Differential-algebraic Equations Research Articles

On Linear Quadratic Optimal Control and Algebraic Riccati Equations for Infinite-Dimensional Differential-Algebraic Equations

We consider linear quadratic optimal control for a very general class of infinite-dimensional differential-algebraic equations (namely, the class of future-resolvable input/state/output nodes) and obtain an algebraic Riccati equation.

MINN: Learning the Dynamics of Differential-Algebraic Equations and Application to Battery Modeling.

The concept of integrating physics-based and data-driven approaches has become popular for modeling sustainable energy systems. However, the existing literature mainly focuses on the data-driven surrogates generated to replace physics-based models. These models often trade accuracy for speed but lack the generalizability, adaptability, and interpretability inherent in physics-based models, which are often indispensable in modeling real-world dynamic systems for optimization and control purposes. We propose a novel machine learning architecture, termed model-integrated neural networks (MINN), that can learn the physics-based dynamics of general autonomous or non-autonomous systems consisting of partial differential-algebraic equations (PDAEs). The obtained architecture systematically solves an unsettled research problem in control-oriented modeling, i.e., how to obtain optimally simplified models that are physically insightful, numerically accurate, and computationally tractable simultaneously. We apply the proposed neural network architecture to model the electrochemical dynamics of lithium-ion batteries and show that MINN is extremely data-efficient to train while being sufficiently generalizable to previously unseen input data, owing to its underlying physical invariants. The MINN battery model has an accuracy comparable to the first principle-based model in predicting both the system outputs and any locally distributed electrochemical behaviors but achieves two orders of magnitude reduction in the solution time.

Accurate and Efficient Numerical Simulation of Land Models Using SUMMA With SUNDIALS.

Numerical simulation of land models without error control can be highly inaccurate. We present the incorporation of the Suite of Nonlinear and Differential-Algebraic Equation Solvers (SUNDIALS) package to solve the equations that simulate thermodynamics and hydrologic processes in the Structure for Unifying Multiple Modeling Alternatives (SUMMA) land model. The algorithmic features of SUNDIALS, such as error estimation and adaptive order and step-size control, result in a SUMMA-SUNDIALS model that delivers substantially improved accuracy and relative computational efficiency compared to integration with the previous SUMMA model, which uses the low-order backward Euler method with no rigorous error control. The results are demonstrated through simulations over the North American continent with more than 500,000 spatial elements. Compared to the previous SUMMA model, we find that the simulations produced by the SUMMA-SUNDIALS model are orders of magnitude closer to converged solutions for the same computational cost. Being able to efficiently perform more reliable simulations makes the SUMMA-SUNDIALS model a powerful tool for improving our understanding of the terrestrial component of the Earth System.

Network algebraization and port relationship for power‐electronic‐dominated power systems

AbstractIn the classical differential‐algebraic equations (DAEs) framework for the traditional power system stability analysis, synchronous generators are depicted by differential equations and network by algebraic equations under the quasi‐steady‐state assumption. Differently, in the power‐electronic‐dominated power system (PEDPS), the dynamics of transmission lines of network for fully differential equations should be considered, due to the rapid response of converters' controls, for example, the alternating current controls. This poses a great challenge for the cognition, modeling, and analysis of the PEDPS. In this article, a nonlinear DAE model framework for the PEDPS is established with differential equations for the source nodes and algebraic equations for the dynamical electrical network, by generalizing the application scenarios of Kron reduction. The internal and terminal voltages of source nodes of converters are chosen as ports of nodes and network. Namely, the internal and terminal voltages of source nodes work as their output and input, respectively, whereas they work as the input and output of the algebraic network, respectively. The impact of dynamical network becomes clear, namely, it serves as a (linear) voltage divider and generates the terminal voltage based on the internal voltage of the sources simultaneously. By keeping only useful independent state variables, all differential equations for the transmission lines can be transferred to algebraic equations. With this simple model, the roles of both nodes and network become apparent, and it enhances the understanding of the PEDPS dynamics. On the other hand, broad simulations are conducted and compared to verify the proposed DAE framework for the PEDPS. As all independent variables have been kept in the model, it is found that they show the same computational accuracy, but better efficiency in computational time, compared to the electromagnetic‐transient simulation results.

High-fidelity analysis and experiments of a wireless sensor node with a built-in supercapacitor powered by piezoelectric vibration energy harvesting

A supercapacitor or an electric double layer capacitor (EDLC) is an essential component of a high-performance wireless sensor node (WSN) that can transmit the three-axis acceleration waveform data of structural vibrations and is powered by a piezoelectric vibration energy harvester (VEH). However, the intrinsically slow charging rate of the piezoelectric VEH triggers charging of extra capacitors formed inside the complex-shaped electrodes of the supercapacitor, which complicates the charging/discharging characteristics and thus estimation of the power generation and energy storage in the supercapacitor. Therefore, in this work, we establish an analytical methodology that allows accurate prediction of the charging/discharging characteristics of the supercapacitor built into the WSN powered by the piezoelectric VEH, considering (1) the nonlinear piezoelectricity, (2) a three-branch circuit model of the supercapacitor, and (3) the current–voltage relation equation for the diode. We develop a parameter identification technique for both the nonlinear piezoelectric VEH and the supercapacitor, and then establish a coupled analysis technique based on the differential algebraic equation (DAE). Using the DAE, we investigate the charging/discharging characteristics of the supercapacitor by considering the actual working steps of the WSN. We revealed that the second branch in the three-branch model contributes not only to an increase in the effective capacitance of the supercapacitor, but also to faster recovery of the voltage across the supercapacitor during WSN operation. We also revealed that a longer charging time led to more energy being stored in the capacitor in the third branch, from which it is difficult to extract energy within a short time period such as the WSN’s transmission process due to its long time constant. The contribution of the second and third branches must be considered to enable accurate prediction of the charging/discharging characteristics of the supercapacitor when built into the WSN powered by the piezoelectric VEH.

Control design for convection‐reaction systems with storage effects and positive relative degree

AbstractOne‐dimensional distributed‐parameter systems consisting of several convection‐reaction equations, which are distributedly coupled with a reaction equation, are considered. To solve the open‐loop control problem for these kind of systems, an inversion‐based approach is proposed. Along the common characteristic projection, the system can be reduced to a single partial differential equation (PDE) whose boundary conditions (BCs) result from the particular choice of the output to be tracked. The solution of this latter equation constitutes the basis of the control design. The particular properties of the scalar boundary value problem (BVP) to be solved depend on the coupling structure of the originally given convection‐reaction system. While in most cases the output has a relative degree of zero, for practical applications a relative degree of one is of particular interest. In this case, the obtained scalar BVP which describes the internal dynamics, possesses the structure of an infinite‐dimensional differential‐algebraic equation (DAE). Within the paper, the properties of the solution of both problems are discussed from a system analytical point of view.

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.

Transient Stability-Constrained Unit Commitment Using Input Convex Neural Network.

This article proposes a transient stability-constrained unit commitment (TSC-UC) model using input convex neural networks (ICNNs). An ICNN is trained to learn the transient function that maps prefault operation conditions (e.g., generator power output) to the transient stability index (TSI), which can be further utilized to identify the transient status (e.g., stable or unstable). Transient stability evaluation is conducted using the learned ICNN, without discretizing differential-algebraic equations (DAEs) or interacting with the time-domain simulation tools. Based on the convexity of ICNNs, the trained ICNN is exactly encoded as a linear programming (LP) model and integrated into conventional UC models to form a TSC-UC model. To impose transient stability constraints and expedite the solution process, the proposed TSC-UC model is decomposed into a UC master problem and two subproblems (i.e., network feasibility check subproblems and transient stability check subproblems). The decomposed problem is then iteratively solved using the Benders decomposition. Simulation tests are conducted in the New England 39-bus test system and IEEE 118-bus test system to verify the validity of the proposed approach.

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