The interaction of system structure, index, and numerical stability in classes of differential/algebraic systems

Examples of differential/algebraic equation (DAE) systems are presented which arise in power systems and nonlinear circuit analysis, and the analytical and numerical challenges that these systems pose are considered. The author discusses the change in the system index due to the inclusion of a parameter limiter, and a numerical instability which may arise when systems of DAEs are simulated at points near dynamic bifurcation. The inclusion of limiters is shown to increase the index of the system. It is shown that, if the system is solvable, there exists at least one stepsize such that the simultaneous Jacobian is invertible and thus no numerical instabilities exist. A stepsize-dependent instability is analyzed. It is shown how the dynamic behavior of the system greatly affects this instability, and a few suggestions to eliminate the instability are presented