Differential-algebraic Equations Research Articles

Quantifying uncertainty in neural network predictions of forced vibrations

AbstractThe prediction of forced vibrations in nonlinear systems is a common task in science and engineering, which can be tackled using various methodologies. A classical approach is based on solving differential (algebraic) equations derived from physical laws ('first principles'). Alternatively, Artificial Neural Networks (ANNs) may be applied, which rely on learning the dynamics of a system from given data. However, a fundamental limitation of ANNs is their lack of transparency, making it difficult to understand and trust the model's predictions. In this contribution, we follow a hybrid modelling approach combining a data‐based prediction using a stabilised Autoregressive Neural Network (s‐ARNN) with a priori knowledge from first principles. Moreover, aleatoric and epistemic uncertainty is quantified by a combination of mean‐variance estimation (MVE) and deep ensembles. Validating this approach for a classical Duffing oscillator suggests that the MVE ensemble is the most accurate and reliable method for prediction accuracy and uncertainty quantification. These findings underscore the significance of understanding uncertainties in deep ANNs and the potential of our method in improving the reliability of predictive nonlinear system modelling. We also demonstrate that including partially known dynamics can further increase accuracy, highlighting the importance of combining ANNs and physical laws.

Mathematical Approach for Directly Solving Air–Water Interfaces in Water Emptying Processes

Emptying processes are operations frequently required in hydraulic installations by water utilities. These processes can result in drops to sub-atmospheric pressure pulses, which may lead to pipeline collapse depending on soil characteristics and the stiffness of a pipe class. One-dimensional mathematical models and 3D computational fluid dynamics (CFD) simulations have been employed to analyse the behaviour of the air–water interface during these events. The numerical resolution of these models is challenging, as 1D models necessitate solving a system of algebraic differential equations. At the same time, 3D CFD simulations can take months to complete depending on the characteristics of the pipeline. This presents a mathematical approach for directly solving air–water interactions in emptying processes involving entrapped air, providing a predictive tool for water utilities. The proposed mathematical approach enables water utilities to predict emptying operations in water pipelines without needing 2D/3D CFD simulations or the resolution of a differential algebraic equations system (1D model). A practical application is demonstrated in a case study of a 350 m long pipe with an internal diameter of 350 mm, investigating the influence of air pocket size, friction factor, polytropic coefficient, pipe diameter, resistance coefficient, and pipe slope. The mathematical approach is validated using an experimental facility that is 7.36 m long, comparing it with 1D mathematical models and 3D CFD simulations. The results confirm that the derived mathematical expression effectively predicts emptying operations in single water installations.

Modeling the Production of Nanoparticles via Detonation—Application to Alumina Production from ANFO Aluminized Emulsions

This paper investigates the production of nanoparticles via detonation. To extract valuable knowledge regarding this route, a phenomenological model of the process is developed and simulated. This framework integrates the mathematical description of the detonation with a model representing the particulate phenomena. The detonation process is simulated using a combination of a thermochemical code to determine the Chapman–Jouguet (C-J) conditions, coupled with an approximate spatially homogeneous model that describes the radial expansion of the detonation matrix. The conditions at the C-J point serve as initial conditions for the detonation dynamic model. The Mie–Grüneisen Equation of State (EoS) is used, with the “cold curve” represented by the Jones–Wilkins–Lee Equation of State. The particulate phenomena, representing the formation of metallic oxide nanoparticles from liquid droplets, are described by a Population Balance Equation (PBE) that accounts for the coalescence and coagulation mechanisms. The variables associated with detonation dynamics interact with the kernels of both phenomena. The numerical approach employed to handle the PBE relies on spatial discretization based on a fixed-pivot scheme. The dynamic solution of the models representing both processes is evolved with time using a Differential-Algebraic Equation (DAE) implicit solver. The strategy is applied to simulate the production of alumina nanoparticles from Ammonium Nitrate Fuel Oil aluminized emulsions. The results show good agreement with the literature and experience-based knowledge, demonstrating the tool’s potential in advancing understanding of the detonation route.

Optimal control of a model for dynamic district heating networks

In the present paper an optimal control problem for a concrete system of differential-algebraic equations (DAEs) is considered. This problem arises in the dynamic optimization of unsteady district heating networks. Based on the Carathéodory theory existence of a unique solution to the specific DAE system is proved using specific properties of the considered district heating network model. Moreover, it is shown that the optimal control problem possesses optimal solutions. For the numerical experiments different networks are considered including also data from a real district heating network.

Development of hybrid first principles – Artificial intelligence models for transient modeling of power plant superheaters under load-following operation

This paper develops different approaches for synergistic integration of physics-based models with data-driven models for modeling of high temperature power plant superheaters. Two types of data-driven models are developed, namely all-nonlinear static-dynamic neural network models as well as models obtained using a Bayesian machine learning approach. Series, integrated, and parallel coupling of first-principles models and data-driven models are proposed. Algorithms for solving the training problems for the data-driven models and for simulation of the hybrid models are developed. The developed approach is applied to high temperature power plant superheaters that face considerable modeling and measurement challenges. Measurement challenges arise due to harsh operating conditions that not only make placement of sensors difficult, but life of the placed sensors very limited. Modeling challenges arise due to complex inhomogeneous spatial distribution of flue gas and steam flowrates, especially under load-following operation and uncertainties in heat transfer characteristics because of multiple factors such as stochastic spatio-temporal variation of ash deposits on the superheater tubes in coal-fired power plants, internal oxide scale formation in the tubes, etc. First-principles two-dimensional differential–algebraic equation models of two industrial superheaters, one from a natural gas combined cycle plant and another from a coal-fired power plant, are developed. The results from the models are compared against industrial operational data. For all cases studied in this work, it is observed that both for steady-state and dynamic data, the hybrid models have much higher accuracy than when only the respective first-principles models are used. For example, during prediction of the average outside tube wall temperature of superheater tubes at the combined cycle power plant, the root mean squared error observed for the standalone first-principles model improved from 12.3 °C to 0.5 °C post hybridization with data-driven models. Similarly, while predicting the main steam outlet temperature in the coal-fired plant, the hybrid models resulted in reduction of the root mean squared error value from 19.5 °C to around 2 °C. In addition, the hybrid models yield highly resolved spatio-temporal temperature profiles that can be helpful for developing monitoring approaches of these critical components under load-following operation.

Measurement and Modeling of Self-Directed Channel (SDC) Memristors: An Extensive Study

This study systematically addresses the challenge of accurately modeling memristors, focusing on four distinct types doped with tungsten, tin, chromium, and carbon, fabricated by Knowm Inc. A comprehensive characterization is performed by subjecting the devices to sinusoidal excitations with varying frequencies and amplitudes, followed by data averaging and high-frequency filtering. The resulting measurements are fitted using three prominent memristor models: VTEAM, MMS, and Yakopcic. Additional bespoke modifications are assessed. These models, typically formulated as coupled algebraic differential equations integrating electrical quantities (voltage and current) with internal state variables governing device dynamics, are optimized using two robust approaches: (1) interior-point optimization with gradient-based search, and (2) Nelder–Mead gradient-free optimization, both with box constraints applied. A thorough comparison and discussion of the optimized model parameters ensue, accompanied by an examination of the sensitivity to diverse frequency and amplitude ranges. The findings inform conclusions and provide a foundation for future refinements, underscoring the importance of multi-model evaluation and advanced optimization strategies in precise memristor modeling. The presented methodology offers a valuable framework for elucidating optimal modeling paradigms tailored to specific memristor architectures and operating regimes, ultimately enhancing their integration in emerging neuromorphic and computational applications.

Model-based workflow for sustainable production of high-quality spirits in packed column stills

This study addresses water scarcity in Chilean distilleries by developing a model-based engineering workflow. Prolonged droughts, likely driven by global warming, have intensified this problem. The primary challenge is maintaining high-quality spirits while reducing cooling water and energy use during batch distillation in packed columns. This process was modeled using mass and energy balances, resulting in a system of partial differential algebraic equations (PDAE). Our workflow includes mechanistic modeling, disturbance modeling, multiobjective optimization, model predictive control, and Monte Carlo simulations. Our findings show that cooling water consumption can be reduced by up to 35 % and energy consumption by up to 14.4 % while maintaining product quality. The proposed system is robust against operational disturbances and model mismatch, ensuring consistent distillate quality. This research demonstrates the integration of model-based optimization and control strategies in batch distillation processes, which can be replicated in other fruit wine distillation processes for improved sustainability.

Index Concepts for Linear Differential-Algebraic Equations in Infinite Dimensions

Different index concepts for regular linear differential-algebraic equations are defined and compared in the general Banach space setting. For regular finite dimensional linear differential-algebraic equations, all these indices exist and are equivalent. For infinite dimensional systems, the situation is more complex. It is proven that although some indices imply others, in general they are not equivalent. The situation is illustrated with a number of examples.

Artificial intelligence techniques for dynamic security assessments - a survey

The increasing uptake of converter-interfaced generation (CIG) is changing power system dynamics, rendering them extremely dependent on fast and complex control systems. Regularly assessing the stability of these systems across a wide range of operating conditions is thus a critical task for ensuring secure operation. However, the simultaneous simulation of both fast and slow (electromechanical) phenomena, along with an increased number of critical operating conditions, pushes traditional dynamic security assessments (DSA) to their limits. While DSA has served its purpose well, it will not be tenable in future electricity systems with thousands of power electronic devices at different voltage levels on the grid. Therefore, reducing both human and computational efforts required for stability studies is more critical than ever. In response to these challenges, several advanced simulation techniques leveraging artificial intelligence (AI) have been proposed in recent years. AI techniques can handle the increased uncertainty and complexity of power systems by capturing the non-linear relationships between the system’s operational conditions and their stability without solving the set of algebraic-differential equations that model the system. Once these relationships are established, system stability can be promptly and accurately evaluated for a wide range of scenarios. While hundreds of research articles confirm that AI techniques are paving the way for fast stability assessments, many questions and issues must still be addressed, especially regarding the pertinence of studying specific types of stability with the existing AI-based methods and their application in real-world scenarios. In this context, this article presents a comprehensive review of AI-based techniques for stability assessments in power systems. Different AI technical implementations, such as learning algorithms and the generation and treatment of input data, are widely discussed and contextualized. Their practical applications, considering the type of stability, system under study, and type of applications, are also addressed. We review the ongoing research efforts and the AI-based techniques put forward thus far for DSA, contextualizing and interrelating them. We also discuss the advantages, limitations, challenges, and future trends of AI techniques for stability studies.

Control of an integrated first and second-generation continuous alcoholic fermentation process with cell recycling using model predictive control

This study explores controlling a first- and second-generation alcoholic fermentation process with cell recycling, modeled through algebraic differential equations (DAE) with constraints. It evaluates seven controller types: PI, PID, Model Predictive Control (MPC), Neural Network Model Predictive Control (NNMPC), Mixed Model Predictive Control (MMPC), Internal Model Control (IMC), and Linear Quadratic Regulator (LQR). Results show that, while PI and PID controllers can track setpoints, they exhibit slow responses and oscillations. In contrast, MPC controllers respond faster, with both MPC and MMPC demonstrating more robust dynamics, achieving a significant reduction in Integral of the Absolute Error (IAE) across various disturbances, notably an 87% reduction in regulatory scenarios. NNMPC outperforms PI and PID but exhibits overshoot and oscillations, and lacks robustness for servo-type problems. However, MMPC showcases comparable or superior performance to MPC, surpassing other controllers in robustness, especially the IMC and LQR in the regulatory problems. NNMPC maintains a simulation time of under 4 seconds, whereas MPC incurs a computational cost 1,000 times higher. Integrating NNMPC’s optimal increment as an initial estimate in MPC reduces computational time by up to 79.7%. These findings highlight the ANN’s effectiveness in addressing complex control challenges, especially when integrated with MPC. MMPC offers a superior balance between accuracy, robustness, and computational efficiency, serving as a promising solution for reducing computational costs in MPC-type controllers.

Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

MultiShape: a spectral element method, with applications to Dynamic Density Functional Theory and PDE-constrained optimization

Abstract A new numerical framework is developed to solve general nonlinear and nonlocal PDEs on complicated two-dimensional domains. This enables the solution of a wide range of both steady and time-dependent problems on nonstandard geometries, as well as providing the ability to impose nonlinear and nonlocal boundary conditions (typical of those arising in the modelling of physical phenomena) in a flexible and automated way. This spectral element methodology, which we called MultiShape, is compatible with other state-of-the-art numerical methods, such as differential–algebraic equation solvers and optimization algorithms. MultiShape is an open-source Matlab library, in which the numerical implementation is designed to be user-friendly: the problem set-up and computations are done automatically through intuitive operator definitions and notation. Validation tests are presented, before we showcase the power and versatility of MultiShape with three motivating examples in Dynamic Density Functional Theory and PDE-constrained optimization.

Deformations of solutions of algebraic differential equations

Deformations of solutions of algebraic differential equations

A numerical framework for modeling evaporation and combustion of isolated, spherically-symmetric, multi-component fuel droplets

This paper presents a comprehensive numerical framework for simulating the evaporation and combustion of isolated, spherically-symmetric, multi-component fuel droplets. The framework incorporates a detailed description of chemical reactions in the gaseous phase and is capable of modeling pure evaporation, autoignition, and hot-wire ignition scenarios.The transport equations for mass, species, and energy are solved in both the liquid (droplet) and gaseous (surrounding atmosphere) phases. Diffusion in the liquid phase is described using the Stefan–Maxwell theory, while in the gaseous phase, both molecular and thermal diffusion are considered. The model also includes the thermophoretic effect for carbonaceous particles and accounts for gas radiation through various models, ranging from optically-thin approximations to more complex methods like the P1 and discrete ordinate methods. Non-gray radiative effects are handled using the Weighted-Sum-of-Gray-Gases Model (WSGGM). Liquid/gas interface conditions are evaluated by imposing flux continuity of mass and energy, along with thermodynamic equilibrium for species. Deviations from ideal thermodynamic behavior in the liquid droplet are managed by incorporating a suitable activity coefficient or by using a proper cubic equation of state. Additionally, the presence of supporting fibers is modeled using a simplified one-dimensional approach. The transport equations are solved using the method of lines, with spatial discretization performed via the finite difference method on a body-fitted grid. The resulting system of Differential–Algebraic Equations (DAEs) is then solved using a fully-coupled approach.Thanks to its generality in terms of kinetic and thermodynamic descriptions and the reduced computational time, the proposed framework offers significant potential for advancing our understanding of the complex combustion processes of multi-component liquid fuels and for enhancing the planning and execution of experiments involving isolated fuel droplets.

Flow dynamics over cylinder in two‐dimensional benchmark problem: A finite element approximation

This research investigates 2D benchmark flow around a circular cylinder, utilizing the incompressible Navier–Stokes equations alongside the continuity and energy equations. The numerical solution is achieved through finite element discretization for space variable, combined with a second‐order Crank–Nicolson scheme for time integration. The computational results are derived using the FEATFLOW finite element based library package. Our study focuses on the dimensionless form of the flow equations and examines three key dimensionless parameters: drag, lift, and pressure drop. Upon applying finite element method (FEM) discretization, the system is converted into a set of linear or nonlinear ordinary differential equations or algebraic equations for steady‐state scenarios. We then apply the Newton–Raphson method as the outer nonlinear solver, and the multigrid method for efficiently resolving the linear subproblems. To ensure numerical accuracy, we evaluated the and errors, confirming that the experimental order of convergence matches the theoretical predictions. Flow profiles were both graphically represented and tabulated, offering a detailed understanding of the simulation results.

Power flow analysis using quantum and digital annealers: a discrete combinatorial optimization approach

Power flow (PF) analysis is a foundational computational method to study the flow of power in an electrical network. This analysis involves solving a set of non-linear and non-convex differential-algebraic equations. State-of-the-art solvers for PF analysis, therefore, face challenges with scalability and convergence, specifically for large-scale and/or ill-conditioned cases characterized by high penetration of renewable energy sources, among others. The adiabatic quantum computing paradigm has been proven to efficiently find solutions for combinatorial problems in the noisy intermediate-scale quantum (NISQ) era, and it can potentially address the limitations posed by state-of-the-art PF solvers. For the first time, we propose a novel adiabatic quantum computing approach for efficient PF analysis. Our key contributions are (i) a combinatorial PF algorithm and a modified version that aligns with the principles of PF analysis, termed the adiabatic quantum PF algorithm (AQPF), both of which use Quadratic Unconstrained Binary Optimization (QUBO) and Ising model formulations; (ii) a scalability study of the AQPF algorithm; and (iii) an extension of the AQPF algorithm to handle larger problem sizes using a partitioned approach. Numerical experiments are conducted using different test system sizes on D-Wave’s Advantage™ quantum annealer, Fujitsu’s digital annealer V3, D-Wave’s quantum-classical hybrid annealer, and two simulated annealers running on classical computer hardware. The reported results demonstrate the effectiveness and high accuracy of the proposed AQPF algorithm and its potential to speed up the PF analysis process while handling ill-conditioned cases using quantum and quantum-inspired algorithms.

Physics-informed neural networks for dynamic process operations with limited physical knowledge and data

In chemical engineering, process data are expensive to acquire, and complex phenomena are difficult to fully model. We explore the use of physics-informed neural networks (PINNs) for modeling dynamic processes with incomplete mechanistic semi-explicit differential–algebraic equation systems and scarce process data. In particular, we focus on estimating states for which neither direct observational data nor constitutive equations are available. We propose an easy-to-apply heuristic to assess whether estimation of such states may be possible. As numerical examples, we consider a continuously stirred tank reactor and a liquid-liquid separator. We find that PINNs can infer immeasurable states with reasonable accuracy, even if respective constitutive equations are unknown. We thus show that PINNs are capable of modeling processes when relatively few experimental data and only partially known mechanistic descriptions are available, and conclude that they constitute a promising avenue that warrants further investigation.

Differential-algebraic equation-based model predictive control for trajectory tracking of spacecraft-mounted continuum manipulators

Spacecraft-mounted continuum manipulators (SMCMs) exhibit great potential for performing dexterous operations in unstructured environments due to their inherent compliance and dexterity. However, the dynamic model of the rigid-flexible coupling SMCM is highly nonlinear and typically formulated as a set of implicit differential-algebraic equations (DAEs), posing significant challenges for precise trajectory tracking control. This paper proposes a novel model predictive control (MPC) framework specifically designed for generic DAEs to achieve precise trajectory tracking of the SMCM under uncertain disturbances and input limitations. The DAE model of the SMCM is discretized into a set of nonlinear algebraic equations. By performing implicit differentiation of these equations with respect to the system state, the state transition matrix (STM) for the DAE model is derived. The optimal control action for the SMCM can be further determined based on the derived STM. Additionally, nonlinear complementary functions are introduced to address the issue of input limitations, allowing the problem of determining the optimal control sequence to be equivalently transformed into a set of nonlinear algebraic equations for solving. Numerical simulations demonstrate that the proposed approach can achieve precise trajectory tracking of the SMCM while strictly adhering to input limitations.

Impact of Phase Angle Jump on a Doubly Fed Induction Generator under Low-Voltage Ride-Through Based on Transfer Function Decomposition

During the fault period, a phase angle jump may occur at the stator or the point of common coupling, which will deteriorate the low-voltage ride-through (LVRT) characteristics of a doubly fed induction generator (DFIG). The existing LVRT studies focus on the impact of a voltage drop on DFIGs but often ignore that of a phase angle jump. The time-domain simulation is accurate in describing the response of a DFIG during the LVRT process, but it is time-consuming for a DFIG with the full-order model. In this paper, by using the voltage magnitude and phase angle of the stator or the point of common coupling as the inputs, and the state variables as the outputs, the transfer function of a DFIG is derived to analyze its response and find the LVRT measures against the voltage drop and, especially, the phase angle jump. Firstly, the differential-algebraic equations of the DFIG are linearized to propose their transfer function model. Secondly, considering its high-order characteristic, a model reduction method for the transfer function of the DFIG using the Schur decomposition is proposed, and the analytical expression of the output variables of the DFIG with the phase angle jump is derived by the inverse Laplace transformation to judge the necessity of the LVRT measures. Finally, the simulation results of the DFIG are provided to verify the accuracy of the transfer function model and its reduced-order form and validate the feasibility of the LVRT against the phase angle jump with the proposed models.

On functions of finite analytical complexity

We construct examples of polynomials and analytic functions of any predetermined finite analytical complexity n n . We obtain an estimate of the order of derivative of the differential-algebraic criteria for membership in the class C l n Cl_{n} of functions of analytical complexity not higher than n n . We find uniform estimates for finite values d n d_{n} of the analytic spectrum { d n } \{d_{n}\} for systems of differential-algebraic equations of fixed order of derivative δ \delta .