Differential-algebraic Equations Research Articles

Grid-Following Inverter-Based Resource: Numerical State–Space Modeling

In the pursuit of a sustainable electric power system, the integration of renewable energy sources and distributed energy resources is gradually replacing traditional power generation. These new resources are integrated into the grid via inverters, which, despite their efficient performance, present dynamic challenges to the power grid when implemented on a large scale. To maintain grid stability and ensure effective regulation during abnormal operations, various modeling techniques are necessary; while the dynamics of inverter-based resources (IBRs) are traditionally modeled by transfer functions, this paper sheds light on differential-algebraic equations (DAEs) modeling and numerical integration methods. The inherent limitations of transfer function modeling stem from its restricted applicability, as it is exclusively suitable for linear and time-invariant systems. In contrast, the nonlinear DAEs of the IBR system can be converted into a state–space form, which offers a versatile framework for modeling, evaluating, and designing a diverse array of systems. In addition to being compatible with time-varying systems and multiple-input multiple-output systems, the state–space technique may incorporate saturation and dead zone characteristics into the dynamic model. Our research focuses on IBR modeling in a grid-following scheme, which is current-controlled and synchronized to the grid by a phase-locked loop (PLL). The presented state–space model consists of the inverter, grid, control, and designed PLL. Beyond the discussion of its application to IBRs, the presented method holds the potential to solve a wide range of DAEs. The proposed model is compared with a benchmarked system.

Modified Extended Lie-Group Method for Hessenberg Differential Algebraic Equations with Index-3

Hessenberg differential algebraic equations (Hessenberg-DAEs) with a high index play a critical role in the modeling of mechanical systems and multibody dynamics. Motivated by the widely used Lie-group differential algebraic equation (LGDAE) method, which handles index-2 systems, we first propose a modified extended Lie-group differential algebraic equation (MELGDAE) method for solving index-3 Hessenberg-DAEs and then provide a theoretical analysis to deepen the foundation of the MELGDAE method. Moreover, the performance of the MELGDAE method is compared with the standard methods RADAU and MEBDF on index-2 and -3 DAE systems, and it is demonstrated that the MELGDAE integrator exhibits a competitive performance in terms of high accuracy and the preservation of algebraic constraints. In particular, all differential variables in index-3 Hessenberg-DAEs achieve second-order convergence using the MELGDAE method, which suggests the potential for extension to Hessenberg-DAEs with an index of 4 or higher.

OpenMKM: An Open-Source C++ Multiscale Modeling Simulator for Homogeneous and Heterogeneous Catalytic Reactions.

Microkinetic modeling is invaluable for coupling "microscale" atomistic data with "macroscale" reactor observables. We introduce an Open-source Microkinetic Modeling (OpenMKM) multiscale mean-field microkinetics modeling toolkit targeting mainly heterogeneous catalytic reactions but applies equally to homogeneous reactions. OpenMKM is a modular, object-oriented, C++ software, built on top of the robust open-source Cantera built mainly for homogeneous reactions. Reaction mechanisms can be input from human-readable files or automatic reaction generators, avoiding tedious work and errors. The governing equations are also built automatically, unlike Matlab and Python manual implementations, providing speed and error-free models. OpenMKM has built-in interfaces with numerical software, SUNDIALS, for solving ordinary differential equations and differential-algebraic equations. Users can choose various ideal reactors and energy balance options, such as isothermal, adiabatic, temperature ramp, and an experimentally measured temperature profile. OpenMKM is tightly integrated with pMuTT for thermochemistry input file generation from density functional theory (DFT), streamlining the workflow from DFT to MKM and eliminating tedious work and human errors. It is also seamlessly integrated with the RenView software for visualizing the reaction pathways and performing the reaction path or flux analysis (RPA). OpenMKM includes local sensitivity analysis (LSA) by solving the augmented system of equations or using the one-at-a-time finite difference (first or second order) method. LSA can identify not only kinetically influential reactions but also species. The software provides two techniques for large reaction mechanisms for which LSA is too expensive to run. One is the Fischer Information Matrix, which is approximate but comes at nearly zero cost. The other is a new method that we term RPA-guided LSA, which is a finite difference-based method but uses RPA to select kinetically relevant reactions instead of exploiting the entire reaction network. Users can quickly set up and conduct microkinetic simulations without writing code. The user inputs are conveniently divided into reactor setup files and thermodynamic and kinetic definition files to set up different reactors. The source code and documentation are openly available at https://github.com/VlachosGroup/openmkm.

Optical axis control system invariant to the character of disturbance

The task of synthesizing an optimal system of automatic control of the optical axis of a camera with guaranteed accuracy, which is invariant to the nature of disturbances, is considered. Disturbances in this context can be different and not limited. The process of creating a mathematical model of the control system and solving the mathematical synthesis problem using the inverse dynamic model approach is shown. The results of the study demonstrate the mechanism of ensuring the quality of the automatic control system during the compensation of disturbances, and also indicate the features of the system structure. A special method for creating the corrective effect necessary to compensate for disturbances is discussed in the context of the approach of the state variable of the automatic control system to the limit of permissible values. An important result is the discovery of a new mathematical structure - an algebraic-differential equation (for a system with one degree of freedom) or a system of such equations (for a system with several degrees of freedom), which corresponds to this structure of the automatic control system.

Interconnected Systems Modelling in Food Industry: General Solution Scheme and Stability Conditions for Linear Time-Invariant Systems

The problem of simulating complex systems, such as production lines, industrial plants, food processing, etc., today represents an opportunity that brings with it the great advantage of limiting design costs. However, nowadays the designer, after defining and implementing the mathematical models of the studied process, may need to rebuild the whole simulation framework because he needs to modify the model of even just one subsystem. It is for this reason that in this paper, a new framework for the use of Individual Subsystem Models (ISM) for the modelling and simulation of interconnected systems has been studied and implemented. Furthermore, the study of the state of the art has revealed the lack of efficient and sufficiently general numerical algorithms, but, at the same time, it is simple to use to solve the algebraic-differential equations deriving from the ISM simulation. The proposed new approach follows the paradigm of co-simulation methods, including graph theory methods, to solve the general ISM simply and efficiently. In this approach, each subsystem is required to have its own representation independently of the other subsystems. In this way, it is always possible to replace any subsystem whenever an updated representation becomes available, making maintenance and evolution of the entire ISM flexible. Our framework calls each subsystem separately in an optimal (suboptimal) order based on the structure of the graph. Each calculated output is transferred to the input of the next subsystem in the chosen. The general procedure has been validated in the context of Linear and Time-Invariant ISMs: in these hypotheses, the stability conditions have been calculated and numerical tests have been performed which show the effectiveness of the proposed approach.

Model Reduction of Linear Differential–Algebraic Equation Systems Using Constructed Spectral Projectors

This contribution introduces linear differential-algebraic equation (DAE) systems and provides the explicit construction steps of spectral projectors for DAEs with index 1 or 2. Further, an interpolatory projection-based model order reduction technique using spectral projectors is applied to an eddy current problem as well as an electrical circuit to compute a reduced order model.

Feasibility Study of Neural ODE and DAE Modules for Power System Dynamic Component Modeling

In the context of high penetration of renewables, the need to build dynamic models of power system components based on accessible measurement data has become urgent. To address this challenge, firstly, a neural ordinary differential equations (ODE) module and a neural differential-algebraic equations (DAE) module are proposed to form a data-driven modeling framework that accurately captures components' dynamic characteristics and flexibly adapts to various interface settings. Secondly, analytical models and data-driven models learned by the neural ODE and DAE modules are integrated together and simulated simultaneously using unified transient stability simulation methods. Finally, the neural ODE and DAE modules are implemented with Python and made public on GitHub. Using the portal measurements, three simple but representative cases of excitation controller modeling, photovoltaic power plant modeling, and equivalent load modeling of a regional power network are carried out in the IEEE-39 system and 2383wp system. Neural dynamic model-integrated simulations are compared with the original model-based ones to verify the feasibility and potentiality of the proposed neural ODE and DAE modules.

Dynamic State Estimation of Nonlinear Differential Algebraic Equation Models of Power Networks

This paper investigates the joint problems of dynamic state estimation of algebraic variables (voltage and phase angle) and generator states (rotor angle and frequency) of nonlinear differential algebraic equation (NDAE) power network models, under uncertainty. Traditionally, these two problems have been decoupled due to complexity of handling NDAE models. In particular, this paper offers the first attempt to solve the aforementioned problem in a coupled approach where the algebraic and generator states estimates are simultaneously computed. The proposed estimation algorithm herein is endowed with the following properties: <i>(i)</i> it is fairly simple to implement and based on well-understood Lyapunov theory; <i>(ii)</i> considers various sources of uncertainty from generator control inputs, loads, renewables, process and measurement noise; <i>(iii)</i> models phasor measurement unit installations at arbitrary buses; and <i>(iv)</i> is computationally less intensive than the decoupled approach in the literature.

Conditions of Solvability of the Problem Inverse to the Cauchy Problem for a Difference-Algebraic Equation

We establish conditions of solvability for the problem inverse to the Cauchy problem for a difference equation. In the case where the problem inverse to the Cauchy problem for the difference equation is not solvable, we propose conditions for the solvability of the problem inverse to the Cauchy problem for a difference-algebraic equation.

Discretization of inherent ODEs and the geometric integration of DAEs with symmetries

Discretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.

Modeling/simulation of transient linear heat conduction problems via integrating a wide variety of space/time methods and choices

We propose an implementation of a general purpose multi-spatial method, multi-time scheme subdomain Differential Algebraic Equations (DAE) framework allowing a mix of different space discretization methods while interfacing altogether different time integration algorithms on a single body analysis for the linear first order transient systems. With the concept of subdomains and DAE framework which allows constraint in time as well as space, we can divide a body into multiple subdomains and implement different spatial methods (finite elements, particles, etc.) in different regions of the body including altogether different time integrators and enable targeting an area with a specific method which can fully utilize its best features. The robust Generalized Single Step Single Solve (GS4) family of Linear Multistep (LMS) framework with second order time accuracy encompasses most of the developments over the last 50 years or so including new and optimal designs, and provides a wide variety of choices available to the analyst in a single analysis setting. We also advance, analyze and present the GS4 parameter studies regarding parameter selection. The GS4 readily allows flexibility in matching different time integration schemes, both existing traditional time schemes and other new and optimal competitive designs while maintaining second order time accuracy. Such implementation of coupling a wide variety of different spatial and time integration algorithms is not possible (especially implicit-implicit algorithm couplings) in the current state of the technology. Existing traditional practices of coupling multiple spatial and time algorithms show limitations in reduced order of accuracy, and consistency and the like in various field variables typical of, and including, temperature, temperature rate, and Lagrange multipliers. Various combinations of spatial methods and time algorithms between subdomains are tested with simple linear first order system problems for readers to readily mimic the results and demonstrate the overall robustness and efficacy for numerical computations.

Canonical Euler splitting method for parabolic partial functional differential algebraic equations

A novel canonical Euler splitting method is presented for semilinear composite stiff parabolic partial functional differential algebraic equations with initial and Dirichlet boundary conditions. The original partial differential problems are transformed into the semi-discrete problems by spatial discretization, and then the canonical Euler splitting method is employed to solve the resulting semi-discrete problems. Under appropriate assumptions, the stability and convergence theories of this method are established. A series of numerical experiments are given to illustrate the effectiveness of this method and the correctness of theoretical results.Numerical results also demonstrate that the constructed method can significantly improve the calculation speed.

A novel dynamic dry friction model for applications in mechanical dynamical systems

This work presents a novel continuously differentiable, dynamic dry friction model, with dependence on normal contact force, slip velocity, and static and dynamic coefficients of friction. The state-of-the-art, Brown and McPhee friction model depends on transitional velocity, which is an empirical parameter. A limitation with this model is that there is no generic approach to select the optimal value for the transitional velocity for a certain application. The simulation results presented in this work highlight the sensitivity of friction force with transitional velocity in Brown and McPhee’s model to obtain smooth solutions, which are supportive in control system applications. It is because control systems require jitter-free signals in order to save costs on low-pass filters. The proposed model overcomes such empirical dependence on transition velocity through a combination of an iterative methodology and an empirical parameterization. In this article, a comprehensive analysis of the proposed friction model and the contributing parameters is carried out. The algorithm to implement the proposed friction model is elaborated. The proposed friction model is then simulated and compared against Brown and McPhee’s model. using a stick-slip benchmark problem. Furthermore, the proposed force model is applied to two spatial multibody systems formulated using index 0 tangent space differential-algebraic equations. The results demonstrate how the proposed friction model can be employed in dynamical mechanical systems. The independence on transitional velocity in the proposed friction model is observed to be effective for obtaining smooth solutions that make it suitable for control system applications.

Numerical Simulation of Nonlinear and Non-Isothermal Liquid Chromatography for Studying Thermal Variations in Columns Packed with Core-Shell Particles

A high-resolution flux-limiting semi-discrete finite volume scheme (HR-FVS) is applied in this study to numerically approximate the nonlinear and non-isothermal flow of one-dimensional lumped kinetic model (1D-LKM), for a fixed-bed column loaded with core-shell particles. The developed model comprise a system of convection-dominated partial differential for mass and energy balances in the mobile phases coupled with differential equation and algebraic equation in the stationary phase. The solution of the model equations is obtained by utilizing a HR-FVS, the scheme has second-order accuracy even on the grid coarse and its explicit nature has the potential to resolve the arisen sharp discontinuities in the solution profiles. A second-order total variation diminishing (TVD) Runge-Kutta technique is used to solve the system of ODEs in time. Several forms of a single-solute mixture are produced to investigate the influences of the fractions of core radius on thermal waves and concentration fronts. Moreover, a particular criterion is introduced for analyzing the performance of the underlying process and to identify the optimal parameter values of the fraction of core radius.

Low-carbon economic dispatch of regional electro-thermal coupled system considering dynamic constraints of CHP units

The flexibility modification of Combined Heat And Power (CHP) units can enable the units to have fast electric output regulation capability, respond to grid auxiliary frequency regulation, and enhance the operational flexibility of regional electric and thermal coupling systems. Compared with the CHP unit constraint model established by the vertex convex combination method, the CHP dynamic constraint in the form of differential equations established by the mechanism modeling can reflect the unit variable states in real time and facilitate the mastering of the unit operation status. With the objective of optimal dispatch cost of the regional electro-thermal coupled system considering the carbon trading process, a low-carbon economic dispatch model of the regional electro-thermal coupled system considering the CHP dynamic constraint is established. Based on the sequential method framework, the finite element orthogonal configuration method is used to discrete the differential algebraic equations of the simulation layer, and the improved adaptive differential evolution algorithm is used to solve the nonlinear planning problem of the optimization layer. The algorithm verifies that the finite element orthogonal configuration method can obtain more accurate results with fewer discrete points, improve the solution speed, and verify that the CHP unit with fast electric output regulation can effectively improve the low-carbon operation of the system, which is also beneficial to the safe and stable operation of the unit.

Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs

In this paper we derive error estimates for Runge–Kutta schemes of optimal control problems subject to index one differential–algebraic equations (DAEs). Usually, Runge–Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge–Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers.

Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package

Rosenbrock–Wanner methods for systems of stiff ordinary differential equations are well known since the seventies. They have been continuously developed and are efficient for differential-algebraic equations of index-1, as well. Their disadvantage that the Jacobian matrix has to be updated in every time step becomes more and more obsolete when automatic differentiation is used. Especially the family of Rodas methods has proven to be a standard in the Julia package DifferentialEquations. However, the fifth-order Rodas5 method undergoes order reduction for certain problem classes. Therefore, the goal of this paper is to compute a new set of coefficients for Rodas5 such that this order reduction is reduced. The procedure is similar to the derivation of the methods Rodas4P and Rodas4P2. In addition, it is possible to provide new dense output formulas for Rodas5 and the new method Rodas5P. Numerical tests show that for higher accuracy requirements Rodas5P always belongs to the best methods within the Rodas family.

A novel Cell-Centered nodal integral method for the Convection-Diffusion equation

The nodal integral methods (NIMs) are coarse mesh methods for solving partial differential equations (PDEs) efficiently and accurately. Here, a new variant of the NIM is proposed for the solution of the two-dimensional convection–diffusion equation. The earlier versions of the NIM for fluid flow problems used surface-averaged variables. In contrast, the current scheme is developed in terms of cell-centered values using different treatments of cell interface conditions. The advantage of this scheme is that coupling with other physics is straightforward. Another distinctive feature of the current approach is that the temporal derivative is explicitly addressed, resulting in a system of differential–algebraic equations (DAEs) of index-1. The system of DAEs is then solved using Backward Difference Formula, which provides higher accuracy in time compared to traditional NIM. A few problems with the known analytical solution are solved to demonstrate the efficacy of the current simplified approach.

Parsimonious physics-informed random projection neural networks for initial value problems of ODEs and index-1 DAEs.

We present a numerical method based on random projections with Gaussian kernels and physics-informed neural networks for the numerical solution of initial value problems (IVPs) of nonlinear stiff ordinary differential equations (ODEs) and index-1 differential algebraic equations (DAEs), which may also arise from spatial discretization of partial differential equations (PDEs). The internal weights are fixed to ones while the unknown weights between the hidden and output layer are computed with Newton's iterations using the Moore-Penrose pseudo-inverse for low to medium scale and sparse QR decomposition with L 2 regularization for medium- to large-scale systems. Building on previous works on random projections, we also prove its approximation accuracy. To deal with stiffness and sharp gradients, we propose an adaptive step-size scheme and address a continuation method for providing good initial guesses for Newton iterations. The "optimal" bounds of the uniform distribution from which the values of the shape parameters of the Gaussian kernels are sampled and the number of basis functions are "parsimoniously" chosen based on bias-variance trade-off decomposition. To assess the performance of the scheme in terms of both numerical approximation accuracy and computational cost, we used eight benchmark problems (three index-1 DAEs problems, and five stiff ODEs problems including the Hindmarsh-Rose neuronal model of chaotic dynamics and the Allen-Cahn phase-field PDE). The efficiency of the scheme was compared against two stiff ODEs/DAEs solvers, namely, ode15s and ode23t solvers of the MATLAB ODE suite as well as against deep learning as implemented in the DeepXDE library for scientific machine learning and physics-informed learning for the solution of the Lotka-Volterra ODEs included in the demos of the library. A software/toolbox in Matlab (that we call RanDiffNet) with demos is also provided.

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero