A stability theory for constrained dynamic systems with applications to electric power systems

This paper develops a stability theory for constrained dynamic systems which are defined as dynamic systems whose state trajectories are restricted to a particular set within the state space called the feasible operating region, Many physical systems may be modeled as constrained dynamic systems because certain variables or functions of variables are often required to remain within acceptable ranges, A restricted stability region for constrained systems is defined as the set of points whose trajectories start and remain within the feasible operating region for all t greater than or equal to 0. The stability analysis is restricted to this region as trajectories hitting the boundary are infeasible and considered to be unstable, We also define the restricted asymptotic stability region ol restricted domain of attraction as the set of points whose trajectories start and remain within the feasible operating region for all t greater than or equal to 0 and converge to the stable equilibrium point as t --> infinity. In [4]-[6], Venkatasubramanian et nl, characterized the stability boundary for differential-algebraic-equation (DAE) systems which are dynamic systems with algebraic equality constraints, In [8], we used their results and some results from bifurcation theory to show that for DAE systems, parts of the stability boundary are formed by trajectories that are tangent to the boundary of the solution sheet on which the algebraic equations have a unique solution, In this paper, using similar techniques as contained in [4], we develop stability results for systems that are constrained to remain within a subset of the state space (i.e., the feasible operating region), This modeling framework is particularly useful for representing systems with inequality constraints, Also, for systems with equality constraints, we can develop an approximate model that is arbitrarily close to the original system, The main theoretical result of this paper is a characterization of the boundary of the restricted asymptotic stability region (i.e., quasistability boundary), Specifically, me show that the quasistability boundary includes trajectories that are tangent to the boundary of the feasible operating region, Our primary application of these results is analyzing electric power system stability following the occurrence of a network fault, We assume that the electric power protection system operates to clear the fault condition but that the post-fault trajectories should remain within the feasible operating region of the new system configuration, A computational procedure was developed in [8] to estimate the critical clearing time for a network fault based on the tangent trajectory results, This method is also extended in this paper to include the more general constrained dynamic system representation, An example is then included to illustrate the use of this computational method in estimating the critical clearing time for a network fault.