Contraction Analysis of Nonlinear DAE Systems

Abstract

This paper studies the contraction properties of nonlinear differential-algebraic equation (DAE) systems. Specifically we develop scalable techniques for constructing the attraction regions associated with a particular stable equilibrium, by establishing the relation between the contraction rates of the original systems and the corresponding virtual extended systems. We show that for a contracting DAE system, the reduced system always contracts faster than the extended ones; furthermore, there always exists an extension with contraction rate arbitrarily close to that of the original system. The proposed construction technique is illustrated with a power system example in the context of transient stability assessment.

A Sufficient Condition for Small-Signal Stability and Construction of Robust Stability Region

“…The proof follows closely the reasoning to prove Theorem 1 for contraction analysis in [13]; however, the generalized reduced and unreduced Jacobian are slightly different from what we define in this paper. Note also that here we focus on the original variables δz = (δx T , δy T ) T instead of their transformed counterparts δw = (δv T , δu T ) T in contraction analysis.…”

“…i ≤ n and zero otherwise. In the power system, the structurepreserving DAE model is gaining importance in eigenvaluebased stability analysis [16], [8] or for contraction region [13]. To obtain the power system equivalent of the DAE system (1) we define a block Jacobian as:…”

“…We work on the Jacobian J, which can be expressed as an affine function of the system variables. This is obtained by expressing the DAEs in quadratic form of variables z [13]. Thus, the system can be expressed as J(z) = J 0 + k z k J k for k = 1 .…”

“…1. A 2-bus test system [13] sensitivity are discussed above. In this section, we illustrate the procedure by constructing the stability region for a twobus system as shown in Figure 1 [13].…”

“…This work is a follow-up to the contraction analysis presented in [13]. In this paper, a relation between the reduced form and the associated DAE system has been established using logarithmic norm.…”