Differential-algebraic Equations Research Articles

Multirate and multi-stage holomorphic embedding for dynamic simulation of systems of differential-algebraic equations

In this paper, we present a novel multirate (MR) extension to the multi-stage holomorphic embedding (MSHE) method and use it to dynamically solve systems of differential-algebraic equations. As a test of real-world applicability, we analyze the algorithm for typical power system cases. When applied to the IEEE 34 bus and IEEE 9 bus benchmark system, the multirate approach significantly reduces the number of recalculations of most variables as compared to the existing MSHE algorithm, thus offering the chance of decreasing simulation time while providing simple means of error control. This integration of a multirate ansatz into the well-established MSHE solver potentially opens up systems for dynamic simulation that were up to now out of numerical reach.

Numerical and experimental investigation on the synthesis of extended Kalman filters for cable-driven parallel robots modeled through DAEs

Cable-driven parallel robots are parallel robots where light-weight cables replace rigid bodies to move an end-effector. Their peculiar design allows obtaining large workspaces, high-dynamic handlings, ease of reconfigurability and, in general, low-cost architecture. Knowing the full state variables of a cable robot may be essential to implement advanced control and monitoring strategies and imposes the development of state observers. In this work a general approach to develop nonlinear state observers based on an extended Kalman filter (EKF) is proposed and validated both numerically and experimentally by referring to a cable-suspended parallel robot. The state observer is based on a system model obtained by converting a set of differential algebraic equations into ordinary differential equations through different formulations: the penalty formulation, the Udwadia–Kalaba formulation, and the Udwadia–Kalaba–Phohomsiri formulation, which have been chosen since they can handle the presence of redundant constraints as often happens in cable-driven parallel robots. In the numerical investigation, the EKF is validated simulating encoders heavily affected by quantization errors to demonstrate the filtering capabilities of EKF. In the experimental investigation, a very challenging validation is proposed: only two sensors measuring the rotations of two motors are used to estimate the actual position and velocity of the end-effector. This result cannot be achieved by sole forward kinematics and clearly proves the effectiveness of the proposed observer.

A robust and generalized effective DAE framework encompassing different methods, algorithms, and model order reduction for linear and nonlinear second order dynamical systems

For linear and nonlinear dynamical problems, we propose the novel development and advancement of the well-known Generalized Single Step Single Solve (GS4) family of second order time-accurate algorithms encompassing the entire class of LMS methods developed over the past 50 years or so with/without controllable numerical dissipation in conjunction with the Differential Algebraic Equation (DAE) framework and Proper Orthogonal Decomposition (POD). Unlike traditional practices which have severe limitations in the ability to provide generality and flexibility and a wide variety of choices to the analyst without losing order of time accuracy, the proposed implementation uniquely not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains in a single body, but also additionally allows the selection of different space discretized methods such as the Finite Element Method (FEM) and particle methods, and other spatial methods as well in a single analysis. Moreover, the addition of the POD further provides reduction in computational times. Furthermore, unlike existing state of the art, the present framework readily permits a wide array of implicit–implicit, implicit–explicit, and explicit–explicit couplings and the integration of such a technology is of interest here. Consequently, the present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration schemes in different subdomains in a selective manner together with Reduced Order Modeling (ROM) to optimize capturing the local and global features of the representative physics. The ROM also employs an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to the prescribed accuracy requirements. Such a novelty, and computational features, are not possible with existing state of art and numerical illustrations validate the claim.

Reactive transport model for calcite-rich caprocks in the context of ge-ological carbon storage

Increased concentrations of greenhouse gases in the atmosphere are known to be the primary cause of the increase in global surface temperature and, consequently, climate change [6]. In order to limit global warming, several decarbonisation strategies have been proposed: among them, geological CO2 storage represents very likely the only short- to medium-term option for substantially improving CO2 sinks and reducing net carbon emissions into the atmosphere.
 To achieve secure and lasting storage of CO2 in underground spaces, it is necessary to have both a reservoir and a low-permeability caprock (seal), that maintains its integrity. Deep saline aquifers, depleted oil and gas fields and unminable coal seams are thus the primary targets for the underground storage of supercritical CO2 [2]. The occurrence of natural reservoirs implies that certain lithotypes have a certain sealing capacity, which can prevent leakage of gas to the atmosphere over long geological time periods (i.e. million of years); however, in order to assess the risk of CO2 leakage through caprock above storage sites, the potential caprock alterations due to the contact with CO2 must be considered. In fact, although certain caprocks can be suitable for hydrocarbons, CO2 in contact with the seal may pose additional risks. As for natural hydrocarbon accumulation, shales and evaporites are potential seals also for carbon storage. With particular reference to shales, their mechanical and transport properties are controlled by the behaviour of clay minerals. The sealing efficiency of intact shale caprocks is in fact dominated by the high specific surface clays, which are characterized by high capillary entry pressures, but are also susceptible to electro-chemical interaction. From the engineering perspective, capillary and electrical phenomena also have relevant effects at the Representative Elementary Volume scale, causing mechanical couplings which could affect porosity, clay fabric, hydraulic conductivity, compressibility, and shear strength [3]. Besides these electro-chemical effects, CO2 dissolution and diffusion in water also result in acidification of the in situ brine, causing chemical reactions with some caprock minerals and potentially affecting the mechanical and transport properties [4].
 This work presents a reactive transport model to assess the effects of pore water acidification on caprock materials. The model includes the water mass balance equation for the saturated porous medium and the mass balance equation for all the primary species dissolved in water, according to the theoretical approach presented in [7]. The modelling approach proposed accounts for both the aqueous (homogeneous) reactions of the CO2 dissolved in water (assumed to be in equilibrium) and the dissolution kinetics of calcite in the acidic environment induced by CO2 injection (see [1] for details). Calcite dissolution is finally linked to porosity changes via the reactive surface area of the mineral and the reaction rate. Chemo-hydraulic coupling is addressed by considering porosity changes in the storage term of the balance equations and by introducing a suitable link between hydraulic conductivity and current porosity. The model requires the solution of an initial chemical speciation problem, which has been performed with the software PHREEQ-C, and then the integration of a set of partial differential equations (the mass balance equations of water and of the master primary species dissolved in water) and non-linear algebraic equations (to obtain the concentration of all the chemical species involved), which in this study has been performed via the Finite Element software Comsol Multiphysics ®, according to the approach presented in [8].
 The numerical model has been validated according to the numerical benchmark proposed in [5], developed to reproduce a geo-chemical scenario where mineral dissolution causes permanent alterations in the transport properties of a porous medium. In partic-ular, the benchmark reproduces a flow-through columns experiment, with the porosity and permeability of a calcite-rich interlayer changing under sulfuric acid attack. Figure 1 shows a comparison between the predictions of the current model and the model proposed in [5] in terms of the spatial evolution of porosity, permeability and dissolved calcite, after 400 hours since the injection of sulfuric acid. The model is able to reproduce that, as the acidic front proceeds into the sample from left to right, calcite is progressively dissolved causing porosity changes and the consequent permeability increase. The agreement between the two models is satisfactory, despite some slight differences that may be attributed to the different numerical mesh and the different numerical solver scheme.

RETRACTED: Analysis of governing equations of multibody dynamic systems through kinematic properties of joints and effects of kinematic chains on closed-form mechanism

In this investigation, the fundamental equations dictating the behavior of dynamic systems were ascertained by means of an innovative approach that amalgamated the kinematic properties of joints and the intricate kinematic chains of multibody systems into a series of governing equations. Subsequently, the equations that govern the behavior of multibody systems were converted into ordinary differential equations (ODEs) utilizing the calculus of matrix-valued functions. This computational technique is adept at swiftly acquiring recursive differential equations of motion for multibody systems. Thus, the expenditure of computational costs associated with the simulation was efficaciously minimized. The mechanisms of Andrew's squeezing and carpet scrapping were employed in conjunction with kinematic constraints to authenticate the validity of the proposed methodology. The findings suggest that, notwithstanding the fact that the root mean square error (RMSE) in the generalized coordinates between the two ODE and differential algebraic equation (DAE) techniques is small in the Andrew's squeezing mechanism (at approximately 0.02) and in the carpet scraping mechanism (at approximately 0.7), there are still discrepancies between the two methods. The degree of convergence in the aforementioned methodologies, despite the slight RMSE difference between them, shows a clear disparity. By way of illustration, when employing the conventional DAE technique, the duration required to resolve the Andrew's squeezing and carpet scraping mechanisms is 5 and 11.34 s, respectively. However, by utilizing the ODE methodology in the aforementioned mechanisms, the time required to determine a solution is truncated to 1.19 and 2.1 s, sequentially.

Impasse Surface of Differential–Algebraic Power System Models: An Interpretation Based on Admittance Matrices

The impasse surface is an important concept in the differential-algebraic equation (DAE) model of power systems, which is associated with short-term voltage collapse. This paper establishes a necessary condition for a system trajectory hitting the impasse surface. The condition is in terms of admittance matrices regarding the power network, generators and loads, which specifies the pattern of interaction between those system components that can induce voltage collapse. It applies to generic DAE models featuring high-order synchronous generators, static loads, induction motor loads and lossy power networks. We also identify a class of static load parameters that prevent power systems from hitting the impasse surface; this proves a conjecture made by Hiskens that has been unsolved for decades. Moreover, the obtained results lead to an early indicator of voltage collapse and a novel viewpoint that inductive compensation to the power network has a positive effect on preventing short-term voltage collapse, which are verified via numerical simulations.

Discrete element models for elasto-plastic wave propagation in nonlinear architected materials

Nonlinear architected materials are known to exhibit a plethora of dynamic phenomena that enhance our capacity to manipulate elastic waves. Since these properties stem from complex, subwavelength geometry, dynamic simulations at high resolutions are often intractable at scales of interest. Therefore, prior studies have turned to effective medium models that capture essential properties in the long-wavelength limit. One class of such models is a periodic structure composed of discrete mass, spring, and damper elements, whose constitutive relationships are computed to match behaviors observed in experiments or fine-scale simulations of representative volume elements. While models of this type have been implemented successfully in many cases, the majority of literature has been focused on recoverable deformation, possibly including linear damping. However, history-dependent effects, such as friction and plasticity, have been much less explored. In this work, we develop a discrete element modeling framework for nonlinear architected materials undergoing plastic deformation. Due to the history- and rate-dependent characteristics of plasticity, the framework generally yields a system of differential-algebraic equations whose computational cost is significantly greater than an elastic system of comparable size. We demonstrate the method using several examples from the literature and explore means to obtain phenomenological plasticity models for general geometries.

Overcoming the modeling bottleneck: A methodology for dynamic gray-box modeling with optimized training data

For model-based control and real-time optimization, the existence of sufficiently reliable process models is a prerequisite. The development of such models often is the bottleneck in developing advanced solutions and particularly challenging in cases where the underlying physico-chemical phenomena are not fully understood. When process data is available, machine learning (ML) approaches can be used to build data-based models but the validity of such models is limited to the conditions for which sufficiently rich training data was available. To combine the flexibility of ML-models and the reliability of purely mechanistic models, gray-box models can be applied. In this work we consider gray-box models in which ML-submodels are embedded into differential–algebraic equations that have been derived from first principles. To identify the ML-submodels is challenging as no explicit input/output data for the submodels is available. Therefore, suitable training data must first be estimated from the available data. We propose a systematic methodology for developing gray-box models in a sequence of steps, employing specific regularizations and numerical techniques. We apply the methodology to two case studies: a distillation column with a reaction occurring in the rectifying section, and a fermentation process of a sporulating bacterium. The value of the fermentation model is demonstrated by its use in dynamic process optimization.

Numerical Simulation of Chemical Propulsion Systems: Survey and Fundamental Mathematical Modeling Approach

This study deals with the mathematical modeling and numerical simulation of chemical propulsion systems (CPSs). For this, we investigate and summarize a comprehensive collection of the simulation modeling developments of CPSs in academic works, applications, and industrial fields. Then, we organize and analyze the simulation modeling approaches in several ways. After that, we organize differential-algebraic Equations (DAEs) for fundamental mathematical modeling consisting of the governing Equations (ordinary differential equations, ODEs) for the components and other equations derived from several physical rules or characteristics (algebraic equations or phenomenological equations, AEs) and then synthesize and summarize the fundamental structures of analytic mathematical modeling by types (liquid-propellant rocket engines, solid-propellant rocket motors, and hybrid-propellant rocket motors) of CPSs.

Qualitative Analysis of Nonregular Differential-Algebraic Equations and the Dynamics of Gas Networks

Qualitative Analysis of Nonregular Differential-Algebraic Equations and the Dynamics of Gas Networks

Discrete nonlinear elastodynamics in a port‐Hamiltonian framework

AbstractWe provide a fully nonlinear port‐Hamiltonian formulation for discrete elastodynamical systems as well as a structure‐preserving time discretization. The governing equations are obtained in a variational manner and represent index‐1 differential algebraic equations. Performing an index reduction, one obtains the port‐Hamiltonian state space model, which features the nonlinear strains as an independent state next to position and velocity. Moreover, hyperelastic material behavior is captured in terms of a nonlinear stored energy function. The model exhibits passivity and losslessness and has an underlying symmetry yielding the conservation of angular momentum. We perform temporal discretization using the midpoint discrete gradient, such that the beneficial properties are inherited by the developed time stepping scheme in a discrete sense. The numerical results obtained in a representative example are demonstrated to validate the findings.

Modeling the thermodynamic behavior of cryo-compressed hydrogen tanks for trucks

The novel set of differential-algebraic equations models the thermodynamic behavior of a cryo-compressed hydrogen on board tank system. The internal energy U is used in the energy balance to be independent of the hydrogen phases in all relevant tank scenarios: refueling, discharge, and dormancy. The occurring phase changes in the tank during refueling from gaseous hydrogen through the two-phase region to fully liquid and finally, supercritical hydrogen can be described rigorously without changing the set of the model equations. The comparison of car and truck tanks shows that smaller car tanks have an at least 40% lower loss-free dormancy time than high volume heavy duty truck tanks with the same insulation. A rapid cool-down of the tank from ambient temperature without loss of hydrogen is shown during regular truck operation. For the cyclic-periodic driving and refueling cycle, an optimal stop density ϱstopopt is found where the refueling shall start such that a maximized usable density Δϱuse of 72.7 g/L is reached.

Electromagnetic Transient (EMT) Simulation Algorithms for Evaluation of Large-Scale Extreme Fast Charging Systems (T& D Models)

Simulation of high-fidelity models of extreme fast charging (XFC) systems and large-area power grids with many XFCs can be time consuming in traditional simulators. Traditional simulators use a single method of discretization for all the components that results in imposing a large computational burden of inverting a large matrix as well as increased computations related to single method of discretization (that is typically a trapezoidal method). To overcome the problem of simulating large-area power grids with many XFCs, in this paper, advanced numerical simulation algorithms are applied for the first time together to reduce the dimension of matrix inversion. The algorithms include numerical stiffness-based segregation, time constant-based segregation, clustering and aggregation on differential algebraic equations (DAEs), and multi-order integration approaches. These algorithms apply multiple discretization algorithms rather than a single discretization algorithm that further reduces the computational burden. The approaches mentioned here have resulted in speed-up of up to 18x in the simulation of a single distribution system with 15 XFCs and of up to 271x in the simulation of a transmission-distribution system with 300 XFCs in multiple distribution feeders with respect to conventional simulators (like power systems computer aided design [PSCAD]).

Gas-lifted systems: A hands-on tutorial on modelling and analysis

Gas lift is deplored when the natural lift energy for matured well provides flow rate below the required value. Casing-heading instability removal, fault-tolerant control (FTC) and general operation of the system require the system model and its analysis. In this article, we present a simple approach to gas lift modelling from a tutorial perspective. We derive these gas-lifted system models comparing the models with existing models in the process. We also derive the pressure profile in the annulus and tubing. A simplified approach to the linearisation of the gas-lifted system that is a differential algebraic equation (DAE) system is also presented. The linearised result is used in simulation and analysis of the gas-lifted system where the phase portrait and the stability of the linearised system were considered. Also considered were behaviour of the linearised system and the nonlinear counterpart. The difference of 4kg, 4kg and 30kg corresponding to percentage difference of 0.16%, 0.29% and 2.38% respectively between the linearised and the nonlinear states observed implies convergence. The key objective of this work is to present a simplified approach to the modelling and analysis of the gas-lifted system.

Non-Iterative Calculation of Quasi-Dynamic Energy Flow in the Heat and Electricity Integrated Energy Systems

Quasi-dynamic energy flow calculation is an indispensable tool for the heat and electricity integrated energy system (HE-IES) analysis. One solves the nonlinear partial differential algebraic equations to obtain thermal, hydraulic and electric variations. However, mainstream iteration solvers face the challenges of inefficiency and bad robustness. For one thing, the frequent update and factorization of Jacobian matrices utilize high CPU time. For another, the per-step iteration numbers grow exponentially as the system loading level creeps up. This paper presents a novel non-iterative algorithm for the quasi-dynamic energy flow calculation. The kernel of the proposed algorithm is to transform these nonlinear equations into linear recursive ones, by solving which, we obtain explicit closed-form solutions of unknown variables. In each step, the proposed algorithm requires only one matrix factorization and fixed times of arithmetic operations regardless of the loading levels, so that it achieves small and consistent per-step time costs. A semi-discrete scheme is used in PDE solution to avoid dissipative and dispersive errors that are often overlooked in previous literature. To ensure convergence, we also propose to control the temporal step sizes adaptively by estimating the simulation errors. Case studies showed that the proposed method manifested efficient and robust time performance compared with the iterative algorithms, and meanwhile preserved high accuracy.

Motor Current Model for Detecting Localized Defects in Planet Rolling Bearings

A motor current model for the diagnosis of localized defects in planet rolling bearings (PRBs) is proposed in this study. A dynamic analysis of the motor-driven planetary gearbox system is conducted, and a translational-torsional vibration model of the system is developed. Both the additional time-varying impact caused by localized defects and the influence of the Hertzian contact and radial clearance on the translational support forces of PRBs are considered in the model. Based on the magnetic equivalent circuit network model, vibrational-electromagnetic coupling analysis is performed with the aid of the air-gap permeance and electromagnetic torque. By iteratively solving the nonlinear algebraic differential equations describing the dynamic characteristics of the coupling system, the motor current variation curve with localized defects in the PRB is obtained. Taking a typical motor-driven planetary gearbox system as the object, electromagnetic finite element calculations and dynamic response tests are performed to verify the proposed motor current model. The results show that the characteristic motor current frequencies are a combination of the fault passing frequency and power supply frequency when the inner race of the PRB has a localized spall fault. When the outer race (or roller) has a localized spall fault, the combination of current characteristic frequencies, which generally consists of the fault passing frequency, power supply frequency, and PRB outer-race rotational frequency (outer-race spall) or cage revolution frequency (roller spall), become complex. By discussing the effect of the spall width on the motor current spectra in detail, the characteristic frequencies sensitive to defects are extracted, providing a theoretical basis for the quantitative diagnosis of PRB-localized defects.

The Use of Model-Predicted Internal States from Locally Linear, Gain-Scheduled, Reduced-Order Physics-Based Models in Model-Predictive Control Strategies

The objective of the research is to develop reduced-order state-space models that can predict internal states that are not directly observable with existing sensors. In doing so, battery management systems (BMS) can make real-time control decisions based upon sensor inferences. For example, estimating local electrode Li and electrolyte Li-ion concentrations or local electrostatic potential differences between electrode and electrolyte phases offer valuable knowledge in predicting behaviors such as Li plating, dendrite growth, or electrode fracture. Such indirect predictive capabilities, for example, can enable fast-charging protocols that avoid damage mechanisms.This talk explores the mathematical feasibility of estimating local Li concentrations and electrostatic potentials using observed current-voltage sequences. The approach for determining internal battery states begins with an estimator that uses observed input-output data from a physics-based reduced-order state-space model. If the model satisfies the property of observability, a sufficiently long window of input-output data can predict unique internal-state trajectories that are compatible with the observable data. The research considers understanding the effects of sampling time and state of charge on internal-state observability.Efficient estimation algorithms must incorporate battery physics and chemistry yet be sufficiently simple as to run in real-time on a battery management system (BMS). The present research first develops a physics-based pseudo-two-dimensional (P2D) model, which describes battery thermodynamics, transport, and kinetics, results in a system of nonlinear, coupled, differential-algebraic equations that can be solved computationally, but too slow for real-time implementation. The P2D model is then reduced to a gain-scheduled, locally linear, state-space model that can be run in real-time. Both models are implemented within the MATLAB/Simulink framework that facilitates development and evaluation of control strategies.

Multiphase Porous Electrode Theory for the Next Generation of SOFC/SOEC Electrodes

The mixed-ionic electronic conduction (MIEC) of gadolinium doped ceria (CGO) under reduced oxygen conditions makes it an excellent fuel electrode material for SOFC/SOEC applications. As part of a composite electrode (Ni/CGO), the nickel phase offers a fast electronic conduction pathway to the current collector and may act as an electrocatalyst at the three-phase boundary. Although the Ni/CGO cermet provides superior electrochemical performance compared with older technologies such as Ni/YSZ, the models used to study MIEC materials do not capture the unique transport and kinetic physics which makes them an excellent choice for the next generation of fuel electrodes. Within the framework of multiphase porous electrode theory, this work provides a novel set of differential-algebraic equations which captures the effects of the activation overpotential on electronic defect concentration, electrostatic surface potential and ionic transport to accurately predict the current-voltage behaviour of the Ni/CGO electrode. Moreover, through concerted electron and proton tunnelling events, we unify the theory of the electrostatic surface potential with proton-coupled electron transfer kinetics.

Delay Differential-Algebraic Equations (DDAEs)

Delay differential-algebraic equations (DDAEs) are an important class of mathematical models that broaden standard differential-algebraic equations (DAEs) to incorporate discrete time delays. The time lag terms pose significant analytical and computational challenges. This paper provides a comprehensive overview of current and emerging methods for solving DDAEs and systems of DDAEs. Generalized Taylor series techniques, linear multistep methods, and reduction to ordinary differential equations are examined for numerically integrating DDAEs. Stability, convergence, and accuracy considerations are discussed to assess solver performance. Software libraries and custom implementation tools are also surveyed. Both theoretical analysis and practical application of algorithms are covered. Through definitions, examples, error analyses, and code demonstrations, this paper equips readers to understand key facets of DDAEs and employ advanced techniques to solve them. The topics presented here represent important progress toward addressing real-world systems across science and engineering that fundamentally include time delays.

Impulse-controllability of system classes of switched differential algebraic equations

In this paper, impulse-controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. For a system class generated by switching signals with a fixed-mode sequence, it is shown that either almost all systems are impulse-controllable, or almost all systems are impulse-uncontrollable. Sufficient conditions for all systems in this system class to be impulse-controllable or impulse-uncontrollable are presented. Furthermore, it is observed that even if all systems in a system class are impulse-controllable, knowledge of all the switching times is generally necessary to construct an input that ensures impulse-free solutions. Consequently, it is impossible to design a controller in real time if the switching times in the future are unknown. This phenomenon can be regarded as a causality issue. Therefore, the concept of (quasi-) causal impulse-controllability is introduced, and system classes which are (quasi-) causal are characterized. Finally, necessary and sufficient conditions for a system class to be causal given some dwell-time are stated.