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An important issue in interconnected microgrids (MGs) is the realization of balance between the generation side and the demand side. Imbalanced generation and load demands lead to security, power quality, and reliability issues. The load frequency control (LFC) is accountable for regulating MG frequency against generation/load disturbances. This paper proposed an optimized fractional order (FO) LFC scheme with cascaded outer and inner control loops. The proposed controller is based on a cascaded one plus tilt derivative (1+TD) in the outer loop and an FO tilt integrator-derivative with a filter (FOTIDF) in the inner loop, forming the cascaded (1+TD/FOTIDF) controller. The proposed 1+TD/FOTIDF achieves better disturbance rejection compared with traditional LFC methods. The proposed 1+TD/FOTIDF scheme is optimally designed using a modified version of the liver cancer optimization algorithm (MLCA). In this paper, a new modified liver cancer optimization algorithm (MLCA) is proposed to overcome the shortcomings of the standard Liver cancer optimization algorithm (LCA), which contains the early convergence to local optima and the debility of its exploration process. The proposed MLCA is based on three improvement mechanisms, including chaotic mutation (CM), quasi-oppositional based learning (QOBL), and the fitness distance balance (FDB). The proposed MLCA method simultaneously adjusts and selects the best 1+TD/FOTIDF parameters to achieve the best control performance of MGs. Obtained results are compared to other designed FOTID, TI/FOTID, and TD/FOTID controllers. Moreover, the contribution of electric vehicles and the high penetration of renewables are considered with power system parameter uncertainty to test the stability of the proposed 1+TD/FOTIDF LFC technique. The obtained results under different possible load/generation disturbance scenarios confirm a superior response and improved performance of the proposed 1+TD/FOTIDF and the proposed MLCA-based optimized LFC controller. [ABSTRACT FROM AUTHOR]

This paper analyses the condition necessary to guarantee no divergence for Caputo's fractional order gradient descent (C-FOG) algorithm on multivariate functions. C-FOG is self-organizing, computationally efficient, simple, and understandable. It converges faster than the classical gradient-based optimization algorithms and converges to slightly different points when the order of the fractional derivative is different. The additional freedom of the order is very useful in situations where the diversity of convergence is required, and it also allows for more precise convergence. Comparative experiments on a typical poor conditioned function and adversarial sample generation frameworks demonstrate the convergence performance of C-FOG, showing that it outperforms currently popular algorithms in terms of convergence speed, and more excitingly, the diversity of convergence allows it to exhibit stronger and more stable attack capability in adversarial sample generation procedures (The code for experiments is available at: https://github.com/mulertan/self%5foptimizing/tree/main , accessed on 30 April 2024). [ABSTRACT FROM AUTHOR]

Metaheuristic optimization algorithms (MHA) play a significant role in obtaining the best (optimal) values of the system's parameters to improve its performance. This role is significantly apparent when dealing with systems where the classical analytical methods fail. Fractional-order (FO) systems have not yet shown an easy procedure to deal with the determination of their optimal parameters through traditional methods. In this paper, a recent, systematic. And comprehensive review is presented to highlight the role of MHA in obtaining the best set of gains and orders for FO controllers. The systematic review starts by exploring the most relevant publications related to the MHA and the FO controllers. The study is focused on the most popular controllers such as the FO-PI, FO-PID, FO Type-1 fuzzy-PID, and FO Type-2 fuzzy-PID. The time domain is restricted in the articles published through the last decade (2014:2023) in the most reputed databases such as Scopus, Web of Science, Science Direct, and Google Scholar. The identified number of papers, from the entire databases, has reached 850 articles. A Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) methodology was applied to the initial set of articles to be screened and filtered to end up with a final list that contains 82 articles. Then, a thorough and comprehensive study was applied to the final list. The results showed that Particle Swarm Optimization (PSO) is the most attractive optimizer to the researchers to be used in the optimal parameters identification of the FO controllers as it attains about 25% of the published papers. In addition, the papers that used PSO as an optimizer have gained a high citation number despite the fact that the Chaotic Atom Search Optimization (ChASO) is the highest one, but it is used only once. Furthermore, the Integral of the Time-Weighted Absolute Error (ITAE) is the best nominated cost function. Based on our comprehensive literature review, this appears to be the first review paper that systematically and comprehensively addresses the optimization of the parameters of the fractional-order PI, PID, Type-1, and Type-2 fuzzy controllers with the use of MHAs. Therefore, the work in this paper can be used as a guide for researchers who are interested in working in this field. [ABSTRACT FROM AUTHOR]

This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in space, which avoids iteration at each time layer. The analysis of mass conservative and convergent properties for this split-step FE scheme is performed. To test its capability, some numerical tests and the simulation of the double solitons intersection and plane wave are carried out. The results and comparisons with the algorithm combined with Newton's iteration illustrate its effectiveness and advantages in computational efficiency. [ABSTRACT FROM AUTHOR]

We propose an adaptive fractional multi-scale optimization optical flow algorithm, which for the first time improves the over-smoothing of optical flow estimation under the total variation model from the perspective of global feature and local texture balance, and solves the problem that the convergence of fractional optical flow algorithms depends on the order parameter. Specifically, a fractional-order discrete L1-regularization Total Variational Optical Flow model is constructed. On this basis, the Ant Lion algorithm is innovatively used to realize the iterative calculation of the optical flow equation, and the fractional order is dynamically adjusted to obtain an adaptive optimization algorithm with strong search accuracy and high efficiency. In this paper, the flexibility of optical flow estimation in weak gradient texture scenes is increased, and the optical flow extraction rate of target features at multiple scales is greatly improved. We show excellent recognition performance and stability under the MPI_Sintel and Middlebury benchmarks. [ABSTRACT FROM AUTHOR]

This paper introduces a new fast image encryption scheme based on a chaotic system and cyclic shift in the integer wavelet domain. In order to increase the effectiveness and security of encryption, we propose a new diffusion scheme by using bidirectional diffusion and cyclic shift and apply it to our encryption scheme. First, a two-level integer wavelet transform is used to split the plaintext picture into four low-frequency components. Second, we use random sequences generated by Chen's hyper-chaotic system to scramble four low-frequency components. The initial value is determined by Secure Hash Algorithm 256-bit (SHA256) and user-defined parameters, which increases the plaintext sensitivity. Then, the new diffusion scheme is applied to the matrix containing most of the information and matrices are transformed by a one-level inverse integer wavelet. Finally, to create the ciphertext image, the diffused matrices are subjected to the one-level inverse integer wavelet transform. In the simulation part, we examine the suggested algorithm's encryption impact. The findings demonstrate that the suggested technique has a sufficient key space and can successfully fend off common attacks. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order control method based on the twin-delayed deep deterministic policy gradient (TD3) algorithm in reinforcement learning is proposed. A fractional-order disturbance observer is designed to estimate the disturbances, and the radial basis function network is selected to approximate system uncertainties in the system. Then, a fractional-order sliding-mode controller is constructed to control the system, and the parameters of the controller are tuned using the TD3 algorithm, which can optimize the control effect. The results show that the fractional-order control method based on the TD3 algorithm can not only improve the closed-loop system performance under different operating conditions but also enhance the signal tracking capability. [ABSTRACT FROM AUTHOR]

In this paper, we propose a secure image encryption method using compressive sensing (CS) and a two-dimensional linear canonical transform (2D LCT). First, the SHA256 of the source image is used to generate encryption security keys. As a result, the suggested technique is able to resist selected plaintext attacks and is highly sensitive to plain images. CS simultaneously encrypts and compresses a plain image. Using a starting value correlated with the sum of the image pixels, the Mersenne Twister (MT) is used to control a measurement matrix in compressive sensing. Then, the scrambled image is permuted by Lorenz's hyper-chaotic systems and encoded by chaotic and random phase masks in the 2D LCT domain. In this case, chaotic systems increase the output complexity, and the independent parameters of the 2D LCT expand the key space of the suggested technique. Ultimately, diffusion based on addition and modulus operations yields a cipher-text image. Simulations showed that this cryptosystem was able to withstand common attacks and had adequate cryptographic features. [ABSTRACT FROM AUTHOR]

To solve the shortcomings of the Otsu image segmentation algorithm based on traditional Moth–Flame Optimization (MFO), such as its poor segmentation accuracy, slow convergence, and tendency to fall into local optimum, this paper proposes fractional order moth–flame optimization with the Otsu image segmentation algorithm. Utilizing the advantages of memorability and heritability in fractional order differentiation, the position updating of moths is controlled by fractional order. Using the adaptive fractional order, the positions of moths are used to adjust the fractional order adaptively to improve the convergence speed. Combining the improved MFO algorithm with the two-dimensional Otsu algorithm, the optimization objective function is achieved by using its dispersion matrix. The experimental results indicate that, compared with traditional MFO, the convergence rate of the proposed algorithm is improved by about 74.62%. Furthermore, it has better segmentation accuracy and a higher fitness value than traditional MFO. [ABSTRACT FROM AUTHOR]

In this paper, a discrete-time fractional flocking control algorithm of multi-agent systems is put forward to address the slow convergence issue of multi-agent systems. Firstly, by introducing Grünwald-Letnikov (G-L) fractional derivatives, the algorithm allows agents to utilize historical information when updating their states. Secondly, based on the Lyapunov stability theory, the convergence of the algorithm is proven. Finally, simulations are conducted to verify the effectiveness of the proposed algorithm. Comparisons are made between the proposed algorithm and other methods. The results show that the proposed algorithm can effectively improve the convergence speed of multi-agent systems. [ABSTRACT FROM AUTHOR]

To solve the drawbacks of the Otsu image segmentation algorithm based on traditional butterfly optimization, such as slow convergence speed and poor segmentation accuracy, this paper proposes hybrid fractional-order butterfly optimization with the Otsu image segmentation algorithm. G-L-type fractional-order differentiation is combined with the algorithm's global search to improve the position-updating method, which enhances the algorithm's convergence speed and prevents it from falling into local optima. The sine-cosine algorithm is introduced in the local search step, and Caputo-type fractional-order differentiation is used to avoid the disadvantages of the sine-cosine algorithm and to improve the optimization accuracy of the algorithm. By dynamically converting the probability, the ratio of global search to local search is changed to attain high-efficiency and high-accuracy optimization. Based on the 2-D grayscale gradient distribution histogram, the trace of discrete matrices between classes is chosen as the fitness function, the best segmentation threshold is searched for, image segmentation is processed, and three categories of images are chosen to proceed with the segmentation test. The experimental results show that, compared with traditional butterfly optimization, the convergence rate of hybrid fractional-order butterfly optimization with the Otsu image segmentation algorithm is improved by about 73.38%; meanwhile, it has better segmentation accuracy than traditional butterfly optimization. [ABSTRACT FROM AUTHOR]

Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς + 2 and 2 ς + 4 , respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time. [ABSTRACT FROM AUTHOR]

With the advancement of information technology, the security of digital images has become increasingly important. To ensure the integrity of images, a novel color image-encryption algorithm based on extended DNA coding, Zig-Zag transform, and a fractional-order laser system is proposed in this paper. First, the dynamic characteristics of the fractional-order laser chaotic system (FLCS) were analyzed using a phase diagram and Lyapunov exponent spectra. The chaotic sequences generated by the system were used to design image-encryption algorithms. Second, a modified Zig-Zag confusing method was adopted to confuse the image. Finally, in the diffusion link, the DNA encoding scheme was extended to allow for a greater number of DNA encoding rules, increasing the randomness of the matrix and improving the security of the encryption scheme. The performance of the designed encryption algorithm is analyzed using key space, a histogram, information entropy, correlation coefficients, differential attack, and robustness analysis. The experimental results demonstrate that the algorithm can withstand multiple decryption methods and has strong encryption capability. The proposed novel color image-encryption scheme enables secure communication of digital images. [ABSTRACT FROM AUTHOR]

Orthogonal generalized Laguerre moments of fractional orders (FrGLMs) are signal and image descriptors. The utilization of the FrGLMs in the analysis of big-size signals encounters three challenges. First, calculating the high-order moments is a time-consuming process. Second, accumulating numerical errors leads to numerical instability and degrades the reconstructed signals' quality. Third, the QR decomposition technique is needed to preserve the orthogonality of the higher-order moments. In this paper, the authors derived a new recurrence formula for calculating the FrGLMs, significantly reducing the computational CPU times. We used the Schwarz–Rutishauser algorithm as an alternative to the QR decomposition technique. The proposed method for computing FrGLMs for big-size signals is accurate, simple, and fast. The proposed algorithm has been tested using the MIT-BIH arrhythmia benchmark dataset. The results show the proposed method's superiority over existing methods in terms of processing time and reconstruction capability. Concerning the reconstructed capability, it has achieved superiority with average values of 25.3233 and 15.6507 with the two metrics PSNR and MSE, respectively. Concerning the elapsed reconstruction time, it also achieved high superiority with an efficiency gain of 0.8. The proposed method is suitable for utilization in the Internet of Healthcare Things. [ABSTRACT FROM AUTHOR]

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo's fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of timefractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo's time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research. [ABSTRACT FROM AUTHOR]

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability. [ABSTRACT FROM AUTHOR]

To address the performance degradation of the conventional linear frequency modulation signal ranging method in the presence of impulse noise, this paper proposes a novel technique that integrates a sliding-window tracking differentiator (TD) with the fractional Fourier transform (FrFT) ranging method. First, the sliding-window TD filtering algorithm is used to suppress the noise in the echo. Subsequently, the filtered signal is subjected to FrFT to calculate the time delay based on the difference in the peak point positions in the fractional domain for realizing target ranging. The simulation results show that the proposed method can effectively suppress impulse noise of different intensities and achieve an accurate and robust ranging of the target. [ABSTRACT FROM AUTHOR]

This paper mainly proposes some improved stochastic gradient descent (SGD) algorithms with a fractional order gradient for the online optimization problem. For three scenarios, including standard learning rate, adaptive gradient learning rate, and momentum learning rate, three new SGD algorithms are designed combining a fractional order gradient and it is shown that the corresponding regret functions are convergent at a sub-linear rate. Then we discuss the impact of the fractional order on the convergence and monotonicity and prove that the better performance can be obtained by adjusting the order of the fractional gradient. Finally, several practical examples are given to verify the superiority and validity of the proposed algorithm. [ABSTRACT FROM AUTHOR]

A control scheme for finite-time stabilization of unstable orbits of the fractional difference logistic map is proposed in this paper. The presented technique is based on isolated perturbation impulses used to correct the evolution of the map's trajectory after it deviates too far from the neighborhood of the unstable orbit, and does not require any feedback control loops. The magnitude of the control impulses is determined by means of H-rank algorithm, which helps to reveal the pseudo-manifold of non-asymptotic convergence of the fractional difference logistic map. Numerical experiments are used to illustrate the effectiveness and the feasibility of the proposed approach, which is applicable beyond the studied fractional difference logistic map. [ABSTRACT FROM AUTHOR]

In this paper, a mixed-order image denoising algorithm containing fractional-order and high-order regularization terms is proposed, which effectively suppresses the staircase effect generated by the TV model and its variants while better preserving the edges and details of the image. Adding different regularization penalties in different regions is fundamental to improving the denoising performance of the model. Therefore, a weight selection function is designed using the structure tensor to achieve a more effective selection of regularization terms in different regions. In each iteration, the regularization parameters are adaptively adjusted according to the Morozov discrepancy principle to promote the performance of the algorithm. Based on the primal–dual theory, the original algorithm is improved by using the predictor–corrector scheme to obtain a more accurate approximate solution while ensuring the convergence of the algorithm. The effectiveness of the proposed algorithm is demonstrated through simulation experiments. [ABSTRACT FROM AUTHOR]

In real-life control problems, such as power systems, there are large-scale high-ranked discrete-time algebraic Riccati equations (DAREs) from fractional systems that require stabilizing solutions. However, these solutions are no longer numerically low-rank, which creates difficulties in computation and storage. Fortunately, the potential structures of the state matrix in these systems (e.g., being banded-plus-low-rank) could be beneficial for large-scale computation. In this paper, a factorized structure-preserving doubling algorithm (FSDA) is developed under the assumptions that the non-linear and constant terms are positive semidefinite and banded-plus-low-rank. The detailed iteration scheme and a deflation process for FSDA are analyzed. Additionally, a technique of partial truncation and compression is introduced to reduce the dimensions of the low-rank factors. The computation of residual and the termination condition of the structured version are also redesigned. Illustrative numerical examples show that the proposed FSDA outperforms SDA with hierarchical matrices toolbox (SDA_HODLR) on CPU time for large-scale problems. [ABSTRACT FROM AUTHOR]

A major problem in power systems is achieving a match between the load demand and generation demand, where security, dependability, and quality are critical factors that need to be provided to power producers. This paper proposes a proportional–integral–derivative (PID) controller that is optimally designed using a novel artificial rabbits algorithm (ARA) for load frequency control (LFC) in multi-area power systems (MAPSs) of two-area non-reheat thermal systems. The PID controller incorporates a filter with such a derivative coefficient to reduce the effects of the accompanied noise. In this regard, single objective function is assessed based on time-domain simulation to minimize the integral time-multiplied absolute error (ITAE). The proposed ARA adjusts the PID settings to their best potential considering three dissimilar test cases with different sets of disturbances, and the results from the designed PID controller based on the ARA are compared with various published techniques, including particle swarm optimization (PSO), differential evolution (DE), JAYA optimizer, and self-adaptive multi-population elitist (SAMPE) JAYA. The comparisons show that the PID controller's design, which is based on the ARA, handles the load frequency regulation in MAPSs for the ITAE minimizations with significant effectiveness and success where the statistical analysis confirms its superiority. Considering the load change in area 1, the proposed ARA can acquire significant percentage improvements in the ITAE values of 1.949%, 3.455%, 2.077% and 1.949%, respectively, with regard to PSO, DE, JAYA and SAMPE-JAYA. Considering the load change in area 2, the proposed ARA can acquire significant percentage improvements in the ITAE values of 7.587%, 8.038%, 3.322% and 2.066%, respectively, with regard to PSO, DE, JAYA and SAMPE-JAYA. Considering simultaneous load changes in areas 1 and 2, the proposed ARA can acquire significant improvements in the ITAE values of 60.89%, 38.13%, 55.29% and 17.97%, respectively, with regard to PSO, DE, JAYA and SAMPE-JAYA. [ABSTRACT FROM AUTHOR]

The instability of the underwater environment and underwater communication brings great challenges to the coordination and cooperation of the multi-Autonomous Underwater Vehicle (AUV). In this paper, a multi-AUV dynamic maneuver countermeasure algorithm is proposed based on the interval information game theory and fractional-order Differential Evolution (DE), in order to highlight the features of the underwater countermeasure. Firstly, an advantage function comprising the situation and energy efficiency advantages is proposed on account of the multi-AUV maneuver strategies. Then, the payoff matrix with interval information is established and the payment interval ranking is achieved based on relative entropy. Subsequently, the maneuver countermeasure model is presented along with the Nash equilibrium condition satisfying the interval information game. The fractional-order DE algorithm is applied for solving the established problem to determine the optimal strategy. Finally, the superiority of the proposed multi-AUV maneuver countermeasure algorithm is verified through an example. [ABSTRACT FROM AUTHOR]

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald difference operator and product trapezoidal formula were utilized to construct a high-order numerical scheme. Also, we analyzed the stability and convergence. The convergence order was O (τ 2 + h x 4 + h y 4) , where τ is the time step size, h x and h y are the spatial step sizes. Furthermore, several examples were provided to verify the correctness of our theoretical reasoning. [ABSTRACT FROM AUTHOR]

A novel morphing activation function is proposed, motivated by the wavelet theory and the use of wavelets as activation functions. Morphing refers to the gradual change of shape to mimic several apparently unrelated activation functions. The shape is controlled by the fractional order derivative, which is a trainable parameter to be optimized in the neural network learning process. Given the morphing activation function, and taking only integer-order derivatives, efficient piecewise polynomial versions of several existing activation functions are obtained. Experiments show that the performance of polynomial versions PolySigmoid, PolySoftplus, PolyGeLU, PolySwish, and PolyMish is similar or better than their counterparts Sigmoid, Softplus, GeLU, Swish, and Mish. Furthermore, it is possible to learn the best shape from the data by optimizing the fractional-order derivative with gradient descent algorithms, leading to the study of a more general formula based on fractional calculus to build and adapt activation functions with properties useful in machine learning. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper comprehensively reviews the spiral dynamics optimization (SDO) algorithm and investigates its characteristics. SDO algorithm is one of the most straightforward physics-based optimization algorithms and is successfully applied in various broad fields. This paper describes the recent advances of the SDO algorithm, including its adaptive, improved, and hybrid approaches. The growth of the SDO algorithm and its application in various areas, theoretical analysis, and comparison with its preceding and other algorithms are also described in detail. A detailed description of different spiral paths, their characteristics, and the application of these spiral approaches in developing and improving other optimization algorithms are comprehensively presented. The review concludes the current works on the SDO algorithm, highlighting its shortcomings and suggesting possible future research perspectives. [ABSTRACT FROM AUTHOR]

Demand response (DR) flexible loads can provide fast regulation and ancillary services as reserve capacity in power systems. This paper proposes a demand response optimization dispatch control strategy for flexible thermostatically controlled loads (TCLs) and plug-in electric vehicles (PEVs) with stochastic renewable power injection. Firstly, a chance constraint look-ahead programming model is proposed to maximize the social welfare of both units and load agents, through which the optimal power scheduling for TCL/PEV agents can be obtained. Secondly, two demand response control algorithms for TCLs and PEVs are proposed, respectively, based on the aggregate control models of the load agents. The TCLs are controlled by its temperature setpoints and PEVs are controlled by its charging power such that the DR control objective can be fulfilled. It has been shown that the proposed dispatch and control strategy can coordinate the flexible load agents and the renewable power injection. Finally, the simulation results on a modified IEEE 39 bus system demonstrate the effectiveness of the proposed demand response strategy. [ABSTRACT FROM AUTHOR]

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy. [ABSTRACT FROM AUTHOR]

This paper proposes new numerical algorithms for calculating the time domain responses of fractional order transfer functions (FOTFs). FOTFs are divided into two categories, explicit fractional order transfer functions (EFOTFs) and implicit fractional order transfer functions (IFOTFs). Transforming an EFOTF into an equivalent fractional order differential equation, its time domain response can be obtained by solving the equation by the difference method. IFOTF cannot be transformed into an equivalent equation, so its time domain response cannot be calculated by existing difference methods. A new numerical algorithm is designed for calculating a convolution and its inverse operation, the time domain response of IFOTF can be calculated based on the algorithm. Error analysis shows that the proposed numerical algorithms are of first-order accuracy. Four calculation examples are presented, and the results are consistent with the theoretical analysis. [ABSTRACT FROM AUTHOR]

The current study discusses a novel approach for numerically solving MTVO-TFDWEs under various conditions, such as IBCs and DBCs. It uses a class of GSJPs that satisfy the given conditions (IBCs or DBCs). One of the important parts of our method is establishing OMs for Ods and VOFDs of GSJPs. The second part is using the SCM by utilizing these OMs. This algorithm enables the extraction of precision and efficacy in numerical solutions. We provide theoretical assurances of the treatment's efficacy by validating its convergent and error investigations. Four examples are offered to clarify the approach's practicability and precision; in each one, the IBCs and DBCs are considered. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors. [ABSTRACT FROM AUTHOR]

The segmentation of crack detection and severity assessment in low-light environments presents a formidable challenge. To address this, we propose a novel dual encoder structure, denoted as DSD-Net, which integrates fast Fourier transform with a convolutional neural network. In this framework, we incorporate an information extraction module and an attention feature fusion module to effectively capture contextual global information and extract pertinent local features. Furthermore, we introduce a fractal dimension estimation method into the network, seamlessly integrated as an end-to-end task, augmenting the proficiency of professionals in detecting crack pathology within low-light settings. Subsequently, we curate a specialized dataset comprising instances of crack pathology in low-light conditions to facilitate the training and evaluation of the DSD-Net algorithm. Comparative experimentation attests to the commendable performance of DSD-Net in low-light environments, exhibiting superlative precision (88.5%), recall (85.3%), and F1 score (86.9%) in the detection task. Notably, DSD-Net exhibits a diminutive Model Size (35.3 MB) and elevated Frame Per Second (80.4 f/s), thereby endowing it with the potential to be seamlessly integrated into edge detection devices, thus amplifying its practical utility. [ABSTRACT FROM AUTHOR]

Rural community population forecasting has important guiding significance to rural construction and development. In this study, a novel grey Bernoulli model combined with an improved Aquila Optimizer (IAO) was used to forecast rural community population in China. Firstly, this study improved the Aquila Optimizer by combining quasi-opposition learning strategy and wavelet mutation strategy, and proposed the new IAO algorithm. By comparing with other algorithms on CEC2017 test functions, the proposed IAO algorithm has the advantages of faster convergence speed and higher convergence accuracy. Secondly, based on the data of China's rural community population from 1990 to 2019, a consistent fractional accumulation nonhomogeneous grey Bernoulli model called CFANGBM(1, 1, b, c) was established for rural population forecasting. The proposed IAO algorithm was used to optimize the parameters of the model, and then the rural population of China was predicted. Four error measures were used to evaluate the model, and by comparing with other forecasting models, the experimental results show that the proposed model had the smallest error between the forecasted value and the real value, which illustrates the effectiveness of using the IAO algorithm to solve CFANGBM(1, 1, b, c). At the end of this paper, the forecast data of China's rural population from 2020 to 2024 are given for reference. [ABSTRACT FROM AUTHOR]

In the current work, a fast θ scheme combined with the Legendre spectral method was developed for solving a fractional Klein–Gordon equation (FKGE). The numerical scheme was provided by the Legendre spectral method in the spatial direction, and for the temporal direction, a θ scheme of order O (τ 2) with a fast algorithm was taken into account. The fast algorithm could decrease the computational cost from O (M 2) to O (M log M) , where M denotes the number of time levels. In addition, correction terms could be employed to improve the convergence rate when the solutions have weak regularity. We proved theoretically that the scheme is unconditionally stable and obtained an error estimate. The numerical experiments demonstrated that our numerical scheme is accurate and efficient. [ABSTRACT FROM AUTHOR]

Depth image enhancement techniques can help to improve image quality and facilitate computer vision tasks. Traditional image-enhancement methods, which are typically based on integer-order calculus, cannot exploit the textural information of an image, and their enhancement effect is limited. To solve this problem, fractional differentiation has been introduced as an innovative image-processing tool. It enables the flexible use of local and non-local information by taking into account the continuous changes between orders, thereby improving the enhancement effect. In this study, a fractional differential is applied in depth image enhancement and used to establish a novel algorithm, named the fractional differential-inverse-distance-weighted depth image enhancement method. Experiments are performed to verify the effectiveness and universality of the algorithm, revealing that it can effectively solve edge and hole interference and significantly enhance textural details. The effects of the order of fractional differentiation and number of iterations on the enhancement performance are examined, and the optimal parameters are obtained. The process data of depth image enhancement associated with the optimal number of iterations and fractional order are expected to facilitate depth image enhancement in actual scenarios. [ABSTRACT FROM AUTHOR]

An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit scheme that does not require solving a nonlinear or linear algebraic system. For the convergence analysis, the discretization error is estimated and proved to converge via a recurrence relation. The discretization error is combined with the Krasnoselskij iteration error to estimate the total approximation error, hence establishing the convergence of the method. Then, numerical experiments are provided, first, to demonstrate the second order convergence of the proposed method, and secondly, to show the better performance of the scheme over the existing nonlinear-based approach. [ABSTRACT FROM AUTHOR]

A numerical technique was developed for solving nonlocal nonlinear stochastic delayed differential equations driven by fractional variable-order Brownian noise. Error analysis of the proposed technique was performed and discussed. The method was applied to the nonlocal stochastic fluctuations of the human body and the Nicholson's blowfly models, and its accuracy and computational time were assessed for different values of the nonlocal order parameters. A comparison with other techniques available in the literature revealed the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the ψ -Hilfer derivative. This allows us to cover several definitions of fractional derivatives that are found in the literature in our study. The convergence of the ψ -Hilfer continuous fractional gradient method was studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we developed an algorithm for the ψ -Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and step size optimization, the ψ -Hilfer fractional gradient method showed better results in terms of speed and accuracy. Our results generalize previous works in the literature. [ABSTRACT FROM AUTHOR]

As the fourth fundamental circuit element, the memristor may execute computations while storing data. Fracmemristor takes advantage of the fractional calculate's long-term memory, non-locality, weak singularity, and the memristor's storage–computational integration. Since the physical structure of the fracmemristor is similar to the topology of the ant transfer probability flow in ACO, we propose the fractional-order memristive ant colony algorithm (FMAC), which uses the fracmemristor physical system to record the probabilistic transfer information of the nodes that the ant will crawl through in the future and pass it to the current node of the ant, so that the ant acquires the ability to predict the future transfer. After instigating the optimization capabilities with TSP, we discovered that FMAC is superior to PACO-3opt, the best integer-order ant colony algorithm currently available. FMAC operates substantially more quickly than the fractional-order memristor ant colony algorithm due to the transfer probability prediction module based on the physical fracmemristor system (FACA). [ABSTRACT FROM AUTHOR]

Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared to the existing doubling algorithm of O (2 k n) flops at the k-th iteration, the newly developed version merely needs O (n) flops for preprocessing and O ((k + 1) 3 m 3) flopsfor iterations and is more proper for large-scale computations when m ≪ n . The convergence and complexity of the algorithm are subsequently analyzed. Illustrative numerical experiments indicate that the presented algorithm, which consists of a dominant time-consuming preprocessing step and a trivially iterative step, is capable of computing the solution efficiently for large-scale DAREs. [ABSTRACT FROM AUTHOR]

In this study, we focus on solving the nonlinear fractional optimization problem in which the numerator is smooth convex and the denominator is smooth concave. To achieve this goal, we develop an algorithm called the adaptive projection gradient method. The main advantage of this method is that it allows the computations for the gradients of the considered functions and the metric projection to take place separately. Moreover, an interesting property that distinguishes the proposed method from some of the existing methods is the nonincreasing property of its step-size sequence. In this study, we also prove that the sequence of iterates that is generated by the method converges to a solution for the considered problem and we derive the rate of convergence. To illustrate the performance and efficiency of our algorithm, some numerical experiments are performed. [ABSTRACT FROM AUTHOR]

This brief investigates the Mittag–Leffler formation bounded control problem for second-order fractional multi-agent systems (FMASs), where the dynamical nodes of followers are modeled to satisfy quadratic (QUAD) condition. Firstly, under the undirected communication topology, for the considered second-order nonlinear FMASs, a distributed event-triggered control scheme (ETCS) is designed to realize the global Mittag–Leffler bounded formation control goal. Secondly, by introducing adaptive weights into triggering condition and control protocol, an adaptive event-triggered formation protocol is presented to achieve the global Mittag–Leffler bounded formation. Thirdly, a five-step algorithm is provided to describe protocol execution steps. Finally, two simulation examples are given to verify the effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this research article, a modified algorithm for the generation of a fractal pattern resulting from the iteration of an algebraic function using the numerical iteration method is presented. This fractal pattern shows the dynamical behavior of the numerical iterations. A nonstandard convergence test of the displayable fractal pattern was applied. [ABSTRACT FROM AUTHOR]