Local approximation of stability boundary of a power system using the real normal form of vector fields

Abstract

The objective of this work is to approximate the stability boundary of a power system without any integration using the normal form of the vector fields. This involves two steps: 1) first t o test which unstable equilibrium point (UEP) lies on the stability boundary, and 2) the second step is to approximate the boundary by the second order approximated manifolds. The approximation is accomplished using the normal forms of vector fields. T h e stability boundary and its behavior under stressed system conditions are examined. T h e method is applied to an 11 generator test system. W"() X E? I. I N T R O D U C T I O N A . Sfability BoundaryThe region of stability of a stable equilibrium point (SEP) is the region within which a system trajectory starting will converge to the SEP. This region is bounded by the union of the stable manifolds of the unstable equilibrium points (UEP's) on the boundary [1, 21. For an autonomous system, this is mathematically stated as follows:where the vector field f maps RN into RN and is continuously differentiable. In our present formulation f is of type C". A point, x is called an equzlibrium point or a fixed point of equation (1) if f(i) = 0. T h e solution curve of equation (1) starting at t = 0 is called a trajectory, and is denoted by $(x, .). Let 2 be a hyperbolic equilibrium point. Its stable and unstable mamfolds, Ws(x) and W"( i) , are defined as W"2) = ( 2 : 4(x1 t ) -+ x as t -+ C O } (2) Figure 1: Eigenspace and local invariant manifolds of a nonlinear system B. Approxzmation of the Invariant Manifolds at the Direct stability analysis involves calculation of a value of critical potential energy against which transient stability assessment is made. T h e UEP of interest is called the controlling UEP and its computation is one of the key steps in power system transient stability assessment by the transient energy function (TEF) method [3]. Thus,we can assume that computation of (at least some of) the UEP's on the stability boundary is feasible.Normal form of vector fields [4] provides the means by which a differential equation may be transformed into a "simplified" form by eliminating higher-order terms. This is done using a nonlinear variable transformation performed on the Taylor series expansion of the power sysUEPs 1 (3) tem's nonlinear differential equations. In this work only second-order terms are used, giving a second order approximation of the nonlinear system, W"(2) = {.c : q 5 ( X l t ) + 2 as t --+ -02

On expansion of estimated stability region: Theory, methodology, and application to power systems

“…In the second iteration, the results of both algorithms have the same tendency except for a slight difference caused by the high-order terms omitted in eq. (6).…”

“…Refs. [6][7][8][9][10] used the normal form transformation to convert the original system to a linear system in a neighborhood of the CUEP. Because the stable and unstable sub-manifolds of the linear system can be obtained easily, the approximation of stability region boundary can be determined by converting the obtained stable manifold to the original coordinate.…”

Transient stability and emergency control

“…There are a good deal of methods for stability margin assessment, such as CUEP [9,22] , BCU [11] , stability region boundary approximation [23] , EEAC [24,25] , etc. Almost all the methods are in the light of the information of the critical point on the stability boundary denoted by x u , SEP x s and the initial state x 0 .…”

Approximation of stability boundary of a power system using the real normal form of vector fields

“…where, 6^ is the rotor euigle of i-machine Now we present the formulation for the real normal form transformation [25].…”