is improved. However, the improvement is very close to that gained when using the proposed controller without UC. Hence, the proposed controller can reduce the cost, weight, and space of the system. Moreover, a Hardware-in-the-Loop (HIL) emulator was implemented using a C2000™ microcontroller LaunchPad™ TMS320F28379D kit (Texas Instruments, Dallas, TX, USA) to evaluate the proposed system and validate the simulation results. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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34 pages, 10150 KiB

Open AccessArticle

Enhancing Power Quality in Decentralized Hybrid Microgrids: Optimized DSTATCOM Performance Using Cascaded Fractional-Order Controllers and Hybrid Optimization Algorithms

by Abdullah M. Alharbi, Sulaiman Z. Almutairi, Ziad M. Ali, Shady H. E. Abdel Aleem and Mohamed M. Refaat

Fractal Fract. 2024, 8(10), 589; https://doi.org/10.3390/fractalfract8100589 - 4 Oct 2024

Cited by 1 | Viewed by 1020

Abstract

At present, the integration of microgrids into power systems presents significant power quality challenges in terms of the rising adoption of nonlinear loads and electric vehicles. Ensuring the stability and efficiency of the electrical network in this evolving landscape is crucial. This paper explores the implementation of cascading Proportional–Integral (PI-PI) and cascading Fractional-Order PI (FOPI-FOPI) controllers for a Distribution Static Compensator (DSTATCOM) in hybrid microgrids that include photovoltaic (PV) systems and fuel cells. A novel hybrid optimization algorithm, WSO-WOA, is introduced to enhance power quality. This algorithm leverages the strengths of the White Shark Optimization (WSO) algorithm and the Whale Optimization Algorithm (WOA), with WSO generating new candidate solutions and WOA exploring alternative search areas when WSO does not converge on optimal results. The proposed approach was rigorously tested through multiple case studies and compared with established metaheuristic algorithms. The findings demonstrate that the WSO-WOA hybrid algorithm significantly outperforms others in optimizing the PI-PI and FOPI-FOPI controllers. The WSO-WOA algorithm showed an improvement in accuracy, surpassing the other algorithms by approximately 7.29% to 14.1% in the tuning of the PI-PI controller and about 8.5% to 21.2% in the tuning of the FOPI-FOPI controller. Additionally, the results confirm the superior performance of the FOPI-FOPI controller over the PI-PI controller in enhancing the effectiveness of the DSTATCOM across various scenarios. The FOPI-FOPI provided controller a reduced settling time by at least 30.5–56.1%, resulting in marked improvements in voltage regulation and overall power quality within the microgrid. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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30 pages, 5303 KiB

Open AccessArticle

State-Space Approach to the Time-Fractional Maxwell’s Equations under Caputo Fractional Derivative of an Electromagnetic Half-Space under Four Different Thermoelastic Theorems

by Eman A. N. Al-Lehaibi and Hamdy M. Youssef

Fractal Fract. 2024, 8(10), 566; https://doi.org/10.3390/fractalfract8100566 - 28 Sep 2024

Viewed by 671

Abstract

This paper introduces a new mathematical modelling method of a thermoelastic and electromagnetic half-space in the context of four different thermoelastic theorems: Green–Naghdi type-I, and type-III; Lord–Shulman; and Moore–Gibson–Thompson. The bunding plane of the half-space surface is subjected to ramp-type heat and traction-free. We consider that Maxwell’s time-fractional equations have been under Caputo’s fractional derivative definition, which is the novelty of this work. Laplace transform techniques are utilized to obtain solutions using the state-space approach. Laplace transform’s inversions were calculated using Tzou’s iteration method. The temperature increment, strain, displacement, stress, induced electric field, and induced magnetic field distributions were obtained numerically and are illustrated in figures. The time-fraction parameter of Maxwell’s equations had a major impact on all the studied functions. The time-fractional parameter of Maxwell’s equations worked as resistant to the changing of temperature, particle movement, and induced magnetic field, while it acted as a catalyst to the induced electric field through the material. Moreover, all the studied functions have different values in the context of the four studied theorems. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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24 pages, 1725 KiB

Open AccessArticle

Leveraging the Performance of Integrated Power Systems with Wind Uncertainty Using Fractional Computing-Based Hybrid Method

by Hani Albalawi, Yasir Muhammad, Abdul Wadood, Babar Sattar Khan, Syeda Taleeha Zainab and Aadel Mohammed Alatwi

Fractal Fract. 2024, 8(9), 532; https://doi.org/10.3390/fractalfract8090532 - 11 Sep 2024

Viewed by 691

Abstract

Reactive power dispatch (RPD) in electric power systems, integrated with renewable energy sources, is gaining popularity among power engineers because of its vital importance in the planning, designing, and operation of advanced power systems. The goal of RPD is to upgrade the power system performance by minimizing the transmission line losses, enhancing voltage profiles, and reducing the total operating costs by tuning the decision variables such as transformer tap setting, generator’s terminal voltages, and capacitor size. But the complex, non-linear, and dynamic characteristics of the power networks, as well as the presence of power demand uncertainties and non-stationary behavior of wind generation, pose a challenging problem that cannot be solved efficiently with traditional numerical techniques. In this study, a new fractional computing strategy, namely, fractional hybrid particle swarm optimization (FHPSO), is proposed to handle RPD issues in electric networks integrated with wind power plants (WPPs) while incorporating the power demand uncertainties. To improve the convergence characteristics of the Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA), the proposed FHPSO incorporates the concepts of Shannon entropy inside the mathematical model of traditional PSOGSA. Extensive experimentation validates FHPSO effectiveness by computing the best value of objective functions, namely, voltage deviation index and line loss minimization in standard power systems. The proposed FHPSO shows an improvement in percentage of 61.62%, 85.44%, 86.51%, 93.15%, 84.37%, 67.31%, 61.64%, 61.13%, 8.44%, and 1.899%, respectively, over ALC_PSO, FAHLCPSO, OGSA, ABC, SGA, CKHA, NGBWCA, KHA, PSOGSA, and FPSOGSA in case of traditional optimal reactive power dispatch(ORPD) for IEEE 30 bus system. Furthermore, the stability, robustness, and precision of the designed FHPSO are determined using statistical interpretations such as cumulative distribution function graphs, quantile-quantile plots, boxplot illustrations, and histograms. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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30 pages, 3304 KiB

Open AccessArticle

Robust Speed Control of Permanent Magnet Synchronous Motor Drive System Using Sliding-Mode Disturbance Observer-Based Variable-Gain Fractional-Order Super-Twisting Sliding-Mode Control

by Ameen Ullah, Jianfei Pan, Safeer Ullah and Zhang Zhang

Fractal Fract. 2024, 8(7), 368; https://doi.org/10.3390/fractalfract8070368 - 24 Jun 2024

Cited by 1 | Viewed by 1668

Abstract

This paper proposes a novel nonlinear speed control method for permanent magnet synchronous motors that enhances their robustness and tracking performance. This technique integrates a sliding-mode disturbance observer and variable-gain fractional-order super-twisting sliding-mode control within a vector-control framework. The proposed control scheme employs a sliding-mode control method to mitigate chattering and improve dynamics by implementing fractional-order theory with a variable-gain super-twisting sliding manifold design while regulating the speed of the considered motor system. The aforementioned observer is suggested to enhance the control accuracy by estimating and compensating for the lumped disturbances. The proposed methodology demonstrates its superiority over other control schemes such as traditional sliding-mode control, super-twisting sliding-mode control, and the proposed technique. MATLAB/Simulink simulations and real-time implementation validate its performance, showing its potential as a reliable and efficient control approach for the system under study in practical applications. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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18 pages, 6627 KiB

Open AccessArticle

The Regulation of Superconducting Magnetic Energy Storages with a Neural-Tuned Fractional Order PID Controller Based on Brain Emotional Learning

by Ashkan Safari, Hoda Sorouri and Arman Oshnoei

Fractal Fract. 2024, 8(7), 365; https://doi.org/10.3390/fractalfract8070365 - 21 Jun 2024

Cited by 3 | Viewed by 1248

Abstract

Intelligent control methodologies and artificial intelligence (AI) are essential components for the efficient management of energy storage modern systems, specifically those utilizing superconducting magnetic energy storage (SMES). Through the implementation of AI algorithms, SMES units are able to optimize their operations in real time, thereby maximizing energy efficiency. To have a more advanced understanding of this issue, DynamoMan is presented in this paper. For use with SMES systems, DynamoMan, an Artificial Neural Network (ANN)-tuned Fractional Order PID Brain Emotional Learning-Based Intelligent Controller (ANN-FOPID-BELBIC), has been developed. ANN tuning is employed to optimize the key settings of the reward/penalty generator of a BELBIC, which are important for its overall efficacy. Following this, DynamoMan is integrated into the SMES control system and compared to scenarios in which a BELBIC, PID, PI, and P are utilized. The findings indicate that DynamoMan performs considerably better than other models, demonstrating robust and control attributes alongside a considerably reduced period of settling time, especially when incorporated with the power grid. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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30 pages, 4990 KiB

Open AccessArticle

Optimizing Economic Dispatch with Renewable Energy and Natural Gas Using Fractional-Order Fish Migration Algorithm

by Abdallah Aldosary

Fractal Fract. 2024, 8(6), 350; https://doi.org/10.3390/fractalfract8060350 - 12 Jun 2024

Viewed by 1219

Abstract

This work presents a model for solving the Economic-Environmental Dispatch (EED) challenge, which addresses the integration of thermal, renewable energy schemes, and natural gas (NG) units, that consider both toxin emission and fuel costs as its primary objectives. Three cases are examined using the IEEE 30-bus system, where thermal units (TUs) are replaced with NGs to minimize toxin emissions and fuel costs. The system constraints include equality and inequality conditions. A detailed modeling of NGs is performed, which also incorporates the pressure pipelines and the flow velocity of gas as procedure limitations. To obtain Pareto optimal solutions for fuel costs and emissions, three optimization algorithms, namely Fractional-Order Fish Migration Optimization (FOFMO), Coati Optimization Algorithm (COA), and Non-Dominated Sorting Genetic Algorithm (NSGA-II) are employed. Three cases are investigated to validate the effectiveness of the proposed model when applied to the IEEE 30-bus system with the integration of renewable energy sources (RESs) and natural gas units. The results from Case III, where NGs are installed in place of two thermal units (TUs), demonstrate that the economic dispatching approach presented in this study significantly reduces emission levels to 0.4232 t/h and achieves a lower fuel cost of 796.478 USD/MWh. Furthermore, the findings indicate that FOFMO outperforms COA and NSGA-II in effectively addressing the EED problem. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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19 pages, 1714 KiB

Open AccessArticle

A High-Performance Fractional Order Controller Based on Chaotic Manta-Ray Foraging and Artificial Ecosystem-Based Optimization Algorithms Applied to Dual Active Bridge Converter

by Felipe Ruiz, Eduardo Pichardo, Mokhtar Aly, Eduardo Vazquez, Juan G. Avalos and Giovanny Sánchez

Fractal Fract. 2024, 8(6), 332; https://doi.org/10.3390/fractalfract8060332 - 31 May 2024

Cited by 1 | Viewed by 922

Abstract

Over the last decade, dual active bridge (DAB) converters have become critical components in high-frequency power conversion systems. Recently, intensive efforts have been directed at optimizing DAB converter design and control. In particular, several strategies have been proposed to improve the performance of DAB control systems. For example, fractional-order (FO) control methods have proven potential in several applications since they offer improved controllability, flexibility, and robustness. However, the FO controller design process is critical for industrializing their use. Conventional FO control design methods use frequency domain-based design schemes, which result in complex and impractical designs. In addition, several nonlinear equations need to be solved to determine the optimum parameters. Currently, metaheuristic algorithms are used to design FO controllers due to their effectiveness in improving system performance and their ability to simultaneously tune possible design parameters. Moreover, metaheuristic algorithms do not require precise and detailed knowledge of the controlled system model. In this paper, a hybrid algorithm based on the chaotic artificial ecosystem-based optimization (AEO) and manta-ray foraging optimization (MRFO) algorithms is proposed with the aim of combining the best features of each. Unlike the conventional MRFO method, the newly proposed hybrid AEO-CMRFO algorithm enables the use of chaotic maps and weighting factors. Moreover, the AEO and CMRFO hybridization process enables better convergence performance and the avoidance of local optima. Therefore, superior FO controller performance was achieved compared to traditional control design methods and other studied metaheuristic algorithms. An exhaustive study is provided, and the proposed control method was compared with traditional control methods to verify its advantages and superiority. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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31 pages, 4661 KiB

Open AccessArticle

A Novel Application of Fractional Order Derivative Moth Flame Optimization Algorithm for Solving the Problem of Optimal Coordination of Directional Overcurrent Relays

by Abdul Wadood and Herie Park

Fractal Fract. 2024, 8(5), 251; https://doi.org/10.3390/fractalfract8050251 - 25 Apr 2024

Cited by 5 | Viewed by 1527

Abstract

The proper coordination of directional overcurrent relays (DOCRs) is crucial in electrical power systems. The coordination of DOCRs in a multi-loop power system is expressed as an optimization problem. The aim of this study focuses on improving the protection system’s performance by minimizing the total operating time of DOCRs via effective coordination with main and backup DOCRs while keeping the coordination constraints within allowable limits. The coordination problem of DOCRs is solved by developing a new application strategy called Fractional Order Derivative Moth Flame Optimizer (FODMFO). This approach involves incorporating the ideas of fractional calculus (FC) into the mathematical model of the conventional moth flame algorithm to improve the characteristics of the optimizer. The FODMFO approach is then tested on the coordination problem of DOCRs in standard power systems, specifically the IEEE 3, 8, and 15 bus systems as well as in 11 benchmark functions including uni- and multimodal functions. The results obtained from the proposed method, as well as its comparison with other recently developed algorithms, demonstrate that the combination of FOD and MFO improves the overall efficiency of the optimizer by utilizing the individual strengths of these tools and identifying the globally optimal solution and minimize the total operating time of DOCRs up to an optimal value. The reliability, strength, and dependability of FODMFO are supported by a thorough statistics study using the box-plot, histograms, empirical cumulative distribution function demonstrations, and the minimal fitness evolution seen in each distinct simulation. Based on these data, it is evident that FODMFO outperforms other modern nature-inspired and conventional algorithms. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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20 pages, 3992 KiB

Open AccessArticle

A Fractional-Order Archimedean Spiral Moth–Flame Optimization Strategy to Solve Optimal Power Flows

by Abdul Wadood, Ejaz Ahmed, Sang Bong Rhee and Babar Sattar Khan

Fractal Fract. 2024, 8(4), 225; https://doi.org/10.3390/fractalfract8040225 - 13 Apr 2024

Cited by 3 | Viewed by 1808

Abstract

This research utilizes the innovative fractional-order Archimedean spiral moth–flame optimization (FO-AMFO) technique to address the issues of the optimal reactive power dispatch (ORPD) problem. The formulated fitness function aims to minimize power losses and determine the ideal flow of reactive power for the IEEE 30- and 57-bus test systems. The extensive functions of the fractional evolutionary computing strategy are utilized to address the minimization problem of ORPD. This involves determining the control variables, such as VAR compensators, bus voltages, and the tap setting of the transformers. The effective incorporation of reactive compensation devices into traditional power grids has greatly reduced power losses; however, it has resulted in an increase in the complexity of optimization problems. A comparison of the findings indicates that swarming fractional intelligence using FO-AMFO surpassed the state-of-the-art competitors in terms of minimizing power losses. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering, 2nd Edition)

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