Iterative Solution Method Research Articles

Distributed optimization of electricity-Gas-Heat integrated energy system with multi-agent deep reinforcement learning

The coordinated optimization problem of the electricity-gas-heat integrated energy system (IES) has the characteristics of strong coupling, non-convexity, and nonlinearity. The centralized optimization method has a high cost of communication and complex modeling. Meanwhile, the traditional numerical iterative solution cannot deal with uncertainty and solution efficiency, which is difficult to apply online. For the coordinated optimization problem of the electricity-gas- heat IES in this study, we constructed a model for the distributed IES with a dynamic distribution factor and transformed the centralized optimization problem into a distributed optimization problem in the multi-agent reinforcement learning environment using multi-agent deep deterministic policy gradient. Introducing the dynamic distribution factor allows the system to consider the impact of changes in real-time supply and demand on system optimization, dynamically coordinating different energy sources for complementary utilization and effectively improving the system economy. Compared with centralized optimization, the distributed model with multiple decision centers can achieve similar results while easing the pressure on system communication. The proposed method considers the dual uncertainty of renewable energy and load in the training. Compared with the traditional iterative solution method, it can better cope with uncertainty and realize real- time decision making of the system, which is conducive to the online application. Finally, we verify the effectiveness of the proposed method using an example of an IES coupled with three energy hub agents.

A topological characterisation of looped drainage networks

Hydrodynamic models are used to analyse water networks (water distribution, drainage, surface water, district heating, etc.). The non-linear nature of water flows necessitates the use of iterative solution methods in hydraulic modelling. This requires a relatively large computational effort. To reduce this effort, networks, network forcing and/or the flow in networks are often simplified and analysed using the Graph Theory. The simplification options depend on the network characteristics. There are many topological features to describe Graph-based networks. In this paper, these characteristics are summarised, applied on 7 urban drainage networks and discussed. As the topological features do not describe the networks in a uniform manner, a new type of topological characterisation of looped drainage networks (Network Linearisation Parameter, NLP) is proposed based on linearized hydraulics and bottlenecks identified in paths to outfalls.

Characteristic Prediction and Temperature-Control Strategy under Constant Power Conditions for Lithium-Ion Batteries

Accurate characteristic prediction under constant power conditions can accurately evaluate the capacity of lithium-ion battery output. It can also ensure safe use for new-energy vehicles and electrochemical energy storage. As the battery voltage continues to drop under constant power conditions, the battery current output will accordingly increase, which brings a risk of thermal runaway in instances of weak heat dissipation. Therefore, knowing how to control the battery temperature is very critical for safe use. At present, the model-based method for characteristic prediction and temperature control has been used by most scholars, and that is also the key to this method. This work firstly extends a cell model to a pack-based electrochemical two-dimensional thermal coupling model, considering the heterogeneity of different cells inside the pack, and obtains the model parameters for a prismatic lithium-ion battery with a rated capacity of 42 Ah. Characteristic prediction under constant power conditions is then conducted based on an iterative solution method. Validations of characteristic prediction indicate the convenience of the developed models, with average absolute errors of voltage and temperature less than 36 mV and 0.4 K, respectively, and power error less than 0.005%. Finally, two model-based temperature feed-forward control strategies with lower cooling costs and shorter prediction times were developed based on the battery characteristic predictions, which leaves room for further controller development.

An integrated finite volume framework for thermal elasto-hydrodynamic lubrication

Recent advances in computational methods and resources have fuelled numerical studies to explore novel techniques for solving thermal elasto-hydrodynamic lubrication (TEHL) problems. Commercial codes such as those that feature the Computational Fluid Dynamics (CFD) approach have gained much attention in the field of tribology due to the ability to tackle the limitations imposed by Reynolds equation assumptions and aptness in modelling complex geometries. However, the computational cost incurred by CFD software is significantly high due to the governing equations involved when solving the most general problem. The present contribution proposes an integrated finite-volume framework to solve TEHL problems to predict the lubrication performance of non-conformal contacts. Fluid flow effects are considered through the generalized Reynolds equation with the p−θ Elrod–Adams mass-conserving cavitation model to ensure the mass flow-rate conservation throughout the lubricated domain. The thermal behaviour of the lubricant film is predicted via the solution of the conservation of energy equation with appropriate boundary conditions for the fluid–solid interfaces. Novel partitioned iterative solution methods are integrated into the framework to handle the strong nonlinearities and improve the convergence of the coupled system of equations. Thermal and isothermal cases are presented and compared to results generated by existing CFD solvers and other experimental data to validate the proposed framework.

Non-Overlapping Domain Decomposition via BURA Preconditioning of the Schur Complement

A new class of high-performance preconditioned iterative solution methods for large-scale finite element method (FEM) elliptic systems is proposed and analyzed. The non-overlapping domain decomposition (DD) naturally introduces coupling operator at the interface γ. In general, γ is a manifold of lower dimensions. At the operator level, a key property is that the energy norm associated with the Steklov-Poincaré operator is spectrally equivalent to the Sobolev norm of index 1/2. We define the new multiplicative non-overlapping DD preconditioner by approximating the Schur complement using the best uniform rational approximation (BURA) of Lγ1/2. Here, Lγ1/2 denotes the discrete Laplacian over the interface γ. The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the BURA-based non-overlapping DD preconditioner has optimal computational complexity O(n), where n is the number of unknowns (degrees of freedom) of the FEM linear system. All theoretical estimates are robust, with respect to the geometry of the interface γ. Results of systematic numerical experiments are given at the end to illustrate the convergence properties of the new method, as well as the choice of the involved parameters.

A new family of optimal fourth-order iterative methods for nonlinear equations

A new two-parameter family of fourth-order iterative methods for the numerical solution of nonlinear equations of the form f(x)=0 has been introduced and their convergence analysis have been performed. The new methods in the family are optimal in the sense of Kung–Traub conjecture. Numerical tests which are made to support the theoretical results also show that our new family of optimal fourth order methods perform well and in many cases some members of the family are superior to other well-known and recent fourth order optimal methods in the literature.

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these methods, nonlinear elimination is an important ingredient to increase the convergence radius of Newton’s method. We prove standard globalization results for SQP-based globalization of Nonlinear FETI-DP, first for the case that the elimination set is empty. We then show how to combine nonlinear elimination and SQP-based globalization. The globalization preserves the block structure of the FETI-DP operator, which is the basis of the computational parallelism.Supporting numerical experiments using homogenous and heterogeneous model problems from nonlinear structural mechanics are provided. In the numerical experiments, we consider four standard choices of different elimination sets and different problem setups including stiff or almost incompressible inclusions in every subdomain. The numerical experiments illustrate that a good elimination set is important. However, the use of the SQP-based globalization approach presented here can improve the convergence of Nonlinear FETI-DP methods further, especially, if combined with a good choice of the elimination set.

A New Iterative Method for the Approximate Solution of Klein-Gordon and Sine-Gordon Equations

This article presents a new iterative method (NIM) for the investigation of the approximate solution of the Klein-Gordon and sine-Gordon equations. This approach is formulated on the combination of the Mohand transform and the homotopy perturbation method. Mohand transform (MT) is capable to handle the linear terms only, thus we introduce homotopy perturbation method (HPM) to tackle the nonlinear terms. This NIM derives the results in the form of a series solution. The proposed method emphasizes the stability of the derived solutions without any linearization, discretization, or hypothesis. Graphical representation and absolute error demonstrate the efficiency and authenticity of this scheme. Some numerical models are illustrated to show the compactness and reliability of this strategy.

On the Application of Successive Over-relaxation Algorithmic and Block Numerical Iterative Solutions for the Stationary Distribution in Markov Chain

The evolution of a system is represented by transitions from one state to the next, and the system's physical or mathematical behavior can also be depicted by defining all of the numerous states it can be in and demonstrating how it moves between them. In this study, the iterative solution methods for the stationary distribution of Markov chains were investigated, which start with an initial estimate of the solution vector and then alter it in such a way that it gets closer and closer to the genuine solution with each step or iteration., and also involved matrices operations such as multiplication with one or more vectors, which leaves the transition matrices unchanged and saves time. Our goal is to use Successive Overrelaxation Algorithmic and Block Numerical Iterative Solution Methods to compute the solutions. With the help of some existing Markov chain laws, theorems, and formulas, the normalization principle and matric operations such as lower, upper, and diagonal matrices are used. The stationary distribution vector’s π^((k+1) )={■(π_1^((k+1) )&π_2^((k+1) )&■(π_3^((k+1) )&π_4^((k+1) )&π_5^((k+1) ) )),k=0,1,2,…,n} are obtained for the illustrative examples, taken the initial stationary solution to be π^((0) )=(■(0.2&0.2&■(0.2&0.2&0.2)))^T and it was observed that all subsequent iterations yield exactly the same result as π^((1) ), and this shows that, the block iterative method requires only a single iteration to obtain the solution to full machine precision.

Optimal Power Quality Compensation of Energy Storage System in Distribution Networks Based on Unified Multi-Phase OPF Model

Energy Storage System (ESS) is a promising solution to suppress the peak-valley difference of residential distribution networks (RDN) with high penetration of distributed photovoltaic generations. Meanwhile, it can also provide certain power quality compensations to RDN due to the flexible adjusting ability of its converter interface. To make full use of this feature, this paper investigates the optimal power quality compensation of the ESS. This is achieved by formulating and solving an optimal power flow (OPF) problem, the objectives of which are to minimize the power loss, harmonic distortion, and voltage unbalance of the network. Noticeably, the established model includes the neutral line and inter-phase loads; thus, it is unified and can be easily extended to networks with different structures. However, the coupling between different phases and the inclusion of harmonic variables significantly increase the complexity of the OPF problem. To address this issue, the optimization model is further transformed into a convex quadratic constrained quadratic program (QCQP) by replacing power flow equations with current injection equations. As a result, the problem can be solved by the primal-dual interior-point method (PDIPM) with guaranteed convergence and computational efficiency. An iterative solution method is further proposed to improve the accuracy of the solution. Finally, the proposed method is verified using a three-phase three-wire 25-node distribution network and a three-phase four-wire 162-node distribution network in North America.

Sobolev gradient type iterative solution methods for a nonlinear 4th order elastic plate equation

This paper considers the numerical solution of an elastic bending model of a thin plate, based on material nonlinearity, which leads to a nonlinear elliptic 4th order boundary value problem. Our goal is to summarize possible efficient iterative solvers, adapted to this problem, based on a Sobolev space background. It is proved that the proposed methods exhibit robust behaviour, that is, convergence rates are bounded independently of the considered Galerkin discretization subspace.

ON THE APPLICATION OF MATRIX SCALING AND POWERING METHODS OF SMALL STATE SPACES FOR SOLVING TRANSIENT DISTRIBUTION IN MARKOV CHAIN

The iterative solution methods for transient distribution in Markov chain is the computation of state probability distributions at an arbitrary point of time, which in the case of a discrete-time Markov chain means, finding the distribution at some arbitrary time step denoted , a row vector whose component is the probability that the Markov chain is in state at time step . In this study, the solutions of transient distribution in Markov chain using matrix scaling and powering methods for small state spaces which produce a significantly more accurate response in less time for some types of situations and also, tries to get to the end result as quickly as possible has been investigated, in order to provide some insight into the solutions of transient distribution of Markov chain. Our goal is to compute solutions and algorithms for tiny state spaces utilizing matrix scaling and powering approaches, which begin with an initial estimate of the solution vector and then comes closer and closer to the true solution with each step or iteration. With the help of several existing Markov chain laws, theorems, and formulas, matrices operations such as multiplication with one or more vectors, Padé variant of the matrix-powering and scaling technique are used. While the algorithms are explained, the transient distribution vector’s Padé approximants , and a backward error analysis of the Padé approximation are obtained for certain illustrative examples..

Efficient multi-partition topology optimization

In topology optimization, the state of structures is typically obtained by numerically evaluating a discretized PDE-based model. The degrees of freedom of such a model can be partitioned in free and prescribed sets to define the boundary conditions. A multi-partition problem involves multiple partitions of the same discretization, typically corresponding to different loading scenarios. As a result, solving multi-partition problems involves multiple factorization/preconditionings of the system matrix, requiring a high computational effort. In this paper, a novel method is proposed to efficiently calculate the responses and accompanying design sensitivities in such multi-partition problems using static condensation for use in gradient-based topology optimization. A main problem class that benefits from the proposed method is the topology optimization of small-displacement multi-input–multi-output compliant mechanisms. However, the method is applicable to any linear problem. We present its formulation and an algorithmic complexity analysis to estimate computational advantages for both direct and iterative solution methods to solve the system of equations, verified by numerical experiments. It is demonstrated that substantial gains are achievable for large-scale multi-partition problems. This is especially true for problems with both a small set of number of degrees of freedom that fully describes the performance of the structure and with large similarities between the different partitions. A major contribution to the gain is the lack of large adjoint analyses required to obtain the sensitivities of the performance measure.

One-dimensional modeling approaches for the piloted ignition of poly (methyl methacrylate)

An unsteady one-dimensional solid-gas model is presented for the spontaneous and the piloted ignition of poly (methyl methacrylate) exposed to thermal irradiance. The mathematical description of the igniter (electrical spark) is given special attention. The simplest approach, namely the heating spark model, describes heat transfer from a pulsating high-temperature plane source (direct heating), together with the heat released by the pre-ignition reactions (indirect heating). A high overestimation of the thermal effects of the spark is observed owing to the one-dimensional approximation. An iterative solution method, indicated as the non-heating spark model, is also implemented to eliminate direct and indirect spark heating which, however, also cuts off possible flashing phenomena. The third approach, indicated as the chemical spark model, disregards the direct spark heating but still describes pre-ignition combustion. This approach is the simplest and the most effective for providing predictions in agreement with literature measurements for both the process dynamics and the ignition delay times.

Operation optimization of electric power - hot water - steam integrated energy system

The distributed integrated energy system can optimize the energy structure of the user side by using natural gas and realize the cascade utilization and low energy consumption of energy. This paper takes the integrated energy system as the research object, establishes the electric power ”source-network-load” model, the combined hydrothermal model and the steam network model, and proposes alternate iterative power flow solution method. On the basis of power flow solution, the integrated energy system optimization model is established with economy and system energy efficiency as the objective function, and the genetic algorithm toolbox is used to optimize the solution. The results show that the proposed operation optimization method is effective and feasible, which can effectively improve the integrated energy efficiency and realize the optimized operation of the integrated energy system.

Refinement of SOR Iterative Method for the Linear Rational Finite Difference Solution of Second-Order Fredholm Integro-Differential Equations

The primary objective of this paper is to develop the Refinement of Successive Over-Relaxation (RSOR) method based on a three-point linear rational finite difference-quadrature discretization scheme for the numerical solution of second-order linear Fredholm integro-differential equation (FIDE). Besides, to illuminate the superior performance of the proposed method, some numerical examples are presented and solved by implementing three approaches which are the Gauss-Seidel (GS), the Successive Over-Relaxation (SOR) and the RSOR methods. Lastly, through the comparison of the results, it is verified that the RSOR method is more effective than the other two methods, especially when considering the aspects of the number of iterations and running time.

The multivisit drone routing problem with edge launches: An iterative approach with discrete and continuous improvements

AbstractIn recent years, the usage of drones in last‐mile logistics has stirred great interest in the operations research community. Many papers have considered schemes of hybrid truck‐and‐drone delivery. In this paper, we focus on the Multivisit Drone Routing Problem with Edge Launches (MVDRP‐EL) which assumes: a heterogenous set of packages, a drone capable of carrying multiple packages at a time and that can be launched and retrieved along an edge, a flexible launch/retrieval site set, and a user‐defined energy depletion function. We believe this paper is the first to exploit edge launch ability through a global continuous approach. In this context, we propose an original formulation based on the Covering Salesman Problem to compute a valid lower bound for the problem and an iterative solution method to determine a MVDRP‐EL solution with quality launch/retrieval sites along the road network edges. Each iteration of the solution method consists of two phases. In the first phase, the road network edges are discretized to obtain launch/retrieval sites, and a first solution is determined. In the second phase, the truck route is set and we reduce the completion time by carefully synchronizing truck and drone routes by solving an original Mixed Integer Second Order Cone Program. The lower bound formulation and the proposed method have been tested on several instances and results indicate the effectiveness of the proposed method and the potential value of launching along an edge, respectively.

Гетероструктура квантово-каскадного детектора частотного диапазона 2.5 ТГц

The design of the heterostructure of a 2.5 THz range quantum-cascade detector is proposed and heterostructure is grown by molecular-beam epitaxy technique. To optimize the thicknesses of the layers of the heterostructure cascades, a numerical method for iterative solution of the Schrodinger ¨ −Poisson equation in the k · p formalism was used. The grown heterostructure of the quantum-cascade detector showed a high structural perfection, confirmed by the small values of the average FWHM of the high-order satellite peaks on the X-ray diffraction rocking curves, which were (8.3 ± 0.5)′′. Analysis of dark-field images obtained by transmission electron microscopy showed that the total thickness of the layers in the cascade is (137.3 ± 6.9) nm, which corresponds to the calculated thickness of the layers in the cascade of the heterostructure of the quantum-cascade detector.

Heterostructure of a 2.5 THz range quantum-cascade detector

The design of the heterostructure of a 2.5 THz range quantum-cascade detector is proposed and heterostructure is grown by molecular-beam epitaxy technique. To optimize the thicknesses of the layers of the heterostructure cascades, a numerical method for iterative solution of the Schrodinger--Poisson equation in the k· p formalism was used. The grown heterostructure of the quantum-cascade detector showed a high structural perfection, confirmed by the small values of the average FWHM of the high-order satellite peaks on the X-ray diffraction rocking curves, which were (8.3±0.5)n. Analysis of dark-field images obtained by transmission electron microscopy showed that the total thickness of the layers in the cascade is (137.3±6.9) nm, which corresponds to the calculated thickness of the layers in the cascade of the heterostructure of the quantum-cascade detector. Keywords: superlattices, quantum-cascade detector, epitaxy, gallium arsenide

Hierarchical stochastic scheduling of multi-community integrated energy systems in uncertain environments via Stackelberg game

An operating entity utilizing community-integrated energy systems with a large number of small-scale distributed energy sources can easily trade with existing distribution markets. To solve the energy management and pricing problem of multi-community integrated energy systems (MCIESs) with multi-energy interaction, this study investigated a hierarchical stochastic optimal scheduling method for uncertain environments. To handle multiple uncertainties, a Wasserstein generative adversarial network with a gradient penalty was used to generate renewable scenarios, and the Kmeans++ clustering algorithm was employed to generate typical scenarios. A Stackelberg-based hierarchical stochastic schedule with an integrated demand response was constructed, where the MCIES operator acted as the leader pursuing the maximum net profit by setting energy prices, while the building users were followers who adjusted their energy consumption plans to minimize their total costs. Finally, a distributed iterative solution method based on a metaheuristic was designed. The effectiveness of the proposed method was verified using practical examples.