# Stability Regions for Differential-Algebraic Systems

Chapter

## Abstract

Many problems in the theory of power systems such as voltage stability can be modeled by a parameter dependent differential-algebraic system of the form In the state space

$$\dot x = f(x,y,p),f:{R^{n + m + p}} \to {R^n}$$

(1)

$$0 = g(x,y,p),g:{R^{n + m + p}} \to {R^m}$$

(2)

$$x \in X \subset {R^n},y \in {R^m},p \in P \in {R^p}$$

*X*×*Y*dynamic state variables*x*and instantaneous state variables*y*are distinguished. Typical dynamic state variables are the time dependent values of generator voltages and rotor phases, instantaneous variables are bus voltages and other load flow variables. The parameter space*P*is composed of system parameters (which describe the system topography, i.e. which lines, buses etc. are energized, and equipment constants such as inductances, transformation ratios etc.), and operating parameters (such as loads, generation, voltage setpoints etc.) Both of these spaces have distinct structural features, sometimes intertwined. Understanding these structures is essential in analysing the voltage dynamics of the power system and of great practical value in the operation [10,12]. The special nature of the dynamics of differential-algebraic equations was recognized and briefly explored by Takens [6,7]. In this paper we make a first attempt at analyzing the structural features of such systems in the state space giving a definition of region of stability for a differential-algebraic system and describing the structure of the stability boundary. Our definition is strongly motivated by qualitative features of power system models for voltage stability, but the results are applicable to any differential-algebraic system with similar features.### Keywords

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