Differential-algebraic Equations Research Articles

Algebraic-differential equations of a nonlinear pass-through quadripole

A method of forming algebraic-differential equations of a nonlinear pass-through active quadripole, which connect its independent pole currents and independent polar voltages, is proposed. The difficulty of the analysis lies in the fact that some of both internal and external unknowns may be under the symbol of differentiation. The common differential equations of the system of internal and external currents and voltages act as starting information for this formation. The method is demonstrated on two cases of the formation of corresponding algebraic-differential equations of systems as formed by nonlinear two-port elements. The analysis is significantly simplified in the case of internal D-degeneracies of the system or purely resistive circuits.

Control‐oriented models of the shallow water equations using energy‐conserving discretization schemes

AbstractIn this study, we introduce higher‐order approximation schemes for a 1D shallow‐water model with a moving boundary and arbitrary cross‐section. The model equations are formulated using Lagrange coordinates to handle the time‐varying spatial domain. By discretizing the action functional on a material‐fixed grid and applying an appropriate quadrature scheme, we derive a finite‐dimensional model. This model, taking mass conservation into account as an auxiliary condition, results in a system of semi‐explicit differential‐algebraic equations (DAE). Unlike previous work, we employ higher‐order quadrature formulae to enhance numerical accuracy, albeit at the cost of more complex nonlinear DAE. In order to compare the performance of the resulting models obtained from using different quadrature schemes, a comprehensive simulation study is conducted.

Impulsive Stability for Positive Linear Differential-Algebraic Equations With Time-Varying Delay

This paper deals with the impulsive stability problem for positive linear differential-algebraic equations (DAEs) with a time-varying delay. First, a sufficient condition for the positivity of the considered system is addressed. Then, a new exponential stability criterion for positive linear DAEs with a time-varying delay is obtained based on a suitable change of variables. According to our results, a stable impulse-free system can keep its original stability property under certain destabilizing impulsive perturbations. It should be noted that it is the first time the exponential stability results for impulsive positive linear DAEs are given. For this purpose, delay-range dependent sufficient conditions for such systems are stated in the paper. An additional numerical example is provided to demonstrate the results' efficacy.

On scaling robust feedback control and state estimation problems in power networks

Many mainstream robust control/estimation algorithms for power networks are designed using the Lyapunov theory as it provides performance guarantees for linear/nonlinear models of uncertain power networks but comes at the expense of scalability and sensitivity. In particular, Lyapunov-based approaches rely on forming semi-definite programs (SDPs) that are (i) not scalable and (ii) extremely sensitive to the choice of the bounding scalar that ensures the strict feasibility of the linear matrix inequalities (LMIs). This paper addresses these two issues by employing a celebrated non-Lyapunov approach (NLA) from the control theory literature. In lieu of linearized models of power grids, we focus on (the more representative) nonlinear differential algebraic equation (DAE) models and showcase the simplicity, scalability, and parameter-resiliency of NLA. For some power systems, the approach is nearly fifty times faster than solving SDPs via standard solvers with almost no impact on the performance. The case studies also demonstrate that NLA can be applied to more realistic scenarios in which (i) only partial state data is available and (ii) sparsity structures are imposed on the feedback gain. The paper also showcases that virtually no degradation in state estimation quality is experienced when applying NLA.

Physical modeling and simulation of electro-mechanical actuator-based TVC systems for reusable launch vehicles

This paper presents the high-fidelity physical modeling of a Thrust Vector Control (TVC) system operated by Electro-Mechanical Actuators (EMAs) for Reusable Launch Vehicles (RLVs). In contrast to simplified, often linear, models, high-fidelity physical models enable a better assessment of the actual system performance and a deeper verification of the on-board software robustness against disturbances, unmodeled dynamics or degradation. This aspect is essential to enable the reusability concept as driven by long-term reliability requirements. Moreover, critical systems like Fault Detection, Isolation and Recovery (FDIR) logic, whose robustness and efficacy are often difficult to prove, can be thoroughly tested without requiring hardware experiments with eventual invasive modifications for fault injections. This work captures the whole dynamics of the EMA and TVC components, including the power drive electronics, the electrical motor, and the mechanical transmission for the EMA, as well as the mechanical properties of the engine nozzle acting as a load. The resulting differential algebraic equation system is modeled in the Modelica acausal object-oriented modeling language. An ad-hoc framework is implemented to guarantee flexibility and modularity, which allows models with different fidelity levels to be easily exchanged to achieve the simulation objectives and needed accuracy level. The impact of the different models on the closed-loop TVC dynamics is analyzed in both the frequency and time domain, and then benchmarked with a realistic RLV mission. The results show that the high-fidelity physical models provide a better understanding of the more complex effects governing the TVC dynamics and can, in turn, effectively improve its requirement definition process.

On the spectral theory of linear differential-algebraic equations with periodic coefficients

On the spectral theory of linear differential-algebraic equations with periodic coefficients

Analytical tangents for arbitrary material laws derived from rheological models at large deformations

AbstractThe development of suitable material laws for various material classes is an essential preliminary task for conducting realistic simulations. Within the framework of large deformations, one recognized approach is the utilization of rheological connections allowing the construction of arbitrary models. A common method to calculate the stress response of such a material model is to formulate a set of algebraic and ordinary differential equations and to solve them numerically. However, in this work, only stress relations between different rheological elements are formulated and directly solved by a numeric algorithm without the need to derive the typical system of algebraic/differential equations. The required derivatives for the solution of these equations for this algorithm and the stiffness of the material model are calculated analytically following the same general principle as the algorithm calculating the stress response. This improves stability and computation effort compared to a forward difference scheme.

A differential-algebraic extended Kalman filter with exact constraint satisfaction

This work proposes an Augmented Manifold Differential-Algebraic Extended Kalman Filter for state and input estimation for multibody systems described by a set of differential algebraic equations. The proposed Kalman Filter allows for exact constraint satisfaction and does not require reformulation of the multibody system equations. The approach consists of two main steps: (i) elimination of the Lagrange multipliers from the system state via a nullspace projection and (ii) an additional correction step that constrains the a-posteriori estimated system state to lie on the constraint manifold. The latter is achieved by solving a constrained optimization problem using a Euclidean descent approach and projection on the constraint manifold, i.e. manifold optimization. The proposed approach is numerically validated on both open and closed kinematic chain rigid multibody models and is shown to be robust and indifferent to the type of kinematic chain being considered. Furthermore, an experimental validation is carried out where the forces and torques acting on the wheel center of a McPherson car suspension test setup are successfully estimated using a flexible multibody model and the proposed Kalman Filter.

Electrified externally heated rotary calciner for calcination of cement raw meal

The cement industry can reduce its CO2 emissions by electrifying the calciner. It can avoid emissions from fuel combustion and produce pure CO2 from the calcination reaction (CaCO3 → CaO + CO2) for direct capture. A differential-algebraic equation (DAE) model of an electrified rotary calciner was developed and validated against experimental results. The heat transfer coefficient was around 30 W/(m2K), with the calciner inclined at 15°. This value increased to 80 W/(m2K) by reducing the inclination to 2°. The rotary calciner for producing 1 Mton/yr clinker with an internal diameter of 5 m needs a length of 485 m to reach a calcination degree of 94 %. The large system size suggests that this calciner may not be suitable for full-scale production. However, it can still be used for small-scale green production of calcined limestone.

Coordinated Optimization for Multi-Infeed LCC-HVDCs Transient Control Considering Short-Term Voltage Stability of Receiving-End Grid

With local synchronous generators in receiving-end grids being replaced by multi-infeed line-commutated converter-based (LCC) high-voltage direct currents (HVDCs), the dynamic voltage supportability of the power grid decreases. Moreover, the reactive power consumption of LCC-HVDCs during the recovery process brings risks to voltage stability. Hence, the transient control of LCC-HVDCs should be optimized coordinately to support short-term voltage stability under credible contingencies. In this paper, the dynamic reactive power characteristics between AC/DC systems are derived mathematically. Based on the coupling mechanism among the voltage-dependent current order limiter (VDCOL), AC dynamic voltage, and HVDC power recovery, an “autonomous-synergic” optimization strategy incorporating short-term voltage stability constraints is proposed for multi-infeed LCC-HVDCs. To handle the problem of large-scale differential-algebraic equations (DAEs), the optimization is iteratively solved based on time-domain simulation (TDS) and trajectory sensitivities. The stability is quantified by an indicator and the trajectory sensitivity analysis is utilized to relax stability constraints. The effectiveness and feasibility of the proposed methods are verified with a benchmark system and a real system in China. The short-term voltage stability under all credible contingencies can be guaranteed, while the total power recovery of LCC-HVDCs is improved.

Robust Stability of Differential-Algebraic Equations under Parametric Uncertainty

Robust Stability of Differential-Algebraic Equations under Parametric Uncertainty

Distributed model for the drill-string system with multiple regenerative effects in the bit-rock interaction

In this paper, an integrated model is presented to study the axial–torsional dynamics of a distributed drill string. The coupled axial and torsional drill string dynamics typically lead to severe phenomena such as bit-bounce and stick–slip, which are the main causes of failure and efficiency reduction. The top-drive dynamics and the Bottom Hole Assembly (BHA) dynamics are interconnected through the drill pipes. In order to preserve the infinite-dimensional feature of the drill pipes in the mathematical modeling, a distributed model in terms of Delay Differential Algebraic Equations (DDAE) is proposed to represent the dynamics of the drill string. The bit-rock interaction, which consists of the cutting and the frictional components, is responsible for the coupling between the axial and torsional dynamics. A rate-independent bit-rock interaction law is employed for both cutting and frictional components. In order to capture the global bit motion, including multiple regenerative effects, the well-bottom depth function is used to determine the depth of cut. The well-bottom pattern evolution is formulated through algebraic equations rather than Partial Differential Equations (PDEs). A new formulation for the linearized drill string system in terms of Neutral-type Delay Differential Integral Equations (NDDIEs) is proposed to investigate the initiation of the oscillations around the nominal operation. By this formulation, the local stability of the distributed system is studied by determining the right-most eigenvalue of the linearized dynamics. A simulation-based case study reflecting real-life scenarios is presented to investigate the dynamical behavior of the system under different operating conditions.

Personalised parameter estimation of the cardiovascular system: Leveraging data assimilation and sensitivity analysis

Detailed models of dynamical systems used in the life sciences may include hundreds of state variables and many input parameters, often with physical meanings. Therefore, efficient and unique input parameter identification, from experimental data, is an essential but challenging task for this class of models. This study presents a comprehensive analysis of a nine-dimensional single ventricle lumped-parameter model, representing the systemic circulation. This model is formulated in terms of differential algebraic equations, often found in other areas of the life sciences. We introduce a novel computational algorithm designed to incorporate patient-specific beat-to-beat variability into model investigations, utilising the Unscented Kalman Filter (UKF) for efficient parameter estimation. Our findings demonstrate the exceptional adaptability of the UKF to severe parameter perturbations, representing significant physiological changes. Furthermore, we provide novel insights into the continuous sensitivity of model input parameters, illustrating the robustness and efficacy of UKF. The monitoring of a patient’s physiological state, with minimal delay, becomes feasible, by incorporating patient-specific measurements and leveraging the UKF. The workflow presented in this paper enables prompt identification of pathophysiological conditions and will improve patient care.

Application of the Constrained Formulation to the Nonlinear Sloshing Problem Based on the Arbitrary Lagrangian–Eulerian Method

Abstract This study deals with an application of constrained formulation to a nonlinear sloshing problem based on the arbitrary Lagrangian–Eulerian finite element method (ALE). The ALE method incorporates a discretized form of equations of motion with mesh updating algorithms in order to prevent a problem of mesh distortion. This paper focuses on an analytical aspect of such treatments as constrained systems in the formulation of the ALE method. Since the mesh updating algorithms give algebraic relations for nodal coordinates, this study treats these relations as constraints. Then, we introduce formulation for constrained systems based on the method of Lagrange multipliers. As a result of this formulation, equations of motion are given by differential algebraic equations (DAEs) consisting of differential equations for time evolution of physical quantities and algebraic equations (constraints). The present method can be classified into a kind of augmented formulation. Moreover, we present a size-reduction technique used in the Newton–Raphson method in order to remove a part of the redundant degrees-of-freedom in the iterative procedures, because the resulting set of DAEs involves a larger number of unknowns than the minimal number of degrees-of-freedom due to the introduction of the constrained formulation. The derived reduced system still holds the physically essential part of equations of motion for the sloshing problem. Even though it is not reduced to the minimal degrees-of-freedom, it does not involve algebraically complicated and inefficient procedures in computation. In addition, this study presents a method to introduce damping effects defined in the modal space into the FEM models. The proposed approach is validated by comparisons with experimental data in the time domain analysis.

Autoignition Problem in Homogeneous Combustion Systems: GQL versus QSSA Combined with DRG

The global quasi-linearization (GQL) is used as a method to study and to reduce the complexity of mathematical models of mechanisms of chemical kinetics. Similar to standard methodologies, such as the quasi-steady-state assumption (QSSA), the GQL method defines the fast and slow invariant subspaces and uses slow manifolds to gain a reduced representation. It does not require empirical inputs and is based on the eigenvalue and eigenvector decomposition of a linear map approximating the nonlinear vector field of the original system. In the present work, the GQL-based slow/fast decomposition is applied for different combustion systems. The results are compared with the standard QSSA approach. For this, an implicit implementation strategy described by differential algebraic equations (DAEs) systems is suggested and used, which allows for treating both approaches within the same computational framework. Hydrogen–air (with 9 species) and ethanol–air (with 57 species) combustion systems are considered representative examples to illustrate and verify the GQL. The results show that 4D GQL for hydrogen–air and 14D GQL ethanol–air slow manifolds outperform the standard QSSA approach based on a DAE-based reduced computation model.

On solving system of differential-algebraic equations using adomian decomposition method

Background: In this paper, we focus on an efficient and easy method for solving the given system of differential-algebraic equations (DAEs) of second order. Methods: The approximate solutions are computed rapidly and efficiently with the help of a semi-analytical method known as Adomian decomposition method (ADM). The logic of this method is simple and straightforward to understand. Results: To demonstrate the proposed method, we presented several examples and the computations are compared with the exact solutions to show the efficient. One can employ this logic to different mathematical software tools such as Maple, SCILab, Mathematica, NCAlgebra, Matlab etc. for the problems in real life applications. Conclusions: In this paper, we offered a method for solving the given system of secondorder nonlinear DAEs with aid of the ADM. We shown that the proposed method is simple and efficient, also one can obtain the approximate solutions quickly using this method. A couple of examples are discussed for illustrating this method and graphical and mathematical assessments are discussed with the analytical solutions of the given problems.

A novel memory stabilization controller design for descriptor Markovian jump systems with time-varying delays

This paper is focused on the stochastic stability and controller design of descriptor Markovian jump systems with time-varying delays. For achieving this goal, systems are decomposed into differential-algebraic equations firstly. A modified mode-dependent Lyapunov-Krasovskii functional (LKF) and a memory state feedback controller are constructed on the basis of the state decomposition. Then, combining the auxiliary functions integral inequality and the extended reciprocally convex matrix inequality (ERCMI), an ameliorative stochastic stability criterion is obtained. Based on this novel stability condition, the designed controller is used to make sure the closed-loop systems are stochastically stable. Finally, the effectiveness of presented results is confirmed with some numerical examples.

Optimal classification tree for frequency security assessment of power systems with inverter‐based resources reinforcement

AbstractFrequency security is the premise of realizing the net‐zero transition. A novel assessment scheme is proposed to quantify the frequency security levels of transmission systems under different inverter‐based resources (IBRs) and their control parameter combinations. A novel system frequency dynamic model is proposed as a parametric optimization method, where the saturation of generators, energy storage systems, and renewable energy sources is incorporated as differential algebra equations. This problem is further reformulated as a mixed‐integer linear programming problem to generate a sufficient amount of data under different IBR integration and control parameters. The frequency security assessment problem is formulated as a data‐driven multivariate classification problem, which is solved by the optimal classification tree (OCT) algorithm with better interptretionability. Simulations are conducted on a transmission system. Numerical results indicate that the proposed system frequency dynamic model can capture the frequency dynamic under different IBR reinforcement plans and the OCT can realize accurate classification of the synthetic data regarding frequency security.

Bounding nonminimality and a conjecture of Borovik–Cherlin

Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of nonminimality is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the U -rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of U -rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.

A time-adaptive FE2-approach within the method of vertical lines

FE2-approaches are chosen to incorporate micro-mechanical information into macroscopical computations. Although the approach is applied for several years, there are some open theoretical aspects in the algorithmic description. In this contribution the entire concept is brought into the method of vertical lines to solve initial boundary-value problems based on the homogenization concept under consideration. This requires a clear description of known and unknown quantities in the space and time discretization procedure. Dependent on the constitutive model, systems of algebraic or differential-algebraic equations (DAE) arise. Here, we concentrate on models of evolutionary type. Frequently, the resulting DAE-systems are solved using the Backward-Euler method, which are embedded in the more general class of diagonally, implicit Runge-Kutta methods. After the time discretization the resulting systems of non-linear equations have to be solved. This can be done using various schemes to obtain more efficient computations. Here, the three-level Newton algorithm, the Newton-Raphson-Schur method, numerical differentiation and a Chord-version are derived in the context of FE2 and discussed. Finally, step-size-controlled FE2-problems are solved, highlighting the importance of time-adaptive computations, particularly, in combination with the non-linear solution scheme, where the Multilevel-Newton-Chord method shows high efficiency. This contributes to an additional reduction of the overall computational times.