Block method of Runge Kutta type for solving differential algebraic equation

2015 ◽  
Author(s):  
Khoo Kai Wen ◽  
Zanariah Abdul Majid ◽  
Norazak Senu
Keyword(s):  
Algebraic Equation ◽  
Differential Algebraic Equation ◽  
Block Method ◽  
Runge Kutta

Solving Differential-Algebraic Equation Systems: Alternative Index-2 and Index-1 Approaches for Constrained Mechanical Systems

2015 ◽  
Vol 11 (4) ◽  
Author(s):  
Bernhard Schweizer ◽  
Pu Li
Keyword(s):  
Algebraic Equation ◽  
Mechanical Systems ◽  
Order Reduction ◽  
Integration Time ◽  
Differential Algebraic Equation ◽  
Constrained Mechanical Systems ◽  
Dae Systems ◽  
Index 2 ◽  
Runge Kutta ◽  
Hidden Constraints

Regarding constrained mechanical systems, we are faced with index-3 differential-algebraic equation (DAE) systems. Direct discretization of the index-3 DAE systems only enforces the position constraints to be fulfilled at the integration-time points, but not the hidden constraints. In addition, order reduction effects are observed in the velocity variables and the Lagrange multipliers. In literature, different numerical techniques have been suggested to reduce the index of the system and to handle the numerical integration of constrained mechanical systems. This paper deals with an alternative concept, called collocated constraints approach. We present index-2 and index-1 formulations in combination with implicit Runge–Kutta methods. Compared with the direct discretization of the index-3 DAE system, the proposed method enforces also the constraints on velocity and—in case of the index-1 formulation—the constraints on acceleration level. The proposed method may very easily be implemented in standard Runge–Kutta solvers. Here, we only discuss mechanical systems. The presented approach can, however, also be applied for solving nonmechanical higher-index DAE systems.

scholarly journals Testing a differential-algebraic equation solver in long-term voltage stability simulation

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
José E. O. Pessanha ◽  
Alex A. Paz
Keyword(s):  
Power System ◽  
Algebraic Equation ◽  
Time Domain ◽  
Voltage Stability ◽  
Power System Stability ◽  
System Stability ◽  
Variable Order ◽  
Differential Algebraic Equation ◽  
Short Period

This work evaluates the performance of a particular differential-algebraic equation solver, referred to as DASSL, in power system voltage stability computer applications. The solver is tested for a time domain long-term voltage stability scenario, including transient disturbances, using a real power system model. Important insights into the mechanisms of the DASSL solver are obtained through the use of this real model, including control devices relevant to the simulated phenomena. The results indicate that if properly used, the solver can be a powerful numerical tool in time domain assessment of long-term power system stability since it comprises, among several important features, suitable and very efficient variable order and variable step-size numerical techniques. These characteristics are very important when CPU time is a great concern, which is the case when the power system operator needs reliable results in a short period of time. Prior to the present work, this solver has never been applied in power system stability computer analysis in time domain considering slow and fast phenomena.

On the Determination of Joint Reactions in Multibody Mechanisms

2004 ◽  
Vol 126 (2) ◽  
pp. 341-350 ◽  
Author(s):  
Wojciech Blajer
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Algebraic Equation ◽  
Matrix Inversion ◽  
Differential Algebraic Equation ◽  
Joint Coordinates ◽  
Reduced Dimension ◽  
Absolute Coordinates ◽  
Coordinate Partitioning

In this paper some existing codes for the determination of joint reactions in multibody mechanisms are first reviewed. The codes relate to the DAE (differential-algebraic equation) dynamics formulations in absolute coordinates and in relative joint coordinates, and to the ODE (ordinary differential equation) formulations obtained by applying the coordinate partitioning method to these both coordinate types. On this background a novel efficient approach to the determination of joint reactions is presented, naturally associated with the reduced-dimension formulations of mechanism dynamics. By introducing open-constraint coordinates to specify the prohibited relative motions in the joints, pseudoinverse matrices to the constraint Jacobian matrices are derived in an automatic way. The involvement of the pseudo-inverses leads to schemes in which the joint reactions are obtained directly in resolved forms—no matrix inversion is needed as it is required in the classical codes. This makes the developed schemes especially well suited for both symbolic manipulators and computer implementations. Illustrative examples are provided.

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