Search Results (30)

Efficient coordination of directional overcurrent relays (DOCRs) is vital for maintaining the stability and reliability of electrical power systems (EPSs). The task of optimizing DOCR coordination in complex power networks is modeled as an optimization problem. This study aims to enhance the performance of protection systems by minimizing the cumulative operating time of DOCRs. This is achieved by effectively synchronizing primary and backup relays while ensuring that coordination time intervals (CTIs) remain within predefined limits (0.2 to 0.5 s). A novel optimization strategy, the fractional-order derivative war optimizer (FODWO), is proposed to address this challenge. This innovative approach integrates the principles of fractional calculus (FC) into the conventional war optimization (WO) algorithm, significantly improving its optimization properties. The incorporation of fractional-order derivatives (FODs) enhances the algorithm’s ability to navigate complex optimization landscapes, avoiding local minima and achieving globally optimal solutions more efficiently. This leads to the reduced cumulative operating time of DOCRs and improved reliability of the protection system. The FODWO method was rigorously tested on standard EPSs, including IEEE three, eight, and fifteen bus systems, as well as on eleven benchmark optimization functions, encompassing unimodal and multimodal problems. The comparative analysis demonstrates that incorporating fractional-order derivatives (FODs) into the WO enhances its efficiency, enabling it to achieve globally optimal solutions and reduce the cumulative operating time of DOCRs by 3%, 6%, and 3% in the case of a three, eight, and fifteen bus system, respectively, compared to the traditional WO algorithm. To validate the effectiveness of FODWO, comprehensive statistical analyses were conducted, including box plots, quantile–quantile (QQ) plots, the empirical cumulative distribution function (ECDF), and minimal fitness evolution across simulations. These analyses confirm the robustness, reliability, and consistency of the FODWO approach. Comparative evaluations reveal that FODWO outperforms other state-of-the-art nature-inspired algorithms and traditional optimization methods, making it a highly effective tool for DOCR coordination in EPSs. Full article

Support vector machine (SVM) models apply the Karush–Kuhn–Tucker (KKT-OC) optimality conditions in the ordinary derivative to the primal optimisation problem, which has a major influence on the weights associated with the dissimilarity between the selected support vectors and subsequently on the quality of the model’s predictions. Recognising the capacity of fractional derivatives to provide machine learning models with more memory through more microscopic differentiations, in this paper we generalise KKT-OC based on ordinary derivatives to KKT-OC using fractional derivatives (Frac-KKT-OC). To mitigate the impact of noise and identify support vectors from noise, we apply the Frac-KKT-OC method to the fuzzy intuitionistic version of SVM (IFSVM). The fractional fuzzy intuitionistic SVM model (Frac-IFSVM) is then evaluated on six sets of data from the UCI and used to predict the sentiments embedded in tweets posted by people with diabetes. Taking into account four performance measures (sensitivity, specificity, F-measure, and G-mean), the Frac-IFSVM version outperforms SVM, FSVM, IFSVM, Frac-SVM, and Frac-FSVM. Full article

Traditional computer vision techniques aim to extract meaningful information from images but often depend on manual feature engineering, making it difficult to handle complex real-world scenarios. Fractional calculus (FC), which extends derivatives to non-integer orders, provides a flexible way to model systems with memory effects and long-term dependencies, making it a powerful tool for capturing fractional rates of variation. Recently, neural networks (NNs) have demonstrated remarkable capabilities in learning complex patterns directly from raw data, automating computer vision tasks and enhancing performance. Therefore, the use of fractional calculus in neural network-based computer vision is a powerful method to address existing challenges by effectively capturing complex spatial and temporal relationships in images and videos. This paper presents a survey of fractional calculus neural network-based (FC NN-based) computer vision techniques for denoising, enhancement, object detection, segmentation, restoration, and NN compression. This survey compiles existing FFC NN-based approaches, elucidates underlying concepts, and identifies open questions and research directions. By leveraging FC’s properties, FC NN-based approaches offer a novel way to improve the robustness and efficiency of computer vision systems. Full article

The continuous Hopfield network (CHN) is a common recurrent neural network. The CHN tool can be used to solve a number of ranking and optimization problems, where the equilibrium states of the ordinary differential equation (ODE) related to the CHN give the solution to any given problem. Because of the non-local characteristic of the “infinite memory” effect, fractional-order (FO) systems have been proved to describe more accurately the behavior of real dynamical systems, compared to the model’s ODE. In this paper, a fractional-order variant of a Hopfield neural network is introduced to solve a Quadratic Knap Sac Problem (QKSP), namely the fractional CHN (FRAC-CHN). Firstly, the system is integrated with the quadratic method for fractional-order equations whose trajectories have shown erratic paths and jumps to other basin attractions. To avoid these drawbacks, a new algorithm for obtaining an equilibrium point for a CHN is introduced in this paper, namely the optimal fractional CHN (OPT-FRAC-CHN). This is a variable time-step method that converges to a good local minima in just a few iterations. Compared with the non-variable time-stepping CHN method, the optimal time-stepping CHN method (OPT-CHN) and the FRAC-CHN method, the OPT-FRAC-CHN method, produce the best local minima for random CHN instances and for the optimal feeding problem. Full article

This article introduces a novel optimization approach known as fractional order whale optimization algorithm (FWOA). The proposed optimizer incorporates the idea of fractional calculus (FC) into the mathematical structure of the conventional whale optimization algorithm (WOA). To validate the efficiency of the proposed FWOA, it is applied to address the challenges associated with the economic load dispatch (ELD) problem, which is a nonconvex, nonlinear, and non-smooth optimization problem. The objectives associated with ELD such as fuel cost and wind power generation cost minimization are achieved by taking into consideration different practical constraints like valve point loading effect (VPLE), transmission line losses, generator constraints, and stochastically variation of renewable energy sources (RES) integration. RES, particularly wind energy, has garnered more attention in recent times due to a range of environmental and economic factors. Stochastic wind (SW) power is also included in the ELD problem formulation. The incomplete gamma function (IGF) quantifies the influence of wind power. To assess its efficacy, the suggested approach is applied to a range of power systems including 3 generating units, 13 generating units and 40 generating units, consisting of 37 thermal units and 3 wind power units. To further strengthen the performance of the optimizer, the FWOA is hybridized with the interior point algorithm (IPA) to further refine the outcomes of the FWOA. The FWOA and IPA are used to address the problem of ELD while including the unpredictable nature of wind power. The simulation results of the suggested technique are compared with the most advanced heuristic optimization methods available, and it has been observed that the proposed optimizer obtained a superior and refined solution when compared to other state of the art optimization techniques. Furthermore, the efficacy of the suggested strategy in enhancing the solution of the ELD issue is validated through statistical analysis in terms of minimum fitness value. Full article

In this paper, we propose a new sliding window edge-oriented filter that computes the output pixels using a cross-correlation-like equation derived from the principles of fractional calculus (FC); thus, we call it the “fractional cross-correlation filter” (FCCF). We assessed the performance of this filter utilizing exclusively edge-preservation-oriented metrics such as the gradient conduction mean square error (GCMSE), the edge-based structural similarity (EBSSIM), and the multi-scale structural similarity (MS-SSIM); we conducted a statistical assessment of the performance of the filter based on those metrics by using the Berkeley segmentation dataset benchmark as a test corpus. Experimental data reveal that our approach achieves higher performance compared to conventional edge filters for all the metrics considered in this study. This is supported by the statistical analysis we carried out; specifically, the FCCF demonstrates a consistent enhancement in edge detection. We also conducted additional experiments for determining the main filter parameters, which we found to be optimal for a broad spectrum of images. The results underscore the FCCF’s potential to make significant contributions to the advancement of image processing techniques since many practical applications such as medical imaging, image enhancement, and computer vision rely heavily on edge detection filters. Full article

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

Due to the widespread application of neural networks (NNs), and considering the respective advantages of fractional calculus (FC), inertial neural networks (INNs), cellular neural networks (CNNs), and fuzzy neural networks (FNNs), this paper investigates the fixed-time synchronization (FDTS) issues for a particular category of fractional-order cellular-inertial fuzzy neural networks (FCIFNNs) that involve mixed time-varying delays (MTDs), including both discrete and distributed delays. Firstly, we establish an appropriate transformation variable to reformulate FCIFNNs with MTD into a differential first-order system. Then, utilizing the finite-time stability (FETS) theory and Lyapunov functionals (LFs), we establish some new effective criteria for achieving FDTS of the response system (RS) and drive system (DS). Eventually, we offer two numerical examples to display the effectiveness of our proposed synchronization strategies. Moreover, we also demonstrate the benefits of our approach through an application in image encryption. Full article

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions ${}_p{\Psi }_q$ and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and ${}_p{F}_q$-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the $\overline{H}$-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and $\overline{H}$-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of $\Gamma$-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and $\overline{H}$-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems. Full article

An imperative application of artificial intelligence (AI) techniques is visual object detection, and the methods of visual object detection available currently need highly equipped datasets preserved in a centralized unit. This usually results in high transmission and large storage overheads. Federated learning (FL) is an eminent machine learning technique to overcome such limitations, and this enables users to train a model together by processing the data in the local devices. In each round, each local device performs processing independently and updates the weights to the global model, which is the server. After that, the weights are aggregated and updated to the local model. In this research, an innovative framework is designed for real-world object recognition in FL using a proposed Deep Q Network (DQN) based on a Fractional Political–Smart Flower Optimization Algorithm (FP-SFOA). In the training model, object detection is performed by employing SegNet, and this classifier is effectively tuned based on the Political–Smart Flower Optimization Algorithm (PSFOA). Moreover, object recognition is performed based on the DQN, and the biases of the classifier are finely optimized based on the FP-SFOA, which is a hybridization of the Fractional Calculus (FC) concept with a Political Optimizer (PO) and a Smart Flower Optimization Algorithm (SFOA). Finally, the aggregation at the global model is accomplished using the Conditional Autoregressive Value at Risk by Regression Quantiles (CAViaRs) model. The designed FP-SFOA obtained a maximum accuracy of 0.950, minimum loss function of 0.104, minimum MSE of 0.122, minimum RMSE of 0.035, minimum FPR of 0.140, maximum average precision of 0.909, and minimum communication cost of 0.078. The proposed model obtained the highest accuracy of 0.950, which is a 14.11%, 6.42%, 7.37%, and 5.68% improvement compared to the existing methods. Full article

Given the intricate nature of contemporary energy systems, addressing the control and stability analysis of these systems necessitates the consideration of highly large-scale models. Transient stability analysis stands as a crucial challenge in enhancing energy system efficiency. Power System Stabilizers (PSSs), integrated within excitation control for synchronous generators, offer a cost-effective means to bolster power systems’ stability and reliability. In this study, we propose an enhanced nonlinear control strategy based on synergetic control theory for PSSs. This strategy aims to mitigate electromechanical oscillations and rectify the limitations associated with linear approximations within large-scale energy systems that incorporate thyristor-controlled series capacitors (TCSCs). To dynamically adjust the coefficients of the nonlinear controller, we employ the Fractional Order Fish Migration Optimization (FOFMO) algorithm, rooted in fractional calculus (FC) theory. The FOFMO algorithm adapts by updating position and velocity within fractional-order structures. To assess the effectiveness of the improved controller, comprehensive numerical simulations are conducted. Initially, we examine its performance in a single machine connected to the infinite bus (SMIB) power system under various fault conditions. Subsequently, we extend the application of the proposed nonlinear stabilizer to a two-area, four-machine power system. Our numerical results reveal highly promising advancements in both control accuracy and the dynamic characteristics of controlled power systems. Full article

Fractional calculus (FC) has recently received increasing attention due to its applications in many fields involving complex and nonlinear systems. However, one of the key challenges in using FC to deal with fractal or multifractal phenomena is how to relate functions to geometries with fractal dimensions. The current paper demonstrates how fractal calculus can be used to represent physical properties such as density defined on fractal geometries that no longer have the Lebesgue additive properties required for ordinary calculus. First, it introduces the recently proposed concept of fractal density, that is, densities defined on fractals and multifractals, and then shows how fractal calculus can be used to describe fractal densities. Finally, the singularity analysis based on fractal density calculation is used to analyze the depth clustering distribution of seismic frequencies around the Moho transition zone in the subduction zone of the Pacific plates and the Tethys collision zones. The results show that three solutions (linear, log-linear, and double log-linear) of a unified differential equation can describe the decay rate of the fractal density of depth clusters at the number (frequencies) of earthquakes. The spatial distribution of the three groups of earthquakes is further divided according to the three attenuation relationships. From north latitude to south latitude, from the North Pacific subduction zone to the Tethys collision zone, and then to the South Pacific subduction zone, the attenuation relationships of the earthquake depth distribution are generally from a linear, to log-linear, to double log-linear pattern. This provides insight into the nonlinearity of the depth distribution of earthquake swarms. Full article

Cancer is a complex disease, responsible for a significant portion of global deaths. The increasing prioritisation of know-why over know-how approaches in biological research has favoured the rising use of both white- and black-box mathematical techniques for cancer modelling, seeking to better grasp the multi-scale mechanistic workings of its complex phenomena (such as tumour-immune interactions, drug resistance, tumour growth and diffusion, etc.). In light of this wide-ranging use of mathematics in cancer modelling, the unique memory and non-local properties of Fractional Calculus (FC) have been sought after in the last decade to replace ordinary differentiation in the hypothesising of FC’s superior modelling of complex oncological phenomena, which has been shown to possess an accumulated knowledge of its past states. As such, this review aims to present a thorough and structured survey about the main guiding trends and modelling categories in cancer research, emphasising in the field of oncology FC’s increasing employment in mathematical modelling as a whole. The most pivotal research questions, challenges and future perspectives are also outlined. Full article

Time dilation (TD) is a principal concept in the special theory of relativity (STR). The Einstein TD formula is the relation between the proper time $t_0$ measured in a moving frame of reference with velocity v and the dilated time t measured by a stationary observer. In this paper, an integral approach is firstly presented to rededuce the Einstein TD formula. Then, the concept of TD is introduced and examined in view of the fractional calculus (FC) by means of the Caputo fractional derivative definition (CFD). In contrast to the explicit standard TD formula, it is found that the fractional TD (FTD) is governed by a transcendental equation in terms of the hyperbolic function and the fractional-order $\alpha$. For small v compared with the speed of light c (i.e., $v\ll c$), our results tend to Newtonian mechanics, i.e., $t\to t_0$. For v comparable to c such as $v=0.9994c$, our numerical results are compared with the experimental ones for the TD of the muon particles ${\mu }^{+}$. Moreover, the influence of the arbitrary-order $\alpha$ on the FTD is analyzed. It is also declared that at a specific $\alpha$, there is an agreement between the present theoretical results and the corresponding experimental ones for the muon particles ${\mu }^{+}$. Full article