Introduction

Purpose and aims of the chapter

As emphasized in the Section 1.1 the equations describing an electric power system and its components are inherently non-linear. The equations contain non-linearities such as the product of voltage and current, functional non-linearities such as sine and cosine, and nonlinear characteristics such as magnetic saturation in machines. The analysis of dynamic systems with non-linearities is complex, particularly for power systems which are large and have a variety of non-linear elements. On the other hand, in the case of linear control systems, there is a comprehensive body of theory and a wide range of techniques and tools for assessing both the performance and stability of dynamic systems.

For small-signal analysis of power systems, the non-linear differential and algebraic equations are linearized about a selected steady-state operating condition. A set of linear equations in a new set of variables, the perturbed variables, result. For example, on linearization, the non-linear equation y = f(x1, x2, …, xn) = f(x) becomes a linear equation in the perturbed variables, Δy = k1Δx1 + k2Δx2 + …+ knΔxn, at the initial steady-state operating condition Y0, X10, X20…, Xn0. The constant coefficients ki depend on the initial condition. The question now is: how does the assessment of stability and dynamic performance based on the analysis of the linearized system relate to those aspects of the non-linear system? As also mentioned earlier, a theorem by Poincaré states that information on the stability of the non-linear system, based on a stability analysis of the linearized equations, is exact at the steady-state operating condition selected. However, information on the variable xi = Δxi + Xi0 becomes exact only as Δxi → 0. That is, for practical purposes, the perturbations must be small - typically a few percent of the steady-state value.

Small-signal analysis of power systems, based on the linearized dynamic equations, provides a means not only of assessing the stability and the damping performance of the system (through eigenanalysis and other techniques), but also for designing controllers and determining their effectiveness. The various applications of small-signal analysis in the field of power systems dynamics and control are the subjects of later chapters. The purpose of this chapter is to introduce and extend some of the concepts in linear control theory, analysis and design which are particularly relevant to understanding of later material.