Small signal stability is the ability of the power system to maintain synchronism when subjected to small disturbances (Kundur, 1994). In this context, a disturbance is considered to be small if the equations that describe the resulting response of the system may be linearized for the purpose of analysis. In order to analyze the effects of a disturbance on a linear system, we can observe its eigenvalues. Although power system is nonlinear system, it can be linearized around a stable operating point, which can give a close approximation to the system to be studied.
The behavior of a dynamic autonomous power system can be modelled by a set of n first order nonlinear ordinary differential equations (ODEs) described as follows (Kundur, 1994)
where x is the state vector; u is the vector of inputs to the system; g is a vector of nonlinear functions relating state and input variables to output variables.
The equilibrium points of system are those points in which all the derivatives x1, x 2,..., xn are simultaneously zero. The system is accordingly at rest since all the variables are constant and unvarying with time. The equilibrium point must therefore satisfy the following equation dx dX0 = f(xo, uo) = 0 (49)
Where (x0, u0) are considered as an equilibrium point, which correspond to a basic operating condition of power system.
Corresponding to a small deviation around the equilibrium point, i.e.
The functions f(x,u) and g(x,u) can be expressed in terms of Taylor's series expansion dx° + dAx = f(x0 ,u0) + AAx + BAu + O(|| Ax, Auf) (53)
dt dt u 0 + Aii = g(x0,u0) + CAx + DAu + O(H Ax, Auf) (54)
With terms involving second and higher order powers in Eqs(53-54) neglected, we have
Where A, B, C and D are called as Jacobian matrices represented in the following
If matrix D is nonsingular, finally we have dAx
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