System Of Differential-algebraic Equations Research Articles

Solving Index-1 Semi Explicit System of Differential Algebraic Equations by Mix-Multistep Method

Solving Index-1 Semi Explicit System of Differential Algebraic Equations by Mix-Multistep Method

Frequency domain integrals for stability preservation in Galerkin-type projection-based model order reduction

We investigate linear dynamical systems consisting of ordinary differential equations with high dimensionality. Model order reduction yields alternative systems of much lower dimensions. However, a reduced system may be unstable, although the original system is asymptotically stable. We consider projection-based model order reduction of Galerkin-type. A transformation of the original system ensures that any reduced system is asymptotically stable. This transformation requires the solution of a high-dimensional Lyapunov inequality. We solve this problem using a specific Lyapunov equation. Its solution can be represented as a matrix-valued integral in the frequency domain.Consequently, quadrature rules yield numerical approximations, where large sparse linear systems of algebraic equations have to be solved. Furthermore, we extend this technique to systems of differential-algebraic equations with strictly proper transfer functions by a regularisation. Finally, results of numerical computations are presented for high-dimensional examples.

IonMonger: a free and fast planar perovskite solar cell simulator with coupled ion vacancy and charge carrier dynamics

Details of an open-source planar perovskite solar cell simulator, which includes ion vacancy migration within the perovskite layer coupled to charge carrier transport throughout the perovskite and adjoining transport layers in one dimension, are presented. The model equations are discretised in space using a finite element scheme, and temporal integration of the resulting system of differential algebraic equations is carried out in MATLAB. The user is free to modify device parameters, as well as the incident illumination and applied voltage. Time-varying voltage and/or illumination protocols can be specified, e.g. to simulate current–voltage sweeps, or to track the open-circuit conditions as the illumination is varied. Typical simulations, e.g. current–voltage sweeps, only require computation times of seconds to minutes on a modern personal computer. An example set of hysteretic current–voltage curves is presented.

A new pot still distillation model approach with parameter estimation by multi-objective optimization

In this work a new approach for pot still distillation modeling is presented, which is based on a general boiler heating and a head cooling functions instead of liquid drain off. The resulting Differential Algebraic Equations System formed by energy and mass balance and equilibrium equations are simplified and split in two separated and independent systems to enable the application of standard efficient numerical methods. Thermodynamic properties of the fluids are calculated by a γ−ϕ model associated with a classic cubic equation of state. In order to validate the process and thermodynamic models, an example of an inverse problem application is also presented. A few experimental data of three independent distillations in a lab-scale pot still are used to estimate process and equipment parameters by Multi-Objective Optimization. After that, process simulation results from the three batches are compared with analytical ones with good agreement.

A Novel Equivalent Model of Active Distribution Networks Based on LSTM.

Dynamic behaviors of distribution networks are of great importance for the power system analysis. Nowadays, due to the integration of the renewable energy generation, energy storage, plug-in electric vehicles, and distribution networks turn from passive systems to active ones. Hence, the dynamic behaviors of active distribution networks (ADNs) are much more complex than the traditional ones. The research interests how to establish an accurate model of ADNs in modern power systems are drawing a great deal of attention. In this paper, motivated by the similarities between power system differential algebraic equations and the forward calculation flows of recurrent neural networks (RNNs), a long short-term memory (LSTM) RNN-based equivalent model is proposed to accurately represent the ADNs. First, the adoption reasons of the proposed LSTM RNN-based equivalent model are explained, and its advantages are analyzed from the mathematical point of view. Then, the accuracy and generalization performance of the proposed model is evaluated using the IEEE 39-Bus New England system integrated with ADNs in the study cases. It reveals that the proposed LSTM RNN-based equivalent model has a generalization capability to capture the dynamic behaviors of ADNs with high accuracy.

Solving hyperbolic conservation laws with active counteraction against numerical errors: Isothermal fixed-bed adsorption

First-order partial differential equations are frequently applied for the simulation of adsorption and reaction processes. The numerous numerical methods available are typically applied without any further modifications despite of various well-known errors caused by, for example, numerical dissipation and crude approximations of certain phenomena. In this work, we analyzed a classical mixing cell model that is capable to simulate isothermal liquid chromatographic separation processes with incompressible mobile phases. This model corresponds to a 1D model of a chromatographic column discretized with a first-order finite volume method. It is our aim to counteract actively two errors, namely numerical dispersion (i.e. the second-order spatial derivative in truncation error) and partition inconsistency related to the nonlinear partition quantified by a competitive adsorption isotherm model. The new numerical method introduced in this article maintains characteristics of the first-order base scheme (non-oscillatory and conditionally stable) and offers enhanced accuracy (smaller numerical errors and improved description of shock waves). Furthermore, the method does not require solving the system of differential-algebraic equations that cause large matrix computations, but rather solves nonlinear equations ‘cell-by-cell’. The numerical routines can be easily parallelized to accelerate computation time with multi-core CPUs. To test the method developed, four-zone simulated moving bed adsorption as a challenging example was considered, which causes dynamically changing complex concentration profiles by periodic operation using several columns.

Multiscale dynamic modeling and simulation of a biorefinery.

A biorefinery comprises a variety of process steps to synthesize products from sustainable natural resources. Dynamic plant‐wide simulation enhances the process understanding, leads to improved cost efficiency and enables model‐based operation and control. It is thereby important for an increased competitiveness to conventional processes. To this end, we developed a Modelica library with replaceable building blocks that allow dynamic modeling of an entire biorefinery. For the microbial conversion step, we built on the dynamic flux balance analysis (DFBA) approach to formulate process models for the simulation of cellular metabolism under changing environmental conditions. The resulting system of differential‐algebraic equations with embedded optimization criteria (DAEO) is solved by a tailor‐made toolbox. In summary, our modeling framework comprises three major pillars: A Modelica library of dynamic unit operations, an easy‐to‐use interface to formulate DFBA process models and a DAEO toolbox that allows simulation with standard environments based on the Modelica modeling language. A biorefinery model for dynamic simulation of the OrganoCat pretreatment process and microbial conversion of the resulting feedstock by Corynebacterium glutamicum serves as case study to demonstrate its practical relevance.

Application of an Intelligent Control on Economics Dynamic System: The Attractive Invariant Ellipsoid Approach

In this paper, we explore the application of attractive ellipsoid methodology (Poznyak et al. in Attractive ellipsoids in robust control. Springer, Berlin, 2014) on a new demand and supply model. The balance between the demand and supply is expressed by the Lin and Yang model (Li and Yang in Int J Bifurc Chaos 27(01):1750016, 2017), described by differential equations, even if in the such a demand-supply model, other factors change and have significant effect on demand-supply dynamics. Analysis is compared with the situation when the prices of most products do not stay close to their equilibrium values and the Li and Yang demand and supply model is described by both ordinary differential equation and differential algebraic equation system. To achieve a specific economic goal, we will be able to design a management strategy based on minimum size of the invariant attractive ellipsoid, associated with the dynamic system, with a good performance in the rejection of external disturbances. We can consider a new transformed problem instead of the original problem with respect to solvability and related questions. Theoretical results are illustrated by an example.

A METHODOLOGY TO OBTAIN ANALYTICAL MODELS THAT REDUCE THE COMPUTATIONAL COMPLEXITY FACED IN REAL TIME IMPLEMENTATION OF NMPC CONTROLLERS

Abstract Model Predictive Control, MPC and NMPC, and real-time optimization, RTO and D-RTO, are known to help plant operability through the mitigation of impacts caused by external disturbances. However, the usage of these tools in industry requires overcoming some challenges, for instance: accurate models of the process, particularly in regard to nonlinearities; suitable computational time for obtaining the solution of large-scale problems and model mismatch between the RTO or D-RTO and NMPC. In this paper, we present a methodology to obtain analytical model predictions based on a Hammerstein structure to represent the process nonlinearities, reducing the computational effort in real-time applications. Unlike most common approaches that transform NMPC internal models, described by differential-algebraic equations (DAE), into an approximate system of nonlinear algebraic (NLA) equations using, for instance, orthogonal collocation, in the proposed approach, the obtained NLA is an exact description of the original DAEs system. The proposed algorithm was applied to a non-isothermal CSTR (continuous stirred tank reactor) integrated with an optimization layer. The results show that the proposed structure presented a significant reduction in computational time without performance loss, when compared with the NMPC using a rigorous model. Moreover, the proposed strategy demonstrated good performance in tracking the targets sent by the optimization layer, without model mismatches between layers.

A non-equilibrium approach to model flash dynamics with interface transport

In this paper, we present a modeling framework for a class of multiphase chemical systems based on non-equilibrium thermodynamics. Compartmental modeling is used to establish the dynamic properties of liquid–vapor systems operating far from thermodynamic equilibrium. In addition to the bulk-phase molar/energetic dynamics, interface transport processes yield to algebraic constraints in the model description. The irreversible system is thus written as a system of differential-algebraic equations (DAEs). The non-equilibrium liquid–vapor DAE system is shown to be of index one. A local stability analysis for the model shows that the equilibrium state is unstable for non-isobaric operation regimes, whereas numerical evidence shows that isobaric operation regimes are stable. To extend the stability analysis, internal entropy production for the irreversible flash-drum is presented as a Lyapunov function candidate.

Theoretical and Experimental Investigation of Thermal Dynamics of Steinhart–Hart Negative Temperature Coefficient Thermistors

The temperature-dependent dynamics of a negative temperature coefficient (NTC) thermistor conducting variable electric current is modeled using the differential approach. The thermistor is assumed to follow the Steinhart–Hart resistance-temperature equation. The developed mathematical model consists of a nonlinear differential-algebraic equations system, and it was analyzed by the Adomian decomposition method (ADM) and its time-marching version known as the multistage Adomian decomposition method (MADM) as well as the Dormand–Prince (DP) numerical method. Five sets of experiments were conducted on five different NTC thermistors and the laboratory measurements were compared with the model predictions. It is demonstrated that the proposed model, when combined with the MADM, can accurately simulate the thermal behavior of the NTC thermistors. The MADM reproduces the experimental temperature dynamics of the five NTC thermistors with an average absolute relative error of about 2.601% while the corresponding errors for the DP method and the classic ADM are 8.122% and 51.255%, respectively. Also, it is shown that the MADM is highly efficient in terms of computational efficiency and it is approximately 6.5 times faster than the classic DP method, when tuned appropriately.

Solving Power System Differential Algebraic Equations Using Differential Transformation

This paper proposes a novel non-iterative method to solve power system differential algebraic equations (DAEs) using the differential transformation, a mathematical tool that can obtain power series coefficients by transformation rules instead of calculating high order derivatives and has proved to be effective in solving state variables of nonlinear differential equations in our previous study. This paper further solves non-state variables, e.g. current injections and bus voltages, directly with a realistic DAE model of power grids. These non-state variables, nonlinearly coupled in network equations, are conventionally solved by numerical methods with time-consuming iterations, but their differential transformations are proved to satisfy formally linear equations in this paper. Thus, a non-iterative algorithm is designed to analytically solve all variables of a power system DAE model with ZIP loads. From test results on a Polish 2383-bus system, the proposed method demonstrates fast and reliable time performance compared to traditional numerical methods.

Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest

We examine nonlinear dynamical systems of ordinary differential equations or differential algebraic equations. In an uncertainty quantification, physical parameters are replaced by random variables. The state or inner variables as well as a quantity of interest are expanded into series with orthogonal basis functions like the polynomial chaos expansions, for example. On the one hand, the stochastic Galerkin method yields a large coupled dynamical system. On the other hand, a stochastic collocation method, which uses a quadrature rule or a sampling scheme, can be written in the form of a large weakly coupled dynamical system. We apply projection-based methods of nonlinear model order reduction to the large systems. A reduced-order model implies a low-dimensional representation of the quantity of interest. We focus on model order reduction by proper orthogonal decomposition. The error of a best approximation located in a low-dimensional subspace is analysed. We illustrate results of numerical computations for test examples.

On an initial-boundary value problem for a semilinear differential-algebraic system of partial differential equations of index (1,0)

We consider a mixed problem for a certain semilinear differential-algebraic system of partial differential equations of index (1, 0) of the first order in a two-dimensional rectangular definition domain. Using the method of characteristics and the method of successive approximations, we prove the unique existence of the classical solution to the mixed problem in the whole definition domain. We also prove that the solution and its first derivatives are bounded in this domain.

Robust Controllability of Nonstationary Differential-Algebraic Equations with Unstructured Uncertainty

We consider a nonstationary system of differential-algebraic equations, i.e., the first order ordinary differential equations with variable coefficients and identically singular matrix at the derivative of the unknown vector-valued function. We construct the structural form for the perturbed system and obtain sufficient conditions for the robust (complete, differential, R-) controllability of such systems of index 1 and 2.

Stability of Difference Scheme for a Semilinear Differential Algebraic System of Index (k, 0)

We consider a semilinear differential-algebraic system of partial differential equations of index (k, 0). We numerically solve this system by applying the spline-collocation method based on splitting the matrix pencil. The method has high accuracy coinciding with the smallest order of the approximating spline.

Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model; it uses a change of variables but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential-algebraic equations, depending on the problem and lifting transformation. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be precomputed. Thus, a key benefit of the approach is that there is no need for additional approximations of nonlinear terms, which is in contrast with existing nonlinear model reduction methods requiring sparse sampling or hyper-reduction. Application of the lifting and POD model reduction to the FitzHugh–Nagumo benchmark problem and to a tubular reactor model with Arrhenius reaction terms shows that the approach is competitive in terms of reduced model accuracy with state-of-the-art model reduction via POD and discrete empirical interpolation while having the added benefits of opening new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.

Power management control strategy using physics-based battery models in standalone PV-battery hybrid systems

Abstract The rechargeable lithium-ion batteries (LiBs) are becoming a technology of choice for energy storage applications, due to its high energy and power density. In this paper, a power management control strategy is proposed for a standalone PV-Battery Energy Storage (BES) hybrid system. The proposed control algorithm tracks the Maximum Power Point (MPP) of the solar-cells while avoiding overcharging of LiBs under different solar radiation and load conditions. The controller has two regulation modes; (a) MPP Tracking mode, (b) battery state of charge limit mode. The standalone PV-BES hybrid system is represented as a system of differential algebraic equations and enables effective implementation of control algorithms. A physics-based single particle model accounting for internal states of a battery is implemented in the simulation and control algorithm. A mixed-order finite difference method with optimal node spacing and strong stability-preserving time-stepping Runge–Kutta scheme is used to solve the battery model. A case study is provided to demonstrate the effectiveness of the proposed strategy using real world data. The study reveals that the proposed power control strategy is robust and meets multiple objectives of standalone PV-BES hybrid systems such as no overcharging, 0% excess output power production, and ensuring no energy is transferred to the dump load.

Stability of a Difference Scheme for a Quasi-Linear Partial Differential Algebraic System of Equations of Index (k, 0)

A quasi-linear partial differential algebraic system of equations of index $$(k,0)$$ is considered. A spline collocation difference scheme with a split matrix pencil is constructed to numerically solve a system of this kind. The difference scheme has a high accuracy coinciding with the lowest order of the spline approximating the required function. Results of numerical calculations are given.

Existence of solution to some mixed problems for linear differential-algebraic systems of partial differential equations

We consider a linear differential-algebraic system of PDEs with special matrix coefficients. Two cases are investigated. In the first one, the system has a small index, and the matrix at unknown vector-function, while the system written in the canonical form, is arbitrary. In the second case, the system has an arbitrary index, while a matrix at the small term is triangular. In both the cases, using the methods of characteristics and successive approximations, we prove the existence of a unique classical solution of mixed problems for the considered differential-algebraic systems of PDEs.