A Generlal Method for Small Signal Stability Analysis Yuri

I. I N T R O D U C T I O N Modern 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 dyPE-153-PWRS-16-09-1997 A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for publication in the IEEE Transactions on Power Systems. Manuscript submlitted May 27, 1997; made available for printing September 30, 1997. namics is required. For example, the models given in [4] [ll] can be used. They include generator and excitation system differential equations, stator and network algebraic equations. These equations build up the set of differentialalgebraic equations (1) X l = F ( Z l 1 2 2 , Y , T ) 0 = G ( X I , Z Z , Y , T ) (1) In the equation (l), 21 is the vector of state (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 (which follows the structure given in [12]), where J l f stands for the load flow Jacobian, J11 = d F / d x l , Jla = d F / d x z , J z ~ = dG/dx l and Jzz = d G / d x z are different parts of J corresponding to differential and algebraic variables. In Fig. 1, Qgen stands for the reactive power at generator buses, Psb is the active power at the swing bus, 6 is the vector of machine rotor angles, w is the vector of machine speeds, K is the vector of the state variables except 6 and w (such as E;, E&, E f d , VR, and RF; load bus voltages K o a d and angles @load should be considered as dynamic state variables in cases where load dynamics is considered [la]), id and i, are vectors of d-axis and q-axis currents; v,,, and N o a d stand for generator and load bus voltages; d s b is the swing bus voltage angle, and B denotes voltage angles at all buses except the swing bus. The prefix A means a small increment in corresponding variables. 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" = J11 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 the algebraic submatrix Jzz see Fig. 1, and 0885-8950/98/$10.00