System Of Differential-algebraic Equations Research Articles

On the Choice of Methods for Numerically Integrating the Equations of Transients in Electric Power Systems

Algorithms for solving the system of differential-algebraic equations describing electromechanical transients in electric power systems are considered along with matters concerned with ensuring the reliability and required accuracy of the solution results. Classical explicit methods, methods for implicit numerical integration and simultaneous solution of differential-algebraic systems of equations are used to model dynamic processes in power systems. On the basis of the methods used, difference models of electric power system components are obtained. The equations of transients in electric power systems are written in a homogeneous coordinate basis with the use of nodal voltage equations. A comparative analysis of algorithms based on the explicit Runge-Kutta method and implicit methods of Euler, trapezoids, Euler with fitting coefficients is carried out. The analysis results have shown that in terms of stability and accuracy, the trapezoid method has the greatest advantages, which is not inferior to the 4th order Runge-Kutta method and allows calculations with a large integration step to be carried out. The effectiveness of the combined method is shown, in which the Euler method with fitting is used to solve the differential equations describing electromagnetic processes, and the trapezoid method is used to solve the equations of electromechanical motion.

EI-NK: A Robust Exponential Integrator Method With Singularity Removal and Newton–Raphson Iterations for Transient Nonlinear Circuit Simulation

In this article, we propose an exponential-integrator-Newton–Krylov (EI-NK) method for transient simulation of large-scale nonlinear circuits. This method aims to address two long-standing problems that affect the performance and robustness of EI in handling nonlinear circuits. First is the numerical instability caused by the singularity in the differential-algebraic equation system. We provide an in-depth analysis of the problem and propose a systematic, algebraic, and sparsity preserving regularization technique to eliminate the unstable modes in the system to be solved. Next, we develop a scheme to enable the iterative Newton–Raphson solution in the EI framework for enhanced nonlinearity handling capability. With the two techniques, we wish to elevate EI’s robustness and performance and make it a competitive alternative to the existing SPICE-type simulators in practical applications.

Diagnostic Model of Aircraft Turbine Engine Governor Pump

Abstract This paper presents a mathematical model for a hydromechanical fuel governor pump, to be used in parametric diagnostics. The design and operation of the governor are described. The main requirements of the model are formulated, its structure is determined, corresponding to the specifics of the diagnostic task, and assumptions to make the model simpler are presented (single-dimensional flow and absence of heat exchange). The presented model consists of idealized elements with lumped parameters (such as pressure and mass consumption of the working fluid), accounting for the compressibility of the substance and the design arrangement of the governor (presence of mechanical rests, metering orifices of complex shapes, relay switchers, etc.). Equations of elements with lumped parameters, linked by hydraulic channels in one node, are presented. The model – a system of first-order differential-algebraic equations – is solved and the parameters of the governor pump are determined for different steady-state and transient operation modes. We compare our results to the requirements for the corresponding parameters outlined in the Engineering Specifications. The model is matched to the specifications by correcting setting parameters (tightening of elastic springs, areas of throttles, etc.), and a method of initial model linearization is developed. Based on the results, we conclude that our model can be used as a diagnostic algorithm for a governor pump, at the testing and development stages, during manufacturing, repair and maintenance.

Asymptotic solutions of boundary value problem for singularly perturbed system of differential-algebraic equations

This paper deals with the boundary value problem for a singularly perturbed system of differential algebraic equations of the second order. The case of simple roots of the characteristic equation is studied. The sufficient conditions for existence and uniqueness of a solution of the boundary value problem for system of differential algebraic equations are found. Technique of constructing the asymptotic solutions is developed.

Compositional Reservoir Simulation: A Review

Summary Compositional reservoir models have many uses, including petroleum reservoir simulation, the simulation of flow and remediation of nonaqueous petroleum liquids, and the study of various carbon-capture technologies. The present work describes these models in a general way but concentrates on isothermal compositional models based on an equation of state (EOS). These models are used to study petroleum reservoirs containing gas condensate or volatile oil fluids and various gas injection processes like CO2 injection. The technology is documented from the earliest rudimentary models which first appeared more than 50 years ago to more recent work. The system of differential-algebraic equations is described along with the physical nature of the parameters. Model formulations are compared with respect to choices of variables and other computational features. Descriptions of efficient and reliable computational procedures are given. In addition to basic algorithms, we detail important side issues of (1) testing for phase appearance, (2) temporal stability, (3) material balance, and (4) other efforts to speed up the calculations and/or reduce the calculation requirements. Information in the literature is supplemented with some work which has not been presented previously. Correct rendering of the physics and chemistry is emphasized which is often intertwined with the numerics. Although many improvements have been made, the work is not finished. We conclude with a discussion of areas needing additional work.

Mathematical modeling of an unmanned object motion control system in SimInTech environment

In today's world robotic devices are more and more often used to help people. They solve both domestic and industrial problems. When designing any object inevitably have to deal with testing under different conditions. To do this can build a test model, but if the object is quite complex and several models need to be built at once, to save labor and material resources can help mathematical modeling. This article presents mathematical modeling of processes based on typical functional blocks in the form of systems of differential-algebraic equations. Mathematical modeling and control algorithm as a set of interrelated structures are considered. The robot motion is simulated for the forward direction of rotation of wheels, the reverse direction of rotation, and the opposite directions of rotation. A model of the control device, which forms the control actions on the wheel motor according to the value of the deviation of the current orientation of the wheeled robot from the preset one, is constructed. These influences allow you to bring the orientation of the robot to the desired orientation. Quality metrics are obtained for various values of the rotational speed of the work. Although this model neglects the action of many forces that arise during the motion, it allows us to identify the influence on the motion and trajectory of the robot of such factors as the radius of the wheels, the distance between them, the magnitude of the voltage applied to the motors during the turn.

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions

AbstractThis paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.

A Reduced and Linearized High Fidelity Waveboard Multibody Model for Stability Analysis

AbstractIn this paper, the robustness of a recently validated linearization approach is demonstrated with the linear stability analysis of a waveboard, a human-propelled two-wheeled vehicle consisting in two rotatable platforms, joined by a torsion bar and supported on two caster wheels. A multibody model with holonomic and nonholonomic constraints is used to describe the system. The nonlinear equations of motion, which constitute a differential-algebraic system of equations (DAE system), are linearized along the steady forward motion. With this approach, the minimal set of linearized equations of motion of the waveboard multibody model with toroidal wheels is derived. The procedure enables the generation of the Jacobian matrix in terms of the geometric and dynamic parameters of the multibody system, and the eigenvalues of the system are parameterized in terms of the design parameters. The resulting minimum set of linear equations leads to the elimination of null eigenvalues, while retaining all the stability information in spite of the reduction of the Jacobian matrix. The linear stability results of the waveboard obtained in previous work are validated with this approach. The procedure shows an excellent computational efficiency with the waveboard, its utilization being highly advisable to linearize the equations of motion of complex constrained multibody systems.

New approach in solving derivative causality problems in the bond-graph method

From the bond-graph model of a dynamic system, the state equations can be obtained. When the system contains only energy storing elements in integral causality, the system of state equations is immediately and directly determined. When the model contains at least one energy storing element in derivative causality, the resulted system of differential-algebraic equations arises some difficulties in finding the final form of the system of state equations. The work presents a new method of deducing the state equations in case of bond-graph models with one, or several energy storing elements in derivative causality. This method is based on the kinetic energy of the system and offers the possibility to avoid a difficult mathematical calculus for the transition from a system of differential algebraic equations to the system of state equations.

Dynamic modeling of a topside process plant with modified black-oil approach

Modeling and simulation of oil and gas facilities has been done by several authors in closed-source commercial applications, with low flexibility for model adaptation. A few authors used free modeling environments, which results in a differential-algebraic equations (DAE) system and lets the model be adaptable for plantwide simulation. Still, it has limitations in obtaining the model's initial conditions and to perform dynamic simulations when the compositional approach is used. The Modified Black-Oil (MBO) approach, which models the thermodynamic equilibria in a simpler way, has been used by several authors for reservoir modeling but not for topside facilities. This work proposes the use of the MBO approach for topside dynamic modeling. This was done in EMSO©, obtaining a large index-1 DAE system and successful obtention of consistent initial conditions. Disturbances on specific variables were applied and shows the potential use of the MBO approach for new dynamic studies.

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems

Navier-Stokes equations are well known in modeling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the discretization parts, we get a descriptor system of index-2 described by a set of differential algebraic equations (DAEs). The two main parts we develop through this paper are focused firstly on constructing an efficient algorithm based on a projection technique onto an extended block Krylov subspace, that appropriately allows us to construct a reduced system of the original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR) problem based on a Riccati feedback approach. This approach uses numerical solutions of large-scale algebraic Riccati equations. To this end, we use the extended Krylov subspace method that allows us to project the initial large matrix problem onto a low order one that is solved by some direct methods. These numerical solutions are used to obtain a feedback matrix that will be used to stabilize the original system. We conclude by providing some numerical results to confirm the performance of our proposed method compared to other known methods.

FORMATION OF A DESIGN MODEL AND ANALYSIS OF THE STRESS-STRAIN STATE OF STRUCTURAL ELEMENTS UNDER ALTERNATING LOADING

The article gives geometric and physical relationships for shell structure elements. To describe the elastoplastic behavior of the shell element, the theory of variable plasticity is used, which allows one to take into account hardening-softening, cyclic anisotropy, and material damage. Applying the linearization method, after a series of transformations, a canonical system of differential-algebraic equations is obtained that describes the elastoplastic behavior of a composite shell structure. Boundary-value problems are solved using the orthogonal sweep method.

An α-Model Parametrization Algorithm for Optimization with Differential-Algebraic Equations

An optimization task with nonlinear differential-algebraic equations (DAEs) was approached. In special cases in heat and mass transfer engineering, a classical direct shooting approach cannot provide a solution of the DAE system, even in a relatively small range. Moreover, available computational procedures for numerical optimization, as well as differential- algebraic systems solvers are characterized by their limitations, such as the problem scale, for which the algorithms can work efficiently, and requirements for appropriate initial conditions. Therefore, an αDAE model optimization algorithm based on an α-model parametrization approach was designed and implemented. The main steps of the proposed methodology are: (1) task discretization by a multiple-shooting approach, (2) the design of an α-parametrized system of the differential-algebraic model, and (3) the numerical optimization of the α-parametrized system. The computations can be performed by a chosen iterative optimization algorithm, which can cooperate with an outer numerical procedure for solving DAE systems. The implemented algorithm was applied to solve a counter-flow exchanger design task, which was modeled by the highly nonlinear differential-algebraic equations. Finally, the new approach enabled the numerical simulations for the higher values of parameters denoting the rate of changes in the state variables of the system. The new approach can carry out accurate simulation tests for systems operating in a wide range of configurations and created from new materials.

Impact of the diffusion coefficient calculation on predicting Fe2B boride layer thickness

Abstract In this study, a single-phase boride layer thickness Fe2B is predicted on two different substrates (Armco iron and XC38 steel) by following the integral method. This method is a mathematical model based on a system of differential algebraic equations that help to deduce the diffusion coefficient, which is the key factor on predicting the layer thickness. Literatures cover different diffusion coefficients for each substrate, albeit researchers usually extract from experimental data, variations of growth rate constants within only one time treatment and deduce the diffusion coefficient from them. This deduction is done via an estimation of a frequency factor and an activation energy from the growth rate constants. Therefore, our main aim is to illustrate the impact of the deduction of the diffusion coefficient on predicting the boride layer thickness. Lastly, the impact with and without incubation time on the boriding kinetics of both substrates was also examined.

System Analysis of Target-Mediated Drug Disposition (TMDD) Models

In this paper we have performed system analysis of target-mediated drug disposition (TDMM) models and study their stability, steady state (long term) behavior, and complexity. The original one- and two- compartment models are represented by systems of nonlinear differential equations. It has been found that these models are unstable at three different sets of their steady state values, which limits their applicability only to short time intervals. As an extension of the two-compartment TMDD model, the quasi-equilibrium model (QE-TMDD) appear to be very complex since it is represented by the system of differential-algebraic equations. Similar conclusion holds for its variant known as the quasi-steady state QSS-TMDD model. Moreover, it was shown analytically that when the initial value of the drug-receptor complex is zero, both QE-TMDD and QSS-TMDD models produce the zero values for the drug-receptor complex at all times, which appear to be a serious drawback for the applicability of these models.

Computer simulation of boronizing kinetics for a TB2 alloy

Abstract The boron diffusion at the surface of a TB2 alloy was simulated via two mathematical models relying on the numerical resolutions of the system of differential algebraic equations (DAE) for the integral method and ordinary differential equations for the mean diffusion coefficient (MDC) method. Both approaches allowed us to compute the boron diffusion coefficients in TiB2 and TiB for a maximum boron content of 31.10 wt.-% in TiB2 at 1223, 1273, 1323 and 1373 K. The boron activation energies in TiB2 and TiB were evaluated and compared with the data published in the literature. Finally, an experimental validation of both models was made through a comparison of the thicknesses of the experimental layers with the predicted values. Consequently, the simulated thicknesses were in line with the experimental values.

Direct single shooting for dynamic optimization of differential-algebraic equation systems with optimization criteria embedded

Differential-algebraic equation systems with embedded optimization criteria (DAEOs) arise in a number of applications, most prominently dynamic models of separation processes based on phase equilibrium and microbial transformation processes modeled via the dynamic flux balance analysis approach.We consider the optimization of DAEOs. To this end, we first reformulate the DAEO using first-order optimality conditions yielding a nonsmooth DAE system. This solution approach tracks a stationary point of the embedded optimization problem that may not be globally or locally optimal on the total time horizon for non-convex problems.We use adjoint sensitivity analysis for the resulting nonsmooth DAEs for gradient-based optimization in a direct single-shooting approach. We apply the solution method to the optimization of a single flash unit, calculation of an optimal start-up scenario of a rectification column, and the optimization of biomass growth in a microbial transformation process.

The boundary value problems for the linear singularly perturbed systems of differential-algebraic equations

The article considers the main stages of the development of the theory of asymptotic integration of boundary-value problems for linear singularly perturbed differential-algebraic systems. The need of developing constructive methods of approximate integration of boundary-value problems for differential-algebraic systems is due to the importance of their practical application in the theory of nonlinear oscillations, stability of motion, control theory, radio engineering, and biology.
 In the present paper the authors offer a review of literary sources, which consider the methods of constructing asymptotic solutions of singularly perturbed systems of differential equations with a degenerate matrix with derivatives under the condition of stability of the spectrum of the limit pencil of matrices. It is noted that the problem of constructing asymptotic solutions of boundary-value problems for systems of this type is poorly studied, and therefore relevant. In particular, the question of the conditions for the existence and uniqueness of the solutions of these problems and the development of methods for constructing their asymptotics in various cases related to the behavior of the spectrum of the limit pencil of matrices has been poorly researched.

Evaluating the performance of an Inexact Newton method with a preconditioner for dynamic building system simulation

Efficiently, robustly, and accurately solving systems of nonlinear differential algebraic equations for dynamic building system simulation is becoming more important due to the increasing need to simulate large-scale problems. The focus of this paper is an investigation of methods for solving the nonlinear algebraic equations encountered in dynamic building system simulations. Dynamic building system simulations employ a myriad of solution techniques, many of which use derivative information. As the problem size grows, so do the memory requirements, leading to challenges in solving large-scale problems. Newton-Krylov methods are a promising candidate for large-scale simulation. The performance of these methods is often improved, sometimes drastically, when employed with a preconditioning technique. Here, a cost-effective preconditioning technique is applied to a Newton-Krylov method and tested on a series of problems. To illustrate the benefits of the approach, comparison is made to the frequently employed Powell’s Hybrid Method.