Newton Raphson Research Articles

Algorithm for determining the operating parameters of an electrical network in the problem of optimal reconfiguration in real time

Relevance. The need to develop technical solutions to reduce electrical energy losses in medium-voltage distribution electrical networks of oil and gas fields. Aim. Development of an algorithm for determining the parameters of the operating mode of an electrical network, applicable in a system of optimal control of the network configuration in order to reduce the level of electricity losses in the network. Methods. Mathematical modeling methods and combinatorial optimization methods. The nodal voltage method and the Newton–Raphson method are used to calculate the network operating mode. The mode is corrected using a sensitivity matrix. Results and conclusions. The authors have developed an algorithm for determining the parameters of the electric network operating mode when determining its optimal configuration. Optimization of the optimal configuration is performed based on the criterion of minimum active power losses in lines, taking into account the limitations on the power reserve of the feeding transformer substations and the number of normal breaks on one line. The developed algorithm is based on solving nonlinear systems of equations of nodal voltages using the Newton–Raphson method, as well as on correcting the parameters of the previously calculated operating mode using the sensitivity matrix. The optimal network configuration is determined by solving the combinatorial optimization problem using the branch and bound method. The operation of the algorithm is illustrated by the example of a network section containing two overhead power lines with two power sources and three feeding transformer substations. Based on the results of the analysis of all possible network configurations determined by combinations of normal breaks, it is concluded that the use of the algorithm will reduce the level of active power losses by up to 23.8%, which will increase the efficiency of the electric grid complex.

A comparison between penalized logistic regressions and classification tree approaches

. The penalized models have been established to be necessary for classifying the emerging complex, high-dimensional datasets. One of these penalized models is the “modified elastic-net” (M-Enet). The special cases of M-Enet (ridge, bridge, Lasso, and elastic-net) can be produced based on its penalty function. By using the fact that the Fisher-Scoring (FS) and Newton-Raphson (NR) algorithms use different approaches to solve numerical problems but are the same in the canonical link, in order to ease implementation, we developed an FS algorithm for the computation of the parameter estimates of the M-Enet. A simulation study of the microbiome as complex, high-dimensional data shows that each special case of the M-Enet penalty function has superiority in the sense of model accuracy, sensitivity, and feature selection technique. A comparison study is performed between the classification of binary responses using M-Enet as a parametric method and the classification tree via “GUIDE” as a non parametric method. The statistical analysis of the numerical results reveals that the special cases of M-Enet models outperform other M-Enet models and GUIDE in terms of average prediction accuracy and sensitivity, with p-values of 0. 0362 and 0. 0214, respectively. Consequently, these models exhibit superior classification performance compared to other classifier models.

Intelligent control algorithms for posture and height control of four-leg hydraulic supports

To address limitations of traditional inclinometers and height sensors in determining the posture and support height of hydraulic supports in coal mining, we propose a novel method predicated on travel measurements of the leg and tail beam cylinders. This method calculates the posture and height of hydraulic supports in mechanized mining. By conducting meticulous kinematic analysis of the hydraulic supports, a skeleton model of the main structural parameters of the hydraulic support was constructed. This approach transforms the traditional geometric relationship solutions of the main structure of the hydraulic supports into solutions based on the coordinate relationships of the main hinge points of the support, resulting in a mathematical expression for solving the support posture and height of the hydraulic supports. Meticulous algorithms for solving the support posture and height of hydraulic supports were developed based on the Newton–Raphson method, secant method, and Broyden’s method. The robustness, stability, calculation accuracy, and speed of these algorithms have been verified through analysis using the skeleton model and field measurement methods. The influence of initial coordinate parameters on the calculation results has been analyzed, and it was determined that the solution method based on the Newton–Raphson method has better robustness, stability, and calculation speed. The results of the research provide a theoretical foundation and technical support for precise, rapid control and intelligent management of hydraulic supports, effectively advancing the development of intelligent control systems for mining equipment.

Vibration-Based Damage Prediction in Composite Concrete–Steel Structures Using Finite Elements

The prediction of structural damage through vibrational analysis is a critical task in the field of composite structures. Structural defects and damage can negatively influence the load-carrying capacity of the beam. Therefore, detecting structural damage early is essential to preventing catastrophic failures. This study addresses the challenge of predicting damage in composite concrete–steel beams using a vibration-based finite element approach. To tackle this complex task, a finite element model to a quasi-static analysis emulating a four-point pure bending experimental test was performed. Notably, the numerical model equations were carefully modified using the Newton–Raphson method to account for the stiffness degradation resulting from material strains. These modified equations were subsequently employed in a modal analysis to compute modal shapes and natural frequencies corresponding to the stressed state. The difference between initial and damaged modal shape curvatures served as the foundation for predicting a damage index. The approach effectively captured stiffness degradation in the model, leading to observable changes in modal responses, including a reduction in natural frequencies and variations in modal shapes. This enabled the accurate prediction of damage instances during construction, service, or accidental load scenarios, thereby enhancing the structural and operational safety of composite system designs. This research contributes to the advancement of vibration-based methods for damage detection, emphasizing the complexities in characterizing damage in composite structural geometries. Further exploration and refinement of this approach are essential for the precise classification of damage types.

A Hybrid Approach Incorporating WSO-HO and the Newton-Raphson Method to Enhancing Photovoltaic Solar Model Parameters Optimisation

Abstract Accurate parameter estimation is vital for optimising the performance and design of photovoltaic (PV) systems. While metaheuristic algorithms (MHAs) offer promising solutions, they often face challenges such as slow convergence and difficulty balancing exploration and exploitation. This study introduces a novel hybrid approach, WSO-HO, which integrates the strengths of the war strategy optimization (WSO) and Hippopotamus Optimization (HO) algorithms, enhanced by the Newton-Raphson (NR) method, to achieve precise parameter estimation for PV models. The effectiveness of the WSO-HO algorithm was rigorously evaluated through intensive testing on three different solar panels, including the RTC France solar cell using the single diode model (SDM) and the double diode model (DDM), over 30 iterations. Comparative analysis highlights the superior performance of WSO-HO against conventional algorithms, which often struggle with accurately identifying PV model parameters. These promising results demonstrate the significant potential of this hybrid approach to improve parameter optimisation in PV systems, enabling more precise design and enhanced overall system efficiency. Furthermore, the simulation result of the performance of the WSO-HO algorithm was benchmarked against other algorithms reported in the literature, further validating its robustness and effectiveness.

A gradient method for solving the boundary value problem of the underwater large deformation cable

A nonlinear large deformation cable equation system based on arc length is employed to determine the configuration of the cable under current and buoyancy loads. Different from a traditional initial method and a shooting method, we have solved this nonlinear problem as a boundary value problem with nonlinear and global boundary conditions directly. A finite difference scheme is proposed to solve the large deformation cable equation, and the Newton–Raphson iteration is used to search for numerical approximate solution. We demonstrate that this system degenerates into the catenary equation in the case of vanishing bending stiffness for the first time. The solution of the catenary equation serves as the initial guess for the three-section cable problem. This method overcomes the disadvantages of the initial value method and step method, avoiding the need to adjust the boundary location. The spatial shapes of the large deformation cable and the effects of the length and position of the buoyancy section are discussed. The impact of ocean currents is also analyzed using Morison's formula.

Basins of Convergence in a Multi-Perturbed CR3BP

The circular restricted three-body problem (CR3BP) is analyzed to introduce additional factors into the dynamic model, such as radiation forces, flattening of the primary bodies, relativity effects, and the presence of natural satellites. The introduction of these factors increases the accuracy when obtaining the position of the Lagrange points and the basins of convergence of the system. The Newton–Raphson methodis used to implement a searching algorithm. Finally, an application to the Sun–Mars system including the presence of Phobos and Deimos is developed.

Parameter Extraction for Photovoltaic Models with Flood-Algorithm-Based Optimization

Accurately modeling photovoltaic (PV) cells is crucial for optimizing PV systems. Researchers have proposed numerous mathematical models of PV cells to facilitate the design and simulation of PV systems. Usually, a PV cell is modeled by equivalent electrical circuit models with specific parameters, which are often unknown; this leads to formulating an optimization problem that is addressed through metaheuristic algorithms to identify the PV cell/module parameters accurately. This paper introduces the flood algorithm (FLA), a novel and efficient optimization approach, to extract parameters for various PV models, including single-diode, double-diode, and three-diode models and PV module configurations. The FLA’s performance is systematically evaluated against nine recently developed optimization algorithms through comprehensive comparative and statistical analyses. The results highlight the FLA’s superior convergence speed, global search capability, and robustness. This study explores two distinct objective functions to enhance accuracy: one based on experimental current–voltage data and another integrating the Newton–Raphson method. Applying metaheuristic algorithms with the Newton–Raphson-based objective function reduced the root-mean-square error (RMSE) more effectively than traditional methods. These findings establish the FLA as a computationally efficient and reliable approach to PV parameter extraction, with promising implications for advancing PV system design and simulation.

Transformer–BiLSTM Fusion Neural Network for Short-Term PV Output Prediction Based on NRBO Algorithm and VMD

In order to solve the difficulties that the uncertain characteristics of PV output, such as volatility and intermittency, will bring to the development of microgrid scheduling plans, this paper proposes a Transformer–Bidirectional Long Short-Term Memory (BiLSTM) neural network PV power generation forecasting fusion model based on the Newton–Raphson optimization algorithm (NRBO) and Variational Modal Decomposition (VMD). Firstly, the principle of the VMD technique and the Gray Wolf Optimization (GWO) algorithm’s key parameter optimization method for VMD are introduced. Then, the Transformer decoder partially fuses the BiLSTM network and retains the encoder to obtain the body of the prediction model, followed by explaining the principle of the NRBO algorithm. And finally, the VMD-NRBO-Transformer-BiLSTM prediction model and hyperparameter selection are evaluated by the NRBO algorithm. The algorithm sets up a multi-model comparison experiment, and the results show that the prediction model proposed in this paper has the best prediction accuracy and the optimal evaluation index.

Modeling and Analysis of Thermoelastic Damping in a Piezoelectro-Magneto-Thermoelastic Imperfect Flexible Beam

This research addresses the phenomena of thermoelastic damping (TED) and frequency shift (FS) of a thin flexible piezoelectro-magneto-thermoelastic (PEMT) composite beam. Its motion is constrained by two linear flexible springs attached to both ends. The novelty behind the proposed study is to mimic the uncertainties during the fabrication of the beam. Therefore, the equation of motion was derived utilizing the linear Euler–Bernoulli theory accounting for the flexible boundary conditions. The beam’s eigenvalues, mode shapes, and the effects of the thermal relaxation time (t1), the dimensions of the beam, the linear spring coefficients (KL0 and KLL), and the critical thickness (CT) on both TED and FS of the PEMT beam were investigated numerically employing the Newton–Raphson method. The results show that the peak value of thermoelastic damping (Qpeak−1) and the frequency shift (Ω) of the beam increase as t1 escalates. Another observation was made for the primary fundamental mode, where an increase in the spring coefficient KLL leads to a further increase in Ω. On the other hand, the opposite trend is noted for the higher modes. Indeed, the results show the possibility of using the proposed design in a variety of applications that involve damping dissipation.

High-cycle constitutive model for traffic loading induced permanent deformation of coarse granular material incorporating particle breakage

Repetitive traffic loads lead to particle rearrangement and breakage in granular trackbeds and roadbeds, resulting in irreversible deformations that hinder normal operations. Current models for cyclic deformation primarily address rearrangement-induced densification but overlook breakage-induced degradation. Particle breakage disrupts inter-particle interlocking and creates finer debris, promoting volumetric contraction and weakening material stiffness. This study introduces a novel cyclic constitutive model for predicting plastic strain in coarse granular materials, emphasizing the role of particle breakage. The model extended from the existing cyclic densification model within the framework named “Shakedown Surface Model,” integrating breakage effects. In this model, shakedown thresholds, associated with material densification, are influenced by both plastic strain and breakage-induced loosening. Parameters for shearing contraction/dilation decrease with breakage, capturing breakage-induced contraction. Additionally, the model accounts for principal stress rotation from moving loads. Implemented via an implicit Euler-backward algorithm and Newton–Raphson iteration, the model was validated against cyclic triaxial and full-scale tests, demonstrating its accuracy in predicting the permanent deformation of granular materials under traffic loads.

Design and analysis of one-pad gas foil bearing with three-part gas film produced by adjusting the length of sliding beams

The one-pad sliding-beam gas foil bearing is studied in this paper. The size of the multiple sliding beams in the bearing is adjusted, which results in dividing the gas film pressure into three parts. The finite element mesh of the bottom foil is established based on rectangle and triangle elements, and the finite element model of sliding beams is established based on the plate element with five degrees of freedom. The separation and friction of the foils are considered, and the model of the foils is established based on minimum potential energy theorem and Newton–Raphson method. The mechanical model of the bearing is finally established by fluid-solid interaction. The calculation scheme is programmed by MATLAB, and the reliability of the model is verified by experiments. The numerical results show that with the increase of preload or the decrease of nominal clearance, the gas film pressure is divided into three parts gradually. Compared with the three-pad structure, the bearing in this paper has better bearing capacity. With nominal clearance being 30 µm, the maximum bearing capacity is obtained with a preload of 10 µm, and the maximum direct damping is obtained with a preload of 20 µm. A proper layout of sliding beams can improve the bearing capacity, but it could also weaken the dynamic performance. The variation of the initial bending stiffness distribution of sliding beams has little impact on the dynamic performance, but it can significantly affect the bearing capacity. When the stiffness of sliding beams of axial middle face is larger than that of the end face, the bearing capacity is obviously improved.

Accurate Dynamic Analysis Method of Cable-Damper System Based on Dynamic Stiffness Method

To suppress large vibrations of the cable in cable-stayed bridges, it is common to install transverse dampers near the end of the cable. This paper focuses on the cable-damper system; based on the dynamic stiffness method, an accurate dynamic analysis method considering cable parameters, damper parameters, and cable forces is proposed. First, a mechanical analysis model is established which is closer to the cable with a transverse damper installed in the bridge. The model considers the cable bending stiffness, sag, inclination angle, cable force, damper stiffness, damping coefficient, and damper installation height. Then, the characteristic frequency equation of the cable-damper system is established, and a solution method that combines the initial value method and Newton–Raphson method is proposed. This method is confirmed to provide more accurate frequency analysis for the cable-damper system. Finally, using this method, the effect of the damper parameters on the dynamic characteristics of the system is investigated.

Analytical study about effects of lubrication on dynamic characteristics of defect-ball ceramic bearing

Due to the self-lubrication and high stiffness, the ceramic bearings are widely applied in mechanical system. However, under some complex working conditions, the happening of defection and uncertain lubrication on the ball's surface is complex. Insufficient oil may increase friction and shorten defect bearing's life. In the paper, the self-lubrication characteristic of defect-ball ceramic bearing is considered to investigate the lubrication performance. Firstly, a novel lubrication model of ceramic bearing is established through Hamrock-Dowson theory. Then, the Newton–Raphson method is employed to analyze the dynamic of defect bearing. Finally, the effect of oil film thickness on the dynamic characteristic of ball-defect ceramic bearing is explored systematically. A vibrometer and the temperature sensors are used in the experiment to validated the accuracy of model. The vibration of defect-ball bearing is highly accord with the experiment, with an error less than 5%. The results indicate that enlarging the volume fraction of oil appropriately improve the effect of lubrication on dynamic of defect bearing. It will decrease the vibration of defect-ball bearing in high-speed condition. Research results have a significance guidance about the oil supply amount on the defect-ball ceramic bearing under a certain condition.

Hermite wavelet-based numerical study on (1 + 1)-dimensional Allen–Cahn and phi-four equations

In recent years, finding precise numerical solutions to nonlinear partial differential equations (NLPDEs) has become a significant research topic. This paper addresses the two well-known nonlinear models, which are [Formula: see text]-dimensional Allen–Cahn (AC) and phi-four (PF) equations, and they have applications in various real-world scenarios. These two models are solved by a numerical technique called Hermite wavelet collocation scheme (HWCS). The operational matrices of integration were developed using the Hermite wavelet basis. The proposed method is based on the collocation process. By employing appropriate grid points, this method converts the nonlinear model into a set of nonlinear algebraic equations. We obtain a solution by employing the Newton–Raphson scheme to solve these nonlinear algebraic equations. Tables and figures demonstrate that the proposed method yields superior results. The proposed examples demonstrate the efficacy of the proposed idea, and the results affirm the practicality of the proposed approach. We compared the present method with the non-polynomial spline method and trigonometric B-spline collocation technique to check the potential of the projected method.

Modeling and Dynamic Analysis of Double-Row Angular Contact Ball Bearing–Rotor–Disk System

This article presents a general numerical method to establish a mathematical model of a bearing–rotor–disk system. This mathematical model consists of two double-row angular contact ball bearings (DRACBBs), a rotor and a rigid disk. The mathematical model of the DRACBB is built on the basis of elastic Hertz contact by adopting the Newton Raphson iteration method, and three different structure forms are taken into account. The rotor is modeled by employing a finite element method in conjunction with Timoshenko beam theory, and the rigid disk is modeled by applying the lumped parameter method. The mathematical model of the bearing–rotor–disk system is constructed by the coupling of the bearing, rotor and disk, and the dynamic response of the bearing–rotor–disk system can be solved by employing the Newmark-β method. The validation of the above mathematical model is demonstrated by comparing the proposed results with the results from the existing literature and finite element software. The dynamic characteristics of the DRACBBs and the dynamic response of the bearing–rotor–disk system are investigated by parametric study. A dynamic characteristic analysis of the DRACBB is conducted to ensure the optimal structure form of the DRACBB under complex external loads, and it can provide a reference for the selection of the structural forms of DRACBBs.

Fired Heaters Optimization by Estimating Real-Time Combustion Products Using Numerical Methods

Fired heaters upstream of distillation towers, despite their optimal thermal efficiency, often suffer from performance decline due to fluctuations in fuel composition and unpredictable operational parameters. These heaters have high energy consumption, as fuel properties vary depending on the source of the crude oil. This study aims to optimize the combustion process of a three-gas mixture, mainly refinery gas, by incorporating more stable fuels such as natural gas and liquefied petroleum gas (LPG) to improve energy efficiency and reduce LPG consumption. Using real-time gas chromatography-mass spectrometry (GC-MS) data, we accurately calculate the mass fractions of individual compounds, allowing for more precise burner flow rate determinations. Thermochemical data are used to calculate equilibrium constants as a function of temperature, with the least squares method, while the Newton–Raphson method solves the resulting nonlinear equations. Four key variables (X4,X6,X8, and X11), representing H2,CO,O2, and N2, respectively, are defined, and a Jacobian matrix is constructed to ensure convergence within a tolerance of 1 ×10−6 over a maximum of 200 iterations, implemented via Python 3.10.4 and the scipy.optimize library. The optimization resulted in a reduction in LPG consumption by over 50%. By tailoring the fuel supply to the specific thermal needs of each processing unit, we achieved substantial energy savings. For instance, furnaces in the hydrocracking unit, which handle cleaner subproducts and benefit from hydrogen’s adiabatic reactions, require much less energy than those in the primary distillation unit, where high-impurity crude oil is processed.

Hybrid Brown-Bear and Hippopotamus Algorithms with Fractional Order Chaos Maps for Precise Solar PV Model Parameter Estimation

The rise in photovoltaic (PV) energy utilization has led to increased research on its functioning, as its accurate modeling is crucial for system simulations. However, capturing nonlinear current–voltage traits is challenging due to limited data from cells’ datasheets. This paper presents a novel enhanced version of the Brown-Bear Optimization Algorithm (EBOA) for determining the ideal parameters for the circuit model. The presented EBOA incorporates several modifications aimed at improving its searching capabilities. It combines Fractional-order Chaos maps (FC maps), which support the BOA settings to be adjusted in an adaptive manner. Additionally, it integrates key mechanisms from the Hippopotamus Optimization (HO) to strengthen the algorithm’s exploitation potential by leveraging surrounding knowledge for more effective position updates while also improving the balance between global and local search processes. The EBOA was subjected to extensive mathematical validation through the application of benchmark functions to rigorously assess its performance. Also, PV parameter estimation was achieved by combining the EBOA with a Newton–Raphson approach. Numerous module and cell varieties, including RTC France, STP6-120/36, and Photowatt-PWP201, were assessed using double-diode and single-diode PV models. The higher performance of the EBOA was shown by a statistical comparison with many well-known metaheuristic techniques. To illustrate this, the root mean-squared error values achieved by our scheme using (SDM, DDM) for RTC France, STP6-120/36, and PWP201 are as follows: (8.183847 × 10−4, 7.478488 × 10−4), (1.430320 × 10−2, 1.427010 × 10−2), and (2.220075 × 10−3, 2.061273 × 10−3), respectively. The experimental results show that the EBOA works better than alternative techniques in terms of accuracy, consistency, and convergence.

On Algorithms and Approximations for Progressively Type‐I Censoring Schemes

ABSTRACTThis research aims to study the estimation of parameters for a Burr‐XII distribution and to investigate optimal sampling plans under progressive Type‐I censoring (PTIC). For point estimation, we employed the maximum‐likelihood estimation (MLE) method using two numerical approaches: Newton–Raphson and the Expectation–Maximization algorithm. We also utilized Bayesian estimation with squared error loss and linear exponential loss functions. Specifically, two approximate Bayesian methods, the Lindley and Tierney‐Kadane methods were examined. Additionally, Bayesian numerical estimation was performed using Markov Chain Monte Carlo with the Metropolis‐Hastings (MH) algorithm. For interval estimation, we constructed asymptotic confidence intervals for MLE and the highest posterior density method within the Bayesian framework. The practical study involved Monte Carlo simulations to assess the efficiency and accuracy of the proposed estimation methods across different PTIC schemes. A real data analysis is also provided to illustrate the practical application of these methodologies in analyzing a clinical trial dataset. Data from a clinical trial using a PTIC scheme reveals patterns in pain relief, aiding in evaluating the antibiotic ointment's effectiveness. The study further investigates optimal sampling plans for the Burr‐XII distribution under PTIC.

An FFT based chemo-mechanical framework with fracture: Application to mesoscopic electrode degradation

An FFT based method is proposed to simulate chemo-mechanical problems at the microscale including fracture, specially suited to predict crack formation during the intercalation process in batteries. The method involves three fields fully coupled, concentration, deformation gradient and damage. The mechanical problem is set in a finite strain framework and solved using Fourier Galerkin for non-linear problems in finite strains. The damage is modeled with Phase Field Fracture using a stress driving force. This problem is solved in Fourier space using conjugate gradient with an ad-hoc preconditioner. The chemical problem is modeled with the second Fick’s law and physically based chemical potentials, is integrated using backward Euler and is solved by Newton–Raphson combined with a conjugate gradient solver. Buffer layers are introduced to break the periodicity and emulate Neumann boundary conditions for incoming mass flux. The framework is validated against Finite Elements the results of both methods are very close in all the cases. Finally, the framework is used to simulate the fracture of active particles of graphite during ion intercalation. The method is able to solve large problems at a reduced computational cost and reproduces the shape of the cracks observed in real particles.