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Keywords = hybrid second-order algorithm

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26 pages, 33081 KB  
Article
Development and DSP Implementation of an Optimized Multi-Channel Active Control System for Vehicle Interior Engine Noise Using Local Secondary Path Equalization
by Jingqiang Liang, Xiaolong Li, Wan Chen, Tao Wang, Shumo He, Zhien Liu and Chihua Lu
Appl. Sci. 2026, 16(17), 8436; https://doi.org/10.3390/app16178436 - 24 Aug 2026
Viewed by 199
Abstract
Engine noise is a predominant source of noise in the cabin of internal combustion engine vehicles and new energy hybrid vehicles. The conventional multi-channel active noise control (ANC) system, based on the adaptive notch filtered-X least mean square algorithm, is commonly employed to [...] Read more.
Engine noise is a predominant source of noise in the cabin of internal combustion engine vehicles and new energy hybrid vehicles. The conventional multi-channel active noise control (ANC) system, based on the adaptive notch filtered-X least mean square algorithm, is commonly employed to mitigate such multi-tonal noise. However, the computational efficiency and convergence performance of this system may be significantly hindered by the large estimated secondary path length and the frequency-dependent convergence behavior. To overcome these limitations, this paper proposes a computationally efficient and fast-converging multi-channel ANC system by incorporating a local secondary path (LSP) equalization method. The proposed method enhances the convergence speed by equalizing the magnitude responses of estimated secondary paths and reduces the computational complexity through an improved LSP modeling approach. Accordingly, a set of low-order equalized LSP models with normalized amplitude-frequency responses is generated and employed for reference filtering. A computational complexity analysis comparing the conventional system, a recent cost-effective system, and the proposed system is presented. Numerical simulations are conducted to evaluate the convergence speed and noise attenuation performance of these three systems. Additionally, real vehicle experiments are performed using a digital signal processing controller. The results demonstrate that the proposed multi-channel ANC system achieves a superior noise reduction effect. Under accelerated conditions, the average attenuation of the second-order noise component at the four error microphones is measured at 4.4 dB(A), 6.2 dB(A), 13.4 dB(A), and 10.0 dB(A). These findings confirm the practical effectiveness of the proposed multi-channel ANC system. Full article
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25 pages, 5624 KB  
Article
Remaining Useful Life Prediction of Retired Lithium-Ion Batteries Under Second-Life Energy Storage Conditions Using Wavelet Packet Energy Entropy
by Lin Chen, Minling Pan, Zihao Liu, Kang Yu, Bing Ji, Yuan Gao and Haihong Pan
Appl. Sci. 2026, 16(16), 8018; https://doi.org/10.3390/app16168018 - 12 Aug 2026
Viewed by 251
Abstract
Retired lithium-ion batteries retain considerable residual value for second-life energy storage applications, but significant variations in health conditions and complex operating scenarios make accurate remaining useful life (RUL) prediction challenging. To address the limited availability of capacity measurements and the poor adaptability of [...] Read more.
Retired lithium-ion batteries retain considerable residual value for second-life energy storage applications, but significant variations in health conditions and complex operating scenarios make accurate remaining useful life (RUL) prediction challenging. To address the limited availability of capacity measurements and the poor adaptability of conventional models to dynamically fluctuating degradation trajectories, a hybrid RUL prediction framework integrating Wavelet Packet Energy Entropy (WPEE), a Fractional-Order Grey Model (FGM), and an Unscented Kalman Filter (UKF) is proposed. WPEE extracted from discharge voltage signals is employed as a degradation indicator, while a Box–Cox transformation enhances its correlation with capacity. An Adaptive Mutation Particle Swarm Optimization (AMPSO) algorithm is used to determine the optimal fractional-order parameter, and the optimized FGM is incorporated into the UKF state-transition process for recursive state correction. Validation was conducted using four retired lithium-ion cells and two series-connected battery packs with different health conditions at prediction starting points of 20, 25, and 30 cycles. The results show that the proposed method effectively tracks degradation evolution, with RUL prediction errors within 7 cycles for retired cells and within 6 cycles for battery packs. Across all 18 prediction cases, FGM–UKF achieved an overall mean AE of 3.111 cycles, lower than those of FGM (5.111 cycles), GM(1, 1) (4.000 cycles), and LR (4.278 cycles). These results demonstrate the effectiveness and robustness of the proposed framework for lifetime assessment in second-life battery energy storage systems. Full article
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30 pages, 5669 KB  
Article
Enhanced Load Frequency Control in Multi-Area Hybrid Power Systems Using a 2-DOF Fractional-Order TID Controller with Artificial Ecosystem Optimization
by Anas F. Abufedda, Momen Alattar, Khalid Masoud and Audih Alfaoury
Electricity 2026, 7(3), 73; https://doi.org/10.3390/electricity7030073 - 23 Jul 2026
Viewed by 442
Abstract
Load frequency control (LFC) plays a critical role in maintaining frequency stability and regulating power transfer between interconnected areas subjected to continuous load variations. In multi-area systems, disturbances tend to propagate through interconnected tie-lines rather than remaining restricted locally, often leading to slower [...] Read more.
Load frequency control (LFC) plays a critical role in maintaining frequency stability and regulating power transfer between interconnected areas subjected to continuous load variations. In multi-area systems, disturbances tend to propagate through interconnected tie-lines rather than remaining restricted locally, often leading to slower responses and weak coordination when conventional controllers are employed. In this paper, a two-degrees-of-freedom fractional-order differential integration (2DOF FO-TID) controller is proposed to improve both frequency regulation and dynamic interaction. The structure enables independent tuning of tracking and disturbance rejection, allowing greater flexibility in shaping system response. The controller parameters are optimally tuned by the Artificial Ecosystem Optimization (AEO) algorithm. The proposed approach is evaluated on a two-area hybrid thermal power system incorporating an SMES unit within the MATLAB/Simulink (R2022b) environment and compared with PID, FOPID, and TID controllers under identical conditions. The results indicate that, although some conventional controllers provide faster stabilization in one area, their performance in the interconnected area remains slower. In contrast, the proposed controller achieves more robust behavior across both areas, with the settling time of the second area reduced from about 25 s to nearly 11 s without degrading the response of the first area. These results highlight the importance of coordination in multi-area systems and demonstrate that the proposed approach enhances overall system performance and damping compared to conventional methods. Full article
(This article belongs to the Topic Power System Dynamics and Stability, 2nd Edition)
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33 pages, 4725 KB  
Article
Performance Comparison of Event-Triggered RLS-EKF, EKF, CKF and SR-CKF for EV Battery SOC Estimation During Interference Bursts: A Simulation-Based Study
by Miin-Jong Hao and Yu-Shuo Yang
Appl. Sci. 2026, 16(14), 7095; https://doi.org/10.3390/app16147095 - 15 Jul 2026
Viewed by 355
Abstract
Accurate state-of-charge (SOC) estimation is essential for preventing battery degradation, improving energy management, and providing reliable driving-range predictions in electric vehicles (EVs). The extended Kalman filter (EKF) is a widely adopted model-based estimation technique and remains an industry-standard approach in EV battery management [...] Read more.
Accurate state-of-charge (SOC) estimation is essential for preventing battery degradation, improving energy management, and providing reliable driving-range predictions in electric vehicles (EVs). The extended Kalman filter (EKF) is a widely adopted model-based estimation technique and remains an industry-standard approach in EV battery management systems (BMS). However, its performance can be degraded by model nonlinearities, parameter uncertainties, measurement noise, and interference bursts commonly encountered in real-world operating environments. To overcome these limitations, this paper proposes an event-triggered adaptive SOC estimation framework that integrates a recursive least squares (RLS) filter with the EKF. In the proposed approach, the RLS filter recursively updates its weighting coefficients in real time to compensate for model uncertainties and measurement disturbances, thereby generating an alternative residual signal for SOC estimation. An event-triggered mechanism dynamically selects the most reliable innovation sequence for updating the EKF state estimate, enhancing estimation robustness under adverse operating conditions. A second-order RC equivalent circuit model (ECM) is employed as the nominal battery model, and a Hybrid Pulse Power Characterization (HPPC)-based current profile is used to evaluate performance over the entire SOC operating range. Extensive simulations are conducted to assess the effectiveness of the proposed event-triggered RLS-EKF algorithm under various noise levels and interference-burst scenarios. The estimation accuracy is compared with that of the conventional EKF, cubature Kalman filter (CKF), and square root cubature Kalman filter (SR-CKF) using root mean square error (RMSE) and mean absolute error (MAE) as performance metrics. Simulation results demonstrate that, under regular noise conditions and short-term interference bursts, the proposed event-triggered RLS-EKF achieves estimation performance comparable to that of the SR-CKF while consistently outperforming the EKF and CKF in both RMSE and MAE. Under long-term interference-burst conditions, the proposed method further surpasses the SR-CKF, achieving approximately 10% improvement in overall estimation accuracy as measured by RMSE and MAE. These results confirm the effectiveness and robustness of the proposed framework, highlighting its potential for practical implementation in advanced EV battery management systems. Full article
(This article belongs to the Special Issue Recent Developments in Electric Vehicles, Second Edition)
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21 pages, 512 KB  
Article
A Reproducible D3Q19 Multiple-Relaxation-Time Lattice Boltzmann Benchmark and Quantum-Operator Audit for Forced Wall-Bounded Flow Simulations
by Muhammad Idrees Khan and Hua-Dong Yao
Fluids 2026, 11(7), 175; https://doi.org/10.3390/fluids11070175 - 10 Jul 2026
Viewed by 618
Abstract
Quantum algorithms for flow simulation are advancing rapidly, but reproducible wall-bounded benchmarks with classical reference data are still needed to evaluate future quantum and hybrid quantum-classical solvers. This work presents a forced D3Q19 multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) benchmark for body-force-driven Poiseuille flow [...] Read more.
Quantum algorithms for flow simulation are advancing rapidly, but reproducible wall-bounded benchmarks with classical reference data are still needed to evaluate future quantum and hybrid quantum-classical solvers. This work presents a forced D3Q19 multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) benchmark for body-force-driven Poiseuille flow in a three-dimensional channel. The solver combines periodic streamwise and spanwise boundaries, halfway bounce-back walls, moment-space relaxation, and body-force forcing with the half-force velocity correction. The solution is verified against the analytical parabolic profile using relative L2 and maximum profile errors, mass conservation, extrapolated wall slip, and wall-normal leakage. A verification study over grid resolution, relaxation time, forcing strength, and initialization demonstrates second-order grid convergence and robust conservation behavior. The verified timestep is then decomposed into quantum-relevant primitives, including streaming, wall reflection, moment transformation, MRT relaxation, equilibrium evaluation, forcing, macroscopic recovery, and measurement. The resulting benchmark connects flow-solver accuracy metrics with operator-level requirements for quantum implementation, providing a compact reference problem for future quantum processing unit (QPU)-assisted, hybrid quantum-classical, and quantum-linear-solver-based computational fluid dynamics (CFD) studies. Performance gains over classical LBM execution are not assessed here. Full article
(This article belongs to the Special Issue Quantum Computing for Flow Simulations)
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26 pages, 1911 KB  
Article
Topology Control in Spherical 3D Sensor Networks
by Nikolaos Zarifis and Dimitrios Katsaros
Sensors 2026, 26(13), 4085; https://doi.org/10.3390/s26134085 - 27 Jun 2026
Viewed by 438
Abstract
The deployment of three-dimensional Wireless Sensor Networks (3D WSNs) in complex environments demands robust topological control to ensure both reliable and fault-tolerant sensing and communication. In order to simultaneously achieve the two objectives over time, an even distribution of the sensors’ energy consumption [...] Read more.
The deployment of three-dimensional Wireless Sensor Networks (3D WSNs) in complex environments demands robust topological control to ensure both reliable and fault-tolerant sensing and communication. In order to simultaneously achieve the two objectives over time, an even distribution of the sensors’ energy consumption is essential. Achieving optimal sensor distribution on non-planar surfaces (3D shapes), such as spheres, while maintaining reliable network routes is a significant algorithmic challenge. While many approaches effectively and efficiently addressed the aforementioned goals in 2D environments, and there exists a significant body of work on coverage, connectivity, or energy efficiency in 3D sensor networks, the solutions for either can not straightforwardly be adapted to the 3D case (e.g., some coverage problems are optimally solved for 2D but are still open problems in the 3D case), or the solutions to the individual problems in the 3D case are not integrated gracefully to solve the entire problem. Moreover, these problems have not been address for the realistic spherical 3D case. This paper presents a novel holistic algorithm designed to generate energy-efficient, optimal sensor topologies over spherical 3D sensor networks that guarantee redundant coverage to deal with sensor failures, connectivity with controlled redundancy support for more efficient communication, and the creation of a hierarchy over the flat network to deal with energy issues, at would be appropriate for real-world tasks. The proposed methodology is executed in three primary phases. First, it approaches the geometric part of the problem to determine the optimal placement of sensor nodes on the surface of a sphere, guaranteeing k-coverage for the target area. Second, it creates a reliable inner-layer backbone network of sensors that establishes k-connectivity ensuring a reliable network for data transmission and distribution of total power in the whole network. Finally, after formulating sensors into clusters, a mathematical formula to change each cluster head is created so that we achieve even distribution of energy consumption across the network. To validate the proposed approach, a 3D WSN software simulator was developed. This tool provides a dynamic visual simulation of the network, enabling the execution, visualization and simulation of the hybrid algorithm and any other 3D WSN. Full article
(This article belongs to the Special Issue Feature Papers in the ‘Sensor Networks’ Section 2026)
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39 pages, 7564 KB  
Article
Sustainable Collection Path Planning for Agricultural Product Cloud Warehouse Under Three-Dimensional Loading and Carbon Emission Constraints
by Huicheng Hao, Yue Zhang, Yihan Liu, Jilai Xun and Cuiping He
Sustainability 2026, 18(12), 6284; https://doi.org/10.3390/su18126284 - 18 Jun 2026
Viewed by 333
Abstract
With the rapid expansion of agricultural e-commerce in China, inefficient cloud warehouse consolidation and high environmental costs have hindered the sustainability of supply chains. To address the challenges of low vehicle loading rates and high carbon emissions, this study proposes an optimization model [...] Read more.
With the rapid expansion of agricultural e-commerce in China, inefficient cloud warehouse consolidation and high environmental costs have hindered the sustainability of supply chains. To address the challenges of low vehicle loading rates and high carbon emissions, this study proposes an optimization model for collection path planning that integrates sales forecasting and three-dimensional loading constraints. First, STL decomposition is employed to identify seasonal sales patterns, and a hybrid SARIMA and ARIMA-BPNN model is constructed to achieve precise forecasting of future orders to provide data support for dynamic demand. Second, a single-objective path planning model is formulated to minimize the fixed vehicle costs, fuel consumption, and carbon emissions while maximizing the load utilization rates. To solve this complex problem, a two-stage solution framework, consisting of path planning and three-dimensional loading verification, was designed. This framework integrates an improved genetic–hill-climbing hybrid algorithm with a constructive heuristic to handle real-time spatial constraints and achieve the efficient optimization of distribution paths. Finally, a case study on the HLYX agricultural cloud warehouse in Harbin, China, demonstrated that the proposed approach significantly enhances space utilization and reduces transportation and carbon emission costs. This study provides a sustainable development path for the cost reduction, economic efficiency improvement, and carbon emission reduction of smart agricultural logistics. Full article
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26 pages, 5536 KB  
Article
Bi-Level Optimal Planning of Soft Open Points Integrated with Energy Storage in Distribution Networks Considering Dynamic Electro-Carbon Factors
by Ke Cheng, Haitao Liu, Yu Ji, Changjun Jiang, Nan Zheng and Geng Niu
Electronics 2026, 15(12), 2693; https://doi.org/10.3390/electronics15122693 - 17 Jun 2026
Viewed by 354
Abstract
To address the deepening electro-carbon coupling and flexibility shortages in active distribution networks with high renewable energy penetration, this paper proposes a bi-level collaborative planning strategy considering dynamic electro-carbon factors. First, considering the spatial–temporal correlation of wind and solar outputs, typical renewable energy [...] Read more.
To address the deepening electro-carbon coupling and flexibility shortages in active distribution networks with high renewable energy penetration, this paper proposes a bi-level collaborative planning strategy considering dynamic electro-carbon factors. First, considering the spatial–temporal correlation of wind and solar outputs, typical renewable energy scenarios are generated using the Frank-Copula function and clustering algorithms. Second, a bi-level planning model for the Soft Open Point integrated with an Energy Storage System (E-SOP) is established: the upper level optimizes the siting and sizing of E-SOPs to minimize the annualized comprehensive cost; the lower level incorporates a dynamic stepped carbon trading mechanism and a continuous price-based demand response (PBDR) mechanism to achieve optimal operational economy. For model solving, a hybrid bi-level decomposition strategy combining the Dhole Optimization Algorithm (DOA) and second-order cone programming (SOCP) is adopted, utilizing a coordinated dual-level solution interaction to favorably support numerical stability. Case studies on a modified IEEE 33-node system demonstrate that the proposed scheme reduces the annualized comprehensive cost by 12.3% and transforms the carbon trading expenditure into a net revenue, thereby significantly enhancing the low-carbon economic efficiency and operational flexibility of the distribution network. Full article
(This article belongs to the Section Power Electronics)
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26 pages, 4926 KB  
Article
An Adaptive Piano-Inspired Memristive Fractional-Order Cryptosystem for Secure Image Protection
by Hayder Najm, Mohammed Salih Mahdi, Noor Redha Alkazaz, Mohammed Nasser Al-Andoli, Mohammad Ahmed Alomari and Amjed Abbas Ahmed
Mathematics 2026, 14(12), 2125; https://doi.org/10.3390/math14122125 - 14 Jun 2026
Cited by 1 | Viewed by 550
Abstract
The growing need for secure image transmission across public networks requires robust encryption algorithms. Traditional chaos-based image ciphers typically have a small key space, weak avalanche behavior, or are susceptible to differential cryptanalysis. To overcome such inadequacies, this paper suggests a new adaptive [...] Read more.
The growing need for secure image transmission across public networks requires robust encryption algorithms. Traditional chaos-based image ciphers typically have a small key space, weak avalanche behavior, or are susceptible to differential cryptanalysis. To overcome such inadequacies, this paper suggests a new adaptive image cryptosystem that combines a fractional-order memristive chaotic engine and a non-linear hybrid encryption kernel. The system uses piano-inspired feedback; the keystream generator dynamically adapts to the previously encrypted pixel, enabling powerful Cipher Block Chaining (CBC)-style chaining and content-dependent diffusion. A four-dimensional memristive system is solved by the use of fractional-order calculus, which gives an ultra-large key space (>1080) and very high sensitivity to initial conditions—confirmed by a positive largest Lyapunov exponent (1.7199). The encryption kernel maps the traditional Exclusive OR (XOR) with the reversible two-step operation: the modular addition of the plaintext with the first keystream byte and the XOR with the second keystream one, both of which increase non-linearity and confusion. Large-scale experiments with six standard 256 × 256 colour images indicate almost ideal entropy (7.9994), Number of Pixel Change Rate (NPCR) which is 99.62, Unified Average Changing Intensity (UACI) which is 33.43, correlation coefficients are near to zero, very low Gray-Level Co-occurrence Matrix (GLCM) homogeneity (≈0.017) and high contrast (≈4843) and low energy (≈0.006 The ciphertext passes seven National Institute of Standards and Technology (NIST) SP-800-22 statistical tests, is extremely sensitive to keys (a perturbation of 1 × 10−14 alters >99.6% of ciphertext) and resists chosen-plaintext and known-plaintext attacks. Decryption has linear time complexity O(N), and average encryption and decryption times are 3.40 s and 2.75 s for 256 × 256 images. The proposed cryptosystem provides an attractive security–performance trade-off that can be used in high-security systems like medical image protection, privacy-preserving multimedia transmission, and secure cloud storage. Full article
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39 pages, 2878 KB  
Article
Multi-Strategy Adaptive Synergistic Exponential Distribution Optimizer for Global Optimization and Cloud Computing Task Scheduling
by Yanyi Jin, Xin Huang and Zhewei Xu
Electronics 2026, 15(11), 2482; https://doi.org/10.3390/electronics15112482 - 5 Jun 2026
Viewed by 403
Abstract
The Exponential Distribution Optimizer (EDO) is a newly developed mathematics-based metaheuristic with a simple structure and high efficiency. However, the EDO faces dilemmas, including poor initial population quality, premature convergence, insufficient population diversity, and low convergence accuracy when addressing complex high-dimensional optimization and [...] Read more.
The Exponential Distribution Optimizer (EDO) is a newly developed mathematics-based metaheuristic with a simple structure and high efficiency. However, the EDO faces dilemmas, including poor initial population quality, premature convergence, insufficient population diversity, and low convergence accuracy when addressing complex high-dimensional optimization and cloud computing task scheduling problems. To overcome these drawbacks, this paper proposes an Adaptive Synergistic Exponential Distribution Optimizer (ASEDO) integrated with three collaborative strategies for global optimization and cloud computing task scheduling. First, a Multi-Source Hybrid Perturbation Initialization is designed using first-order differential mutation and high-order Bernstein polynomial perturbation to expand the initial search space and boost population diversity. Second, a Bipolar Adaptive Search Mechanism is presented to enable bidirectional learning from elite and inferior individuals, effectively preventing local optima trapping. Third, an Oscillating Random Mapping Learning Mechanism is introduced to strengthen local search ability and convergence precision via random learning and second-order oscillation mapping. The proposed ASEDO is verified on CEC2022 benchmark functions and cloud computing task scheduling under small-scale, large-scale, and dynamic task scenarios. Ablation experiments and comparison results demonstrate that the synergistic effect of the three strategies significantly improves the performance of EDO. Meanwhile, the ASEDO shows stronger global search capability, higher solution accuracy, and better stability than several state-of-the-art algorithms in both global optimization and cloud task scheduling applications. Full article
(This article belongs to the Special Issue AI-Driven Edge and Cloud Computing for IoT)
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47 pages, 14821 KB  
Article
Multi-Strategy Improved Love Evolutionary Algorithm for Global Optimization and Art Image Segmentation
by Zhengxing Yang, Liwei Liu and Junjun Li
Symmetry 2026, 18(6), 961; https://doi.org/10.3390/sym18060961 - 2 Jun 2026
Cited by 1 | Viewed by 349
Abstract
Although the Love Evolution Algorithm (LEA) has achieved encouraging results in optimization tasks, several shortcomings still limit its effectiveness when solving high-dimensional multimodal problems. In particular, the fixed interaction threshold, stochastic reflection mechanism, and convergence-biased role evolution process may weaken population diversity and [...] Read more.
Although the Love Evolution Algorithm (LEA) has achieved encouraging results in optimization tasks, several shortcomings still limit its effectiveness when solving high-dimensional multimodal problems. In particular, the fixed interaction threshold, stochastic reflection mechanism, and convergence-biased role evolution process may weaken population diversity and reduce the coordination between exploration and exploitation during evolution. To overcome these issues, this paper develops a Multi-Strategy Improved Love Evolution Algorithm (MILEA) under a phase-oriented cooperative evolutionary framework. First, a diversity-enhanced reflection mechanism is incorporated to enlarge the search region and dynamically regulate evolutionary dispersion during the early search stage. Second, an adaptive acceptance threshold strategy is introduced to adjust pairwise interaction behaviors according to the evolutionary state, thereby improving search flexibility and adaptability. Third, an elite-guided role evolution mechanism is designed to strengthen local exploitation and guide the population toward promising regions more efficiently. Furthermore, a probability-based collaborative update scheme is employed to coordinate multiple search behaviors adaptively while preserving the same computational complexity order as the original LEA framework. To evaluate the effectiveness of the proposed algorithm, extensive experiments are conducted on the CEC2017 and CEC2022 benchmark suites. The experimental results indicate that MILEA exhibits competitive optimization performance with respect to convergence behavior, solution accuracy, and optimization stability when compared with several advanced metaheuristic algorithms. Relative to the original LEA, the proposed method obtains improved average fitness values on most benchmark functions and significantly suppresses result fluctuations on several multimodal and hybrid optimization problems, indicating enhanced robustness during repeated independent runs. In addition, statistical evaluations based on the Wilcoxon signed-rank test and Friedman ranking analysis further support the reliability of the proposed optimization framework. To verify its practical applicability, MILEA is also applied to Otsu-based multi-threshold image segmentation tasks. Experimental results evaluated by PSNR, SSIM, and FSIM demonstrate that the proposed algorithm can generate high-quality segmentation results and preserve important structural image information. Overall, the proposed MILEA provides an effective optimization framework for both benchmark optimization and practical image segmentation applications. Full article
(This article belongs to the Special Issue Symmetry in Numerical Analysis and Applied Mathematics)
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27 pages, 2024 KB  
Article
Complex-Order Gold Rush Optimizer Algorithm
by Sixuan Chen, Xiaobo Wu, Tao Wang, Hongli Ma, Xiang Li and Lisheng Yin
Computation 2026, 14(6), 124; https://doi.org/10.3390/computation14060124 - 27 May 2026
Viewed by 313
Abstract
This study proposes an enhanced variant of the gold rush optimizer (GRO) algorithm, termed the complex-order gold rush optimizer (CoGRO) algorithm, to address two inherent theoretical limitations of the original GRO. First, GRO employs a random initialization strategy that lacks ergodicity and uniform [...] Read more.
This study proposes an enhanced variant of the gold rush optimizer (GRO) algorithm, termed the complex-order gold rush optimizer (CoGRO) algorithm, to address two inherent theoretical limitations of the original GRO. First, GRO employs a random initialization strategy that lacks ergodicity and uniform coverage, leading to insufficient population diversity and a higher risk of premature convergence. Second, its position update mechanism relies solely on current-time information without incorporating historical search experience, which restricts the algorithm’s ability to model long-term dependencies and escape local optima in complex multimodal landscapes. To overcome these deficiencies, we introduce a chaotic LCS1 initialization to enhance population diversity through improved ergodic coverage, and we embed a complex-order derivative mechanism into the migration and collaboration updates to provide infinite memory capability. A comprehensive sensitivity analysis is conducted to examine the influence of control parameters on CoGRO’s performance, leading to the identification of an optimal parameter configuration. The effectiveness of the proposed algorithm is evaluated using the CEC2022 benchmark suite through ablation studies and comparative analyses with state-of-the-art algorithms. Experimental results on the CEC2022 benchmark suite comprising 12 test functions demonstrate that CoGRO significantly outperforms the original GRO, achieving an average solution accuracy improvement of 0.84% and an average standard deviation reduction of 67.6 across all 12 functions, with particularly notable improvements on hybrid and composition functions. Wilcoxon signed-rank tests confirm the statistical significance of these improvements (p<0.05). These results confirm the feasibility and effectiveness of CoGRO as an improved optimization method for complex engineering problems. Full article
(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)
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19 pages, 4186 KB  
Article
Identification of Complex Nonlinear Fractional-Order System Based on Fuzzy Multi-Model
by Qian Zhang, Chunlei Liu, Jingwen Chen, Hongwei Wang and Zheng Zhang
Fractal Fract. 2026, 10(6), 362; https://doi.org/10.3390/fractalfract10060362 - 27 May 2026
Viewed by 468
Abstract
Fractional-order dynamic systems, due to their long memory and nonlocality, have significant advantages in describing the dynamic behavior of complex engineering systems. However, existing identification methods often struggle to balance modeling accuracy and model structural complexity under conditions of strong nonlinearity, strong coupling, [...] Read more.
Fractional-order dynamic systems, due to their long memory and nonlocality, have significant advantages in describing the dynamic behavior of complex engineering systems. However, existing identification methods often struggle to balance modeling accuracy and model structural complexity under conditions of strong nonlinearity, strong coupling, and multiple operating conditions. To address the challenge of modeling complex fractional-order systems with strong nonlinearity, strong coupling, and multiple operating conditions, this paper proposes a fuzzy multi-model modeling and identification method based on the decomposition-synthesis approach. First, a fractional-order fuzzy multi-model structure is constructed to characterize the dynamic characteristics of such complex systems. Second, an improved SKFCM hybrid clustering algorithm is proposed, combining K-means clustering and satisfactory fuzzy C-means clustering. This optimizes the cluster center selection strategy and overcomes the shortcomings of traditional satisfactory FCM algorithms, such as random initial membership and unreasonable cluster center selection, thus achieving a reasonable determination of the number of local models. Finally, the least-squares and Levenberg–Marquardt algorithms are integrated to interactively identify local model parameters, system fractional-order, and fuzzy scheduling function parameters, solving key difficulties such as unknown fractional-order s in antecedent variables, numerous parameter couplings, and difficulty in determining the antecedent space. Through academic examples and simulations of a robotic arm system, the proposed method effectively achieves high-precision modeling of complex fractional-order systems, demonstrating strong feasibility and superiority. Full article
(This article belongs to the Section Engineering)
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24 pages, 3883 KB  
Article
Research on FOPID Controller and CMOPSO Optimization for Prevention and Control of Oscillatory Instability at the PCC in a Hydro–Wind–Photovoltaic Grid-Connected System
by Bojin Tang, Weiwei Yao, Teng Yi, Rui Lv, Zhi Wang and Chaoshun Li
Electronics 2026, 15(10), 2104; https://doi.org/10.3390/electronics15102104 - 14 May 2026
Viewed by 325
Abstract
To address the key problems of low-frequency oscillation and insufficient regulation accuracy at the Point of Common Coupling (PCC) in hydro–wind–photovoltaic hybrid systems, which are caused by the randomness of wind and photovoltaic output, the water-hammer effect of hydropower units, and multi-source power [...] Read more.
To address the key problems of low-frequency oscillation and insufficient regulation accuracy at the Point of Common Coupling (PCC) in hydro–wind–photovoltaic hybrid systems, which are caused by the randomness of wind and photovoltaic output, the water-hammer effect of hydropower units, and multi-source power coupling, a joint control strategy based on Fractional-Order Proportional Integral Derivative (FOPID) and Co-evolutionary Multi-objective Particle Swarm Optimization (CMOPSO) is proposed. First, a small-signal transfer function model of the system covering photovoltaic inverters, doubly fed induction generators (DFIGs), hydropower units and voltage-source converter-based high-voltage direct current (VSC-HVDC) converter stations is established to accurately characterize the water-hammer effect and multi-source dynamic coupling characteristics. Second, a Caputo-type FOPID controller is designed. Compared with traditional integer-order controllers with limited tuning flexibility, the FOPID controller utilizes its five degrees of freedom to address specific multi-source coupling challenges. This precisely compensates for the non-minimum phase lag caused by the water-hammer effect in hydropower units via the fractional derivative link, and effectively smooths the impact of stochastic wind–solar fluctuations on PCC voltage through the memory characteristics of the fractional integral link. This multi-parameter regulation mechanism prevents a trade-off between response speed and overshoot suppression, achieving effective decoupling of complex multi-source dynamic interactions. Third, a dual-objective optimization framework with the Integral of Time-weighted Absolute Error (ITAE) and Oscillatory Disturbance Risk Index (ODRI) as the objectives is constructed. The multi-population co-evolution mechanism of the CMOPSO algorithm is adopted to solve the Pareto-optimal solution set, realizing the coordinated optimization of dynamic response accuracy and oscillation instability risk. Finally, comparative simulations are carried out on the Simulink platform with traditional PI/FOPI controllers and optimization algorithms such as Multi-objective Particle Swarm Optimization based on the Decomposition/Simple Indicator-Based Evolutionary Algorithm (MPSOD/SIBEA). The results show that the proposed strategy can effectively suppress low-frequency oscillations in the range of 0~30 Hz. Compared with the traditional PI controller, the PCC voltage overshoot is reduced by more than 40%, the oscillation decay time is shortened by 33%, the ITAE and ODRI indices are decreased by 12.58% and 2.47%, respectively, and the stability of DC bus voltage is significantly improved. Its robustness and comprehensive control performance are superior to existing methods, providing an efficient and stable control scheme for power electronics-dominated complex new energy grid-connected systems. Full article
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36 pages, 2607 KB  
Article
A Coupled Mathematical Model of Groundwater Dynamics and Salt Transport in a Two-Layer Porous Medium
by Ergashevich Halimjon Khujamatov, Sherzod Daliev, Sherzod Urakov, Sirojiddin Elmonov, Abdinabi Mukhamadiyev and Razvan Craciunescu
Mathematics 2026, 14(10), 1593; https://doi.org/10.3390/math14101593 - 8 May 2026
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Abstract
Understanding the coupled dynamics of groundwater flow and salinity transport is essential for the sustainable management of aquifer systems, particularly in irrigated and semi-arid regions where evaporation, recharge variability, and groundwater abstraction strongly influence hydrogeological regimes. In multilayer porous media, groundwater-level fluctuations and [...] Read more.
Understanding the coupled dynamics of groundwater flow and salinity transport is essential for the sustainable management of aquifer systems, particularly in irrigated and semi-arid regions where evaporation, recharge variability, and groundwater abstraction strongly influence hydrogeological regimes. In multilayer porous media, groundwater-level fluctuations and salt migration processes are closely interconnected, since hydraulic gradients control solute transport while salinity variations may affect flow behaviour through density-related mechanisms. In this study, a nonlinear mathematical model is developed to describe groundwater-level evolution and salt transport within a two-layer porous medium consisting of a phreatic layer and an underlying confined aquifer. The model accounts for filtration processes, interlayer hydraulic exchange, density-dependent effects, and external forcing factors including surface recharge, evaporation, and pumping. For numerical implementation, the governing equations are discretized using a finite-difference scheme with central spatial approximations and an implicit Crank–Nicolson-type temporal formulation. A hybrid second-order time approximation is introduced for the main-layer equation to improve numerical smoothness and stability. The resulting tridiagonal algebraic systems are solved using the Thomas algorithm within an iterative quasi-linearization framework, ensuring both computational efficiency and numerical robustness. Simulation results reveal a clear difference in the dynamical behaviour of the two layers. The phreatic aquifer exhibits rapid and high-amplitude responses to external forcing, whereas the confined aquifer demonstrates slower and smoother hydraulic and geochemical adjustments. Sensitivity analysis further identifies the filtration coefficient, transmissivity, porosity, density-related parameters, surface flux, and pumping intensity as the dominant factors governing groundwater dynamics and salinity redistribution. The proposed modelling framework provides a reliable tool for analysing coupled groundwater–salinity processes and offers a scientifically grounded basis for groundwater monitoring, salinization risk assessment, and sustainable aquifer management. Full article
(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems, 3rd Edition)
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