Differential And Algebraic Equations Research Articles

A Frequency Domain Approach to Predict Power System Transients

The dynamics of power grids are governed by a large number of nonlinear differential and algebraic equations (DAEs). To safely operate the system, operators need to check that the states described by these DAEs stay within prescribed limits after various potential faults. However, current numerical solvers of DAEs are often too slow for real-time system operations. In addition, detailed system parameters are often not exactly known. Machine learning approaches have been proposed to reduce the computational efforts, but existing methods generally suffer from overfitting and failures to predict unstable behaviors. This paper proposes a novel framework to predict power system transients by learning in the frequency domain. The intuition is that although the system behavior is complex in the time domain, there are relatively few dominant modes in the frequency domain. Therefore, we learn to predict by constructing neural networks with Fourier transform and filtering layers. System topology and fault information are encoded by taking a multi-dimensional Fourier transform, allowing us to leverage the fact that the trajectories are sparse both in time and spatial frequencies. We show that the proposed approach does not need detailed system parameters, greatly speeds up prediction computations and is highly accurate for different fault types.

Constrained profile estimation for distributed parameter system in one dimension using orthogonal collocation

Several industrial processes fall under the category of Distributed parameter systems (DPS) which are represented as coupled partial differential equations (PDEs). Obtaining accurate and physically meaningful profile estimates of states despite only a limited number of spatially distributed measurements, is a pre-requisite for adequately controlling such systems. In this work, we propose an online constrained state profile estimation approach for DPS. The approach is based on reduced-dimension model represented by differential and algebraic equations (DAE) which in turn is developed by discretization of the PDEs along the spatial coordinate using the orthogonal collocation (OC) method. A profile prediction error based approach is proposed for systematically fixing the dimension of the OC model to be used. Further, to compensate for the inevitable model plant mismatch (MPM) arising in the process of the dimensionality reduction, additive zero mean Gaussian white noise processes are introduced in both state dynamics as well as measurement model. Statistics of these noises are obtained based on simulation based realizations of approximation errors at collocation points and measurement locations. To incorporate the identified correlation between the resulting state and measurement noises in the estimation method, we propose a constrained DAE-EKF (extended Kalman Filter) extension to deal with such correlations. The estimation approach also involves an appropriately formulated optimization based update step to ensure constraint satisfaction over the entire spatial profile leading to physically meaningful state profile estimates. The constrained profile estimator is also integrated with an online moving window based parameter estimation, which enables tracking of both time varying parameters and state profiles. The efficacy of the proposed approach is demonstrated by conducting stochastic simulation studies using two tubular reactor systems. The results show that the proposed constrained online state estimation approach is able to obtain accurate profile estimates at low computational cost, while ensuring that the profile estimates satisfy the given constraints.

Optimal energy flow in integrated heat and electricity system considering multiple dynamics

Wide promotion of combined heat and power (CHP) units necessitates the combined operation of the power and heating system. However, the dynamics and nonlinearity in integrated heat and electricity systems (IHES) remain an obstacle to efficient and accurate analysis. To handle this issue, this paper constructs an optimal energy flow (OEF) model for the coordinated operation of the IHES considering the multiple dynamics. The dynamic heating system model is formulated as a set of nonlinear partial differential and algebraic equations (PDAE). The dynamic CHP model is formulated as a set of nonlinear differential and algebraic equations (DAEs). Then, the finite difference method (FDM) is adopted to make the dynamics tractable in the OEF. On this basis, a comprehensive OEF model for IHES is proposed. Simulations in two cases verify the effectiveness of the proposed method and highlight the significance of the dynamics in IHES.

Modeling and Simulation Challenges and Solutions in Cooling Systems for Nanoscale Integrated Circuits [Feature

The power density in modern Integrated Circuits (ICs) is tremendous. For example, Multi-Processor Systems-on-Chip (MPSoCs) nowadays undergo temperature swings of 40 degrees in 100 milliseconds or less, with rapidly emerging and vanishing sub-millimeter hot spots. As such, not only a simulation-based cooling assessment is vital, but one has to simulate the on-chip thermal phenomena jointly with the heat dissipation system - historically, a challenge. In recent years, however, the idea of coupling traditional 3D chip simulators with heat dissipation models written in Equation-Based Modeling (EBM) languages has proven to be a game changer. EBM languages allow one to compose a model by assembling components described in terms of Differential and Algebraic Equations (DAE) and have the simulation code generated automatically. In this article, we take a tutorial viewpoint on the matter just sketched, to put the reader in the position of exploiting the above technology. We also present the first nucleus of a model library for cooling systems, that we release as free software for the scientific and engineering community.

Approach to Energy Objects’ Dynamics Modelling Based on Singular Systems’ Elements

Over the last decades there has been substantial progress on the development of theory and numerical methods for singular systems (known also as descriptor systems, semistate systems, differential alge-braic systems, generalized state-space systems, etc.). The need for such methodsarisen primarily from the increased practical interest for a more general system description which takes the intrinsic physical sys-tem model structure into account. Besides that, many physical process-es are most naturally and easily modelled as mixed systems of differen-tial and algebraic equations (DAE). As the title implies the paper de-scribes the singular systems theory application in power systems dy-namics simulation, particularly considered an alternative method for energy systems’ mathematical models formulation based on the singu-lar systems theory elements with some indicative examples illustrating feasibility and efficiency of this approach

Optimal control of a rate-based modelled batch distillation column: Initialization strategy

Batch distillation columns are dynamic processes represented by a system of differential and algebraic equations (DAE). Their optimal control problem can be solved through orthogonal collocation, which is an effective discretization method with accurate approximations to the behavior of state and control variables. Convergence of such technique is greatly improved by providing good initial values and bounds of the variables. Thus, this work proposes a strategy to estimate the initial values for a sieve trays batch distillation column modeled with a rate-based approach. During the initialization procedure, the modeling equations are solved under a steady state assumption by using the GAMS modeling environment. An experimental pilot scale batch distillation column was used as the case-study to evaluate the proposed initialization algorithm. This column was used to separate a mixture of methanol and ethanol. The initialization results for the state profiles show good agreement with the experimental results.

A mechanistic model of direct forsterite carbonation

Mineral carbonation is a promising method to sequester large amounts of carbon dioxide and to produce value-added substitutes for the cement, paper, and plastic industries. In understanding carbonation reaction mechanisms in batch operation, dynamic models play a crucial role, and they allow to evaluate the effects of important process quantities such as temperature, pressure, particle size of solid phases and additives on reaction kinetics. We develop a mechanistic, dynamic forsterite carbonation model that accounts for gas, liquid, and multiple solid phases. In this model, gas-liquid and dissociation equilibria and surface-controlled reactions between solids and liquid phase are based on nonideal thermodynamics. We account for particle size distribution of raw material and product phases by formulating population balances considering nucleation and growth of particles. We model gas and liquid phases each as a homogeneous phase and we use isopotential conditions to describe equilibrium. The resulting high index system of differential and algebraic equations (DAE) is reformulated to obtain a DAE of differential index 1. Model predictions qualitatively match experimental data taken from literature and thus, we can quantify influences of key process conditions by simulation.

Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation

We develop two linear, second order energy stable schemes for solving the governing system of partial differential equations of a hydrodynamic phase field model of binary fluid mixtures. We first apply the Fourier pseudo-spectral approximation to the partial differential equations in space to obtain a semi-discrete, time-dependent, ordinary differential and algebraic equation (DAE) system, which preserves the energy dissipation law at the semi-discrete level. Then, we discretize the DAE system by the Crank-Nicolson (CN) and the second-order backward differentiation/extrapolation (BDF/EP) method in time, respectively, to obtain two fully discrete systems. We show that the CN method preserves the energy dissipation law while the BDF/EP method does not preserve it exactly but respects the energy dissipation property of the hydrodynamic model. The two new fully discrete schemes are linear, unconditional stable, second order accurate in time and high order in space, and uniquely solvable as linear systems. Numerical examples are presented to show the convergence property as well as the efficiency and accuracy of the new schemes in simulating mixing dynamics of binary polymeric solutions.

Computer Simulation of the Ternary Problem – Technical Aspects and Possibilities

Interaction between groundwater contaminants (e.g. radioactive elements), soil and humic matter plays a crucial role in transport prognoses. This type of interaction, denoted as the ternary problem, may speed up propagation of a contaminant in comparison with the case in the absence of humic compounds. It is very difficult (almost impossible) to find a simple description of the ternary problem in the form of chemical equations. This difficulty is caused by the complex nature of humic substances. Rather than with particular chemical equations, the ternary problem is commonly described by schematic expressions which consider groups of species. The usual approach for the problem formulation is grouping of species resulting in a system of differential and algebraic equations (DAE). This article introduces a different approach allowing a semi-automatic formulation of a system of ordinary differential equations (ODE) for the same problem. The proposed method enables to avoid an operator-splitting already during the model formulation process. The approach for the model formulation and its solution has been implemented using two different open-source software packages. The results have been compared with reference results of a traditional solution approach using DAE systems. The implementation in Python verified the possibility of automation of the formulation and solution. A sensitivity analysis has been performed to evaluate the behaviour of the system with respect to parameter variation.

An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities

A quasi-incompressible hydrodynamic phase field model for flows of fluid mixtures of two incompressible viscous fluids of distinct densities and viscosities is derived by using the generalized Onsager principle, which warrants the variational structure, the mass conservation and energy dissipation law. We recast the model in an equivalent form and discretize the equivalent system in space firstly to arrive at a time-dependent ordinary differential and algebraic equation (DAE) system, which preserves the mass conservation and energy dissipation law at the semi-discrete level. Then, we develop a temporal discretization scheme for the DAE system, where the mass conservation and the energy dissipation law are once again preserved at the fully discretized level. We prove that the fully discretized algorithm is unconditionally energy stable. Several numerical examples, including drop dynamics of viscous fluid drops immersed in another viscous fluid matrix and mixing dynamics of binary polymeric solutions, are presented to show the convergence property as well as the accuracy and efficiency of the new scheme.

Removal of phenol from wastewater using spiral-wound reverse osmosis process: Model development based on experiment and simulation

The removal of the ubiquitous phenol and phenolic compounds in industrial wastes is a critical environmental issue due to their harmful threats to wildlife and potential adverse human health effects. The removal of such compounds is therefore of significant importance in water treatment and reuse. In recent years, reverse osmosis (RO) has been successfully utilised in several industrial processes and wastewater treatment including phenol removal. In this paper, a new model based on a spiral-wound RO process is developed for the removal of phenol from wastewater. A simplified mathematical algorithm using an irreversible thermodynamic approach is developed. This results in a set of non-linear Differential and Algebraic Equations (DAEs), which are solved based on a number of optimised model parameters using a combined methodology of parameter estimation and experimental phenol-water data derived from the literature. The effects of several operational parameters on the performance (in terms of removal of phenol) of the process are explored using the model.

Wastewater treatment by spiral wound reverse osmosis: Development and validation of a two dimensional process model

Reverse osmosis (RO) has become a significant method for removing salts and organic compounds from seawater and wastewater in recent decades. Spiral-wound module has been widely used due to a number of special features such as high packing density, premium separation and low operating cost. In this paper, a two-dimensional mathematical model is developed for the transport of dilute aqueous solutions through a spiral-wound RO module and the operational characteristics of the process under steady state conditions are analysed. The model is based on the solution-diffusion model coupled with the concentration polarization mechanism. This model yields a set of Differential and Algebraic Equations (DAEs), which are solved using the gPROMS software. The model is validated using experimental data from the literature for the rejection of dimethylphenol as solute in aqueous solutions. The model is then used to simulate the process under steady state conditions to gain deeper insight of the process.

Multibody Modelling of Wave Energy Converters Using Pseudo-Spectral Methods With Application to a Three-Body Hinge-Barge Device

Multibody wave energy converters are composed of several bodies interconnected by joints. Two different formulations are adopted to describe the dynamics of multibody systems: the differential and algebraic equations (DAEs) formulation, and the ordinary differential equations (ODEs) formulation. While the number of variables required for the description of the dynamics of a multibody system is greater in the DAE formulation than in the ODE formulation, the ODE formulation involves an extra computational effort in order to describe the dynamics of the system with a smaller number of variables. In this paper, pseudo-spectral (PS) methods are applied in order to solve the dynamics of multibody wave energy converters using both DAE and ODE formulations. Apart from providing a solution to the dynamics of multibody systems, pseudo-spectral methods provide an accurate and efficient formulation for the control of multibody wave energy converters. As an application example, this paper focuses on the dynamic modeling of a three-body hinge-barge device, where wave-tank tests are carried out in order to validate the DAE and ODE models against experimental data. Comparison of the ODE and DAE PS methods against a reference model based on the straightforward (Runge-Kutta) integration of the equations of motion shows that pseudo-spectral methods are computationally more stable and require less computational effort for short time steps.

Effects of Power System Models on Angle Stability Margin in Transient Stability Analysis

Transient stability is the study of synchronism when a power system is subjected to a severe disturbance such as a short circuit fault. This study involves determining the rotor angles of the machines in a power system before, during and after a disturbance through dynamic simulations. Various power system instabilities that had occurred in the past could be linked to the use of too simplified model of power system for stability assessments. The accuracy of rotor angle dynamic responses determines how well transient stability of a multi-machine power system can be examined in dynamic security assessment. This accuracy is predetermined by the model of the power system used in dynamic simulations because the rotor angle stability margin can be compromised by using a simplified model for dynamic assessment. In this work, a more detailed model of synchronous machine, two-axis model, was used and compared with classical model which is popularly used for power system transient stability studies. Differential and Algebraic Equations (DAE) model of a multi-machine power system derived using two-axis and classical models of synchronous machines were used; with machine control dynamics (excitation and governor) added for both models. These equations were solved in MATLAB and taking a full advantage of time-domain simulation. IEEE 30-Bus test system was used for the analysis to compare the rotor angle responses for two-axis model with responses obtained for classical model. The stability margin of the system was found compromised with the use of the classical model, while the two-axis model gave a response that is more dependable for dynamic security assessment.

Fully Discretized Energy Stable Schemes for Hydrodynamic Equations Governing Two-Phase Viscous Fluid Flows

We develop systematically a numerical approximation strategy to discretize a hydrodynamic phase field model for a binary fluid mixture of two immiscible viscous fluids, derived using the generalized Onsager principle that warrants not only the variational structure but also the energy dissipation property. We first discretize the governing equations in space to arrive at a semi-discretized, time-dependent ordinary differential and algebraic equation (DAE) system in which a corresponding discrete energy dissipation law is preserved. Then, we discretize the DAE system in time to obtain a fully discretized system using a structure preserving finite difference method like the Crank---Nicolson method, which satisfies a fully discretized energy dissipation law. Alternatively, we solve the first order DAE system using the integration factor method after the algebraic equation is solved firstly. The integration factor method, which treats the linear, spatial derivative terms explicitly. Finally, two numerical examples are presented to compare the efficiency and accuracy of the two proposed methods.

A Partitioning Technique for a Waveform Relaxation Method Using Eigenvectors in the Transient Stability Analysis of Power Systems

It is of paramount importance that power system operators be able to assess transient stability in order to realize a reliable and stable power supply. Transient stability analysis can be formulated as a large-scale system of differential and algebraic equations (DAE). However, as power systems are becoming larger and more complex, it is becoming difficult to solve DAE in a practical amount of time for system operations. Parallel computing based on the waveform relaxation method is an effective solution to achieve faster calculations for transient stability analysis. To enhance the performance of the waveform relaxation method, a proper partitioning of the original problem is essential. Although various partitioning approaches have been used, those approaches might not be effective when analyzing a weakly damped low-frequency oscillation. In particular, in the Japanese 60-Hz power system, this oscillation becomes an important problem. To resolve this issue, in this paper we have developed a new partitioning method that is better suited to analyzing a weakly damped low-frequency oscillation based on eigenvalue analysis. Specifically, effective partitioning can be automatically determined by the proposed index, which can evaluate the validity of the partitioning. The proposed method was tested using the Japanese standards of the IEEJ WEST10 system model and the WEST30 system model.

A new approach to physics-based reduction of power system dynamic models

This paper proposes a new approach to physics-based reduction of large-scale power system dynamic models, described by a set of differential and algebraic equations (DAEs). The proposed technique, based on the balanced realization theory (BRT), replaces the original model by one of reduced order, while preserving the physical meaning of retained state variables. Input and output variables of the model are selected in accordance with the goal of a particular study. State variables removed from the model are those that do not dominantly affect the nature of the phenomenon to be analyzed, as evidenced by the magnitude of the frequency response. The proposed technique provides insight into steps of the mathematical description of reduction for the considered phenomenon. In a given study, the number of state variables that can be eliminated depends on the power system model itself and on the required accuracy. Described approach is applied to two (benchmark and real-world) power systems.

A rigorous computational model for hydrogen production from bio-ethanol to feed a fuel cell stack

A pseudo dynamic rigorous model of a bio-ethanol processor system (BPS) to produce hydrogen for feeding a Proton Exchange Membrane Fuel Cell (PEM-FC) is presented. The main contribution of this work is to give the overall set of differential and algebraic equations (DAE), assumptions and the way to computationally implement it. This model is able for testing the dynamic behavior of this integrated process, obtaining a reduced order linear model and checking any plant-wide control structure design. It is implemented in two programs, HYSYS and MATLAB, properly communicated to coordinate the calculations performed on each one. A part of the process considered with a faster dynamic than the rest of the units of the plant are simulated in HYSYS environment working in steady state mode. Then, auxiliary equipments and the heat exchangers network are in HYSYS which is called by MATLAB every integration interval for doing the simulation of the complete system. On the other side, the PEM-FC and the dynamic models of the plug flow reactors are developed in MATLAB workspace. Hence, strictly speaking this model must be considered as “pseudo” dynamic. The linearized and reduced order model is developed by applying system identification techniques on the rigorous model. Therefore, accounting the main objectives of the process and the most critical disturbances, a preliminary control structure can be well-tested here. Several results are presented in this work analyzing the obtained performances for opened and closed loop modes.

Modelling in the Documentation Level Using Mosaic and Numerical Libraries

In the today’s constellation of globally operating enterprises, the workforce is locally decentralized. Engineering tasks are increasingly divided up under several work groups that are situated at different premises. Information technology is used to coordinate the collaboration and to merge the results. Due to its versatility and adaptability, the cooperation on the Internet becomes increasingly important. One key issue in this development is the placement of applications in the Internet (cloud computing). Another important issue is the centralized access of cooperative results. Such centralized information may be stored in internet-based knowledge databases. With regard to the above developments, it seems to make sense to have a modelling environment working as a web application connected to a model database. Such a modelling environment should be linked to a strong numeric library. Such a library should contain reliable and very performing algorithms for solving ordinary differential equation (ODE) systems and differential and algebraic equation (DAE) systems. It should also be able to automatically recognize the sparsity and the structure of the Jacobians and to automatically select the best set of algorithms to integrate the differential system. By the help of these measures the computational time should be reduced significantly. In this contribution the MOSAIC modelling environment and the BzzMath library are presented and some main characteristics are described.

Observer‐based control of linear complementarity systems

AbstractIn this paper we present observer and output‐based controller design methods for linear complementarity systems (LCS) employing a passivity approach. Due to various inherent properties of LCS, such as the presence of state jumps, mode dynamics described by differential and algebraic equations (DAEs), and regions for certain modes being lower dimensional, various observer and control design schemes that have been proposed for other classes of (hybrid) dynamical systems do not apply to LCS. In particular, we present an observer design method for LCS which is effective even in the presence of state jumps. We show the well‐posedness of the observer, in the sense of existence and uniqueness of solution trajectories for the estimated state, and prove the global exponential stability of the observation error. These two properties guarantee that the estimated state exponentially recovers the state of the system. For the problem of stabilization based on output measurements only, we adopt an observer‐based control approach in which we apply a state feedback law to the estimated state obtained from the observer. We prove that the resulting closed‐loop system is well‐posed and globally exponentially stable. In order to show the well‐posedness of the closed loop, novel well‐posedness results for LCS based on low‐index properties are presented. Copyright © 2010 John Wiley & Sons, Ltd.