Index-analysis for a method of lines discretising multirate partial differential algebraic equations

Sliding motion of discontinuous dynamical systems described by semi-implicit index one differential algebraic equations

Symbolic preprocessing for simulation of PDE models of chemical processes

This paper proposes the implementation of dynamic power systems as a system of differential algebraic equations (DAEs) and to solve it by means of DAE solvers, because it is easier and more natural to write the equations as a system of DAEs than to write them as a system of ordinary differential equations (ODEs). Problem-solving environments have recently included routines in their packages to solve DAEs. One of these routines has been used to simulate a power system composed of a nonlinear, three-phase, three-legged transformer, a load and all their possible connections. The magnetic transformer relations and the connections give some of the algebraic equations of the DAE system.

Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

This paper proposes notions and a tool for passivity properties of non-homogeneous switched Differential Algebraic Equation (DAE) systems and their relationships with stability and control design. Motivated by the lack of results on input-output analysis (such as passivity) for switched DAE systems and their interconnections, we propose to model non-homogeneous switched DAE systems as a class of hybrid systems, modeled here as hybrid DAE systems with linear flows. Passivity and its variations are defined for switched DAE systems and methods relying on storage functions are proposed. The main contributions of this paper are: 1) passivity and detectability concepts for switched DAE systems, 2) links of the aforementioned passivity and detectability properties to stabilization via static output-feedback. Our results are illustrated in a power system, namely, the DC-DC boost converter, whose model involves DAEs and requires feedback control.

Reachability analysis is a fundamental problem for safety verification and falsification of Cyber-Physical Systems (CPS) whose dynamics follow physical laws usually represented as differential equations. In the last two decades, numerous reachability analysis methods and tools have been proposed for a common class of dynamics in CPS known as ordinary differential equations (ODE). However, there is lack of methods dealing with differential algebraic equations (DAE), which is a more general class of dynamics that is widely used to describe a variety of problems from engineering and science, such as multibody mechanics, electrical circuit design, incompressible fluids, molecular dynamics, and chemical process control. Reachability analysis for DAE systems is more complex than ODE systems, especially for high-index DAEs because they contain both a differential part (i.e., ODE) and algebraic constraints (AC). In this paper, we propose a scalable reachability analysis for a class of high-index large linear DAEs. In our approach, a high-index linear DAE is first decoupled into one ODE and one or several AC subsystems based on the well-known Marz decoupling method utilizing admissible projectors. Then, the discrete reachable set of the DAE, represented as a list of star-sets, is computed using simulation. Unlike ODE reachability analysis where the initial condition is freely defined by a user, in DAE cases, the consistency of the initial condition is an essential requirement to guarantee a feasible solution. Therefore, a thorough check for the consistency is invoked before computing the discrete reachable set. Our approach successfully verifies (or falsifies) a wide range of practical, high-index linear DAE systems in which the number of state variables varies from several to thousands.

System of fractional differential algebraic equations with applications

A method is proposed to transform a system of differential algebraic equations (D.A.E.) to a system of ordinary differential equations (O.D.E.) which can be solved relatively easily by standard numerical techniques.Two examples, including a model of an absorption tower, are given to illustrate the utility of the method. The example problems reveal that this easily implemented technique offers significant savings in CPU time compared to the numerical solution of the untransformed D.A.E.s particularly when the algebraic equations are nonlinear.Furthermore, it appears to be faster and/or more reliable than other numerical schemes which have been recently developed for equations of this type.

The first attempt to improve battery performance through model-based design optimization was made by W. Tiedemann and J. Newman in 19751. They varied the electrode thickness and porosity and then observed the effects on the effective capacity. Newman later applied the reaction-zone model to maximize the specific energy of the system using the same varying and observing method2. V. Ramadesigan et al.3developed optimal spatially varying porosity for porous electrodes by considering ohmic resistance model in the electrode. They minimized the ohmic resistance in the positive electrode by varying porosity using control vector parameterization (CVP) method. They also introduced the idea of using graded electrode to further improve the performace. Since the introduction of the pseudo-2D (P2D) model by the Newman group, many efforts have been made in utilizing the P2D model in battery design. W. Du et al. applied an optimization framework based on surrogate method4. S. Golmon et al. performed the multi-objective and multi-design-parameter optimization problem with adjoint sensitivity analysis5 incorporating mechanical stress-strain relationship. S. De and his coworkers6 used a reformulated model, to perform simultaneous optimization of multiple design parameters. N. Xue et al. 7 applied the gradient-based algorithm framework to optimize the cell design. More recently, Y. Dai and V. Srinivasan8used a gradient-free direct search method to maximize the specific energy. Z. Du et al. studies the effects of various design parameters on thick electrode performance9. Most of the aforementioned optimization research, used the sequential optimization approach, also known as CVP. CVP takes the differential algebraic equation (DAE) system describing the battery physics, applies a certain nonlinear programming (NLP) solver to them, discretizes only the control variables, and solves the model. The only exception being the effort by Golmon et al which used an indirect approach. Note that as of today, global optimization cannot be guaranteed when CVP type approaches are used with P2D type models. In general optimization approaches can be classified as indirect and direct approach. In the direct approach there are two main categories – sequential and simultaneous approach. The simultaneous approach10, on the other hand, discretizes both the control variables and the design variables before solving the problem. When used for optimization, the DAE system will only be solved once at the optimal point, compared to the repeated numerical integration needed for sequential optimization. By using high order discretization scheme on both the state and the design variables, simultaneous optimization approach results in faster determination of the optimum with fewer time steps. Furthermore, this approach offers more flexibility over constraints on the state variables. Since the state variables are also discretized, it is possible to apply equality/inequality constraints on their internal values. In this work, we will first apply the simultaneous optimization approach to a resistance model similar to the one mentioned earlier3, and compare its performance with the sequential approach. We will also explore the application of the same optimization approach to the P2D model, and identify the challenges. It is viewed as a controversial topic in the literature wherein different authors disagree on the relevance of graded porosity for improved performance8. By using a simultaneous optimization approach and guaranteeing global optimization for convex problem statements, we hope to provide additional insights. Acknowledgements The authors are thankful for the financial support by the Clean Energy Institute at the University of Washington and the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium).

The modeling of a system composed by a gas phase and organic aerosol particles, and its numerical resolution are studied. The gas-aerosol system is modeled by ordinary differential equations coupled with a mixed-constrained optimization problem. This coupling induces discontinuities when inequality constraints are activated or deactivated. Two approaches for the solution of the optimization-constrained differential equations are presented. The first approach is a time splitting scheme together with a fixed-point method that alternates between the differential and optimization parts. The ordinary differential equations are approximated by the Crank-Nicolson scheme and a primal-dual interior-point method combined with a warm-start strategy is used to solve the minimization problem. The second approach considers the set of equations as a system of differential algebraic equations after replacing the minimization problem by its first order optimality conditions. An implicit 5th-order Runge-Kutta method (RADAU5) is then used. Both approaches are completed by numerical techniques for the detection and computation of the events (activation and deactivation of inequality constraints) when the system evolves in time. The computation of the events is based on continuation techniques and geometric arguments. Moreover the first approach completes the computation with extrapolation polynomials and sensitivity analysis, whereas the second approach uses dense output formulas. Numerical results for gas-aerosol system made of several chemical species are proposed for both approaches. These examples show the efficiency and accuracy of each method. They also indicate that the second approach is more efficient than the first one. Furthermore theoretical examples show that the method for the computation of the activation is of second order for the first approach and exact for the second one.

In his algebraic theory of differential equations, J. F. Rittf has developed a decomposition theory for systems of algebraic differential equations by introducing the idea of irreducible systems and proving that every system is equivalent to one and essentially only one finite set of irreducible systems.The analogous theorem for algebraic difference equations was given by Ritt and Doob.fThe purpose of the present paper is to derive a decomposition theorem for algebraic mixed difference equations; i.e., equations which contain algebraically one or more unknown functions y(x), their "transforms" y(x), y(x+l), y(x+2), ---, and the derivatives of those transforms.

Modelica is a new object-oriented multi-domain modelling and simulation language used for solving large, complex and heterogeneous physical systems with differential algebraic equations (DAEs). Presently, there is no simulation support available in Modelica for solving partial differential equations (PDEs) problems. In this paper, based on the object-oriented idea, we present a method of consistent representation of PDE and DAE problems. The description of PDE problems in Modelica is introduced. Then the PDE model is transferred into DAE model with the method of lines, and is solved in MWorks simulation platform based on Modelica. To illustrate the application of our method, a heat conduction problem is modelled and simulated in Modelica.

Supporting the generation of a state space model by adding tearing information to the bond graph

The multi-agent optimal control problem involves a decision process with multiple agents, where each agent solves an optimal control problem with the individual cost functional and strategy set, and the cost functional is dependent on all the other agents’ state and/or control variables. Here the “agent” can be understood as a true decision maker, or as an abstract optimization criterion. The strategy sets, along with admissible control set, are often described by a system of parameterized ordinary differential/difference equations (the state dynamic) or partial differential equations, and in realistic settings they may be dependent on the rivals’s variables due to, for example, certain constraints from the common resources. This chapter describes the multi-agent optimal control problem, and studies the reformulation of a system of differential equations constrained by parameterized variational inequalities, along with some initial and/or boundary conditions. This reformulation presents differential equations, variational inequalities, and equilibrium conditions in a systematic way, and is advantageous since it can be treated as a system of differential algebraic equations, for which abundant theory and algorithms are available.KeywordsVariational InequalityOptimal Control ProblemDifferential Algebraic EquationQuasi Variational InequalityNash Equilibrium ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the modelling of actuator dynamics and the computation of prescribed trajectories