Introduction
1.2 The purpose and features of the book
The main purpose of this book is to introduce the graduate engineer to the concepts and applications of small-signal analysis and controller design for the enhancement of the dy- namic performance of multi-machine power systems. To this end, the analyses of the con- trol and dynamic performance are illustrated by examples based on an interconnected high- voltage system comprising fourteen generating stations and various types of FACTS devic- es. An emphasis in the book is on more recent theoretical developments and application to practical issues which are amenable to small-signal analysis using a comprehensive software package. In addition, the tools in - and features of - such a software package for analysis and controller design are illustrated.
The aim and features of the book are illustrated in the following summary.
1. In the following chapters it is assumed that the reader has already been introduced to the basics of: (i) the steady-state and dynamic performance of power systems [2], [3], [4], [5], [6], [7], [8] and (ii) control system theory [9], [10]. However, because control- ler design and tuning are described in later chapters of this book, Chapters 2 and 3 are devoted particularly to those aspects of basic control theory and the associated analysis and design techniques which are employed in later. Practical insights and lim- itations in control theory and analysis are emphasized in order to isolate that material which is important for application in later chapters.
2. For the practical design of robust power system stabilizers (PSSs), a tuning approach based on the generator P-Vr characteristics, which are a development of the GEP Method for single generator applications, is applied to multi-machine systems [11], [12]. The uses of the P-Vr characteristics are (a) analysed in detail in Chapter 5 for a single generator power system in order to explain the features of the P-Vr Method
Sec. 1.2 The purpose and features of the book 3 and its limitations; (b) applied in Chapters 9 and 10 to the multi-machine system over a range of operating conditions.
3. It is shown in Chapters 5 and 10 that the rationale in the tuning of PSSs based on the P-Vr method is that:
(i) there are two important components in the PSS transfer function which are essentially decoupled for practical purposes1:
(a) the rotor modes are more-or-less directly left-shifted2 by the PSS compensating transfer function with increase in the PSS damp- ing gain, k3;
(b) the extent of the left-shift of the rotor modes is determined by the damping gain, k;
(c) the incremental left-shifts of the rotor modes are linearly related to increments in damping gain for changes about the nominal values (typ- ically to pu on generator MVA rating) [11];
(ii) the tuning of the PSS is based on a more extreme set of encompassing operat- ing conditions;
(iii) the PSS damping gain has special significance: it is also the damping torque coefficient induced by the PSS on the shaft of the generator. It forms the basis for the theoretical developments in Chapters 5, 10, 12 to 14.
(iv) the PSS damping gain can be adjusted to ‘swamp out’ any inherent negative damping torques;
(v) as a result of (i), (ii) and (iv) above, the PSS transfer function is robust over the encompassing range of normal and outage operating conditions [12];
(vi) the PSS damping gain, when expressed in per unit on generator MVA rating, is a meaningful quantity, unlike the term “PSS gain” currently used. PSS damping gains less than 10 pu are low, are normal between 10 and 30 pu, and greater than 30 pu tend to be high.
As opposed to the application of advanced control techniques, the signifi- cance of the above rationale is that the natural characteristics of the generator and the system are employed and thus meaningful insight and explanations for the dynamic behaviour of the system can be established.
1. The features described in items (i)(a) and (i)(b) are illustrated for six operating conditions in Figure 10.26
2. By ‘direct left-shift’ is implied that the mode shift is , . As explained in Chapter 13, deviations from the ‘direct left-shift’ of modes are mainly due to interactions between multi-machine PSSs and non-real generator participation factors.
3. The ‘damping’ gain of the PSS is defined in Section 5.4.
kG s
– j0 0
G s
5 10
kG s
4 Introduction Ch. 1 4. Methods other than the P-Vr method for the design of PSSs, namely the commonly- used Method of Residues [13] and the GEP Method [14], are described in Chapter 6.
By means of an example the merits, deficiencies and limitations of the latter Meth- ods and the P-Vr approach are examined [15]. (See item 4 in Section 6.7.)
5. Various concepts and methods for the tuning of automatic voltage regulators (AVRs) are introduced and examined in Chapter 7. Some simplifications in the approaches to the commonly-used techniques are suggested.
6. A more fundamental and detailed examination is undertaken - than previously con- ducted - to explain, and understand more fully, not only the performance of certain devices but also the theory behind certain tools. Examples are: (a) the performance of the ‘Integral of accelerating power PSS’ in Chapter 8; (b) the characteristics of two tools, Mode Shapes and Participation Factors in Chapter 9 (these are used in the analysis of the performance of multi-machine systems).
7. The tuning of power oscillation dampers for FACTS devices (PODs, also referred to as FACTS Device Stabilizers (FDSs)) is described in Chapter 11. Some of the prob- lems encountered in the design are revealed in the case of a multi-machine system for a wide range of operating conditions [16], [17].
Due to the short-comings of existing techniques for the tuning of FACTS Device Sta- bilizers, their robustness is more difficult to achieve compared the tuning of PSSs for robustness (see item 3 above).
8. The concept, theory, and calculation of Modal Induced Torque Coefficients (MITCs) are described in Chapter 12. In this chapter the synchronizing and damping torques induced on a generator by both PSSs and FDSs at the modal frequencies are explained; this chapter provides the theoretical basis for Chapter 13 [16], [17].
9. The interactions between, and effectiveness of, PSSs and FDSs in a multi-machine power system are analysed in Chapter 13. A valuable aid in establishing the relative effectiveness of stabilizers are the Stabilizer Damping Contribution Diagrams (SDCDs) [18]. Extending the concepts introduced in Chapter 13 the SDCDs form a basis for the heuristic coordination of power system stabilizers and FACTs device stabilizers. Both the latter approach as well as an optimization approach based on linear programming are illustrated in Chapter 14 and [20].
10. A comprehensive set of the various small-signal models of synchronous generators and FACTS devices are provided in Chapter 4. These models are intended to be embedded in a set of differential and algebraic equations (DAEs) which are employed to take advantage of sparsity in the system equations [21], [22].
11. The practical theme throughout this book is based on consulting projects for indus- try and queries raised by industry on practical problems that they have encountered.
Many of the queries relate to some lack of understanding of the theoretical or practi- cal backgrounds to the issues raised.
Sec. 1.3 Synchronizing and damping torques 5 An aim of the design of controllers is to enhance the damping of the rotor modes of oscil- lation, either to stabilize unstable oscillations or to ensure that the damping criteria for the power system are satisfied. Therefore the concepts of synchronizing and damping torques [23] - which operate on the shafts of generating units - are introduced in Section 1.3, fol- lowed by the concepts and definitions of stability in Section 1.4.