Differential Algebraic Equation Model Research Articles

Nonlinear state estimation of a power plant superheater by using the extended Kalman filter for differential algebraic equation systems

A nonlinear estimator was developed for process monitoring of a power plant superheater system to estimate the transient temperature profiles of the critical measured and unmeasured variables under load-following operations. The model used in the estimator was a detailed, distributed, first-principles, differential-algebraic equation model. The superheater geometry is characterized based on an operating natural gas combined cycle power plant. The model included mass and energy balances for the operating fluids, i.e., steam and flue gas, with due consideration of tube wall temperature dynamics. The dynamic model was used for state and parameter estimation by using a modified extended Kalman filter. When results are compared with the data from a natural gas power plant, the estimator yielded root mean squared error of 0.2 °C for the main steam temperature compared to 1 °C obtained from the first-principles model. While the first-principles model yielded as high as 2.5 °C absolute instantaneous error for the tube temperatures, the estimator reduced the absolute instantaneous error below 0.3 °C for most variables. Performance of the estimator for estimating critical variables like the main steam temperature and flue gas temperature was evaluated in the presence of high model mismatch and noisy measurement data by simulating load-following operation of the plant. The estimator yielded maximum absolute instantaneous error of 3.4 °C for the flue gas and 1 °C for the main steam temperature.

Generalization to differential–algebraic equations of Lyapunov–Schmidt type reduction at Hopf bifurcations

The Lyapunov–Schmidt procedure, a well-known and powerful tool for the local reduction of nonlinear systems at bifurcation points or for ordinary differential equations (ODEs) at Hopf bifurcations, is extended to the context of strangeness-free differential–algebraic equations (DAEs), by generalizing the comprehensive presentation of the method for ODEs provided in the classical textbook by Golubitsky and Schaeffer (1985). The appropriate setting in the context of DAEs at Hopf bifurcations is first detailed, introducing suitable operators and addressing the question of appropriate numerical algorithms for their construction as well. The different steps of the reduction procedure are carefully reinterpreted in the light of the DAE context and detailed formulas are provided for systematic and rational construction of the bifurcating local periodic solution, whose stability is shown, likely to the ODE context, to be predicted by the reduced equations. As an illustrative example, a classical DAE model for an electric power system is considered, exhibiting both supercritical and subcritical Hopf bifurcations, demonstrating the prediction capability of the reduced system with regard to the global dynamics.

Distributed model predictive control based on the alternating directions method of multipliers applied to voltage and frequency control in power systems

AbstractWe present a straightforward way to solve a model predictive control problem for a power network system given as a nonlinear differential‐algebraic equation (DAE) in a distributed way using the consensus alternating directions method of multipliers (consensus ADMM) algorithm. While no convergence‐ or stability results are available for fully nonlinear DAE models, this gives unprecedented experimental evidence that power network systems of the presented structure allow to be controlled in this way, unlocking the numerous combined advantages of distributed and predictive control schemes in the context of energy distribution networks, as well as broadening the field of use for the consensus ADMM algorithm to nonlinear DAE models.

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.

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.

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.

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.

Stability analysis of a hybrid DC-DC buck converter model using dissipation inequality and convex optimization

The stability analysis of a DC-DC buck converter is a challenging problem due to the hybrid systems characteristic of its dynamics. Such a challenge arises from the buck converter operation which depends upon the ON/OFF logical transitions of its electronic switch component to correspondingly activate different continuous vector fields of the converter’s temporal dynamics. This paper presents a sum of squares (SOS) polynomial optimization approach for stability analysis of a hybrid model of buck converter which explicitly takes into account the converter’s electronic switching behavior. The proposed method first transforms the converter’s hybrid dynamics model into an equivalent polynomial differential algebraic equation (DAE) model. An SOS programming algorithm is then proposed to computationally prove the stability of the obtained DAE model using Lyapunov’s stability concept. Based on simulation results, it was found that the proposed method requires only 8.5 seconds for proving the stability of a buck converter model. In contrast, exhaustive simulations based on numerical integration scheme require 15.6 seconds to evaluate the stability of the same model. These results thus show the effectiveness of the proposed method as it can prove the converter stability in shorter computational times without requiring exhaustive simulations using numerical integration.

Load- and Renewable-Following Control of Linearization-Free Differential Algebraic Equation Power System Models

Electromechanical transients in power networks are mostly caused by a mismatch between power consumption and production, causing generators to deviate from the nominal frequency. To that end, feedback control algorithms have been designed to perform frequency and load/renewable-following control. In particular, the literature addressed a plethora of grid-and frequency-control challenges with a focus on linearized, differential equation models whereby algebraic constraints i.e., power flows (PFs) are eliminated. This is in contrast to the more realistic nonlinear differential algebraic equation (NDAE) models. Yet, as grids are increasingly pushed to their limits via intermittent renewables and varying loads, their physical states risk escaping operating regions due to either a poor prediction or sudden changes in renewables or demands—deeming a feedback controller based on a linearization point virtually unusable. In lieu of linearized differential equation models, the objective of this article is to design a simple, purely decentralized, linearization-free, feedback control law for the NDAE models of power networks. The aim of such a controller is to primarily stabilize frequency oscillations after a significant, unknown disturbance in renewables or loads. Although the controller design involves advanced NDAE system theory, the controller itself is as simple as a decentralized proportional or linear quadratic regulator (LQR) in its implementation. Case studies demonstrate that the proposed controller is able to stabilize dynamic and algebraic states under significant disturbances.

Dynamic State Estimation of Nonlinear Differential Algebraic Equation Models of Power Networks

This paper investigates the joint problems of dynamic state estimation of algebraic variables (voltage and phase angle) and generator states (rotor angle and frequency) of nonlinear differential algebraic equation (NDAE) power network models, under uncertainty. Traditionally, these two problems have been decoupled due to complexity of handling NDAE models. In particular, this paper offers the first attempt to solve the aforementioned problem in a coupled approach where the algebraic and generator states estimates are simultaneously computed. The proposed estimation algorithm herein is endowed with the following properties: <i>(i)</i> it is fairly simple to implement and based on well-understood Lyapunov theory; <i>(ii)</i> considers various sources of uncertainty from generator control inputs, loads, renewables, process and measurement noise; <i>(iii)</i> models phasor measurement unit installations at arbitrary buses; and <i>(iv)</i> is computationally less intensive than the decoupled approach in the literature.

A space of filters for the consistent initialization of differential-algebraic equations in optimization tasks

In this article, a new approach to determining consistent initial conditions for an optimization task with differential-algebraic equations (DAEs) is presented. The work is motivated by the fact that initial conditions have a significant impact on both the obtained state trajectories, as well as the progress of the optimization procedure. In the first step of the designed approach, an α-parametrization of the DAE model takes place. This gives the possibility of influencing the dynamics of the system, and in particular its total reduction. When the initial values of the differential state variables are given, then the solution of the DAE system depends only on the algebraic state variables and the input function, which should have the character of the consistent initial conditions. To obtain the consistent initial conditions and, at the same time, to optimize the considered process, in the second step of the procedure a filter space is defined and applied. The filter space is an innovative extension of Fletcher's filter method. It is built from a finite number of filters, and the dimension of the filter space depends on the input function parametrization approach. As a result of using the filter space, it is possible to obtain a reasonable initial solution to the whole optimization problem, which can be improved by available exact local algorithms. The operation of the new method is tested in the heat and mass transfer optimization task, which is described by a system of nonlinear differential-algebraic equations.

Observers for Differential Algebraic Equation Models of Power Networks: Jointly Estimating Dynamic and Algebraic States

Phasor measurement units (PMUs) have become instrumental in modern power systems for enabling real time, wide-area monitoring and control. Accordingly, many studies have investigated efficient and robust dynamic state estimation (DSE) methods in order to accurately compute the dynamic states of generation units. Nonetheless, most of them forego the dynamic-algebraic nature of power networks and only consider their nonlinear dynamic representations. Motivated by the lack of DSE methods based on power network’s differential-algebraic equations (DAEs), this article develops a novel observer-based DSE framework in order to perform simultaneous estimation of the dynamic and algebraic states of multimachine power networks. Specifically, we leverage the DAE dynamics of a power network around an operating point and combine them with a PMU-based measurement model capable of capturing bus voltages and line currents. The proposed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> observer, which only requires detectability and impulse observability conditions that are satisfied for various power networks, is designed to handle various noise, unknown inputs, and input sensor failures. The results obtained from performing extensive numerical simulations on IEEE 9-bus and 39-bus systems showcase the effectiveness of the proposed approach for DSE purposes.

An analysis of structural and practical identifiability applied to a transient generator model

This article addresses issues of the structural and practical identifiability of a transient model of a synchronous generator. A structural identifiability analysis was conducted for nonlinear models of differential-algebraic equations (DAEs) and then applied to the DAE model of the synchronous generator, whose parameters were estimated by a practical identifiability analysis employing a trajectory sensitivity method. An optimal time window of measurement data was selected for improving the accuracy of the estimated parameters, which were validated from different sets of measurements collected in two different experiments.

Dynamic Modeling and Nonlinear Model Predictive Control of a Moving Bed Chemical Looping Combustion Reactor

A partial differential-algebraic equation (PDAE) model of a dynamic moving bed chemical looping combustion reduction reactor is developed in the IDAES process modeling framework for nonlinear model predictive control (NMPC). Numerical stability of the PDAE discretization and the index-1 property of the DAE are validated. An NMPC case study is performed, which manipulates inlet flow rates to control the system to a steady-state setpoint over approximately 6,000 s of simulated time, while respecting a lower bound on outlet oxygen carrier conversion. The dynamic optimization problems are solved for each NMPC cycle with an average of 39 CPUs, indicating that this model has the potential for use in a real-time scenario.

H∞ Control of a Planar 3-DOF Flexible-Cable Manipulator

This paper deals with the dynamic modeling and the H∞ control of a cable-driven parallel robot with flexible cables. The modeling of planar 3-DOF (two translation and one rotation) robot pulled by four-cables has been performed using the assumed modes approach and the Euler-Lagrange equation, leading to a differential-algebraic equations model (DAE model). This model has been then transformed into an ordinary-differential equations model (ODE model) whose linearization is observable and controllable. An H∞ controller has been synthesized, considering a four-bloc synthesis scheme, in order to control the translation and the rotation of the platform and the mean tension of the cables. An original tuning strategy of the weighting functions is proposed, in order to ensure different performances on the different DOF, that suits to the physical nature of the process. The approach is evaluated in simulation on the nonlinear model and shows good disturbance rejection and reference tracking.

Hierarchical Structural Analysis Method for Complex Equation-Oriented Models

Structural analysis is a method for verifying equation-oriented models in the design of industrial systems. Existing structural analysis methods need flattening of the hierarchical models into an equation system for analysis. However, the large-scale equations in complex models make structural analysis difficult. Aimed to address the issue, this study proposes a hierarchical structural analysis method by exploring the relationship between the singularities of the hierarchical equation-oriented model and its components. This method obtains the singularity of a hierarchical equation-oriented model by analyzing a dummy model constructed with the parts from the decomposing results of its components. Based on this, the structural singularity of a complex model can be obtained by layer-by-layer analysis according to their natural hierarchy. The hierarchical structural analysis method can reduce the equation scale in each analysis and achieve efficient structural analysis of very complex models. This method can be adaptively applied to nonlinear-algebraic and differential-algebraic equation models. The main algorithms, application cases and comparison with the existing methods are present in this paper. The complexity analysis results show the enhanced efficiency of the proposed method in the structural analysis of complex equation-oriented models. Compared with the existing methods, the time complexity of the proposed method is improved significantly.

A mathematical model of cardiovascular dynamics for the diagnosis and prognosis of hemorrhagic shock.

A variety of mathematical models of the cardiovascular system have been suggested over several years in order to describe the time-course of a series of physiological variables (i.e. heart rate, cardiac output, arterial pressure) relevant for the compensation mechanisms to perturbations, such as severe haemorrhage. The current study provides a simple but realistic mathematical description of cardiovascular dynamics that may be useful in the assessment and prognosis of hemorrhagic shock. The present work proposes a first version of a differential-algebraic equations model, the model dynamical ODE model for haemorrhage (dODEg). The model consists of 10 differential and 14 algebraic equations, incorporating 61 model parameters. This model is capable of replicating the changes in heart rate, mean arterial pressure and cardiac output after the onset of bleeding observed in four experimental animal preparations and fits well to the experimental data. By predicting the time-course of the physiological response after haemorrhage, the dODEg model presented here may be of significant value for the quantitative assessment of conventional or novel therapeutic regimens. The model may be applied to the prediction of survivability and to the determination of the urgency of evacuation towards definitive surgical treatment in the operational setting.

Continuous reactor network design for multiple rigid polyol productions

AbstractThis study focuses on developing continuous reactor network models able to produce multiple rigid polyol products under strict product and safety specifications. We first determine reactor networks and operating decisions for minimum capital cost, and then find reactor networks with optimal profit through multi‐criteria optimization, using a Pareto chart to analyze the relationship between capital cost and net sales. Based on our previous study, we narrow down the types of continuous reactors that can be part of the network to two: plug flow reactor (PFR) with multiple feed injection points and a network of continuous stirred tank reactors (CSTR). The CSTR model can be written as a mixed integer nonlinear programming (MINLP) problem. The PFR model is a differential algebraic equation (DAE) optimization problem. The simultaneous collocation method is applied to transform the resulting DAE model into an MINLP, which is solved as a NLP by fixing the binary terms. Both capital cost and multi‐criteria problems are formulated as multi‐scenario optimization problems to determine the best single network design to produce multiple polymer products.

A Direct Optimization Algorithm for Problems with Differential-Algebraic Constraints: Application to Heat and Mass Transfer

In this article, an optimization task with nonlinear differential-algebraic equations (DAEs) is considered. As a main result, a new solution procedure is designed. The computational procedure represents the sequential optimization approach. The proposed algorithm is based on a multiple shooting parametrization method. Two main aspects of a generalized parametrization approach are analyzed in detail: a control function and DAE model parametrization. A comparison between the original and modified DAEs is made. The new algorithm is applied to solve an optimization task in heat and mass transfer engineering.

Critical Clearing Time Sensitivity for Differential-Algebraic Power System Model

Standard power systems are modeled using differential-algebraic equations (DAE). Following a transient event, voltage collapse can occur as a bifurcation of the transient load flow solutions. This results from the states' trajectories reaching a singular surface where the voltage causality is lost. Preventive controls such as changes in AVR setpoints need to be taken to enhance stability. To determine which actions are most effective, the sensitivity of the critical clearing time to controllable parameters can be exploited. This work derives the expressions for critical clearing time sensitivity for a generic DAE model using trajectory sensitivities. The results are illustrated for multiple test systems that are then validated against computationally intensive time-domain simulations.