A general method for small signal stability analysis

Abstract

A b s t r a c t -This paper presents a new general method for computing the different specific power system small signal stability conditions. The conditions include the points of minimum and maximum damping of oscillations, saddle node and Hopf bifurcations, and load flow feasibility boundaries. All these characteristic points are located by optimizing an eigenvalue objective function along the rays specified in the space of system parameters. The set of constraints consists of the load flow (equations, and requirements applied to the dynamic state matrix eigenvalues and eigenvectors. Solutions of the optimization problem correspond to specific points of interest mentioned above. So, the proposed general method gives a Comprehensive characterization of the power system small signal stability properties. The specific point obtained depends upon the initial guess of variables and numerical methods used to solve the constrained optimization problem. The technique is tested by analyzing the small signal stability properties for well-known example systems. I. I N T R O D U C T I O NModern power grid:; are becoming more and more stressed with the load demands increasing rapidly. The voltage collapses which occurred recently have again drawn much attention to the issue of stability security margins in power systems [I]. The small signal stability margins are highly dependent upon such system factors as load flow feasibility boundaries, minimum and maximum damping conditions, saddle node and Hopf bifurcations, etc. Unfortunately, it is very difficult to say in advance which of these factors will make a decisive contribution to instability. Despite the progress achieved recently, the existing approaches deal with these factors independently -see [a], [3] for example, and additional attempts are needed to get a more comprehensive view on small-signal stability problem.To study the power system small signal stability problem, an appropriate model for the machine and load dy- (differential) variables, x~ is the vector of algebraic variables, y is the vector of specified system parameters, and T is a parameter chosen for bifurcation analysis. In many cases y is a function of T.In the small signal stability analysis, the set (1) is then linearized at an equilibrium point to get the system Jacobian and state matrix. The structure of the system Jacobian J is shown in Fig. 1 The problem addressed here is that these different small signal stability conditions correspond to different physical phenomena and mathematical descriptions [13]. Saddle node bifurcations happen where the state matrix J" = J 1 1 -JlzJ,-,lJzl becomes singular and, for example, a static (aperiodic) type of voltage collapse or angle instability may be observed as a result. Hopf bifurcations occur when the system state iiiatrix J has a pair of conjugate eigenvalues passing the imaginary axis while the other eigenvalues have negative real parts, and the unstable oscillatory behavior may be seen. Singularity induced bifurcations are caused by singularity of t...

Solving Multi-Objective Voltage Stability Constrained Power Transfer Capability Problem using Evolutionary Computation

“…In general, the equilibrium points of the dynamic model, as shown by Equation (1), can be represented by a simplified model, which can then be expressed as follows (IEEE/PES 2002;Goh et al 2007;Dong et al 2005;Makarov et al 1998;Makarov et al 2000):…”

“…The voltage collapse margin can be obtained by computing the distance towards voltage collapse boundaries from a stable equilibrium operating point as denoted by  , i.e. Dong et al 1998;Dong et al 2005;Makarov et al 1998;Makarov et al 2000;Canizares 1998). In this paper, this voltage stability margin,   , will be used to evaluate voltage stability when quantifying inter-area power transfer capability, i.e.…”

“…These equations can be solved using the Newton-Raphson-Seydel method or optimization approaches (Dong et al 2005;Makarov et al 1998;Makarov et al 2000;Canizares 1998), or even EC techniques as shown in Goh et al 2007). The voltage collapse margin can be obtained by computing the distance towards voltage collapse boundaries from a stable equilibrium operating point as denoted by  , i.e.…”

“…The power system model used in stability studies can be expressed by the following differential-algebraic equations (DAE) (IEEE/PES 2002;Goh et al 2007;Dong et al 2005;Makarov et al 1998;Makarov et al 2000):…”

Interior point based optimal voltage stability, oscillatory stability and ATC margin boundary tracing

“…Although there are many papers related to oscillatory stability problems, most of this literature is concentrated either on identifying dominant eigenvalues [50] [51] or predicting the Hopf bifurcation point directly [43,52,53,54,55]. To the best of our knowledge, nobody has yet made an attempt to develop an optimal control strategy to achieve a specific oscillatory stability margin beyond the initial Hopf bifurcation point.…”