Iteration Method For Solving Research Articles

Research and Analysis of the Small Disturbance Equation in Subsonic, Transonic, and Supersonic Regimes

This paper explores the application of the Small Disturbance Equation (SDE) across subsonic, transonic, and supersonic flow regimes. Derived from the Euler and Navier-Stokes equations, the SDE offers an efficient framework for analyzing aerodynamic behaviors, particularly through the utilization of discretization techniques and iterative solving methods executed in Python. The study assesses the accuracy and limitations of the SDE in detailing essential flow characteristics, revealing that while the equation performs effectively in subsonic and transonic flows, it encounters challenges in supersonic regimes. Nonlinear effects such as shock waves significantly hinder its performance at high speeds. Compared with conventional computational fluid dynamics (CFD) methods, the SDE stands out in scenarios where computational efficiency is paramount. However, its limitations in handling high-speed flows must be carefully considered, highlighting the need for further refinement in its application to supersonic dynamics. This analysis suggests that while the SDE is beneficial for certain aerodynamic studies, its scope and utility are constrained by the inherent complexities of high-speed fluid dynamics.

Optimization of Organic Rankine Cycle for Hot Dry Rock Power System: A Stackelberg Game Approach

Due to its simple structure and stable operation, the Organic Rankine Cycle (ORC) has gained significant attention as a primary solution for low-grade thermal power generation. However, the economic challenges associated with development difficulties in hot dry rock (HDR) geothermal power systems have necessitated a better balance between performance and cost effectiveness within ORC systems. This paper establishes a game pattern of the Organic Rankine Cycle with performance as the master layer and economy as the slave layer, based on the Stackelberg game theory. The optimal working fluid for the ORC is identified as R600. At the R600 mass flow rate of 50 kg/s, the net system cycle work is 4186 kW, the generation efficiency is 14.52%, and the levelized cost of energy is 0.0176 USD/kWh. The research establishes an optimization method for the Organic Rankine Cycle based on the Stackelberg game framework, where the network of the system is the primary optimization objective, and the heat transfer areas of the evaporator and condenser serve as the secondary optimization objective. An iterative solving method is utilized to achieve equilibrium between the performance and economy of the ORC system. The proposed method is validated through a case study utilizing hot dry rock data from Qinghai Gonghe, allowing for a thorough analysis of the working fluid and system parameters. The findings indicate that the proposed approach effectively balances ORC performance with economic considerations, thereby enhancing the overall revenue of the HDR power system.

Assessment of spatiotemporally coordinated cyberattacks on renewable energy forecasting in smart energy system

The employment of deep learning (DL) in renewable energy forecasting (REF) may bring novel cyberthreats. However, this fact has not drawn sufficient attention in the existing literature. To fill the gap, this article, among the first, investigates the latent cyberthreat incurred by the DL-based REF model. First, a spatiotemporally coordinated cyberattack strategy is proposed by considering attackable geo-distributed renewable energy resources (RESs) and periods. It is fulfilled by compromising meteorological data transmitted from external online weather forecast interfaces which are required for REF but also exposed to potential adversaries. In this manner, the malicious adversary is able to deteriorate the REF performance, thereby jeopardizing the operation of the smart energy system. Second, the cyberattack optimization models under the white-box and black-box scenarios are designed. In the white-box scenario, the adversary has full knowledge of these operator-own REF models to obtain false data that should be injected into the meteorological data for launching cyberattacks; in the black-box scenario, the adversary will train surrogate models to replace these operator-own models for attack guidance. Third, an iterative solving method is proposed to solve the proposed white-box and black-box cyberattack optimization models. As DL models (sealed, non-differentiable, and highly non-linear) are introduced to the optimization problem, conventional optimization solvers are unable to handle the cyberattack optimization problem. The proposed solving method utilizes the proximal gradient descent principle, and is feasible to explore the near-optimal solution to the problem. At last, the efficacy of the proposed cyberattack strategy under different RES penetration levels and attack strengths and its catastrophic impacts are comprehensively assessed on the IEEE 39-bus benchmark. The results reveal that the cyberattack would cause enormous economic losses and even collapse the smart energy system.

An Iterative Algorithm for Predicting Seafloor Topography from Gravity Anomalies

As high-resolution global coverage cannot easily be achieved by direct bathymetry, the use of gravity data is an alternative method to predict seafloor topography. Currently, the commonly used algorithms for predicting seafloor topography are mainly based on the approximate linear relationship between topography and gravity anomaly. In actual application, it is also necessary to process the corresponding data according to some empirical methods, which can cause uncertainty in predicting topography. In this paper, we established analytical observation equations between the gravity anomaly and topography, and obtained the corresponding iterative solving method based on the least square method after linearizing the equations. Furthermore, the regularization method and piecewise bilinear interpolation function are introduced into the observation equations to effectively suppress the high-frequency effect of the boundary sea region and the low-frequency effect of the far sea region. Finally, the seafloor topography beneath a sea region (117.25°–118.25°E, 13.85°–14.85°N) in the South China Sea is predicted as an actual application, where gravity anomaly data of the study area with a resolution of 1′ × 1′ are from the DTU17 model. Comparing the prediction results with the data of ship soundings from the National Geophysical Data Center (NGDC), the root-mean-square (RMS) error and relative error can be up to 127.4 m and approximately 3.4%, respectively.

Minimize Tracking Occlusion in Collaborative Pick-and-Place Tasks: An Analytical Approach for Non-Wrist-Partitioned Manipulators.

Several industrial pick-and-place applications, such as collaborative assembly lines, rely on visual tracking of the parts. Recurrent occlusions are caused by the manipulator motion decrease line productivity and can provoke failures. This work provides a complete solution for maintaining the occlusion-free line of sight between a variable-pose camera and the object to be picked by a 6R manipulator that is not wrist-partitioned. We consider potential occlusions by the manipulator as well as the operator working at the assembly station. An actuated camera detects the object goal (part to pick) and keeps track of the operator. The approach consists of using the complete set of solutions obtained from the derivation of the univariate polynomial equation solution to the inverse kinematics (IK). Compared to numerical iterative solving methods, our strategy grants us a set of joint positions (posture) for each root of the equation from which we extract the best (minimizing the risks of occlusion). Our analytical-based method, integrating collision and occlusion avoidance optimizations, can contribute to greatly enhancing the efficiency and safety of collaborative assembly workstations. We validate our approach with simulations as well as with physical deployments on commercial hardware.

An iterative solution for improving the generalization ability of unsupervised skeleton motion retargeting

In this paper, a deep learning-based iterative solving method for unsupervised motion retargeting is proposed. Unlike existing methods, we do not utilize generative-adversarial-network-based (GAN-based) models as a solution for unsupervised motion retargeting, but utilize an iterative solution to yield retargeted motions based on a trained motion autoencoder. The proposed method includes a training phase and an execution phase. In the training phase, an autoencoder named the iterative motion retargeting network (ItMRnet) learns a continuous latent space of natural motion data. At the execution phase, the autoencoder-based hidden units are iteratively adjusted to yield retargeted motion sequences by minimizing the loss functions defined in motion space. Existing GAN-based methods need to train discriminators to distinguish whether the output of the generator is real or fake motion data. Obviously, the discriminator would have a poor discrimination ability for characters and motions that are not included in the training dataset. The proposed method does not use adversarial training which hurt generalization ability, but yields motion from a continuous latent space that represents a motion manifold, and can generate motions for those characters not included in the training dataset. That is, our method generalizes well to unseen motions and characters. The experimental results demonstrate that our method clearly outperforms state-of-the-art methods by reducing the motion retargeting error and motion deformation phenomenon, especially when the source or target character is not seen in the training dataset.

A Robust Scheduling Methodology for Integrated Electric-Gas System Considering Dynamics of Natural Gas Pipeline and Blending Hydrogen

As smart grid develops and renewables advance, challenges caused by uncertainties of renewables have been seriously threatening the energy system’s safe operation. Nowadays, the integrated electric-gas system (IEGS) plays a significant role in promoting the flexibility of modern grid owing to its great characteristic in accommodating renewable energy and coping with fluctuation and uncertainty of the system. And hydrogen, as an emerging and clean energy carrier, can further enhance the energy coupling of the IEGS and promote carbon neutralization with the development of power-to-hydrogen (P2H) technology and technology of blending hydrogen in the natural gas system. Dealing with the uncertainty of renewables, a robust schedule optimization model for the integrated electric and gas systems with blending hydrogen (IEGSH) considering the dynamics of gas is proposed and the iterative solving method based on column-and-constraint generation (C&CG) algorithm is implemented to solve the problem. Case studies on the IEGSH consisting of IEEE 39-bus power system and 27-node natural gas system validate the effectiveness of the dynamic energy flow model in depicting the transient process of gas transmission. The effectiveness of the proposed robust day-ahead scheduling model in dealing with the intra-day uncertainty of wind power is also verified. Additionally, the carbon emission reduction resulting from the blending of hydrogen is evaluated.

Power law modeling of functionally graded Adomian’s decomposition method for thermoelastic materials in heat transfer and thermal stress analysis

This work deals with applying the iteration method for the numerical solutions of a one-dimensional thermoelastic material. The governing equations have been constructed in the context of generalized thermoelasticity with one relaxation time model of Lord and Shulman (L-S). Some properties have been considered functional graded based on power law, and the lattice design parameter has been assumed. Adomian's decomposition method (ADM) and iteration techniques have been applied to solve the governing differential equations with an algorithm method. The numerical results have been represented in figures. The numerical results affirm that Adomian's decomposition method is a successful iteration method for solving and modeling thermoelastic problems. The lattice design parameter, empirical parameter, and ratio of the material properties significantly impact the temperature increment, strain, stress, displacement distributions, where they play vital roles in transferring the thermo-mechanical waves through the thermoelastic materials.

Practical implementation of genetic groups in single-step genomic evaluations with Woodbury matrix identity–based genomic relationship inverse

The growing amount of genomic information in dairy cattle has increased computational and modeling challenges in the single-step evaluations. The computational challenges are due to the dense inverses of genomic (G) and pedigree (A22) relationship matrices of genotyped animals in the single-step mixed model equations. An equivalent mixed model equation is given by single-step genomic BLUP that are based on the T matrix (ssGTBLUP), where these inverses are avoided by expressing G-1 through a product of 2 rectangular matrices, and (A22)-1 through sparse matrix blocks of the inverse of full relationship matrix A-1. A proper way to account genetic groups through unknown parent groups (UPG) after the Quaas-Pollak transformation (QP) is one key factor in a single-step model. When the UPG effects are incompletely accounted, the iterative solving method may have convergence problems. In this study, we investigated computational and predictive performance of ssGTBLUP with residual polygenic (RPG) effect and UPG. The QP transformation used A-1 and, in the complete form, T and (A22)-1 matrices as well. The models were tested with official Nordic Holstein milk production test-day data and model. The results show that UPG can be easily implemented in ssGTBLUP having RPG. The complete QP transformation was computationally feasible when preconditioned conjugate gradient iteration and iteration on data without explicitly setting up G or A22 matrices were used. Furthermore, for good convergence of the preconditioned conjugate gradient method, a complete QP transformation was necessary.

Capacity credit evaluation of wind energy using a robust secant method incorporating improved importance sampling

Capacity credit (CC) evaluation is a conception used in power system planning, which quantifies the contribution of various generating resources to the power system reliability. CC evaluation is essentially an iterative process of solving equations, where the reliability index of the power system is computed in every round of iteration. Therefore, fast and accurate evaluation of CC requires both a competent reliability assessment method and a compatible iterative method for equation solving. From this perspective, this paper presents a new methodology of CC evaluation for wind energy, by combining an improved cross-entropy-based importance sampling (ICE-IS) method for reliability assessment and a robust secant method for equation solving. The ICE-IS method is able to substantially accelerate the computation of reliability indices, especially when the wind power output and the system load are highly correlated. The robust secant method provides an unbiased estimate of the real CC value and ensures fast convergence, despite the error in the reliability index computed by ICE-IS in each round of iteration. Numerical tests are designed based on the historical data of two actual wind farms to prove the correctness of the proposed methodology. Besides, the results are discussed to analyze the impact of different factors on the CC of wind energy.

A THREE-STEP NINTH ORDER ITERATIVE METHOD FOR SOLVING NON-LINEAR EQUATIONS

A THREE-STEP NINTH ORDER ITERATIVE METHOD FOR SOLVING NON-LINEAR EQUATIONS

Investigation of Fractional Diffusion Equation via QSGS iterations

We investigate the application of Quarter-Sweep Gauss-Seidel (QSGS) iterative method for solving (SFPDE’s) space-fractional partial diffusion equations with Dirichlet boundary condition. To do this, implicit finite difference scheme and Caputo’s derivative operator are used to discretize one-dimensional linear space-fractional equation to form system of linear equations. Then basic ideas formulation and application of the suggested iterative method are also introduced. Numerical examples of tested problems were carried out to demonstrate advantages of the proposed iterative method beside to the HSGS and FSGS as control method. Based on computational results, the QSGS method is shown to be the most superior than the HSGS and FSGS iterative methods.

A new approach to electricity market clearing with uniform purchase price and curtailable block orders

The European market clearing problem is characterized by a set of heterogeneous orders and rules that force the implementation of heuristic and iterative solving methods. In particular, curtailable block orders and the uniform purchase price pose serious difficulties. A block order spans over multiple hours, and can be either fully accepted or fully rejected. The uniform purchase price prescribes that all consumers pay a common price in all the zones, while producers receive zonal prices, which can differ from one zone to another.The market clearing problem in the presence of both the uniform purchase price and block orders is a major open issue in the European context. The uniform purchase price scheme leads to a non-linear optimization problem involving both primal and dual variables, whereas block orders introduce multi-temporal constraints and binary variables into the problem. As a consequence, the market clearing problem in the presence of both block orders and the uniform purchase price can be regarded as a non-linear integer programming problem involving both primal and dual variables with complementary and multi-temporal constraints.The aim of this paper is to present a non-iterative and heuristic-free approach for solving the market clearing problem in the presence of both curtailable block orders and the uniform purchase price scheme. The solution is exact, with no approximation up to the level of resolution of current market data. By resorting to an equivalent uniform purchase price formulation, the proposed approach results in a mixed-integer linear program, which is built starting from a non-linear integer bilevel programming problem. Numerical results using real market data are reported to show the effectiveness of the proposed approach. The model has been implemented in Python, and the code is freely available on a public repository.

Optimization under non-convex Quadratic Matrix Inequality constraints with application to design of optimal sparse controller

Matrix optimization problems that contain one or more non-convex quadratic matrix constraints are considered. An iterative solving method is proposed; at each iteration convex matrix subproblem is formulated and solved using standard Convex Optimization algorithms. Global convergence of the method is proven. Implementation of the method is especially simple if non-convex matrix constraints are concave. The method is applied for solving long-standing problem of the H∞ design with additional sparsity constraint or objective on the controller.

TWO CSCS-BASED ITERATION METHODS FOR SOLVING ABSOLUTE VALUE EQUATIONS

Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration method for solving AVEs. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iterationsolver for AVEs. The advantage of these methods is that they do not require the storage of coefficient matrices at all, and the sub-system of linear equations can be solved efficiently via the fast Fourier transforms (FFTs). Therefore, computational cost and storage can be saved in practical implementations. Numerical examples including numerical solutions of nonlinear fractional diffusion equations are reported to show the effectiveness of the proposed methods in comparison with some existing methods.

An Iterative Method for Solving of Coupled Equations for Conductive-Radiative Heat Transfer in Dielectric Layers

The mathematical model for describing combined conductive-radiative heat transfer in a dielectric layer, which emits, absorbs, and scatters IR radiation both in its volume and on the boundary, has been considered. A nonlinear stationary boundary-value problem for coupled heat and radiation transfer equations for the layer, which exchanges by energy with external medium by convection and radiation, has been formulated. In the case of optically thick layer, when its thickness is much more of photon-free path, the problem becomes a singularly perturbed one. In the inverse case of optically thin layer, the problem is regularly perturbed, and it becomes a regular (unperturbed) one, when the layer’s thickness is of order of several photon-free paths. An iterative method for solving of the unperturbed problem has been developed and its convergence has been tested numerically. With the use of the method, the temperature field and radiation fluxes have been studied. The model and method can be used for development of noncontact methods for temperature testing in dielectrics and for nondestructive determination of its radiation properties on the base of the data obtained by remote measuring of IR radiation emitted by the layer.

Self-balancing robust scheduling with flexible batch loads for energy intensive corporate microgrid

With the development of microgrid technology, for an energy intensive corporate such as an iron and steel plant energy consumptions and costs can be saved significantly by achieving the balance between self-generation and consumption. In this paper, we present a self-balancing and robust scheduling model with flexible batch loads for an energy intensive corporate. The model is a multi-level optimization model with the objective to minimize the unbalance cost in the worst case. Load following properties are given to determine whether the uncertain loads can be followed or not, and the self-balancing capability of an energy intensive corporate is analyzed. The problem is equivalently converted from the multi-level model to a single-level optimization model with a set of constraints. In this way, the iteration between the outer and inner level can be avoided. Case study based on an energy intensive corporate microgrid is tested and the results show that the unbalance cost can be significantly reduced by using the robust self-balancing model. In addition, compared the approach method with iterative solving method, computational efficiency can be improved and local optimum can be avoided.

On convergence of three iterative methods for solving of the matrix equation $$X+A^{*}X^{-1}A+B^{*}X^{-1}B=Q$$ X + A ∗ X - 1 A + B ∗ X - 1 B = Q

In this paper, we give new convergence results for the basic fixed point iteration and its two inversion-free variants for finding the maximal positive definite solution of the matrix equation $$X+A^{*}X^{-1}A+B^{*}X^{-1}B=Q$$ , proposed by Long et al. (Bull Braz Math Soc 39:371–386, 2008) and Vaezzadeh et al. (Adv Differ Equ 2013). The new results are illustrated by numerical examples.

Fully Implicit Iterative Solving Method for the Fokker-Planck Equation in Tokamak Plasmas

A three dimensional bounce-averaged Fokker-Planck (FP) numerical code has been newly developed based on fully implicit iterative solving method, and relativistic effect is also included in the code. The code has been tested against various benchmark cases: Ohmic conductivity in the presence of weak Ohmic electric field, runaway losses of electrons in the presence of strong Ohmic electric field, lower hybrid current drive and electron cyclotron current drive via two- or three-dimensional simulation. All the test cases run fast and correctly during calculations. As a result, the code provides a set of powerful tools for studying radio frequency wave heating and current drive in tokamak plasmas.

LINTRAN v2.0: A linearised vector radiative transfer model for efficient simulation of satellite-born nadir-viewing reflection measurements of cloudy atmospheres

Radiance measurements of solar radiation that is backscattered by the Earth׳s atmosphere or surface contain information about the atmospheric composition and the state of the Earth׳s surface. Retrieving such information from satellite-based observations in nadir geometry employs a radiative transfer forward model. The forward model simulates the observed quantity, aiming to reproduce the observation.LINTRAN v2.0 is a linearised vector radiative transfer forward model, employing forward-adjoint theory, that is capable of modelling cloud contaminated satellite observations and their derivatives with respect to the state of the atmosphere and the Earth׳s surface in a numerically efficient manner. A significant gain in efficiency with respect to its predecessor (LINTRAN v1.0) is achieved through a mathematical framework that combines an approximate iterative solving method using the forward-adjoint perturbation theory with separation of the first N orders of scattering from the diffuse intensity vector field. Contributions to the observable up to order of scattering N are recursively solved in an analytical manner. Contributions from higher orders of scattering are subsequently solved in a numerical manner, assuming that the intensity field varies linearly with the vertical coordinate within an optically homogeneous model layer. This method is implemented in LINTRAN v2.0, choosing N=2, within the general framework of forward-adjoint perturbation theory.This new approach allows us to decrease the number of model layers and the degree of angular quadrature within the numerical solver by a factor of 10 and 1.4 respectively, compared to the previous model version, assuming a homogeneous atmosphere loaded with scattering Mie particles (size parameter χ≈35). In this homogeneous atmosphere, the reduced discretisation sampling in turn reduces the numerical effort associated with the numerical matrix solver by a factor of 42 relative to the previous model version, without a loss in model accuracy.