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Stochastic Fractal Search: A Decade Comprehensive Review on Its Theory, Variants, and Applications

CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES 2025年第3期142卷 2339-2404页

作者： El-Shorbagy, Mohammed A. Bouaouda, Anas Abualigah, Laith Hashim, Fatma A. Prince Sattam Bin Abdulaziz Univ Coll Sci & Humanities Al Kharj Dept Math Al Kharj 11942 Saudi Arabia Hassan II Univ Casablanca Fac Sci & Tech Mohammadia 28806 Morocco Sunway Univ Sch Engn & Technol Petaling Jaya 27500 Selangor Malaysia Chitkara Univ Inst Engn & Technol Ctr Res Impact & Outcome Rajpura 140401 Punjab India Helwan Univ Fac Engn Cairo 11792 Egypt Appl Sci Private Univ Appl Sci Res Ctr Amman 11937 Jordan

With the rapid advancements in technology and science, optimization theory and algorithms have become increasingly important. A wide range of real-world problems is classified as optimization challenges, and meta-heuristic algorithms have shown remarkable effectiveness in solving these challenges across diverse domains, such as machine learning, process control, and engineering design, showcasing their capability to address complex optimization problems. The Stochastic Fractal Search (SFS) algorithm is one of the most popular meta-heuristic optimization methods inspired by the fractal growth patterns of natural materials. Since its introduction by Hamid Salimi in 2015, SFS has garnered significant attention from researchers and has been applied to diverse optimization problems across multiple disciplines. Its popularity can be attributed to several factors, including its simplicity, practical computational efficiency, ease of implementation, rapid convergence, high effectiveness, and ability to address single- and multi-objective optimization problems, often outperforming other established algorithms. This review paper offers a comprehensive and detailed analysis of the SFS algorithm, covering its standard version, modifications, hybridization, and multi-objective implementations. The paper also examines several SFS applications across diverse domains, including power and energy systems, image processing, machine learning, wireless sensor networks, environmental modeling, economics and finance, and numerous engineering challenges. Furthermore, the paper critically evaluates the SFS algorithm's performance, benchmarking its effectiveness against recently published meta-heuristic algorithms. In conclusion, the review highlights key findings and suggests potential directions for future developments and modifications of the SFS algorithm.

关键词： meta-heuristic algorithms stochastic fractal search evolutionary computation engineering applications swarm intelligence optimization

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Hybrid strategy collaborative enhancement of white shark optimization algorithm

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JOURNAL OF SUPERCOMPUTING 2025年第5期81卷 1-38页

作者： Liu, Junchang Liu, Yu Yang, Yahao Zhao, Zhenlong Univ Sci & Technol Liaoning Sch Elect & Informat Engn Anshan 114051 Liaoning Peoples R China Yingkou Inst Technol Sch Elct Engn Yingkou 115014 Liaoning Peoples R China

The white shark optimizer (WSO), a novel meta-heuristic algorithm proposed in recent years, is inspired by the predatory behaviors of great white sharks. While WSO has demonstrated superior performance over other algorithms in numerous optimization problems, the no free lunch theorem asserts that no single swarm intelligence optimization algorithm can universally address all optimization challenges, as each method inherently possesses limitations. Specifically, WSO requires excessive iterations to converge to optimal solutions for complex problems and exhibits susceptibility to local optima. To address these deficiencies, this study proposes an enhanced white shark optimization algorithm (EWSO) incorporating multi-strategy enhancements across distinct evolutionary phases. First, an ICMIC chaotic mapping strategy is employed for population initialization to enhance diversity and mitigate premature convergence. Second, a Cauchy noise escape mechanism is introduced during the iteration phase to strengthen local optima avoidance capabilities. Finally, the Levy flight strategy is integrated into the velocity update phase to amplify global exploration potential. Comprehensive evaluations using CEC-2022 benchmark functions and practical engineering optimization problems demonstrate EWSO's efficacy through comparisons with classical and state-of-the-art algorithms. Experimental results confirm that EWSO achieves significant improvements in convergence speed, solution accuracy, and global search performance.

关键词： White shark optimizer meta-heuristic algorithms ICMIC map Cauchy noise escape mechanism Levy flight

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Late acceptance hill climbing based algorithm for Unmanned Aerial Vehicles (UAV) path planning problem

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APPLIED SOFT COMPUTING 2025年 170卷

作者： Salehi, Emad Deilam Fazli, Mohammadamin Sharif Univ Technol Tehran *** Iran

This research innovatively applies the Late Acceptance Hill-Climbing (LAHC) algorithm to unmanned aerial vehicle (UAV) path planning in complex urban environments, marked by irregularly shaped threat areas. Moving beyond conventional models that simplify threat representations, our approach meticulously calculates collision costs, factoring in both the size and proximity of threat areas to waypoints, thereby enhancing path safety and feasibility. The standout feature of the LAHC algorithm is its memory-based strategy, which is simple yet remarkably effective, as evidenced by its superior performance over leading meta-heuristic algorithms across various urban flight scenarios. The study's findings are significant, revealing that LAHC not only excels in optimizing path costs but also outperforms in terms of the number of iterations to convergence and average execution time. Additionally, a detailed convergence and complexity analysis of LAHC is conducted, providing deeper insights into its operational efficiency. Key contributions of this study include the development of realistic benchmark flight environments, the introduction of a novel method for collision cost calculation in urban settings, and the successful demonstration of LAHC's rapid optimization capabilities and high-quality solutions.

关键词： heuristic algorithms Late Acceptance Hill Climbing (LAHC) meta-heuristic algorithms path planning Unmanned Aerial Vehicle (UAV)

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Enhancing Immune Plasma Algorithm for Coverage Optimization Problem in Wireless Sensor Networks

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ARABIAN JOURNAL FOR SCIENCE AND ENGINEERING 2025年 1-22页

作者： Tasdemir, Adnan Demirci, Sercan Aslan, Selcuk Ondokuz Mayis Univ Dept Comp Engn TR-55139 Samsun Turkiye Erciyes Univ Dept Aeronaut Engn TR-38280 Kayseri Turkiye

Wireless sensor networks (WSNs) have gained paramount importance in diverse applications, resulting in extensive research efforts. Among the pivotal challenges facing WSNs is the strategic deployment of nodes, which are critical for efficient data processing and dissemination. Maximizing the coverage area of the sensor nodes emerges as a key determinant of optimal performance in various application domains. Leveraging advanced node deployment algorithms holds the promise of significantly enhancing sensor node coverage within monitoring regions, thereby yielding benefits such as reduced energy consumption, prolonged network lifespan, and streamlined sensor operations. This study endeavors to address the coverage area challenge by employing two variants of the immune plasma algorithm (IP), augmented by sophisticated modeling techniques and tools. Inspired by the biological transfer of plasma or antibodies between patients, the IP algorithm offers a robust framework to optimize WSN deployment. Rigorous experimentation showcases the efficacy of the proposed algorithm in effectively addressing the multifaceted challenges inherent in WSN deployment, thereby presenting compelling avenues for future research and implementation.

关键词： Wireless sensor networks Node deployment Coverage optimization Immune plasma algorithm meta-heuristic algorithms Sensor coverage models Binary detection model Probabilistic detection model Optimization techniques

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Energy efficiency in building: Entropy-based Grey Wolf Optimization for improved MLP performance

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ENERGY REPORTS 2025年 13卷 4247-4260页

作者： Tugal, Hsan Mus Alparslan Univ Dept Software Engn F-49250 Mus Turkiye

Energy efficiency in building HVAC systems is essential for sustainable development, requiring accurate predictions of heating and cooling loads. Traditional hyperparameter tuning methods often lead to suboptimal performance in artificial neural network models due to local optima and premature convergence issues. This study proposes an Entropy-Based Grey Wolf Optimization to enhance the hyperparameter tuning of Multi-Layer Perceptron models for improved load prediction. Unlike traditional Grey Wolf Optimization, which relies on a fixed linear decay for search balancing, the Entropy-Based Grey Wolf Optimization dynamically adjusts the search strategy based on entropy levels. This prevents premature convergence, enhances global search capabilities, and improves fine-tuning of Multi-Layer Perceptron hyperparameters for HVAC load prediction. The proposed optimization approach is compared with ten different optimization techniques, and results indicate that Entropy-Based Grey Wolf Optimization achieves superior accuracy with lower mean squared error and faster convergence. The study contributes to both the improvement of energy efficiency in HVAC systems and the development of a generalizable entropy-based optimization framework that can be applied to various machine learning tasks.

关键词： Building HVAC systems Energy efficiency Multi-layer perceptron Hyperparameter optimization meta-heuristic algorithms Entropy

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An efficient framework for proton exchange membrane fuel cell parameter estimation using numerous MH algorithms

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RENEWABLE & SUSTAINABLE ENERGY REVIEWS 2025年 216卷

作者： Rathod, Asmita Ajay Sharma, Pankaj Choudhary, Arun Raju, Saravanakumar Subramanian, Balaji Vellore Inst Technol Sch Elect Engn Vellore Tamil Nadu India Vellore Inst Technol Sch Adv Sci Dept Math Vellore Tamil Nadu India

Proton Exchange Membrane Fuel Cells (PEMFCs) are essential for the progress of environmentally friendly hydrogen automobiles. Due to their ability to transform hydrogen into power, they are very promising alternatives for replacing traditional engines. Fuel cell (FC) systems have a complex and non-linear structure. Therefore, it is essential to accurately model the system for the purpose of simulation, design, as well as analysis. The primary aim of this paper is to provide a better meta-heuristic (MH) algorithm for estimating the values of unknown variables in the PEMFC model. In this paper, 25 state of art MH algorithms are utilized to calculate the unknown parameters of the PEMFC stacks (Ballard Mark V, BCS 500 W, Stack 250 W, NedStack PS6, Horizon H-12 as well as Temasek stack). The objective is to minimize the sum of square error (SSE) between the estimated data obtained using the MH algorithms and the actual data. Also, the obtained results are compared with each other to validate their effectiveness. Furthermore, qualitative assessments such as statistical, convergence characteristics, box plots, correlation, and radar chart analysis are carried out to evaluate the effectiveness of the 25 state-of-the-art MH algorithms. Additionally, the (I-V and I-P) polarization curves obtained from the applied 25 MH algorithms exactly match the manufacturing polarization curves across all case study outcomes. The findings enhance the understanding of the MH algorithms and offer significant insights for the effective design of PEMFC.

关键词： Proton exchange membrane fuel cell Sum of Square Error meta-heuristic algorithms Statistical analysis Correlation analysis

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Using Moving Least Square With Particle Swarm Optimization to Solve Nonlinear Transient Convective-Radiative Heat Transfer Problems in the Existence of a Magnetic Field

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第5期48卷 5987-5997页

作者： Mahmoodabadi, M. J. Atashafrooz, M. Ibrahim, M. Yousef Sirjan Univ Technol Dept Mech Engn Sirjan Iran Federat Univ Australia Sch Engn & Informat Technol Churchill Australia

In the current research, a novel hybrid scheme is proposed to solve nonlinear equations arising in heat transfer through the combination of meta-heuristic algorithms and interpolation methods. In order to define a proper objective function for minimization by particle swarm optimization (PSO), the constrained problem is converted into an unconstrained one through the penalty method. Furthermore, the moving least square (MLS) technique is implemented to interpolate and approximate the derivatives appeared in the equation. The main problem for challenging this combined scheme is nonlinear transient convective-radiative heat transfer in existence of a magnetic field. To study the efficiency of the MLS, the results would be contrasted with those extracted by a finite difference method (FDM) based PSO approach. Through five distinctive examples, the evolutionary diagrams as well as temperature distributions found by different methods are displayed, and the effects of the constant parameters are investigated. Besides, the simulations of this research work clearly depict good agreements of the numerical results obtained by the suggested idea with those reported in literature.

关键词： convection-radiation interpolation methods magnetic fields meta-heuristic algorithms nonlinear equation transient heat transfer

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Optimized deep neural network architectures for energy consumption and PV production forecasting

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ENERGY STRATEGY REVIEWS 2025年 59卷

作者： Hosseini, Eghbal Saeedpour, Barzan Banaei, Mohsen Ebrahimy, Razgar Tech Univ Denmark Dept Appl Math & Comp Sci Copenhagen Denmark Univ Kurdistan Dept Comp Engn Erbil Iran

Accurate time-series forecasting of energy consumption and photovoltaic (PV) production is essential for effective energy management and sustainability. Deep Neural Networks (DNNs) are effective tools for learning complex patterns in such data;however, optimizing their architecture remains a significant challenge. This paper introduces a novel hybrid optimization approach that integrates Genetic algorithms (GA) and Particle Swarm Optimization (PSO) to enhance the DNN architecture for more accurate energy forecasting. The performance of GA-PSO is compared with leading hyperparameter optimization techniques, such as Bayesian Optimization and Evolutionary Strategy, across various optimization benchmarks and DNN hyperparameter tuning tasks. The study evaluates the GA-PSO-enhanced Optimized Deep Neural Network (ODNN) against traditional DNNs and state-of-the-art machine learning methods on multiple real-world energy forecasting tasks. The results demonstrate that ODNN outperforms the average performance of other methods, achieving a 27% improvement in forecasting accuracy and a 22% reduction in error across various metrics. These findings demonstrate the significant potential of GA-PSO as an effective tool to optimize DNN models in energy forecasting applications.

关键词： Photovoltaic production Deep neural networks meta-heuristic algorithms Time series forecasting

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Energy Loss Reduction for Distribution Electric Power Systems with Renewable Power Sources, Reactive Power Compensators, and Electric Vehicle Charge Stations

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SENSORS 2025年第7期25卷 1997页

作者： Kien, Le Chi Loi, Tran Duc Duong, Minh Phuc Nguyen, Thang Trung Ho Chi Minh City Univ Technol & Educ Fac Elect & Elect Engn Ho Chi Minh City 700000 Vietnam Ton Duc Thang Univ Fac Elect & Elect Engn Power Syst Optimizat Res Grp Ho Chi Minh City 700000 Vietnam

This paper applies the Chameleon Swarm Algorithm (CSA) and Snow Geese Algorithm (SGA) for optimizing the placement of electric vehicle charge stations (EVCSs), renewable energy sources (RESs), and shunt capacitors (SCs). The actual power ranges of the EVCSs of the Vinfast company in Vietnam are used to check the stabilization of the IEEE 85-node distribution power grid by considering four penetration levels of EVCSs, namely 25%, 50%, 75%, and 100%. All penetration levels of EVCSs violate the operating load voltage limits, and the grid cannot work for all the penetration levels. Different scenarios are performed to find the minimum RES penetration level and the most possible SC penetration level to satisfy the operating voltage limits. The use of only SCs cannot satisfy the voltage limits even for the 25% EVCS penetration level. The placement of RESs provides the capability to maintain voltage within the allowed range for 25% and 50% EVCS penetration but not for 75% and 100%. Using both RESs and SCs, the operating voltage limits are satisfied by using RESs with 1385 kW (about 30.44% of loads and EVCSs) and SCs with 2640 kVAr for the 75% EVCS penetration level and using RESs with 2010 kW (about 38.58% of loads and EVCSs) and SCs with 2640 kVAr (100% of loads) for the 100% EVCS penetration level. The study indicates that the installation of EVCSs should be calculated for stable operation of the distribution power grid, and the combination of both RESs and SCs can satisfy the maximum penetration level of EVCSs in the distribution power grids.

关键词： electrical charging station distribution electric power system renewable energy sources shunt capacitors meta-heuristic algorithms Chameleon Swarm Algorithm Snow Geese Algorithm

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Design and optimization of various hybrid renewable energy systems using advanced algorithms for powering rural areas

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JOURNAL OF CLEANER PRODUCTION 2025年 501卷

作者： Rathod, Asmita Ajay Subramanian, Balaji Vellore Inst Technol Sch Elect Engn Vellore Tamil Nadu India

This work presents the design of an off-grid Hybrid Energy System (HES) that consists of solar panels, wind turbines, batteries, hydrogen energy storage systems, and diesel generators to meet the electrical load demands of rural communities. Four newly developed meta-heuristics algorithms, namely, GOOSE algorithm, Liver Cancer Algorithm (LCA), Blood-Sucking Leech Optimizer (BSLO), and the Osprey Optimization Algorithm (OOA) are first time applied in the history of HES design problems. The four off-grid systems such as solar PV/Wind Turbine/Battery Energy Storage System/Diesel Generator/Hydrogen Energy Storage System (SPV/WnT/BESS/DG/HESS), SPV/WnT/BESS/DG, SPV/BESS/DG, and WnT/BESS/DG are compared on the basis of techno-economical, social, and environmental analysis. After comparing the search capabilities of the stated algorithms on the sizing issue, it has been determined that the BSLO algorithm outperforms the other algorithms. The results also demonstrated that the SPV/WnT/BESS/DG configuration is the most economical choice for fulfilling the load demand of the 3 different geographical areas represented as case-1 (Kanyakumari), case-2 (Trichy), and case-3 (Chennai). The obtained optimal findings include Annual System Cost (ASC), Net Present Cost (NPC), and Cost of Energy (COE) of the SPV/WnT/BESS/DG system. For case-1, the values of ASC, NPC, and COE are 89972 $/year, 1278700 $, 0.1095 $/kWh. Similarly, for case-2, the values are 113840 $/kWh, 1617900 $, 0.1385 $/kWh, and for case-3, the values are found to be 76019 $/year, 1080400 $, 0.0925 $/kWh. Besides optimizing the techno-economic parameters, the algorithm yields minimal Greenhouse Gas (GHG) emission and it is found to be 157.282 kg/year, 239.3530 kg/year, and 3.4116 kg/year for the respective three cases. Finally, a sensitivity analysis is conducted to assess the impact of Loss of Power Supply Probability (LPSPmax) on the performance of the SPV/WnT/BESS/DG system, to choose a more appropriate design there

关键词： Hybrid renewable energy system Optimal sizing meta-heuristic algorithms Techno-economic analysis Cost of energy

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