Chapter 7
Construction of Numerical Energy Functions for Lossy Transient Stability Models
7.1 INTRODUCTION
The existing numerical energy functions for lossy power system stability models are not analytical in the sense that they are not well-defined functions. They all contain two major terms: the analytical terms (i.e., path-independent terms) and the path-dependent terms. The path-dependent terms are not well-defined functions and require numerical approximations to become well-defined. The so-called ray approximation scheme and the trapezoidal approximation scheme are popular schemes for numerically approximating the path-dependent terms (Fouad and Vittal, 1991; Pai, 1989; Pavella and Murthy, 1994; Qiang and Zhong, 2005; Sasaki, 1979).
In this chapter, methodologies for constructing numerical energy functions for lossy power system stability models will be presented. At present, only two methods are available. One is based on the so-called first integral principle, while the other is based on a two-step procedure. Numerical energy functions derived by these two methods all contain path-dependent terms which need to be evaluated by numerical approximation schemes.
It will be shown that the so-called ray approximation scheme leads to a numerical ill-conditioned problem during the numerical approximation of path-dependent terms. A scheme to eliminate this numerical ill-conditioning will be presented. However, even after this elimination, the ray approximation scheme can give ...
