Introduction

Although this chapter is concerned with the application of a power system stabilizer (PSS) to a single-machine system, the concepts for the most part are applicable to multi-machine systems: such applications will be discussed in Chapters 9 and 10. Various important aspects of the tuning of the PSS can therefore considered in some detail because the analysis involves a simple system only.

The reasons for the wide-spread deployment of PSSs in power systems today are twofold, (i) to stabilize the unstable electro-mechanical modes in the system, (ii) to ensure that there is an adequate margin of stability for these modes over a wide range of operating conditions and contingencies, that is, the electro-mechanical modes are adequately damped. Some systems, such as the Eastern Australian grid, would be unstable without the use of both PSSs and stabilizers installed on certain FACTS devices.

A marginally stable electro-mechanical mode is oscillatory in nature and is very lightly damped. The frequency of rotor oscillations is typically between 1.5 to 15 rad/s, and the 5% settling time may be many tens of seconds. Typically a mode of a lengthy duration would not satisfy the system operator's criterion for modal damping. A stable mode is said to be ‘positively’ damped, whereas an unstable mode is referred to as being ‘negatively’ damped.

With the growth of power systems, and the need to transmit power over long distances by means of high-voltage transmission lines, the problems of instability following a major fault or disturbance have increased. Instability in such events is typically the result of a generator falling out of step due to insufficient synchronizing torques being available to hold generators in synchronism. In order to increase the synchronizing torques between generators, high-gain fast-acting excitation systems were developed with the objective of increasing field flux linkages rapidly during and following the fault. However, such high-gain excitation systems may introduce negative damping on certain electro-mechanical modes.

In linear control systems design, rate feedback is employed not only to stabilize an unstable system but also to enhance the system's damping performance. A PSS that uses generator speed (i.e. the rate of change of rotor angle) as a stabilizing signal is such a rate-feedback controller.

Introduction

In the 1990s the development of high power semiconductor devices found application in power electronic equipment in power systems. Such transmission systems and associated devices are generally known as Flexible AC Transmission Systems (FACTS); a comprehensive description of the technology, the devices and references to the literature are given in [1] (published in 2000).

In this chapter the tuning of stabilizers is outlined for FACTS devices such as Static Var Compensators (SVCs), the converters at the ends of High Voltage Direct Current (HVDC) transmission lines, Thyristor-Controlled Series Capacitor (TCSC), and other similar FACTS devices. Such stabilizers are generally known as Power Oscillation Dampers (PODs), however, the role of PSSs is also to act as power oscillation dampers - hence we will refer to PODs as FACTS Device Stabilizers (FDSs) to emphasize the application to FACTS devices.

Consider the studies for Cases 1 to 6 presented in the previous chapter. Referring to Tables 10.11, 10.15 and 10.16 it is noted that, for all PSSs in service with the damping gain set to 20 pu on machine MVA rating, the real parts of the mode shifts for the local-area modes typically vary from -1.3 to -2.5 Np/s over the encompassing range of operating conditions covered by the six cases. However, the real parts of the mode shifts for the inter-area modes, modes K, L and M, roughly vary over a much smaller range, from -0.4 to -1.1 Np/s for the same operating conditions. The damping of all modes in these cases is good, the lowest damping ratio being about 15%. However, because the damping of some modes may be poor, stabilizers installed on FACTS devices can provide a significant improvement in the damping of targeted modes. By reducing PSS damping gains to 5 and/or 10 pu on machine MVA ratings, cases of poorer damping are also examined in which the damping ratios of the inter-area modes are in the range 2 to 8%.

The common configuration of the FACTS device and controllers is shown in Figure 11.1. In the case of a Static Var Compensator (SVC), for example, the controller regulates the voltage at its terminals or at an electrically close, high-voltage busbar where voltage support is required [1], [2]. The location of the SVC in the network may be such that a stabilizer installed on the SVC is effective in improving the damping of certain inter-area modes.

Introduction

Purposes

Given a model and the parameters of the generator and its exciter, there is little published in the literature describing the various methods for the tuning of automatic voltage regulators (AVRs) to achieve certain performance specifications for the generator off- and on-line.

An aim of this chapter is to introduce and provide an analytical basis for various tuning methodologies, which provide a set of parameters for the particular AVR model. Further analysis may depend on the type and form of the AVR supplied by a manufacturer. However, even for complex AVR structures, the proposed methodologies may provide an initial set of parameters based on a simplified model of the AVR. Subsequent fine-tuning, based on the complex structure, can then yield an appropriate final set of parameters.

It should be emphasized that the tuning methodologies considered here are based on the concept of transient gain reduction, though various other design approaches are employed [1]. Depending on the type of AVR, rate-feedback may also be used to essentially effect a similar behaviour as transient gain reduction. Furthermore, more modern systems which employ proportional-integral-derivative (PID) controls can be tuned to give a response akin to transient gain reduction. It is recognized that manufacturers of the equipment have their own, effective procedures for tuning. However, when tuning, it is important in a number of scenarios to account for the power system characteristics over an encompassing range of normal and outage conditions. The latter considerations are often of concern to the transmission service provider (TSP) who may be responsible for system security. It is therefore desirable that staff in such TSPs understand the relevant methodologies and can undertake or validate, if necessary, the tuning of AVRs.

A further objective in the description of the methodologies is to provide for young engineers an introductory and a reference text which not only covers the relevant control systems background but also highlights the power system requirements and performance.

Coverage of the topic

Because powerful methods of analysis are available in linear control systems theory, the tuning of AVRs is based on small-signal analysis and the linearized models of the power system and associated devices. The performance of the resulting tuned AVR, and the other elements of the power system, should then be subject to simulation studies for an appropriate set of large-signal disturbances over the range of operating conditions.

Introduction

In this chapter the theoretical basis and a case study are used to illustrate the concepts of interactions between, and effectiveness of, PSSs and FDSs in a multi-machine power system. The theoretical relationships between the incremental modal induced torque coefficients (MITCs), the associated mode shifts, and increments in stabilizer gains are outlined. The case study will illustrate how the method developed for estimating rotor mode (eigenvalue) shifts can be used to assess the relative effectiveness of stabilizers and, thereby, gain some important insights which form a basis for the coordination of stabilizers [1], [2], [3], [4].

Techniques have been described in the literature for determining shifts in the modes of rotor oscillation due to changes in stabilizer parameters [5], [6], [7], [8]. These techniques have been used not only for determining optimal locations for PSSs and FACTS devices [6], [7], [8] but also for tuning PSS parameters [9], [10].

In this chapter the theory and analysis is used to:

1. • develop a new method, based on incremental MITCs, for estimating the mode or eigenvalue shifts;

2. • develop, for a given rotor mode, a method for estimating the contributions to damping of selected stabilizers for a selected increment in stabilizer gain, be they PSSs or FDSs;

3. • deduce the relative effectiveness of selected stabilizers in contributing to damping, say, of an inter-area mode;

4. • assess the effect of interactions between PSSs, particularly for inter-area and local modes;

5. • provide a basis for the systematic coordination of both PSSs and FDSs in multimachine systems [4].

Relationship between rotor mode shifts and stabilizer gainIncrements

Recall from Section 12.6 that the MTIC Tijh is the modal (complex) torque coefficient for the hth mode, λh, this torque coefficient being induced on the shaft of the ith generator by the jth stabilizer. It is established in (12.59) for both PSSs and FDSs that Tijh is dependent on the gain setting kj (in pu on device base) of stabilizer j. In Section 12.6.2 a relationship is developed between the incremental MITCs and increments in stabilizer gains; this is then employed in the following analysis to determine the eigenvalue shifts due to increments in any or all of the gains of the n PSSs and z FDSs.

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.

Introduction

The previous chapter introduced some important concepts in the tuning of PSSs in multi-machine power systems. The purpose of this chapter is to demonstrate the application of the associated techniques for the analysis and tuning of PSSs in a fourteen-generator power system which, without continuously acting PSSs, is inherently unstable. Each ‘generator’ in this system, in fact, represents a power station which accommodates between one and twelve units; the number of units in-service (nu) depends on the particular operating condition. The units in a power station are assumed to be identical, therefore the rating of the equivalent generator for a station is nu times the rating of a single unit. It is assumed that the individual generators in each power station are fitted with identical excitation systems and PSSs.

In a later chapter a class of stabilizers known as Power Oscillation Dampers (PODs) are discussed; these are stabilizers that can be fitted to power-electronic based transmission devices such as FACTS (e.g. Static Var Compensators) and HVDC transmission. The analysis and tuning of POD stabilizers are demonstrated by means of examples in Chapter 11. In the fourteen-generator power system described in this chapter the Static Var Compensators (SVCs) are fitted with continuously acting voltage regulators controlling bus voltage, but are not fitted with stabilizers.

The steps in the tuning of PSSs of machines in a multi-machine system are explored, commencing with (i) the eigen-analysis of the system with all PSSs out of service, and (ii) the associated analysis based on Mode Shapes and Participation Factors. The PSSs are then tuned using the P-Vr approach discussed in Section 9.4. Having completed the determination of the PSS parameters, the effect on the shifts of eigenvalues associated with the rotor modes are assessed as the damping gains of the PSSs are increased; ideally over the range of operating conditions such shifts are directly to the left in the complex s-plane.

In practice a new power station is built to supply energy to an existing power system in which many of the existing generators may already be fitted with PSSs. The latter PSSs would have been tuned and their parameters set to fixed values. The PSSs in a new power station have to be tuned to satisfy the damping and other performance criteria of the system operators over the range of system operating conditions and contingencies.

Introduction

In Section 5.8 the P-Vr method for the tuning of the PSS for a generator in a single-machine infinite-bus (SMIB) system is described. Several other methods, which will be shown to be somewhat related to the P-Vr method, are described in the literature. Two other methods will be discussed here, the first is based on Transfer-Function Residues, the second on the so-called GEP Method. The P-Vr method, the Method of Residues and the GEP Method are reconciled for a practical, multi-machine system in [1]. However, for illustrative purposes in this chapter we will examine only the application of the Residues and GEP Methods to a generator in a SMIB system.

The background to the Method of Residues is provided in [2] and its application to PSSs is illustrated in Appendix A of [3]. The method is also used in practice for the design of Power Oscillation Dampers (PODs) which are fitted to FACTS devices such as SVCs, typically to enhance the damping of inter-area modes. The design of PODs using the Method of Residues is described in [4], however, this topic is considered in more detail in Chapter 11.

Method of Residues

Theoretical basis for the Method

The theoretical basis, calculation and significance of the residues of a transfer function are discussed earlier in Section 2.5. In essence, for a set of distinct poles ri is the residue of the pole at s = pi. A transfer function G(s) is described by its partial fraction expansion equation (2.14), or by

The derivation of residues from the state equations is outlined in Section 3.7.

Consider a SMIB system for which a PSS is to be designed and installed. The transfer function from the reference voltage input to the speed output signal of the generator is GS(s)= Δ ω/ ΔVref. The PSS, with transfer function F(s), is a speed-input PSS (although other stabilizing signals can be employed). When operating in closed-loop the PSS output is connected to the AVR summing junction, as shown in Figure 6.1.

It is emphasized that the following simple approach to the determination of the compensation transfer function of the PSS is based on the change of the rotor mode of oscillation when the PSS feedback path is switched from open to closed loop.

Introduction

In Chapter 5 a speed-PSS based on the P-Vr design approach is described. The purpose of this chapter is to describe in detail the theoretical basis for some of the widely deployed types of PSSs and the associated practical implications. For some other types of PSSs, including the multi-path, multi-band PSS developed by Hydro-Québec, only a brief overview is provided. Furthermore, the details of a number of other types of PSSs and their development are omitted from this book, for example: delta-omega stabilizers (without and with torsional filters) [1]; the use of notch filters to attenuate the first torsional mode [2]; the application of the coordinated AVR/PSS, called the “Desensitized Four Loops Regulator” [3].

The input to the PSS in Chapter 5 is assumed to be the ‘true’ rotor speed as measured directly by a high-fidelity tacho-generator, a toothed wheel, or some other device mounted on the shaft of the turbine-generator unit. In practice there may be physical difficulties in positioning any such device on the shaft as well as locating it to minimize the introduction of the torsional modes of the shaft into the speed signal. Moreover, other difficulties such as noise, lateral shaft movement (runout or ‘wobble’ [4]) in vertical units, may present themselves. In this chapter, however, synthesized speed perturbations, which are assumed to accurately represent the true rotor speed perturbations, are used as the input to the PSS. This means that the same basis and procedure as that outlined in Chapter 5 can be employed for the design and tuning of the PSS.

The major factor in the selection of a stabilizing signal for input to the PSS is the requirement that the modes of concern, which may be the local-, inter-area, and possibly the intra-station modes, must be observable by the signal over a wide range of operating conditions. Typically, perturbations in rotor speed, the electric power output, and the frequency at the generator terminals are the commonly-used local signals.

Various types of pre-filters are in use which convert one or more signals derived from variable(s) other than speed into a synthesized speed signal. Such variables are electric power, bus-voltage angle, frequency, terminal voltage and current; some manufacturers develop a ‘speed’ signal from such variables using various techniques.

Introduction

In this chapter various models of synchronous generators, FACTS devices and of the power system are developed in forms which are employed in software for the analysis of the small-signal dynamic performance of multi-machine systems. Small-signal models for the synchronous generator are formulated in Section 4.2. An essential feature of this analysis is that the higher order coupled-circuit representation of the generator electromagnetic dynamic behaviour is formulated in Section 4.2.3. This is treated as the fundamental model from which the following two alternative but equivalent formulations of the electromagnetic model are derived. The first is the Operational Parameter formulation described in Section 4.2.12. The second, described in Section 4.2.13, is referred to as the Classical Parameter formulation and is expressed directly in terms of the classically-defined standard parameters of the generator. The Classical Parameter formulation is presented because it is employed in widely used power system simulation software packages such as Siemens PTI PSS®E [1] and GE PSLF™ [2]. The parameters for the fundamental coupled-circuit formulation are the resistances and inductances of the d- and q-axis circuits. The parameters for the Operational Parameter representation of the electromagnetic equations are the gains and time constants of the transfer-function representations of the respective axes and are collectively referred to as the ‘exactly-defined standard parameters’. The Classical Parameter formulation requires the classically-defined standard parameters. The relationship and conversion between the three parameter sets are outlined in Section 4.2.14.

Small-signal models of a range of FACTS devices are formulated in Section 4.3 and include those of the Static VAR Compensator (SVC), Voltage Sourced Converter (VSC), Static Synchronous Compensator (STATCOM), and HVDC transmission links. The general purpose VSC model formulated in Section 4.3.3 is used as a component in the simplified STATCOM model in Section 4.3.4 as well as for the rectifier and inverter in the model of the VSC HVDC transmission link in Section 4.3.7. A general model for a voltage-commutated thyristor- controlled AC/DC converter is formulated Section 4.3.8; this model is then used in a modular fashion to represent the rectifier and inverter of a line-commutated HVDC transmission link. A methodology to formulate the small-signal equations of the power system is described in Section 4.4. Finally, in Section 4.5 a general purpose small-signal representation of a static load model is described.

Introduction

Various techniques have been reported in the literature for the coordination of PSSs in multi- machine power systems [1], [2], [3], [4]. Some of these techniques have used linear programming solutions for coordinating PSS gains [5], [6]. However, little attention has been given to the simultaneous coordination of PSSs and FDSs [7], [8], [9] [10]; this aspect is the subject of this chapter. It must be emphasized that in the current context the term ‘coordination’ is used to mean coordinating the gains of stabilizers installed on generators and FACTS devices, say, in an area of interest for the purpose of improving the damping of rotor modes. This is as opposed to coordination in the context of coordinating controllers, e.g. AVR-PSS coordination, within a single generating unit [11]. In the following text the damping gains of PSSs and the gains of FDSs are collectively referred to as stabilizer gains.

It has been emphasized that the predominately left shift of the modes with increasing stabilizer gain is the objective of the design procedures outlined in Chapters 5 and 10 for PSSs and Chapter 11 for FDSs. In essence, because the stabilizer transfer functions are of the form kG(s), where is k a real gain and the transfer function G(s) provides the phase compensation, then ideally, (i) G(s) ensures the left shift of all modes over the selected range of modal frequencies, and (ii) the gain k determines the extent of the left-shift of the mode. This basic approach to the tuning of stabilizers provides the following rationale for the methods of heuristic and automated coordination.

1. • In both the heuristic and automated coordination procedures the stabilizer gain and the phase compensation are the two important components which are essentially decoupled for practical purposes. Therefore, in the coordination procedures that follow, the stabilizer gains are the adjustable quantities and the parameters of the compensation transfer functions remain G(s) unchanged.

2. • For the process of stabilizers coordination the PSSs and FDSs should be robust over an encompassing range of operating conditions, normal and outage (see Section 1.2 item 3 and Section 11.8.2, respectively).

3. • Ideally, the incremental left-shifts of the rotor modes should be more-or-less linearly related to increments in stabilizer gain for small changes about the nominal values.

We have written this book in the hope that the following engineers, or potential engineers, will benefit from it:

1. • Recent graduates in electrical engineering who need to understand the tools and techniques currently available in the analysis of small-signal dynamic performance and design.

2. • Practicing electrical engineers who need to understand the significance of more recent developments and techniques in the field of small-signal dynamic performance.

3. • Postgraduate students in electrical engineering who need to understand current developments in the field and the need to orient their research to achieve practical, useful outcomes.

4. • Undergraduate electrical engineering students in courses oriented towards electric power engineering in which there is an introductory subject in power system dynamics (for access to basic material).

5. • Managerial staff with responsibilities in power system planning, and system stability and control.

An aim of the book is to provide a bridge between the mathematical/theoretical and physical/ practical significance to the topic. Some of the fundamental background relevant to the main topics of the book is presented in the early chapters so that the necessary material is readily available to the reader in the one book.

1. • Because the emphasis is on controllers for generators, for FACTS and other devices, the pertinent topics in classical control and eigenanalysis techniques are provided in Chapters 2 and 3.

2. • The authors have covered in Chapter 4 a wide range of small-signal generator models, equations, and associated material. Third- to eighth-order generator models in their coupled-circuit and operational parameter versions are described. The following features are also included in the generator models: (i) the ‘classical’ and ‘exact’ definitions of the operational parameters; (ii) the various approaches to the modelling of saturation; (iii) the formulation of the differential-algebraic generator equations to exploit sparsity. These models and features are employed in the Mudpack software package. Small-signal equations and models of FACTS devices employed in the software are also described. Devices covered include SVCs, STATCOMS, Thyristor Controlled Series Compensators, HVDC links with Voltage Source Converters or with line-commutated converters.

3. • In Chapters 5, 9 and 10 there is an emphasis on practical robust techniques, based on the P-Vr method, for the design of robust stabilizers for generators in multi-machine systems.

Introduction

The objective of the application of stabilizers in multi-machine power systems is to stabilize the system by providing adequate damping for the critical rotor modes of oscillation. These modes typically involve several power stations and their machines. In the case of inter-area modes many power stations, geographically widely separated, may participate in both the local and inter-area modes. It is therefore necessary that the stabilizer which, when fitted to a generator, contributes with stabilizers on other machines to the damping of the relevant modes. Furthermore, because operating conditions on the system continuously change, the performance of a fixed-parameter stabilizer should be robust to any such changes.

By employing the P-Vr method in the tuning of the PSS, as demonstrated in Chapter 5, the inherent magnitude and phase characteristics of the generator and power system are being utilized; for practical purposes these characteristics consistently lie in a relatively narrow band. Not only can the method account for variations over a wide range of loading conditions on the system, line outages, etc., but the resulting PSS is most effective and beneficial at the higher generator real power outputs as revealed in Table 5.6, and discussed in the associated text.

Prior to considering the application of the P-Vr method to the tuning of PSSs in multi-machine power systems, the use and significance of two valuable tools in the small-signal analysis of the dynamic performance of such systems are discussed. These tools concern the so-called “Mode Shape” and “Participation Factor” analyses of the system for a selected operating condition. Such analyses reveal the nature and significance of the various modes (both rotor or other modes), the involvement - and extent of involvement - of generators in the modes, and other insights such as the nature of the dynamic behaviour of other devices in the system (e.g. FACTS devices and their controls).

The application of other PSS tuning methods, namely the GEP Method and the Method of Residues, is discussed in Chapter 6. While these approaches can be adapted to the multi-machine system, for the reasons explained in the latter chapter the P-Vr method is considered to possess some significant advantages.

Introduction

The description of the dynamics of large systems, such as power systems, by their transfer functions is unsatisfactory for a number of reasons. For example, for a system of order n, say 100, the characteristic polynomial has degree 100 and 101 coefficients of s. Moreover, such systems typically have more than one output variable and more than one input signal. modelling based on the multi-input multi-output state equations of the system is simpler and problems of loss of accuracy are reduced. Moreover, such modelling has a number of advantages and features some of which are described in the following sections. To illustrate the formation of the state equations of a plant or an electro-mechanical system, let us consider two examples.

Much of the material on linear systems analysis in the later sections is covered by [1].

Example 3.1.

Find a set of state and output equations for the simple RLC circuit shown in Figure 3.1. The voltage supplied by an ideal source is vs(t), and the required outputs are the capacitor voltage are vC(t) and iL(t) inductor current.

Loop voltages:

where p is the differential operator d/dt.

Current flow:

Note that each of the right-hand equations is a first-order differential equation with the derivative specifically sited on the left-hand side of the equation.

There are two independent energy storage elements, C and L. Because the instantaneous energy stored in C and L is and, respectively, the variables x1 = vC and x2 = iL are ‘natural’ selections for states. Hence, the state equations are formed as follows:

The two output equations required are y1= vC = x1 and y2= iL = x2.

The state and output equations can thus be written in matrix form as follows:

Where

Example 3.2

A drive system shown in Figure 3.2 consists of a DC motor driving an inertial load through a speed-reducing gearbox. The controlled DC supply voltage to the armature is supplied by a power amplifier. The motor field current is maintained constant (i.e. the flux/pole is constant). Write down the equations of motion for this system.

Introduction

In this chapter the concept, the theory, and calculation of modal induced torque coefficients (MITCs) in multi-machine power systems are introduced. The concept of a modal induced torque coefficient is new [1], [2]. It forms the basis for calculation of the shifts in rotor modes when the stabilizer gains of one or more PSSs and/or FDSs are incremented by Δk (pu) on device base. Based on the concept of MITCs, the background theory of the rotor modes shifts, together with analysis of the effectiveness of, and interactions between, PSSs and FDSs in multi-machine systems are described in Chapter 13.

The theoretical development of MITCs in this chapter is fairly detailed and can be omitted if the practical applications of the analysis of rotor modes shifts are of primary interest. Where relevant, references are made in Chapter 13 to the results and equations that are developed in this chapter. A case study in the latter chapter demonstrates the significance of the MITCs and the insights that they provide into the dynamic performance of a multi-machine power system.

In essence, the concept of a modal induced torque coefficient is a further development of the concepts of damping and synchronising torque coefficients based on frequency response analysis (i.e. s = Jωf) [1]. In this chapter the torque coefficients are evaluated at the complex rotor modes (λh = α±jω). The frequency-analysis-based torque coefficients are introduced in Section 5.3 for a SMIB system and in Sections 9.5 and 10.6 for the multi-machine case. With shaft dynamics enabled, the modal induced torque coefficient, Tijh, for the complex rotor mode λh is defined as a complex torque coefficient which is induced on generator i due to a perturbation in the stabilizing signal of stabilizer j. The stabilizer in question may be a power system stabilizer (PSS) or a FACT device stabilizer (FDS) installed on a FACTS device. Such FACTS devices are static var compensators (SVCs), high voltage DC links, thyristor controlled series capacitors (TCSCs) among others. It will be shown that the effect of any such stabilizer on the damping of any mode of rotor oscillation can be quantified. The concept can be extended to other devices such as wind turbine generators, photovoltaics and any other power-converter based transmission or generation equipment.

Why analyse the small-signal dynamic performance of power systems?

We shall be concerned mainly with the analysis of the dynamic performance and control of large, interconnected electric power systems in the following chapters. The differential-algebraic equations which describe the behaviour of a power system are inherently non-linear. Among the non-linearities are functional types (e.g. sinδ), product types (e.g. voltage × current), limits on controller action, saturation in magnetic circuits, etc. The general method of assessing the performance of the system, with all its non-linearities, is through a time-domain simulation which reveals the response of the system to a specific disturbance, e.g. a fault, the loss of a generating unit, line switching. Typically, it may be necessary to conduct many such studies with disturbances applied in various locations in the system to ascertain its stability and dynamic characteristics. Even with many such studies, many of the characteristics of the dynamic behaviour may be missed and insights into system performance lost. In small-signal analysis of dynamic performance of multi-machine systems the stability and characteristics of the system are readily derived from eigenanalysis and other tools. Further-more, in such linear analysis the design of controllers and their integration into the dynamics of plant are facilitated.

Modern linear control system theory contains many powerful techniques, not only for determining the stability and dynamic characteristics of large linear systems, but also for tuning controllers that satisfy steady-state and dynamic performance specifications. Fortunately and importantly, Henri Poincaré [1] showed that if the linearized form of the non-linear system is stable, so is the non-linear system stable at the steady-state operating condition at which the system is linearized. Moreover, the dynamic characteristics of the system at the selected operating condition can be established from linear control system theory and, as long as the perturbations are small, the time-domain responses can be calculated. With such information the design of linear controllers may be undertaken and the resulting controls embedded in the non-linear system. In practice, if the modelling of the devices is adequate, small-signal tests involving generator controls, for example, have revealed close agreement between simulation and test results. Continuously-acting controllers of interest for synchronous generating units are Automatic Voltage Regulators (AVRs), Power System Stabilizers (PSSs) and speed governors.