Differential And Algebraic Equations Research Articles

Efficient Numerical Solver for Partially Structured Differential and Algebraic Equation Systems

Given a sparse set of differential and algebraic equations (DAEs), it is always recommended to exploit the structure of the system’s sparsity (e.g., tridiagonal blocks matrix, band matrix, and staircase matrix, etc.), thus to use tailored numerical solvers in order to reduce the computation time. Very frequently, though, while highly structured, a couple of elements enter the description which make it difficult for the solvers to reach a solution. They are common in process control applications, where the states added to the plant description by the integral parts of the controllers introduce unstructured elements in the otherwise very structured Jacobian of the mathematical model. Such systems are characterized by a partially structured Jacobian, which inhibits the use of the solvers tailored to fit problems with fully structured matrices. In such cases, one can either use a solver with lower performance, resulting in larger computation times, or alternatively one seeks an approximation for the unstructu...

Structural Analysis of Process Models Using Their Representation Graph

A graph-theoretical method for the structural analysis of dynamic lumped process models described by differential and algebraic equations (DAEs) is applied in this paper in order to determine the most important solvability properties (degree of freedom, structural solvability, model decomposition, dynamic degree of freedom, differential index, e.g.) of these models by using the so-called dynamic representation graph. The structure of the dynamic representation graph is suitable for the determination of the mentioned solvability properties. The most common methods in the modelling practice for the construction of models of complex systems are the union of submodels and hierarchical modelling. Our goal is to investigate the effect of the model union to the solvability properties, especially to the differential index. We show how the representation graph of a complex model can be built up from the representation graphs of submodels. The effect of the structure of submodels and their joining points to the structure of the complex graph and the conclusions drawn from the complex graph structure to the solvability properties are also investigated. The representation graph of the complex model can be easily built up from the representation graphs of the simple models according to the linking of the technological subsystems. If one of the submodels has greater than one differential index then the under and overspecified subgraphs referring to this higher index can be found in the representation graph of the complex model, too. The change in the relative position of the underspecified and the overspecified subgraphs has an effect to the value of differential index. If these subgraphs move further from their original positions the value of the differential index increases. If their relative positions do not change during the built up process then the value of the differential index of the complex system is equal to the value of the differential index of the subsystem having the higher value.

A Variable Partitioning Strategy for the Multirate Method in Power Systems

An application of the multirate method to differential and algebraic equations (DAEs) is discussed in this paper. Based on the specific formulation of the power systems algebraic equations, an effective algebraic variable partitioning strategy is developed and implemented. The application of the partitioning strategy to an example power system provides validation of the proposed partitioning strategy in terms of speed-up and accuracy.

Simulation of constrained mechanical systems

AbstractMotion equations of constrained mechanical Multi Body Systems (MBS) are described by differential and algebraic equations (DAE). Various modeling methods are implemented, while the simulation is based on commercial DAE‐solvers. A comparison study concerning the quality of their results is presented. Issues, such as the importance of DAE‐index, finding a consistent set of initial conditions, the drift‐off phenomenon as well as the application of projection and stabilization techniques are discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Integrated Multilevel Optimization in Large-Scale Poly(Ethylene Terephthalate) Plants

The present paper discusses the vertical integration of advanced predictive control, businesswide optimization and markets laws, based on a large-scale first-principles model. The case study is the poly(ethylene terephthalate) (PET) plant, with esterifier sections, finisher reactors, a crystallizer, and a solid-state polymerizer, and the overall mathematical model consists of more than 1500 differential and algebraic equations (DAEs). The complex integration among supply chain layers is reached using a Windows−Unix hybrid system; the methodology adopted is described, and a real-time strategy is implemented by using a secure shell connection for linking a PentiumIV, which simulates the plant, and a cluster Opteron, where the optimizer operates.

Dynamic Simulation of Reactive Batch Distillation Column for Ethyl Acetate Synthesis

A detailed mathematical dynamic model of reactive batch distillation column is formulated for ethyl acetate synthesis and presented in terms of differential and algebraic equations (DAEs). These DAEs are solved using fourth order Runge-Kutta method in MATLAB® to obtain the detailed column dynamics. The simulation results provide the dynamics of reboiler and distillate compositions, reboiler temperature, ethyl acetate purity in the accumulated distillate and conversion of the reactants. These results are analyzed to derive the optimum operating policy, i.e., reflux ratio and batch time.

A novel tool for transient stability analysis of large-scale power systems: Its application to the KEPCO system

Time-domain simulation is an important tool for power system dynamic analysis. We solve a set of differential and algebraic equations (DAE) in order to study the dynamic behavior of power systems. These power systems include thousands of generators, exciters, turbine-governors, loads, and other devices. The resultant large-scale DAEs are very difficult to handle and solve. Nevertheless, solution techniques are needed to not just guarantee accuracy but have computational efficiency. In this paper, we report on a novel tool that we developed to deal with time-domain simulation for dynamic analysis and operation of large-scale power systems. The tool has several major features related to transient stability analysis, contingency screening, and ranking. We mainly discuss the structure of this tool and the accuracy of the included dynamic models. Also, the paper proposes a new load model to overcome the low-voltage problem. The proposed technique provides a good performance and convergence when the terminal voltage is below some predefined value. Compared to the commercial tools, the developed tool is numerically well conditioned by introducing the ZIP model algorithm. This tool has been used to support and enhance power engineering education at both the undergraduate and graduate levels. In the case study, simulation results were validated through comparative simulations with the Power System Simulator for Engineering (PSS/E) and Transient Security Assessment Tool (TSAT).

MOVING HORIZON ESTIMATION FOR AN INDUSTRIAL GAS PHASE POLYMERIZATION REACTOR

A large-scale set of differential and algebraic equations (DAEs) is used to model and control an industrial gas phase polymerization reactor. Moving horizon estimation, preceding the nonlinear control optimization, provides an estimate of the current states and unmeasured disturbances. MHE is compared to Implicit Dynamic Feedback (IDFTM)*. With MHE, there is improved estimation of unmodeled disturbances in the UNIPOLTM** polyethylene plant. The polymerization model is converted to algebraic equations by orthogonal collocation and solved with the MHE objective function in a simultaneous optimization. NOVATM, an active-set sparse NLP solver, is used to converge the problem that has 46,870 variables. This large, sparse optimization problem is initiated every 5 minutes to update the model as new plant measurements become available and prior to the control optimization. The same plant model is used for nonlinear model predictive control (MPC) with 10 manipulated variables (MVs) and 26 controlled variables (CVs). In this case, MHE improves the control by better estimating reactor compositions of hydrogen, the condensing agent, and other measured state variables.

An output feedback precompensator for nonlinear DAE systems with control-dependent state-space

This note considers singular systems of nonlinear differential and algebraic equations (DAEs) whose constrained state space depends on the control inputs. A state-space realization of such systems cannot be derived independently of the controller design. An output feedback precompensator is derived, which results in a modified DAE system whose state-space is invariant under any feedback control law and can be used for output feedback controller synthesis. Its application is illustrated by a nonlinear electrical circuit example.

Numerical Method-Independent Structural Solvability Analysis of DAE Models

A graph-theoretical method for the structural analysis of dynamic lumped process models described by differential and algebraic equations (DAEs) is applied in this paper in order to determine the most important solvability properties of these models by using the so-called dynamic representation graph. The construction of the dynamic representation graphs that was originally proposed for the most simple, single-step, explicit numerical methods, has been extended in this paper to higher order explicit and implicit solution methods, that are used more frequently and efficiently for numerical solution of DAE-systems. It is shown here that the representation graph for both higher order explicit and implicit solution methods has similar properties to the case of explicit numerical solution procedures both for index one and higher index models. Thus it is proven that the important properties of the representation graph including the differential index of the models are independent of the assumption whether a single-step, explicit or implicit numerical method is used for the solution of the differential equations.

Dynamic optimization of bioprocesses: Efficient and robust numerical strategies

The dynamic optimization (open loop optimal control) of non-linear bioprocesses is considered in this contribution. These processes can be described by sets of non-linear differential and algebraic equations (DAEs), usually subject to constraints in the state and control variables. A review of the available solution techniques for this class of problems is presented, highlighting the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these systems. In order to surmount these difficulties, we present several alternative stochastic and hybrid techniques based on the control vector parameterization (CVP) approach. The CVP approach is a direct method which transforms the original problem into a non-linear programming (NLP) problem, which must be solved by a suitable (efficient and robust) solver. In particular, a hybrid technique uses a first global optimization phase followed by a fast second phase based on a local deterministic method, so it can handle the nonconvexity of many of these NLPs. The efficiency and robustness of these techniques is illustrated by solving several challenging case studies regarding the optimal control of fed-batch bioreactors and other bioprocesses. In order to fairly evaluate their advantages, a careful and critical comparison with several other direct approaches is provided. The results indicate that the two-phase hybrid approach presents the best compromise between robustness and efficiency.

A Novel Approach to Trace Time-Domain Trajectories of Power Systems in Multiple Time Scales

This work presents a novel approach for multitime-scale power system dynamic simulation based on differential and algebraic equations (DAEs). By properly selecting the continuation (local) parameter, the approach can avoid the numerical divergence encountered by a conventional approach, due to the singularity of the algebraic part of the equations. Meanwhile, by setting step size in a local parameterization, the approach varies its time step size according to the first- or second-order sensitivity of the solution to the variation of local parameter. It employs larger time step sizes for slow dynamics and smaller ones for fast dynamics. The approach presents an advanced tool to study voltage dynamics in multiple time scales and is easily adaptable for quasisteady-state analysis (QSS). The applicability of the approach to multiple time scales and QSS is demonstrated through a 39-bus New England system.

Dynamic Optimization of Single- and Multi-Stage Systems Using a Hybrid Stochastic−Deterministic Method

A hybrid stochastic−deterministic method, based on the control vector parameterization (CVP) approach, is presented as a reliable and efficient alternative for the solution of dynamic optimization (or open loop optimal control) problems.The problems under consideration are related to free final time single-stage systems and more general multi-stage procecesses that are described by different sets of differential and algebraic equations (DAEs), one for each stage. The operating conditions and the duration of each stage must be computed in order to achieve an overall optimal result for the process subject to constraints in the state and control variables. The solution of three challenging dynamic optimization problems is presented, including a large-scale case study, showing the capabilities of this new strategy.

Walking pattern generation employing DAE integration method

A stable walking pattern generation method for a biped robot is presented in this paper In general, the ZMP (zero moment point) equations, which are expressed as differential equations, are solved to obtain a stable walking pattern However, the number of differential equations is less than that of unknown coordinates in the ZMP equations It is impossible to integrate the ZMP equations directly since one or more constraint equations are involved in the ZMP equations To overcome this difficulty, DAE (differential and algebraic equation) solution method is employed The proposed method has enough flexibility for various kinematic structures Walking simulation for a virtual biped robot is performed to demonstrate the effectiveness and validity of the proposed method The method can be applied to the biped robot for stable walking pattern generation

Hybrid models for hardware-in-the-loop simulation of hydraulic systems Part 1: Theory

Physical modelling of hydraulic systems results in systems of differential and algebraic equations (DAEs) that are normally stiff. This is because they often include multiple temporal scales and present complex and non-linear behaviour. The simulation of these stiff DAEs in real time demands small time steps, in order to achieve algorithm stability, because explicit fixed step solvers are usually applied. Modelling some fast dynamics as instantaneous changes in order to reduce the model's stiffness has been an area of research in the last few years. The description of the dynamic behaviour as piecewise continuous modes, interspersed with discrete transitions, together with the reduction in the model complexity, facilitates the use of fixed time step integration methods and thus allows real-time simulation. The main goal of this work was to obtain not too complex models in order to allow the models to be used in hardware-in-the-loop experiments. This paper proposes the use of the statecharts formalism to describe hybrid behaviour in hydraulic systems and presents semiempirical models of a valve controlled hydraulic cylinder intended to test real controller performance on a hardware-in-the-loop setup. These semiempirical models require less computing power, and it is easier to adjust the model parameters.

Dynamic Modeling and Simulation of Impact in Tether Net/Gripper systems

The objective of this paper is to study the dynamic modeling andsimulation of a tether-net/gripper system during an impact, while it isbeing deployed or retrieved by a winch on a satellite orbiting aroundearth. We stick to Tether-Net system but the analysis is applicable toTether-Gripper systems too. We assume that the net is deployed from thesatellite in orbit and the motion is restricted to the orbital plane.This net captures a second satellite and tows it. The motion of atether-net system can be broken down into the following phases: (i)Phase 1: Net is shot out from the satellite with the tether completelyslack, (ii) Phase 2: Net comes to a location where the tether is tautwhile the drum on the orbiter is locked, (iii) Phase 3: Drum is unlockedand the net moves with the tether, (iv) Phase 4: Net captures a body.The continua (tether) is modeled using mode functions and coordinates.The theory of impulse and momentum can be used to model Phases 1, 2, and4 of motion of the tether-net system. The dynamics of the motion of thesystem in phase 3 is characterized by differential and algebraicequations (DAEs). Matlab ODE solvers were used to solve these DAEs.

On the Design of Optimally Informative Dynamic Experiments for Model Discrimination in Multiresponse Nonlinear Situations

We present a new method for determining optimally informative dynamic experiments for the purpose of model discrimination among several rival multiresponse nonlinear structured dynamic models generally described by systems of differential and algebraic equations (DAEs). A robust and efficient algorithm based on an extension to the dynamic case of the discrimination criterion put forth by Buzzi-Ferraris and Forzatti (Chem. Eng. Sci. 1984, 39, 81) is developed to calculate dynamic input trajectories by reformulation of the experiment design problem as an optimal control problem. We show that the new approach, by taking parametric uncertainty into account, can provide significant improvements in the ability to distinguish among a series of rival dynamic models over previous attempts to design dynamic experiments primarily based on parameter point estimates and thus maximizes the divergence of the model predictions without regard for uncertainty (Espie, D. M.; Macchietto, S. AIChE J. 1989, 35, 223). We illust...

The IWA Anaerobic Digestion Model No 1 (ADM1)

The IWA Anaerobic Digestion Modelling Task Group was established in 1997 at the 8th World Congress on Anaerobic Digestion (Sendai, Japan) with the goal of developing a generalised anaerobic digestion model. The structured model includes multiple steps describing biochemical as well as physicochemical processes. The biochemical steps include disintegration from homogeneous particulates to carbohydrates, proteins and lipids; extracellular hydrolysis of these particulate substrates to sugars, amino acids, and long chain fatty acids (LCFA), respectively; acidogenesis from sugars and amino acids to volatile fatty acids (VFAs) and hydrogen; acetogenesis of LCFA and VFAs to acetate; and separate methanogenesis steps from acetate and hydrogen/CO2. The physico-chemical equations describe ion association and dissociation, and gas-liquid transfer. Implemented as a differential and algebraic equation (DAE) set, there are 26 dynamic state concentration variables, and 8 implicit algebraic variables per reactor vessel or element. Implemented as differential equations (DE) only, there are 32 dynamic concentration state variables.

A novel, efficient and reliable method for thermal process design and optimization. Part I: theory

The design and optimization of thermal processing of foods needs accurate dynamic models through which to systematically explore new operation policies. Unfortunately, the governing and constitutive equations of thermal processing models usually lead to complex sets of highly nonlinear partial differential equations (PDEs), which are difficult and costly to solve, especially in terms of computation time. We overcome such limitation by using a powerful model reduction technique based on proper orthogonal decomposition (POD) which yields simple, yet accurate, dynamic models still based on sound first principles. Model reduction is carried out by projecting the original set of PDEs on a low dimensional subspace which retains most of the relevant features of the original system. The resulting model consists of a small set of differential and algebraic equations (DAEs) suitable for real-time industrial applications (optimization and control). Further, this approach can be easily adapted to handle complex nonlinear convection-diffusion processes regardless of how irregular the domain geometry might be.

Model decomposition based method for solving general dynamic optimization problems

The simultaneous strategy and control parameterization are the two most widely used direct methods for solving dynamic optimization problems. The control parameterization approach generally results in a comparatively smaller nonlinear program (NLP) but has difficulties in dealing with the path constraints. The advantage of the simultaneous approach lies in its ability to handle path constraints, thus eliminating the need to obtain expensive and possibly infeasible intermediate solutions. The disadvantage is that it requires solution of a potentially very large dimensional NLP. This work presents a decomposition strategy, which combines the advantages of the control parameterization and simultaneous approaches for solving dynamic optimization problems with path constraints. Using the proposed strategy, first the set of state variables x is divided into two sets x 1 and x 2, and the system is partitioned into two corresponding sub-systems. The criterion used to partition the state variables and the system model are that the equations which define the state variables involved in the path constraints should be in the same sub-system. For the resulting sub-systems, one is enforced in the master NLP through collocation method as in simultaneous approach, the other is solved together with the sensitivities in a differential and algebraic equations (DAE) solver. This strategy constitutes a general approach in the sense that for problems with a specific structure the method is equivalent to the control parameterization method, while for other problems with special structures the approach is the same as the simultaneous approach. In general it possesses the advantage of the simultaneous method in handling the path constraints since it is directly enforced in the master NLP as well as the advantage of control parameterization in resulting a small master NLP because only a fraction of the state variables are directly discretized. The method is demonstrated through several numerical examples.