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IEEE Power & Energy Society

IEEE Tutorial Course Power System Stabilization

Via Excitation Control

09TP250

Copyright IEEE 2009 978-1-4244-5069-5

IEEE Tutorial Course – Power System Stabilization via Excitation Control – June 2007

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IEEE TUTORIAL COURSE

POWER SYSTEM STABILIZATION

VIA EXCITATION CONTROL

Sponsored by

IEEE Power Engineering Society Life Long Learning Committee

and the

Excitation Systems Subcommittee

of the

Energy Development and Power Generation Committee

Presented at

the IEEE Power Engineering Society General Meeting

Tampa

Florida

28th June 2007


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CONTENTS CHAPTER 1 REVIEW OF FEEDBACK CONTROL CONCEPTS ........................................................................................ 7

I. INTRODUCTION ..................................................................................................................................................................... 7 II. LAPLACE TRANSFORMS .................................................................................................................................................... 7 III. TRANSFER FUNCTIONS AND BLOCK DIAGRAMS.................................................................................................................. 9

A. Transfer Function Example.......................................................................................................................................... 9 B. Block Diagrams............................................................................................................................................................ 9 C. Interconnection Of Systems........................................................................................................................................ 10 D. B.D. Manipulation Example....................................................................................................................................... 10 E. Transfer Function Example........................................................................................................................................ 10

IV. FREQUENCY RESPONSE MODELS .................................................................................................................................... 11 V. STABILITY CRITERIA FOR FEEDBACK CONTROL SYSTEMS .............................................................................................. 11

A. General Comments..................................................................................................................................................... 11 B. Nyquist Criterion (Gain And Phase Margins) ........................................................................................................... 12 C. Example...................................................................................................................................................................... 14 D. Root Locus Example................................................................................................................................................... 15

VI. STATE-SPACE TECHNIQUES............................................................................................................................................. 16 A. General Comments..................................................................................................................................................... 16 B. State-Space Models .................................................................................................................................................... 17 C. State Space model Example ....................................................................................................................................... 17

VII. SYSTEM SIMULATION...................................................................................................................................................... 18 CHAPTER 2 OVERVIEW OF POWER SYSTEM STABILITY CONCEPTS ................................................................... 19

I. INTRODUCTION ................................................................................................................................................................... 19 II. POWER SYSTEM STABILITY CLASSIFICATION.................................................................................................................. 19

A. Definition.................................................................................................................................................................... 19 B. Categories Of Stability ............................................................................................................................................... 19

1) Voltage Stability..................................................................................................................................................... 19 2) Frequency Stability ................................................................................................................................................ 19 3) Rotor Angle Stability ............................................................................................................................................. 19

III. BACKGROUND - GENERATOR CONNECTED TO INFINITE BUS ........................................................................................... 20 IV. TRANSIENT STABILITY.................................................................................................................................................... 20 V. TRANSIENT VS. OSCILLATORY STABILITY ...................................................................................................................... 21 VI. OSCILLATORY STABILITY ............................................................................................................................................... 21

A. Characteristic Dynamic Equation.............................................................................................................................. 21 B. Local Vs. Inter Area Oscillations............................................................................................................................... 22 C. Negative Damping Due To Voltage Regulator .......................................................................................................... 22 D. PSS For Improved Oscillatory Stability..................................................................................................................... 24

VII. GENERATOR MODELS ..................................................................................................................................................... 25 VIII. CONCLUSIONS............................................................................................................................................................. 25

CHAPTER 3 PERFORMANCE CRITERIA AND TUNING TECHNIQUES..................................................................... 26 I. INTRODUCTION ................................................................................................................................................................... 26 II. PERFORMANCE OBJECTIVES............................................................................................................................................ 26

A. Oscillatory Stability Limits......................................................................................................................................... 26 B. System Modes Of Oscillation ..................................................................................................................................... 26 C. Tuning Concepts......................................................................................................................................................... 27 D. Speed Input Stabilizers............................................................................................................................................... 28 E. Frequency Input Stabilizers ....................................................................................................................................... 28 F. Power Input Stabilizers .............................................................................................................................................. 29

III. TUNING EXAMPLE........................................................................................................................................................... 29 A. Phase Compensation.................................................................................................................................................. 30 B. Root Locus.................................................................................................................................................................. 30 C. Step Test and Fault Simulations................................................................................................................................. 31 D. PSS – Torsional Interaction ....................................................................................................................................... 32 E. Use of Modified Lead-lag Compensation................................................................................................................... 33


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F. Inter-area Mode Damping ......................................................................................................................................... 33 IV. SUMMARY AND CONCLUSIONS........................................................................................................................................ 34

CHAPTER 4 INTEGRAL OF ACCELERATING POWER TYPE STABILIZERS............................................................ 35 I. INTRODUCTION ................................................................................................................................................................... 35 II. OVERVIEW OF PSS STRUCTURES.................................................................................................................................... 35

A. Speed-Based (Dw) Stabilizer...................................................................................................................................... 35 B. Frequency-Based (∆f) Stabilizer................................................................................................................................. 35 C. Power-Based (∆P) Stabilizer...................................................................................................................................... 36 D. Integral-of-Accelerating Power (∆Pω) Stabilizer ...................................................................................................... 36

III. PRACTICAL APPLICATION ISSUES.................................................................................................................................... 37 A. Signal Mixing ............................................................................................................................................................. 37 B. Mechanical Power Variations.................................................................................................................................... 38 C. Input Signals .............................................................................................................................................................. 39 D. Compensated Frequency............................................................................................................................................ 40

IV. HARDWARE CONSIDERATIONS ........................................................................................................................................ 41 V. PSS COMMISSIONING & FIELD VERIFICATION ................................................................................................................ 41 VI. APPENDIX - DERIVATION OF FILTER RESPONSES ............................................................................................................ 42

A. Background ................................................................................................................................................................ 42 B. Conventional Low Pass Filter.................................................................................................................................... 42 C. Ramp-Tracking Filter................................................................................................................................................. 42

CHAPTER 5 FIELD TESTING TECHNIQUES...................................................................................................................... 44 I. INTRODUCTION ................................................................................................................................................................... 44 II. MEASUREMENT TECHNIQUES AND INSTRUMENTATION .................................................................................................. 44

A. Signal Transducers and Conditioning ....................................................................................................................... 44 B. Terminal Voltage........................................................................................................................................................ 44 C. Electrical Power......................................................................................................................................................... 44 D. Field Voltage.............................................................................................................................................................. 45 E. Generator Speed......................................................................................................................................................... 45 F. Terminal or Internal Frequency................................................................................................................................. 46 G. Power System Stabilizer Output................................................................................................................................. 46 H. Generator Torque Angle ............................................................................................................................................ 46 I. Signal Recording............................................................................................................................................................ 46

III. TESTING TECHNIQUES..................................................................................................................................................... 46 A. Step and Impulse Response Testing ........................................................................................................................... 46 B. Frequency Response Testing...................................................................................................................................... 47 C. Equipment and Techniques for Frequency Domain Analysis .................................................................................... 48 D. General Comments..................................................................................................................................................... 48

IV. ON SITE TUNING AND STABILITY ASSESSMENT.............................................................................................................. 48 A. The Excitation System ................................................................................................................................................ 48 B. Tuning Criteria........................................................................................................................................................... 49 C. PSS Testing ................................................................................................................................................................ 49

V. SHAFT TORSIONAL OSCILLATION.................................................................................................................................... 49 CHAPTER 6 APPLICATION CONSIDERATIONS.............................................................................................................. 51

I. INTRODUCTION ................................................................................................................................................................... 51 II. UNIDIRECTIONAL EXCITERS............................................................................................................................................ 51 III. DIGITAL EXCITERS.......................................................................................................................................................... 51

A. Processor Cycle Time................................................................................................................................................. 51 B. Integer Or Floating Point Arithmetic......................................................................................................................... 51 C. Passwords And Security ............................................................................................................................................. 51

IV. MINIMUM EXCITATION LIMITERS ................................................................................................................................... 52 CHAPTER 7 FUTURE DIRECTIONS IN PSS DESIGN....................................................................................................... 53

I. INTRODUCTION ................................................................................................................................................................... 53 II. ANALYTICAL ADAPTIVE CONTROL BASED APSS ........................................................................................................... 53

A. Direct Adaptive Control ............................................................................................................................................. 53 B. Indirect Adaptive Control........................................................................................................................................... 54


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C. System Model ............................................................................................................................................................. 55 D. System Parameter Estimation .................................................................................................................................... 55

III. INDIRECT ADAPTIVE CONTROL STRATEGIES................................................................................................................... 55 A. Linear Quadratic Control .......................................................................................................................................... 55 B. Minimum Variance (MV) Control .............................................................................................................................. 56 C. Pole-Zero And Pole Assigned Control ....................................................................................................................... 56 D. Pole Shift Control....................................................................................................................................................... 56

IV. PS CONTROL BASED ADAPTIVE PSS............................................................................................................................... 56 A. Self-Adjusting Pole-Shift Control Strategy................................................................................................................. 56

V. PERFORMANCE STUDIES WITH POLE-SHIFTING CONTROL PSS........................................................................................ 58 VI. ARTIFICIAL INTELLIGENCE BASED PSS........................................................................................................................... 59

A. Adaptive PSS With NN Predictor And NN Controller................................................................................................ 59 B. Adaptive Network Based Fuzzy Logic Controller ...................................................................................................... 59 C. Architecture................................................................................................................................................................ 59 D. Training And Performance ........................................................................................................................................ 60 E. Self-Learning ANF PSS.............................................................................................................................................. 60 F. Neuro-Fuzzy Controller Architecture Optimization................................................................................................... 60

VII. AMALGAMATED ANALYTICAL AND AI BASED PSS ........................................................................................................ 61 A. Adaptive PSS With NN Identifier And Pole-Shift Control.......................................................................................... 61 B. Adaptive PSS with Fuzzy Logic Identifier and Pole-Shift controller ......................................................................... 62 C. Adaptive PSS With RLS Identifier And Fuzzy Logic Control ..................................................................................... 62

VIII. MULTI BAND PSS ....................................................................................................................................................... 63 A. Tuning Methodology .................................................................................................................................................. 64 B. Application Experience .............................................................................................................................................. 64 C. Multiband PSS Conclusions ....................................................................................................................................... 65

IX. CONCLUDING REMARKS.................................................................................................................................................. 66 CHAPTER 8 REFERENCES ..................................................................................................................................................... 67

CHAPTER 9 BIOGRAPHIES.................................................................................................................................................... 70

PSS Tutorial - March 2007 B.doc


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FOREWORD In 1981 the first IEEE tutorial course on power system stabilization via excitation control was presented by

Ken Bollinger, Joe Hurley, Frederick Keay, Einar Larsen and David Lee and the notes from that tutorial became a widely used reference for generation engineers working to improve power system stability.

Ideas for power system stabilization using excitation control originated as a result of electric power oscillations

occurring on interties between large power pools and stability problems associated with single generators, or banks of generators connected to large power systems. Before the 1981 tutorial increasing use of high gain excitation systems and increased use of transmission systems had been leading to decreased stability margins and power system stability problems. A considerable amount of effort had been spent on research projects, sitework and the development of control electronics to stabilize multi-machine systems.

Since the 1981 tutorial, research work and product development has continued. Electronic control units now

utilize digital technology to provide repeatability, more features and easier use, and control algorithms have been improved to eliminate some earlier shortcomings.

The development of the integral of accelerating power type stabilizer (PSS2B type as described in IEEE

Standard 421.5 2005) allowed stabilizers to operate successfully with minimal terminal voltage fluctuation even during very rapid loading and unloading of generators. This type has now become the de-facto standard and this type of stabilizer is now a requirement in many parts of North America.

Increasing numbers of power system stabilizers have been installed as grid codes around the world and North

American requirements such as the WECC guidelines have demanded that generators be equipped with stabilizers. In addition the power system disturbances in Western North America during July and August 1996 caused increased effort to be focused on testing and validating of generating units including their excitation control systems and associated stabilizers. These factors have resulted in many more engineers being introduced to power system stabilizers for the first time.

Whilst excellent papers are available on many aspects of power system stabilizer design, implementation and

testing, this tutorial is intended to provide engineers and technicians with a set of key insights into problems related to power system oscillations and the currently available solutions.

It is expected that the course participants will have a basic understanding of power system analysis and control

concepts. The tutorial includes introductory material to provide a basis for understanding of the terminology used in the latter part of the course.

This tutorial includes contributions from present day experts in the field of power system stability in addition to

material from the authors of the original tutorial. The first two chapters review feedback control and power system stability concepts. Following chapters describe more detail on performance criteria, tuning techniques, accelerating power type stabilizers, field testing techniques and application considerations. The last chapter describes some future directions in stabilizer design.

The authors and presenters of this tutorial are grateful to the authors of original tutorial and to the experts in

this industry who have written many excellent papers referenced here. Robert Thornton-Jones March 2007


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Abstract—The document comprises chapters supporting a tutorial course on power system stabilization via excitation control to be presented at the IEEE PES general meeting in June 2007. Feedback control and power system stability theory is presented to explain the foundation concepts of power system stabilization by excitation control. Tuning and testing techniques and application considerations are described. Both present day stabilizer types and future directions in stabilizer design are explained.

Index Terms - Control, Excitation, Generator, Regulator, Stabilizer.

IEEE Tutorial Course Power System Stabilization

Via Excitation Control J.C.Agee and Shawn Patterson – Bureau of Reclamation, Denver

Roger Beaulieu and Murray Coultes – Goldfinch Power Engineering, Toronto Robert Grondin, Innocent Kamwa, Gilles Trudel – Hydro-Québec

Arjun Godhwani – Southern Illinois University, Edwardsville Roger Bérubé and Les Hajagos – Kestrel Power Engineering, Toronto

Om Malik – University of Calgary Alexander Murdoch and George Boukarim – GE, Schenectady

José Taborda – ABB, Switzerland Robert Thornton-Jones – Brush, United Kingdom


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CHAPTER 1 REVIEW OF FEEDBACK CONTROL CONCEPTS

Arjun Godhwani

I. INTRODUCTION Study of feedback control requires knowledge of various

mathematical tools that consist of Laplace transforms, Bode plots, Nyquist plots, and Root locus. These tools apply to single-input-single-output systems (SISO) that are linear and time invariant - LTI. The state space approach is used to analyze not only (SISO) systems but also multiple-input-multiple-output (MIMO) systems. Some of the systems tend to be nonlinear and are linearized around an operating point for small excursions of the signals before applying these tools. Knowledge of these tools is critical to understanding power system control and stability. Therefore a review of these tools is presented in the following.

It will be impossible to cover all relevant mathematics in the brief period available yet the Tutorial Course organizers felt that some review was needed. This section of the course will survey those areas of mathematics that are part of the repertoire of analysts who are working in the power system control area.

There are two approaches to handling power system control problems and, in particular, power system stabilization problems.

a) The analyst can use a combination of intuition and common sense to deduce proper control strategy and settings.

b) The problem may be approached from the theoretical viewpoint and computer techniques employed to synthesize the controller.

Most engineers involved in tuning controllers will tend to agree that the best approach lies somewhere between 'a' and 'b'. If one relies too heavily on intuition, situations will arise when the process, or system, does not respond according to plan and there is nothing to fall back on as an alternate control strategy. On the other hand, theoretical studies often lead to elegant solutions with no practical way of implementing the solution. This latter approach does not yield a feel for the physical system and sometimes masks an obvious solution.

A compromise between intuition, or common sense, and theory is usually the best approach. In many situations Engineering judgment must be used to cull out second-order effects and reduce the problem to the point where mathematical concepts will yield practical solutions. A simple power system model will be used to illustrate this compromise and to highlight certain analytical techniques. Analyzing plant (or process) stability can be done simply by observing the mechanical linkage, or gauges, in a control loop or doing an in depth study on a mathematical model of the system. Synthesizing, or tuning controllers can involve everything from "tweaking" the knobs with an "experienced" hand to applying computerized algorithms to calculate

controller parameters in offline studies on a mathematical model of the system.

The subsequent sections of this chapter will attempt to concentrate on those aspects of feedback control theory that are pertinent to power system stabilizer (PSS) tuning. Emphasis will be placed on concepts that have been used in offline studies and subsequently used for tuning hardware that is now in place at power plants. This should not imply that other methods are not equally applicable but it was felt that a comprehensive survey of all controller synthesis techniques would be outside the scope of this tutorial. In many situations advanced controller synthesis concepts must be applied when design constraints on the controller are quite stringent.

The next section describes basic concepts relating time, frequency, and Complex-S-Plane Parameters. This is followed by a brief review of basic control concepts with emphasis on stability of feedback control systems. State space techniques are discussed next followed by introductory concepts related to simulation techniques.

II. LAPLACE TRANSFORMS Fourier analysis is extremely useful in analyzing many

practical problems involving signals and LTI systems. Fourier series can represent periodic signals, while aperiodic signals can be represented by Fourier transforms. In each case the signals are represented by an infinite set of complex exponentials est with s = jω. In addition when a complex exponential passes through an LTI system the output is also a complex exponential of same frequency with a change in amplitude and change in phase. This concept is one of the reasons for usefulness of frequency response techniques such as Bode plots and Nyquist plots. The above-mentioned property of complex exponentials not only applies to pure imaginary values of s but also arbitrary values of s. This leads to generalization of Fourier transform to Laplace transform. Laplace transforms have many of the same properties that make Fourier transforms useful. But usefulness of Laplace transforms goes beyond that of Fourier transforms. For example Laplace transforms can be used to investigate stability. Using Laplace transforms we will develop the concepts of transfer function, natural and forced modes of response, poles and zeros etc. In addition block-diagram reduction will allow us to manipulate systems that possess one or more feedback loops.

The starting point for studying a control system is to represent it by its differential and algebraic equations. The study is usually done to answer one or both of the following questions:

a) How stable is the system? b) What has to be added, or changed, to achieve a desired

level of stability? One approach to answering these questions is to derive a

compact model of the process (system) relating the input of the process to the output. This is shown in "block diagram" form in Fig.1.1. The output y(t) is a function of the input x(t)


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and the system model.

Figure 1.1 Block Diagram Of Input Output Model

A convenient way to represent the system model is to use

Laplace Transforms [1]. The Laplace transform model is usually derived as a special case of general transform theory, but an intuitive approach will be used for expediency.

A periodic signal can be represented by a series of sinusoidal signals. The analysis for calculating the amplitudes of the sine and cosine terms of the series is referred to as Fourier series [2]. The Fourier series for periodic functions, and Fourier transform for aperiodic functions, transform a time domain signal into its frequency components.

An extension of this transform concept to the more general case allows the analyst to transform functions of time into the "Complex Frequency" domain.

Laplace transform of a time function f(t) is denoted as F(s) and is defined as follows:

∫∞

−=0

dte)t(f F(s) st (1.1)

The set of commonly used Laplace transform pairs and

properties are listed in Table I. Laplace transforms for commonly used time functions are rational functions i.e. ratio of two polynomials.

One of the important applications of the Laplace transform is in the analysis of systems. The system transfer function is defined as the ratio of the output transform and the input transform, assuming all initial conditions as 0.

)s(X)s(Y H(s) = (1.2)

The roots of the numerator of the transfer function are

called the zeros of the system and the roots of the denominator of the transfer function are called the poles of the system. An example of a transfer function and how we display poles and zeros on the s-plane is shown in Fig.1.2. In this example K is the gain, and the system has a zero at s = -2, and poles at s = 0 and s = -10.

)s(s)s(K H(s)

102

++= (1.3)

Figure 1.2 Pole-Zero Plot

Differential equations (D.E.) can be transformed into

algebraic equations using Laplace transforms. The resulting Laplace expression can then be manipulated algebraically to obtain stability information and other relevant information about the physical system described by the D.E. In some cases the Laplace transform is used to obtain the time domain solution of the D.E. but in most cases it is used to obtain stability information about a process.

Table I contains the Laplace transform of relevant time domain expressions including differential and integration operations. The entries in Table I are determined using (1). Table I also includes initial value and final value properties.

TABLE I

LAPLACE TRANSFORM PAIRS AND PROPERTIES ( )tf ( )sF ( )tδ 1

( )tu s1

t 21s

ate− as +1

( )tcos ω 22 ss+ω

( )tsin ω 22 s+ωω

( )tcose at ω− 22 )as(as++ω

+

( )tsine at ω− 22 )as( ++ωω

dt)t(df ( ) ( )0fssF −

2

2

dt)t(fd ( ) ( ) ( )

dtdfsfsFs 002 −−

∫t

dt)t(f0

s

)s(F

( )tflimt ∞→

( )ssFlims 0→

( )tflimt 0→

( )ssFlims ∞→

System Model

Input x(t) Output y(t)


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III. TRANSFER FUNCTIONS AND BLOCK DIAGRAMS A D.E. can be solved using Laplace transforms as shown

below. A general D.E. is shown in (1.4).

)(3)()(2)(3)(2

2

tfdt

tdftydt

tdydt

tyd +=++ (1.4)

The Laplace Transform of (1.4) yields

I(s) F(s) 3)(s Y(s) 2) s 3 (s2 ++=++ (1.5)

where, F(s) = Laplace Transform of f(t), and I(s) = Laplace term associated with initial conditions.

Equation (1.5) can be written as

)s(D)s(I)s(F

)s(D)s(N Y(s) += (1.6)

Where

3)(s N(s) += (1.7)

and

2) s 3 (s D(s) 2 ++= (1.8)

It is readily seen that the polynomial D(s) appears both in

the denominator of the "forcing function" term and also the "initial condition" term. It is further seen that if f(t) is specified, the time domain solution for y(t) can be obtained by factoring (finding the roots of) D(s) and using partial fractions.

Then each of the roots of D(s) contributes a term to the time domain response with the roots having a direct relationship with the time constants and frequency of oscillation of the time domain terms. The polynomial D(s) is called the characteristic polynomial of the system.

Since the forcing function term contains the characteristic polynomial (and thus all information pertaining to stability and the form of the time domain transients), the initial condition term is dropped and (1.6) reduces to

)s(F)s(D)s(N Y(s) = (1.9)

The term N(s)/D(s) as discussed before is referred to as the

"Transfer Function" of the system.

A. Transfer Function Example Determine the transfer function, Y(s)/F(s) of the following

mechanical system.

Figure 1.3 Mechanical System

The Differential Equation that describes this system is

)t(f)t(Kydt

)t(dyDdt

)t(ydM =++2

2 (1.10)

Using entries of Laplace Transform Table I, with initial

conditions set to zero,

F(s) Y(s)K Y(s) s D Y(s) s M 2 =++ (1.11)

or

KDsMs)s(F)s(Y

++= 2

1 (1.12)

The main advantage of deriving a transfer function relating

system output to system input is that it enables the analyst to assess system stability from the roots of the characteristic equation of the system.

One of the main problems with using Laplace transforms to assess system stability is that it usually requires a considerable amount of algebra to obtain the transfer function (T.F.) and the corresponding characteristic equation. This is partly expedited by using block diagrams to represent the system and then manipulating the block diagram to obtain the characteristic equation. This concept is discussed in the next section.

B. Block Diagrams Block diagram representation of a physical system enables

the analyst to obtain overall input/output relationships by using "block-diagram" manipulation rather than apply matrix methods or other reduction techniques [3] on the set of Laplace equations describing the physical system. Of greater significance is the fact that block diagram representation of the physical system indicates a cause and effect, or signal flow pattern, of the physical system. When set up properly, the block diagram (B.D.) is a one-to-one representation of the


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elements of the physical system and the blocks of the Laplace B.D.

The basic element of B.D. representation is the transfer function. The input/output relation,

F(s) (s)G Y(s) 1= (1.13)

is represented by the block diagram shown in Fig.1.4 with

Model represented by the transfer function G1(s).

C. Interconnection Of Systems Many real systems are built as interconnection of

subsystems. Fig.1.4 below shows a series (also called cascade) and a parallel interconnection. In addition to the transfer function blocks, parallel connection uses the summing block.

The series connection is represented by the following equations,

X(s) (s)G Y(s) 2= (1.14)

And

F(s) (s)G X(s) 1= (1.15) (1.14) and (1.15) can be combined as

F(s) (s)G (s)G Y(s) 21= (1.16) Thus the series connection can be equivalently represented

by a single block with transfer function G1(s) G2(s). Similarly the parallel connection can be represented by a single block with transfer function G1(s) + G2(s).

Figure 1.4 Series And Parallel Connections It is possible to manipulate block diagram models using the

elementary operations described previously. An example of the application of B.D. manipulation to obtain a system transfer function is shown below.

D. B.D. Manipulation Example Derive the T.F., C(s)/R(s), for the feedback system shown

below in Fig.1.5.

Figure 1.5 Feedback Control System

From the B.D.

Z(s)- R(s) E(s) = (1.17)

E(s) G(s) C(s) = (1.18)

C(s) H(s) Z(s) = (1.19) Substituting for E from (1.17) and Z from (1.19) in (1.18),

and simplifying gives

)s(H)s(G)s(G

)s(R)s(C

+=

1 (1.20)

Where 1 + GH = 0 is the Characteristic Equation of this

system.

E. Transfer Function Example Derive the T.F. E(s)/R(s) for the system shown in the

previous example.

E(s) G(s)H(s) - R(s) E(s) = (1.21) Therefore

)s(H)s(G)s(R)s(E

+=

11 (1.22)

It can be readily seen from the previous two examples that

both transfer functions have identical characteristic equations but that they have different numerator terms in the transfer functions. The roots of characteristic equation dictate the types of modes of response, and the numerator terms affect the magnitudes of the response terms. This basic concept is worth noting as it explains why system transients have different characteristics when viewed from different points on a system even though the T.F.s have identical characteristic equations and the same input signal is applied in both instances.


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IV. FREQUENCY RESPONSE MODELS By definition, the frequency response of a system is the

ratio of the output over the input when the input to a system is a sinusoidal forcing function of radian frequency ω and steady state conditions have been reached.

We are already familiar with sinusoidal steady-state methods of analysis from circuit analysis.

Theoretically, the frequency response of a system can be obtained from the transfer function by setting s = jω in the transfer function. The result is,

)(jG )j(X)j(Y ω=

ωω (1.23)

Let us consider a general frequency response function that

includes all types of terms as follows:

)j )j()(jT()j(

)j )j(( )j T(K)j(G

npnpp

n

nznzZ

2

2

211

211

ωω+

ωωζ+ω+ω

ωω+

ωωζ+ω+

=ω (1.24)

The term G(jω) can be completely represented graphically

by plotting the magnitude |G| and the angle ∠ G against ω on two separate graphs. It is customary to plot gain in dB given by 20 Log |G| versus Log(ω). Hence the general frequency expression in decibels can be obtained from (1.24).

P

Z2

nz nz

2

np np

20Log | G |20LogK n 20Log 20Log |1 T j |

20Log |1 T j |

j20Log | (1 2 ( ) j ) |

j20Log | (1 2 ( ) j ) |

=− ω − + ω

+ + ω

ω ω+ + ζ + ω ω

ω ω− + ζ + ω ω

(1.25)

The expression for the angle of the frequency response is

given by

∑=∠ G Angles of numerator terms ∑− Angles of denominator terms (1.26)

Where angles of numerator and denominator terms are

determined by specifying a particular value of ω and algebraically summing the angles of the terms in the numerator and denominator of (1.24).

Each of the terms in (1.25) contribute to the magnitude plot (Bode Plot) of the total magnitude plot of 20 log |G| (this is written in shorthand as | G | dB). Fig.1.6 illustrates the contribution of each term to the Bode Plot.

When the transfer function is given, a complete Bode plot

of the transfer function can be obtained by summing up the contribution to the Bode plot of each term as seen in Fig.1.6. It can further be noted from Fig.1.6 that the complex pair can give very sharp attenuation or magnification in the magnitude plot at frequencies in the vicinity of the "resonant humps" of these terms. This feature is important in the design of "pass" and "stop" band filters; a concept that is used in the design of filters to block generator shaft "torsional" frequencies from feeding back into the AVR through the PSS.

Bode diagrams are also used to plot experimental frequency response data when measurements have been made using a sinusoidal generator and recording equipment. In these cases a transfer function can be obtained from the frequency response data by doing a "manual" or "computerized" analysis to best fit the data. In some situations frequency response data is used directly to obtain data for assessing stability of existing systems or for designing controllers to enhance the stability of the control loop. These concepts will be discussed later.

It will be seen in the next section that transfer functions and frequency response concepts form a vital basis for assessing stability of certain types of control systems.

V. STABILITY CRITERIA FOR FEEDBACK CONTROL SYSTEMS

A. General Comments Classical control theory is based on single-input-single-

output (SISO) control systems [4] [5]. Stability of these systems is assessed by investigating the effect of adjustable parameters on the roots of the characteristic equations. The characteristic equation is obtained by manipulating the set of equations describing the dynamic system, or the system block diagram.

Once an expression for the characteristic equation is obtained, several techniques can be applied to this equation to assess stability. These so-called classical tools for stability assessment include [6] - [8]:

• Routh-Hurwitz Criterion • Nyquist Criterion • Bode Plots • Root Locus Technique The latter two will be described here as they relate directly

to techniques that are used for PSS tuning. Many feedback control systems have adjustable gain K

either in the forward or feedback path. The locations of closed-loop poles change as the gain K is varied. The closed loop pole locations decide key control system performance characteristics such as stability, transient response, and closed-loop bandwidth.


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Figure 1.6 Graph Showing Contribution Of Each Term Of (1.24) To Bode Plot |G|

The root-locus method is a graphical technique for plotting

closed-loop poles as a function of some system parameter commonly the gain K.

In contrast to the root-locus method, the Nyquist criterion gives information on the stability of a feedback system. The Nyquist method can be applied not only to systems for which mathematical description is available but also to systems for which experimental frequency response information is available. The Nyquist criterion gives relative stability information in the form of gain and phase margins. The Nyquist criterion uses a polar plot of the open-loop transfer function.

Another frequency response method that is widely used is the Bode diagram. In the Bode gain plot, gain in decibels - dB is plotted against ω on logarithmic scale. In the Bode phase plot the phase in degrees is plotted against ω on logarithmic scale. The stability information of Nyquist criterion is easily transferred to the Bode plots. In addition magnitude Bode curve can be easily plotted using asymptotic approximation. Since Bode plots are much more widely used than the Nyquist plots, we will discuss Bode plots in more detail.

Reader should note that Root-locus and Bode plots complement each other. While root-locus gives the exact closed loop pole locations, the Bode plots give relative stability information in the form of gain and phase margins. The root-locus approach gives information about the stability and the transient characteristics of the system. The frequency response approach can be used to design systems to minimize

the effects of noise. The correlation between the transient and frequency responses is indirect, except for the case of second-order systems.

There are many other sophisticated computerized tuning techniques described in the literature that have been applied to simulated power system models or Scaled-down laboratory systems. These include optimal control, optimal output feedback control [9] - [11], pole placement techniques [12], minimum variance methods [13], [14] and frequency domain techniques [15]. The need for imposing additional design constraints (such as minimizing complex performance indices related to system transient swings) on the controller does not appear to be of prime concern to Utilities. Presently most common method for tuning PSS consists of improving damping (i.e. moving characteristic roots).

B. Nyquist Criterion (Gain And Phase Margins) The Nyquist criterion utilizes either the transfer function or

experimentally obtained frequency response data and gives a measure of the relative stability of the roots of the characteristic equation.

Consider the closed-loop system shown in Fig.1.7. If a sinusoidal signal, R(s), is applied it will be modified and phase-shifted by G(jω) and H( jω) so that the feedback signal Z(jω) will subtract vectorially from the input R(jω).

) Z(j)R(j )E(j ωωω −= (1.27)


- 13 -

Figure 1.7 Feedback Block Diagram

If GH phase shifts the input sinusoid by 180° and amplifies

it by an amount greater than unity, the feedback signal will enhance the original input sinusoid and instability will result. This same intuitive argument can be applied to a wide range of sinusoidal signals applied to the input. It is apparent that if GH has frequency response characteristics such that,

o180- GH when 1 |GH| =∠< (1.28)

And

1 |GH| when 180 |GH| o =<∠ (1.29) Over a range of frequencies then the system should be

stable. There are exceptions to this rule for some systems and in those cases the generalized "Nyquist criterion" must be applied.The relative stability of a feedback system follows

directly from (1.28) and (1.29). Phase margin, PM, is defined as the phase angle of GH relative to 180° when |GH| = 1 or,

o180 GH PM −∠= (1.30)

when |GH| = 1 and Gain Margin, G.M., is the relative

magnitude of GH when the angle of GH is 180°, or,

||1 GM

GH= (1.31)

when oGH 180−=∠ . The gain and phase margin stability parameters are shown

in the Bode plot of Fig.1.8. Gain and phase margins are used to assess the stability of

power systems control loops. This can be illustrated by considering the following power system application. The Bode plot contains the same information as the polar plot and the former is normally used as gain and phase margin for the AVR system can be more readily extracted from the Bode plot.

Figure 1.8 Bode Plots Illustrating Gain And Phase Margins


- 14 -

C. Example A block diagram is shown below for a hypothetical

terminal voltage control loop for a synchronous generator operating at rated speed on open circuit.

Figure 1.9 Voltage Control Loop

A typical problem would be to assess the stability of this

control loop for varying AVR gains. For this case GH = AVR x Gen x Transducer

sTK

sTK

sTKGH

v

v

a

g

a

a

+×

+×

+=

111 (1.32)

The corresponding Bode plot of GH is shown below in

Fig.1.10 for three different values of AVR gain. It is assumed that all other gains and time constants are known.

Bode Plot of GH(jω) shows the magnitude and phase of GH(jω) for 3 values of AVR gain. The gain and phase

margin information is readily available from the Bode plots. Root-Locus Theory The root-locus method involves a graphical approach to

evaluating the excursion of system roots (loci) as a function of the change in magnitude of one or more of the coefficients of the characteristic equation. This graphical technique yields the approximate root locations of the characteristic equation and hence gives information about the "form" of time domain transients of the system for a particular combination of controller and plant parameters.

Most single-input-single-output (SISO) control systems can be manipulated into the standard form shown in Fig.1.7. For our application the transfer function G(s) can be related to the power system transfer function and the feedback transfer function H(s) to the power system stabilizer.

The closed loop transfer function for the system is given by,

)()(1)(

)()(

sGsHsG

sRsC

+= (1.33)

H(s) G(s) 1+ (1.34)

Figure 1.10. Bode Plot For 3 Different Values Of AVR Gain


- 15 -

The characteristic equation is,

0 H(s) G(s) 1 =+ (1.35) Recall that the roots of the characteristic equation give a

direct indication of the frequency modes and time constants of the time domain transients. This can be deduced by first finding the roots of (1.35). After finding the roots, (1.35) can be written as

0 = )jQ P+(s)p+(s mmmnn ΠΠ (1.36)

where -pn are the real roots of (1.35) and -Pm ± jQm are

the complex conjugate roots of (1.35).

)jQ P+(s )p+(ss)Numerator( C(s)

mmmnn ±ΠΠ= (1.37)

The natural modes of response of the closed loop system

are shown below,

)tQsin(ekek mmtp

mm

tpnn

mn θ±Σ+Σ −− (1.38)

Where it is apparent that the time constant in the

exponential terms of (1.38) are identical to the real parts of the roots shown in (1.36). It should also be noted that the imaginary part of the roots shown in (1.36) give a direct indication of the damped radian frequency of the decaying sinusoids as represented in the second set of terms in (1.38).

The root-locus method provides a graphical approach to

finding the roots of a characteristic equation. Consider a slightly modified form of (1.35) given by,

1- H(s)G(s) = (1.39)

It follows that any value of s, say si, which satisfies

1=)s(H)s(G ii (1.40) and

q360° 180°=)s(H)s(G ii∠ (1.41) is a root of the characteristic equation. The transfer

function G(s) H(s) (commonly referred to as the "open loop" transfer function) can be written as,

1−=+Π+Π

)Ps()Zs(K

j

i (1.42)

Where –Zi and –Pj are referred to as the zeros and poles of GH. It follows that the roots of the characteristic equation must satisfy,

1=+Π+Π

|)Ps(||)Zs(|K

j

i (1.43)

and

( ) ( ) q360 180°PsZs ji =+∠∑−+∠∑ (1.44)

For K= 0 to ∞

By initially locating the zeros and poles on a graph with

an imaginary vertical axis and a real horizontal axis (referred to as the complex s-plane), it is possible to establish some rules using (1.43), and (1.44) to sketch the characteristic root excursions (referred to as root loci) while varying K from zero to infinity.

For the Characteristic Equation,

0 KGH(s)1 =+ (1.45)

TABLE 2

RULES FOR ROOT-LOCUS CONSTRUCTION FOR K > 0 1. Loci originate on poles of GH(s) and terminate on the

zeros of GH(s) 2. The root locus on the real axis always lies in a section

of the real axis to the left of an odd number of poles and zeros.

3. The root locus is symmetrical with respect to the real axis.

4. The number of asymptotes na is equal to the number of poles of GH(s), np, minus the number of zeros of GH(s), nz, with angles given by [(2m+1)180°] / (np-nz), where m is an integer between 0 and (na-1).

5. The asymptotes intersect the real axis at σ where σ = (Σ poles of GH – Σ zeros of GH) / (np-nz) 6. The breakaway points are given by the roots of

0=dsdK

Wherevag

vga'

TTTKKK

K = (1.46)

If all parameters are specified except the AVR gain Ka, then a root locus plot for varying Ka would be constructed as shown in Fig.1.11.

D. Root Locus Example Consider the voltage regulator loop of the previous

example shown in Fig.1.9. A typical application for applying root locus techniques would be to study the effects of Ka on the roots of the characteristic equation.

For this case the characteristic equation is given by


- 16 -

0111

1 =+

×+

×+

+sT

KsT

KsT

K

v

v

a

g

a

a (1.47)

This can be written as

1111

−=+++ )

Ts)(

Ts)(

Ts(

K

vga

' (1.48)

Figure1.11 Root Locus Plot Of The System The open-loop poles and zeros of the system are shown

on the s-plane. As the AVR gain increases two of the roots of the characteristic equation move towards the right half-plane (unstable side of the s-plane). Typical time domain characteristics for three values of gain Ka are shown in Fig.1.12. For instance the roots of the characteristic equation for gain Ka1 where the AVR is at a low gain, give rise to reasonably damped oscillations.

This is compared with the root locations for a gain Ka2

where the system is oscillatory and those for gain Ka3

where the system is unstable. The next section describes the State-Space approach to

modeling and assessing stability of linearized systems. This technique allows the analyst to formulate the system model using first order differential equations and then use computer programs to assess stability and synthesize controllers.

VI. STATE-SPACE TECHNIQUES

A. General Comments Root-locus and frequency-response methods are quite

useful for dealing with single-input single- output-systems. For example, by means of open-loop frequency-response tests, we can predict the dynamic behavior of the closed-loop system. If necessary, the dynamic behavior of

a complex system may be improved by inserting a simple lead or lag compensator. The techniques of conventional control theory are conceptually simple and require only a reasonable amount of computation.

In conventional control theory, only the input, output, and error signals are considered important; the analysis and design of control systems are carried out using transfer functions, together with a variety of graphical techniques such as root-locus plots and Bode plots. The unique characteristic of conventional control theory is that it is based on the input-output (transfer function) relationship of the system.

The main disadvantage of conventional control theory is that, generally speaking, it is applicable only to low order linear time-invariant systems having a single input and a single output. It is cumbersome for time-varying, nonlinear larger systems with multiple-inputs multiple-outputs (MIMO).

Figure 1.12 Time Domain Sketches For 3 Values Of Ka

Because of the necessity of meeting increasingly

stringent requirements on the performance of complex control systems, modern control theory is gaining in popularity. Modern control theory is based on the concept of state [7], [9]. The concept of state by itself is not new since it has been in existence for a long time in the field of classical dynamics and other fields.

System design using modern control theory enables the engineer to design optimal control systems with respect to given performance indices.

The state-space approach can be used to study various aspects of control systems. These include:


- 17 -

• obtaining root contours, or root loci, or eigenvalue plots using digital computer programming techniques,

• optimal controller design for large systems, • system reduction and equivalent models of large

systems, • sensitivity studies to ascertain the influence of

system parameters on roots and time domain transients.

We will be mostly dealing with the eigenvalue plots. A physical system described by a set of differential

equations of any order can be transformed into a set of first order D.E.'s using simple substitutions. The variables of the new set of D.E.'s define the "state" of the system at any instant of time "t".

The transformed set in matrix form is given by,

)()()(.

tBFtAXtX += (1.49)

X(t) C Y(t)= (1.50) Where, X is an n x 1 vector containing the state variables .

X is an n x 1 vector containing the first derivative of X

F is an m x 1 vector containing the independent forcing functions to the system

A is an n x n characteristic matrix B is an n x m coupling matrix Y is an r x 1 output vector C is an r x n coupling matrix

B. State-Space Models It is possible to derive the standard state-space form,

.X = A X + BF(t) of a physical system from any of the

different mathematical models of the system. State space equations can be written from a block

diagram of a system as illustrated by considering the power system "inertia" model shown below. In this model, the turbo-generator inertia is denoted "H”; sources of damping are represented by the term "D”; and the synchronizing coefficient is denoted "K1”. States ω and δ are the generator speed and rotor angle, respectively.

C. State Space model Example Determine the state-space model and the eigenvalues for

the system shown in Fig.1.13.

Figure1.13 Inertia Model

The steps used in finding the state space model are

described below. i) Assign a state variable ix to the output of each

integrator block (l/s).

ii) Since the input to each integrator block must be ix.

, the state space equations can be written directly.

For this example,

distPH

xKDxH

x21)(

21

211

.

1 +−−=

(1.51)

12

.xx =

(1.52) The matrix form of these two equations is,

dist2

11

.

2

.

1 P0H21

xx

0H2K

1H2D

x

x

+

−

−=

(1.53)

which is of the form,

)t(BFAXX.

+= (1.54) The output equation is

1xy = (1.55) The matrix form of this equation is,

=

2

1 0] [1 Yxx

(1.56)

We next show the connection between roots of

characteristic equation and of the characteristic matrix [A]. In this example the transfer function linking shaft speed,

ωs, to power disturbance, Pdist, is derived as follows,


- 18 -

EHss 21=ω (1.57)

ssdist sKDPE ωω 1−−= (1.58)

Substituting (1.58) into (1.57) and simplifying yields

s

dist 2 1

1 s2HD KP s s

2H 2H

ω =+ +

(1.59)

Where the roots of the characteristic equation are given

by

2/1

12

21 244,

−

±−=

HK

HD

HDss (1.60)

The eigenvalues, λ, of matrix [A] of the corresponding

state space model are given by the solution to,

[ ] 0=− AIDet λ (1.61) Where, Det is the determinant of the matrix, and

=

1001

I (1.62)

It follows that

=

λλ

λ0

0I (1.63)

For this example,

−

−=

02

12

1

HK

HD

A (1.64)

Equation (1.61) applied to this example yields the

following,

−

+

λ

λH

KHD

21

21

(1.65)

Expanding the determinant expression of (1.65) yields,

022

12 =++H

KHD λλ (1.66)

It can be noted that (1.66) is identical to the

Characteristic Equation shown in (1.59). Several powerful digital computer subroutines are

available for calculating eigenvalues (characteristic roots) directly from the matrix [A]. This allows the analyst to obtain roots of very large systems by generating the [A] matrix and inserting the elements of [A] directly into a computer program.

It is useful to compare classical methods for assessing system stability with the state-space eigenvalue approach. For classical methods a considerable amount of algebra is required in order to get the equations in a form that will allow approximate graphical methods or root-finding algorithms to be applied. In other words, the dynamic equations have to be compressed down to an input-output expression. With the state-space approach, the state-space model is formulated directly from the separate blocks or equations describing the physical system. Computer techniques are applied directly to the [A] matrix and thus manual algebra is virtually eliminated.

VII. SYSTEM SIMULATION An important part of investigating system stability and

designing controllers is to simulate the complete system with nonlinearities. State-space and classical methods for assessing stability are applicable to linearized representation of the physical plant and to low order nonlinear models.

If one is concerned about the operation of the system under large signal operation, then the nonlinear equations could be simulated on the digital computer and plots of time domain transients obtained for pre-selected parameters and disturbances to the system. Transient stability programs are used extensively for this purpose in the electric power industry. Although these nonlinear simulation programs are needed for large-signal investigations they are very inefficient when used for trial-and-error investigations in the design of controllers for linear systems. The Matlab/Simulink [16] package is very widely used for this purpose. The next chapter outlines basic power system dynamics and identifies factors contributing to negative damping of synchronous generator rotor oscillations.


- 19 -

CHAPTER 2 OVERVIEW OF POWER SYSTEM

STABILITY CONCEPTS Arjun Godhwani, Jose Taborda, Les Hajagos

I. INTRODUCTION Power system stability is considered by first presenting

the definition followed by the classification of stability in to Voltage, Frequency and Rotor angle stability. It is then argued that the subject of this tutorial is ‘Power System Stabilization via Excitation Control’ and the rotor angle stability is the one that plays the critical role. We consider a synchronous machine connected to infinite bus and develop an expression for electric power as it is related to rotor angle. We go on to discuss the transient (large signal) and oscillatory rotor angle (small signal) stability and their dependence on synchronizing and damping torques. It is established that the synchronizing torque governs the transient stability whereas the oscillatory stability is affected by the damping torque. We next show that a fast acting, high gain AVR improves the synchronizing coefficient but may reduce the damping coefficient. The reduction in damping can be compensated by power system stabilizer (PSS), acting through the voltage regulator, to provide a supplementary signal to increase the damping and thus stabilize the system against oscillatory instability. We conclude the paper by briefly discussing different kinds of PSS in use and their basic structure.

The overview of power system stability concepts was part of the IEEE tutorial of 1981 [17]. An IEEE committee report [18] just preceded the tutorial. Although the basic concepts of the power system stability have not changed that much, the technology of power system stabilizers has gone through significant evolution. In this paper we present the status of power system stability concepts at present and discuss the status of some issues such as stability classification, new types of power system stabilizers etc. We start with stability classification.

II. POWER SYSTEM STABILITY CLASSIFICATION

A. Definition Power System Stability is the ability of an electric power

system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with system variables bounded so that system integrity is preserved. Integrity of the system is preserved when practically the entire power system remains intact with no tripping of generators or loads, except for those disconnected by the isolation of the faulted elements or intentionally tripped to preserve the continuity of operation of the rest of the system.

B. Categories Of Stability Analysis of stability, including identifying key factors

that contribute to instability and devising methods of improving stable operation, is greatly facilitated by classification of stability into appropriate categories. There are mainly three categories of stability: voltage [19], frequency [20], and rotor angle stability. The following are descriptions of the corresponding forms of stability phenomena.

1) Voltage Stability

Voltage stability is concerned with the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition. Instability that may result occurs in the form of a progressive fall or rise of voltages of some buses. A possible outcome of voltage instability is loss of load in an area, or tripping of transmission lines and other elements by their protection equipment leading to cascading outages.

2) Frequency Stability

Frequency stability is concerned with the ability of a power system to maintain steady frequency within a nominal range following a severe system upset resulting in a significant imbalance between generation and load. It depends on the ability to restore balance between system generation and load, with minimum loss of load. Instability that may result occurs in the form of sustained frequency swings leading to tripping of generating units and/or loads.

3) Rotor Angle Stability

Rotor angle stability is concerned with the ability of interconnected synchronous machines of a power system to remain in synchronism after being subjected to a disturbance from a given initial operating condition. It depends on the ability to maintain/restore equilibrium between electromagnetic torque and mechanical torque of each synchronous machine in the system. Instability that may result occurs in the form of increasing angular swings of some generators leading to their loss of synchronism with other generators.

Note: The subject of this tutorial is ‘Power System

Stabilization via Excitation Control’. While Frequency stability and Voltage stability are important in the overall power system stability problem, it is the rotor angle stability that has a critical role to play in ‘Power System Stabilization via Excitation Control’. Therefore we have limited ourselves to a short description of the Frequency and Voltage stabilities.

We next discuss basic issues involved in the power

system stability. We begin with the consideration of a generator connected to infinite bus configuration and develop the concepts of transient and oscillatory stability.


- 20 -

III. BACKGROUND - GENERATOR CONNECTED TO INFINITE BUS

Since power systems rely on synchronous machines for the generation of electrical power, a necessary condition for the transmission and exchange of power is that all generators rotate in synchronism. Consider a remote generator connected radially to a major substation of a very large system, as shown in Fig. 2.1. The transmission system feeds into the large system or infinite bus, which assumes that its voltage is so strongly influenced by large nearby generation that it is independent of events at the remote generator.

Gen.

ET XT XS

EOEHV

XE=XT+XS

InfiniteBus

Figure 2.1 Single-Machine Power System Configuration

A phasor diagram of the voltages on such a system is

shown in Fig.2.2. These voltages are: Eo infinite bus voltage ET generator terminal voltage E’q generator internal voltage behind transient reactance Eq generator internal voltage behind synchronous

reactance

I

I.Xd

EO

ET

Eq

Eq'

I.Xd'

I.XE

δSS

δ

Figure 2.2 Generator/ Infinite Bus Phasor Diagram

The angle difference in the phasors indicates a real

power flow from the generator into the infinite bus. In the analysis of classical steady-state stability, the

voltage Eq and the power angle δss are used in the characteristic electrical power flow equation. However, for simple transient and dynamic stability analyses, it is more appropriate to use the voltage E behind transient reactance and the angle δ between this voltage and the infinite bus voltage. In this way, the generator electrical power can be expressed as:

δ+

= sinXdX

EqEPe

'o

'

E (2.1)

This power-angle characteristic is shown as the sine

function plotted in Fig. 2.3.

Figure 2.3 Power-Angle Curve

IV. TRANSIENT STABILITY Transient stability analysis is primarily concerned with

the immediate effects of transmission line disturbances on generator synchronism. Fig.2.4 illustrates the typical behavior of a generator in response to a fault condition. Starting from the initial operating condition (point 1), a close-in transmission line fault causes the generator electrical output power PE to be drastically reduced. The resultant difference between electrical power and mechanical turbine power causes the generator rotor to accelerate with respect to the system, increasing the power angle (point 2). When the fault is cleared, the electrical power is restored to a level corresponding to the appropriate point on the power-angle curve (point 3). Clearing the fault necessarily removes one or more transmission elements from service and at least temporarily weakens the transmission system. For simplicity, this effect is not shown in Fig.2.4.

After clearing the fault, the electrical power out of the generator becomes greater than the turbine power. This causes the unit to decelerate (point 4), reducing the momentum the rotor gained during the fault. If there is enough retarding torque after fault clearing to make up for the acceleration during the fault, the generator will be transiently stable on the first swing and will move back toward its operating point in approximately 0.5 second from the inception of the fault. If the retarding, torque is insufficient, the power angle will continue to increase until synchronism with the power system is lost.


- 21 -

180o0o 90o

Unstable

Stable

PACCEL

PDCEL

1

3

2

4

Power Angle -o

PMAX

PMTurbinePower

Power

Figure 2.4 Power-Angle Curve Illustrating Transient Stability.

Excitation system forcing during and following the fault

attempts to increase the electrical power output by raising the generator internal voltage Eq, thus increasing PMax. Fast and powerful excitation systems can improve transient stability, although the effect is limited due mainly to the large field inductance of the generator which prevents a sudden change in E’q for a sudden increase in exciter output voltage.

The steady-state stability refers to the ability of a power system to maintain synchronism at all points for incremental slow-moving changes in power output of units or power transmission facilities. Steady-state stability a small signal phenomenon is governed by the synchronizing coefficient. Transient stability a large signal phenomenon is also governed by the synchronizing coefficient. A fast acting, high gain AVR in general increases the synchronizing coefficient but may decrease the damping coefficient. Thus a high gain AVR helps the steady state and transient stabilities but may reduce the oscillatory stability.

V. TRANSIENT VS. OSCILLATORY STABILITY In present-day systems, a machine being transiently

stable on the first swing does not guarantee that it will return to its steady-state operating point in a well-damped manner and thus be stable in an oscillatory mode. Significant improvement in transient stability has been achieved through very rapid fault detection and circuit breaker operation. System effects such as sudden changes in load, short circuits, and transmission line switching not only introduce transient disturbances on machines, but also may give rise to less stable operating conditions. For example, if a transmission line must be tripped due to a fault, the resulting system may be much weaker than that existing prior to the fault and oscillatory instability may result.

One solution to improve the dynamic performance of this system and large scale systems in general could be to

add more parallel transmission lines in order to lower the reactance between the generator and the load center. Such a solution may be quite costly as well as unfeasible to implement. In the presence of a weak transmission system, control means, such as a power system stabilizer (PSS), acting through the voltage regulator, can provide significant stabilization of such oscillations if properly implemented.

VI. OSCILLATORY STABILITY

A. Characteristic Dynamic Equation Following a system disturbance, whether it is a large

disturbance or just a minimal load change on the system, a generating unit will characteristically tend to oscillate around its operating point until it again reaches steady state. The characteristics of these oscillations are analogous to the motion of the spring mass systems.

For a synchronous machine under constant field excitation, an approximation of its dynamic motion is obtained by relating the angular acceleration of the generator rotor to the torques imposed on the rotor, in the same manner that linear acceleration relates to force in a spring mass system. This relationship for a synchronous machine is:

(Inertia) (Angular Acceleration) + (Damping Torque) + (Electrical Torque - Turbine Torque) = 0. For small changes, the behavior is described by a

characteristic equation having the same structure as that of the mechanical spring mass system. This is the "swing" equation:

0KdtdD

dtdH2

1s

2

2

s=δ∆+δ

ω+δ

ω (2.2)

Where the parameters are defined as:

∆δ (radians) rotor angle deviation from the steady state operating point

H (kW sec/kVA) inertia constant of the rotor of the generating unit (or group of units)

D (p.u. power/p.u. freq. change) damping coefficient representing friction and windage, prime mover and load damping, etc.

ωs (radians/sec) synchronous frequency = 377 r/s on 60 Hz systems

K1 (p.u. ∆P/radian) synchronizing coefficient. Acting like the restoring force of the spring in the

mechanical spring mass system, the term K1∆δ called the synchronizing power, acts to accelerate or decelerate the rotating inertia back toward the synchronous operating point. For small deviations from the operating point, the synchronizing coefficient K1 is the slope of the transient


- 22 -

power angle curve at the particular steady state operating point, as shown in Fig. 2.5.

If δo is the steady state angle between Eo and E q& , then the slope is simply the derivative of the power angle function, or:

o

e

'o

'

oE cos

XdX

EqEd

dPK δ+

=δδ1 (2.3)

Where:

E'

q is the internal voltage behind transient reactance in p.u., Eo is the infinite bus voltage in p.u., X

'd is the generator transient reactance in p.u.,

Xe is the external reactance in p.u., and δ is the angle between E and Eo.

Stronger transmission systems, with lower values of Xe,

have a larger value of K1 and thus provide more synchronizing power to the generator.

The characteristic "swing" equation in (2.2) governs the power system dynamic response with a damped oscillatory behavior, having an oscillation frequency of approximately

.sec/radiansH2

K s1n

ω≅ω (2.4)

The inherent modal frequency that is exhibited is

dependent mainly on such factors as unit inertia(s), machine and transmission system reactances, and load level.

There are two distinct types of dynamic oscillations that have been known to present problems on power systems. One type occurs when a generating unit (or group of units) at a station is swinging with respect to the rest of the system. Such oscillations are called "local mode" oscillations. The oscillations are termed "local" because the behavior is mainly localized at one plant, with the rest of the system experiencing much less of the effects. Spontaneous local oscillations tend to occur when a very weak transmission link exists between a machine and its load center, such as for an isolated power station sending power across a single long transmission line. Such systems can usually be accurately modeled by a machine, single transmission line, and infinite bus.

B. Local Vs. Inter Area Oscillations

δ⋅⋅

= sinXE'E

P OqE

PM

Power

Power angle 0o 180o90o

OperatingPoint

PE

0

Figure 2.5 Power Angle Curve Showing Derivation of Synchronizing Coefficient K1.

The frequency of the characteristic local mode is

generally in the 1 2 Hz range, depending mainly on the impedance of the transmission system. Stronger transmission systems generally have the higher local mode frequencies along with less of a tendency toward spontaneous or undamped oscillations.

A second type of oscillations, known as "interarea" modes, is more complex because they usually involve combinations of many machines on one part of a system swinging against machines on another part of the system.

C. Negative Damping Due To Voltage Regulator Just as in a mechanical spring mass system, a power

system contains inherent damping effects that tend to damp out dynamic oscillations. Even when the proper conditions exist for dynamic instability (i.e., high network reactances, line outages, high load levels, etc.), the natural damping of the system, represented by the positive D term in equation (2.2), will prevent any sustained oscillations unless a source of negative damping is introduced.

It is generally recognized that the normal feedback control actions of voltage regulators and speed governors on generating units have the potential of contributing negative damping which can cause undamped modes of dynamic oscillations. Direct evidence of this has been seen by the fact that sustained oscillations on power systems have been stopped simply by switching voltage regulators from automatic to manual control. However, removing voltage regulators from service is not a realistic solution to the problem, because the beneficial features of the voltage regulator would be lost. The fortunate aspect of the problem is that the same voltage regulator control that causes negative damping can be supplied with supplementary controls to contribute positive damping for oscillatory stabilization.

The major function of the voltage regulator is to continually adjust the generator excitation level in response to changes in generator terminal voltage. The voltage


- 23 -

regulator acts to accurately maintain a desired generator voltage and change the excitation level in response to disturbances on the system.

Fig. 2.6 shows a block diagram of the major elements associated with a generating unit under voltage regulator control. Any change in the terminal voltage magnitude ET from the reference set point provides an error signal (∆e) to the voltage regulator, which calls for a change in excitation level. The major delay in this voltage feedback loop is due to the response in machine flux (E’q) for a change in generator field voltage (EFD). This delay is due to the large inductance of the generator field winding. For a generator on line, this delay can be represented by a time constant T’q which is usually about 2 seconds.

As an illustration of how the action of voltage regulator control affects dynamic oscillations, the "swing" equation is rewritten to show the effect of changes in machine flux, ∆E’q

02212

2=∆+δ∆+δ

ω+δ

ω

'

ssqEKK

dtdD

dtdH (2.5)

The term K2∆E’q is determined mainly by changes in

excitation level, as determined by the control action of the voltage regulator with phase lags due to the exciter and generator field circuit.

Figure 2.6 Block Diagram Of Generator Under Voltage Regulator Control

The undamping effect of the regulator can be illustrated

by the phase relationship of the various rotor torque components of dynamic oscillations. The phasor diagram in Fig. 2.7 shows the relative phase of various generator signals for small deviations around the operating point. For local machine oscillations the angle deviation (∆δ) lags the speed deviation (∆ω) by 90 degrees and the terminal voltage deviation (∆ET) by 180 degrees. Note that the phasor plots of Fig. 2.7, Fig. 2.8, and Fig. 2.10 are used to illustrate the concepts of oscillatory stability and are not considered a useful analytical tool for assessing stability.

The vertical axis of Fig. 2.7 is the synchronizing axis, and components of restoring power along this axis in the positive direction tend to increase the frequency of dynamic oscillations. The horizontal axis is the damping axis. Components of restoring power, which are in phase with the oscillations of machine speed or frequency, provide damping to these oscillations.

Figure 2.7 Phasor Relationship Of Signals And Torque Components; Unit

With Fixed Excitation However, components that are out of phase with

machine speed and point toward the right tend to cancel the natural damping provided by the unit. The components of restoring power shown in Fig. 2.7 correspond to that of a machine under manual voltage control (i.e., constant excitation). The resultant forcing action for oscillations points to the left of the vertical axis, indicating that dynamic oscillations will be damped.

The effect of voltage regulator control on oscillations can be seen by analyzing the phase components of the restoring power caused by changes in excitation. Generator terminal voltage deviation (∆ET) is sensed by the error detector of the voltage regulator with reversed polarity due to negative feedback voltage control. It is this signal which initiates control action of the generator excitation system.

Fig. 2.8 shows the effect of the voltage feedback signal on the resultant restoring power for the case of a plant with a very weak transmission system. The phase lags Ø1 and Ø2 between the error signal (∆e or -∆ET) and the generator flux (∆E’q) correspond to the time delays associated with the regulator/exciter action and the time constant of the generator field, respectively. The major delay occurs in the generator, with the delay of the regulator/exciter of secondary effect. The restoring power due to ∆E’q has a component, which acts to reduce the damping of dynamic oscillations. The resultant dynamic forcing action may point into the unstable region and result in undamped oscillations that grow in magnitude.

An excitation system with a very fast response and a high effective gain will act to magnify the negative damping contribution. With a slightly lower phase lag in the excitation system, the contribution due to regulator action is more directly in line with the negative damping axis, and has a larger magnitude because of the higher effective gain. The result is more of a tendency toward oscillatory instability with high response and high initial response excitation systems, and generally a greater need for, supplementary stabilizing devices.


- 24 -

t

DampingAxis

ResultingResponse

Et

Et

D s

Positive Damping(Stable Region)

Negative Damping(Unstable Region)

Synchronizing Axis

E'qRegulator

Forcing Action

12

Figure 2.8 Phasor Relationship Of Signals And Torque Components; Unit

Under Voltage Regulator Control, Large Xe. It should be made clear that the undamping component

K2∆E’q is generally only large enough to cause growing oscillations when the unit is under conditions that cause large power angles, such as a weak transmission system, large load, or low terminal voltage. Test experience has shown that a unit with a steady state power angle above 70 degrees (between the generator internal voltage and the infinite bus voltage) tends to have potential local mode stability problems under voltage regulator control.

D. PSS For Improved Oscillatory Stability Since voltage regulator control can act to reduce the

damping of unit oscillations by sensing terminal voltage, it seems reasonable that a supplementary signal to the voltage regulator can increase damping by sensing some additional measurable quantity. In doing so, not only can the undamping effect of voltage regulator control be cancelled, but damping can be increased so as to allow operation even beyond the steady state stability limit. This is the basic idea behind the power system stabilizer (PSS). The supplementary signal of a PSS may be derived from such quantities as changes in shaft speed (∆ω), generator electrical frequency (∆f)), or electrical power (∆PE).

There are a number of considerations in selecting the right input quantity. The factors that play a role are requisite gain and phase compensation, the susceptibility to other interactions such as torsional oscillations, and the noise level in transducers. The speed and frequency inputs have been widely used. The trend is more towards PSS design based on integral of accelerating power. This type of PSS provides satisfactory damping and reduced torsional interactions. The basic theory of a PSS control based on integral of accelerating power input signal, its advantages and the methods of tuning the stabilizers are discussed by Murdoch et al [21].

A block diagram showing the major elements of a typical PSS is shown in Fig. 2.9. A special speed, frequency, or power transducer converts the stabilizing signal to a control voltage. The transducer output is then phase shifted by an adjustable lead lag network, which acts to compensate for time delays in the generator and

excitation system. The resulting signal is amplified to a desired level and sent through a signal wash out module. This module functions to continuously balance the stabilizer output and thus prevent it from biasing the generator voltage for prolonged frequency or power excursions. The output limiter serves to prevent the stabilizer output signal from causing excessive voltage changes upon load rejection and to retain the beneficial action of regulator forcing during severe system disturbances [22].

Transducer SignalWashout

SignalLimiter

Lead/LagNetwork Amplifier

Term. Freq.Shaft Speed

or Power

Outputto Regulator

SensingCircuit

Figure 2.9 Major Elements Of Power System Stabilizer

For proper damping action, PSS control settings

consisting of the lead, the lag, and the gain adjustments of the stabilizer have to be judiciously determined. Since the dynamic response of a unit involves the machine and the external system, such settings may vary from unit to unit. Also, particular PSS settings designed to suppress intertie oscillations may not be effective in damping local machine/system oscillations. Therefore, setting procedures for PSS's generally involve either a field test, a study of the machine and system, or both.

A PSS setting procedure involving frequency response tests in the field has been widely accepted by utilities in the Upper Midwest and Western United States for damping inter area oscillations. From the measurement of terminal voltage deviation in response to sinusoidal inputs to the voltage regulator reference, phase information is obtained upon which PSS lead lag settings are determined. The PSS amplifier gain is empirically found. This is achieved by monitoring the dynamic response of the unit with the PSS in service and slowly increasing the PSS gain until small rapid oscillations appear, usually in the frequency range of 1 to 4 Hz. The PSS gain is then set to approximately one third this value.

Since local mode stability problems often involve a single machine and the system, modeling procedures can be used as an aid in determining PSS settings. Such modeling techniques include representation of the PSS, excitation system, generator, and external system. Both time response and frequency domain analysis can be used in evaluating PSS settings.


- 25 -

DampingAxis

ResultingResponse t

∆ω

Positive Damping(Stable Region)

Negative Damping(Unstable Region)

Synchronizing Axis

φ4φ3

PSS Forcingaction

PSS output

Figure 2.10 Phasor Relationship Of Signals; Unit Under Voltage Regulator Control With PSS In Service.

A typical phase relationship of the signals associated

with the PSS control sensing frequency or speed is shown in Fig. 2.10. The quantity ∆ω sensed by the PSS is phase shifted by an amount Ø3 in the lead lag network. This PSS output signal is sent into the voltage regulator. The time delays in the excitation system and generator result in an actual forcing action that has a large damping component. By increasing the PSS amplifier gain, this damping action can be increased. However, higher order effects limit this gain. The resultant dynamic forcing action due to the PSS cancels the undamping effect of voltage feedback control and extends the stability limit.

VII. GENERATOR MODELS Linearized generator models are obtained by considering

small variations about an operating point. These linearized models are then used for analyzing simple oscillatory stability problems. DeMello and Concordia [23] have described such a model with the aid of the block diagram of Fig. 2.11, relating the pertinent variables of electrical torque (∆Te), speed (∆ω), angle (∆δ), terminal voltage (∆ET), and field voltage (∆EFD). This model represents a

generator as it behaves when connected through an external reactance to an infinite bus. This model is used to illustrate generator dynamic behavior as well as to analyze actual local mode dynamic stability problems. The model applies to a D-Q axis generator representation with a field circuit in the direct axis, but without sub-transient amortisseur or solid iron eddy current effects in either axis.

The effects of the voltage regulator can be included in the model by adding a feedback branch between the output ∆ET and the input ∆EFD. Similarly, a power system stabilizer branch can be included in Fig. 2.11 between the signal ∆ω and the summing junction of the voltage regulator. A speed-governor can also be represented as a path between ∆ω and the mechanical turbine torque ∆Tm.

Kundur [22] shows the effect of a high gain voltage regulator and the role played by a PSS.

Specifically the value of K5 has a significant bearing on the influence of the AVR on the damping of system oscillations. Kundur [22] shows analytically as well as via examples that for high value of external system reactance and high generator outputs, K5 is negative. In practice, the situation where K5 is negative is commonly encountered. For such cases, a high response exciter is beneficial in increasing sysnchronizing torque. However, in so doing it introduces negative damping. In order to meet these conflicting requirements one then uses the high response exciter and a PSS.

VIII. CONCLUSIONS Power system stability concepts are presented. It is

shown that a PSS can be used to compensate for the detrimental effects of a high response exciter. Basic structure of a PSS is shown.

Figure 2.11 Small Signal Model Of Generator, Transmission Line And Infinite Busbar


- 26 -

CHAPTER 3 PERFORMANCE CRITERIA AND

TUNING TECHNIQUES Alexander Murdoch and George Boukarim

I. INTRODUCTION This chapter addresses the application of power system

stabilizers, beginning with a discussion of the required contribution to improve system small signal (dynamic) stability, and ending with criteria for tuning a stabilizer to meet these objectives.

Tuning of supplementary excitation controls for stabilizing system modes of oscillation has been the subject of much and continuing work during the past 30 years, and a number of different tuning approaches have been proposed and used during this period, each with its own set of benefits and limitations. These approaches include phase compensation, root locus, optimal and robust control, as well as others. The approach that will be discussed in this tutorial utilizes a combination of phase compensation, root locus, and time domain analyses to tune and evaluate the performance of the PSS. Phase compensation is the process of selecting PSS lead/lag settings to compensate for phase lags introduced by the generator, excitation system, and power system. The PSS lead/lag settings introduce the necessary phase shift to the PSS control signal to ensure that generator electrical torque modulations resulting from the PSS action are in phase with generator speed oscillations [23]to[33]. Root locus analysis is used to select the PSS gain, and to determine the PSS instability gain and ensure adequate gain margin in the PSS control loop. Finally, time domain analysis showing the PSS response to small and large system disturbances is used to validate the performance of the PSS in the non-linear power system under different system and operating conditions.

Independent of the techniques utilized in tuning stabilizer equipment, it is necessary to recognize the non-linear nature of power systems and that the objective of adding power system stabilizers is to extend power transfer limits by stabilizing system oscillations. Adding damping is not an end in itself, but as a means to extending power transfer limits. This chapter addresses the performance characteristics of power system stabilizers with respect to extending power transfer stability limits for both remote generation and situations where inter-tie mode oscillations may be critical to system stability. Both small and large disturbance aspects of performance are included, resulting in a definition of desired stabilizer performance to ensure a robust design meeting the system requirements. In addition, a relationship is established between desired performance and the phase compensation characteristics that are useful for stabilizer tuning.

II. PERFORMANCE OBJECTIVES

A. Oscillatory Stability Limits Applying power system stabilizers can extend power

transfer stability limits by improving small signal stability which is characterized by lightly damped or spontaneously growing oscillations in the 0.1 to 4 Hz frequency range. This is accomplished via excitation control to contribute damping to the system modes of oscillation. Consequently, it is the stabilizer's ability to enhance damping under the least stable conditions, i.e., the "critical conditions", which is important. Additional damping is primarily required under conditions of weak transmission and heavy load as occurs, for example, when attempting to transmit power over long transmission lines from remote generating plants or over relatively weak ties between systems. Contingencies, such as line or unit outages, often precipitate such conditions. Hence, systems that normally have adequate damping can often benefit from stabilizers during such abnormal conditions. In most cases the PSS is not required for normal operation but may be critical during contingencies or islanding operation.

It is important to realize that the stabilizer is intended to provide damping for small excursions about a steady-state operating point, and not to enhance transient stability, i.e., the ability to recover from a severe disturbance. In fact, the stabilizer will often have a deleterious effect on transient stability by attempting to pull the generator field voltage out of ceiling too early in response to a fault. The stabilizer output is generally limited to prevent serious impact on transient stability, but stabilizer tuning also has a significant impact upon system performance following a large disturbance, as will be discussed.

B. System Modes Of Oscillation As described in previous chapters, the power system

oscillations of concern to stability occur generally in the 0.2 to 2.5 Hz frequency range. There are occasionally inter-tie modes in the region of 0.1 Hz and slightly below, and some aero-derivative turbines have local mode frequencies near 4Hz operating into strong systems. The rotors of machines, behaving as rigid bodies, oscillate with respect to one another using the electrical transmission path between them to exchange energy. There are many different modes in which such oscillations may occur, often simultaneously.

Experience suggests that it is not unusual for a generating unit to participate in both local and inter-area or inter-tie modes of oscillation, the damping and frequency of which vary with system operating conditions. The Power system stabilizers must therefore be able to accommodate both of these modes, and be robust to changes in the system operating conditions. Since a single unit or power plant is dominant in local modes, its stabilizer can have a very large impact on damping those oscillations. By contrast, a single unit participates in only a portion of the total magnitude of power oscillation in the inter-tie mode.


- 27 -

Therefore, a power system stabilizer applied to a single unit can only contribute to the damping of an inter-tie mode in proportion to the power generation capacity of the unit relative to the total capacity of the area of which it is a part. As a consequence, a stabilizer should be designed to provide adequate local mode damping under all operating conditions, with particular attention to conditions of heavy load and weak transmission, and simultaneously to provide a high contribution to damping of inter-tie modes. These criteria ensure good performance for a wide range of power system contingencies.

C. Tuning Concepts Stabilizers must be tuned to provide the desired system

performance under the condition which requires stabilization, typically weak systems with heavy power transfer, while at the same time being robust in that undesirable interactions are avoided for all system conditions.

To provide damping, the stabilizer must produce a component of electrical torque on the rotor that is in phase with speed variations. The implementation details differ, depending upon the stabilizer input signal employed. However, for any input signal the transfer function of the stabilizer must compensate for the gain and phase characteristics of the excitation system, the generator, and the power system [31] to [33]. These elements collectively determine the transfer function from the stabilizer output (Epss) to the component of electrical torque, which can be modulated via excitation control (Tep). It must be noted that Tep is only a portion of the total torque acting on the rotor, and that it cannot be measured directly. Since it is related to flux changes in the machine, it can be measured indirectly via terminal voltage, as will be discussed in later sections. The transfer function relating ∆Tep to ∆Epss, denoted here as GEP(s), is strongly influenced by voltage regulator gain, generator power level, and ac system strength.

Figure 3.1 Stabilizer With Speed Input System Block Diagram

The block diagram in Fig. 3.1 illustrates, in terms of a

few basic small-signal transfer functions, the relationship between the applied torques on the turbine-generator shaft and the resulting generator rotor speed, Gϖ and rotor angular displacement, δ. The electrical torque may be

considered to have two components, viz. (3.2) that which is produced by the power system stabilizer solely by modulation of generator flux, Tep, and (3.2) that which results from all other sources, including shaft motion, Teo The functional relationship between speed and torque is shown for a stabilizer employing generator speed as an input signal. The contribution of torque due to the stabilizer path is given by:

)s(P)s(GEP)s(PSST

G

ep ∆

ω ==ϖ∆

∆ (3.1)

where GEP(s) = the plant through which the stabilizer must operate. (Generator, Exciter, and Power System)

= pssep ET ∆∆ / (3.2)

PSSω(s) = speed-input stabilizer = Gpss ϖ∆∆Ε / (3.3)

Tep = component of electrical torque due solely to stabilizer path

Epss = stabilizer output signal The plant GEP(s) has the highest gain and greatest phase

lag under conditions of full load on the unit, hence a target value of base load for the tuning process. The tuning will also be affected by the transmission system. Expected values from short circuit studies give starting point and various contingency cases will define likely weaker transmission cases. A wide range should be covered to insure robustness in the tuning. The PSS may not be critical for normal operation but critical if the system become weaker. Choosing high AVR gains may limit operation without PSS for maintaining small signal stability margins.

The de Mello Concordia model [23] in Fig 3.2 is similar to the basic block diagram in Fig 3.1 and allows us to illustrate the process in conceptual terms. The top part of this figure shows the relationship between the accelerating torque and the rotor speed and angle. These are the basic electromechanical equations for the generator. The terms "D" and "K1" represent the equivalent damping and synchronizing torque coefficients. The lower part of this figure shows the effects of the AVR and the flux dynamics in the generator. The effect of the AVR is to increase the effective "K1" term, but unfortunately at the same time this reduces the damping term. To maintain or increase the dynamic stability margin it is necessary to increase the effective "D" term in the model. If we consider the addition of the PSS loop which acts through the AVR and generator field circuit, what we would like the PSS control loop to do is to provide changes in torque which are exactly in phase with changes in rotor speed.


- 28 -

12Hs

ω0s∆ω

∆δ

D

K1

∆Tm

K2

∆Te

Σ

K31 + sK3T’do

AVR Σ∆Eref∆E’q

PSS InputSection

PSS Gainand Phase

Input(s)

K4 K5

Σ

K6

-+

-

-+

+

+

-

-+ “y”Σ

∆ω Input

+

+

Σ+

Fig. 3.2. deMello-Concordia Block Diagram

The stabilizer design may have one or more inputs, and

from a control point of view any measurable signal in which the modes that are to be controlled are “observable” would be a candidate for application. The signals of common use are speed, frequency, or electrical power. They may appear singly or in combination to form a PSS input signal as indicated in the first lower block of Figure 3.2. After appropriate phase and gain compensation there is one output signal most commonly applied to the reference summing point in the excitation system, but may be added later with the output of the AVR in some systems.

A great deal of material was presented in the 1981 tutorial [17] to outline the characteristics of the various input signal choices. At that time, there were many commercial offerings that used single input speed, frequency or power. Early designs of PTI for accelerating power and later work by Ontario Hydro for an integral of accelerating power design have led to today’s commercial offerings that are the subject of the next section of this document.

It is more relevant here to focus on the PSS tuning process or methodology, and only briefly focus on the design choices of input signals. Most of the tuning process is independent of the choice of input signal in the sense that it will of course influence the tuning of parameters but not the methods used in analysis. The material in the 1981 tutorial on the characteristics of the various PSS designs is still useful for understanding various designs and rather than repeat it here we will summarize briefly the commonly applied designs.

D. Speed Input Stabilizers A commonly applied design has a speed input based on

measurement directly at the turbine front standard toothed

wheel with magnetic proximity probe. The speed measurement provided a reliable measure of local mode and inter-tie oscillation induced from electromechanical tie to the power system. Unfortunately, the speed measurement also provided a rich spectrum of oscillations from the turbine-generator torsional dynamics and on units with low frequency torsional modes (and light damping – i.e. large steam turbines), multi-stage band reject filters were applied to mitigate any PSS-torsional interaction. Speed input stabilizers require 60-70 degrees of phase lead at local mode and the corresponding control mode gain margins result in limited damping, particularly with the torsional filter phase lags. Note that a torsional filter equivalent was placed in the PSS1A model to include these effects on system performance. Also, speed input PSS designs are especially susceptible to intra-plant oscillations in multi-unit plants that limit the allowable gain margins and available performance [34].

A new variation on the speed input PSS is what is called the multi-band PSS [35]. Measured speed is discriminated into three frequency bands, then compensated by three lead-lag stages in each band and algebraically combined with weighting factors into a PSS output. This model and design concept are presented in 421.5-2005 [36] and [35].

E. Frequency Input Stabilizers The ac bus frequency input stabilizer is another

commonly used design. This design has two major differences with speed input with respect to tuning. First, the frequency signal is less sensitive to intra-plant oscillations than either speed or power. These modes have higher frequencies than the local mode of the power plant to the power system, and the phase lag of the stabilizer loop is therefore greater. Hence, with speed or power input these modes will become unstable and impose a limitation upon stabilizer performance. Secondly, the sensitivity of the frequency signal to speed variations increases as the connected transmission system becomes weaker [33]. This offsets the reduction in gain from voltage reference to electrical torque, GEP(s), due to a weaker transmission system. As a consequence, the frequency input stabilizer can be tuned for the best performance under weak transmission conditions where the stabilizer contribution is most required.

Although these differences in tuning criteria will generally result in less high frequency gain for frequency input stabilizers than for speed input stabilizers, significant attenuation is still required at the torsional frequencies to prevent excessive torsional interaction. Frequency input PSS was often used in place of speed for four pole turbine-generators where the band-reject torsional filters for speed input PSS would result in more significant phase lags due to the lower frequency torsional modes. The phase compensation for frequency input PSS is similar to speed input designs


- 29 -

F. Power Input Stabilizers Other designs are based on electrical power by itself, or

accelerating power based on measurements of electrical power and speed or frequency. The next chapter has detail of the current accelerating power PSS design. The interest in using accelerating power as a stabilizer input signal results from the inherently low level of torsional interaction [33]. It must be noted that a practical power-based stabilizer must utilize some form of compensation for mechanical power changes, as well as the washout stage to remove steady-state value of input signal.

As a point of reference, it is instructive to compare the power-based stabilizer design to a speed-input stabilizer. This can be accomplished by recognizing that speed is related to accelerating power by the inertia, and hence power has an inherent 90 degree phase lead relative to speed with, correspondingly reduced phase-lead requirements in the PSS.

The integral of accelerating power design results in canceling the phase lead from power and the phase compensation is similar to those required for speed input designs. The reduced filtering requirements of the power-based system allow much greater small-signal damping to be obtained with the power-based system [21], [38].

A final preface to the tuning example is a note on a key aspect of a rigorous application of the tuning process is the ability to function in both frequency domain and time domain studies [40]. While there are many software tools able to function with explicit forms of transfer functions and systems, it has been long recognized that it is not only possible but straightforward to mechanize the output of state matrices from the dynamic equations linearized around an operating point. Unfortunately, it has not been common practice in the past 30 years to include this into commercially available stability software. While it is possible to write the closed-form equations for a small system, the mechanization of the work to include quickly developing the state matrices for any system size and configuration of dynamic models that is a cornerstone of the process.

III. TUNING EXAMPLE An example will be used to illustrate the combined use

of phase compensation and root locus techniques to meet the tuning objectives. The tuning process is done using a single machine infinite bus (SMIB) system. Although the SMIB system only has local mode response, we will see later that the frequency response characteristics can be used to design for inter-area mode damping. The example machine is a 600MW two pole steam turbine with integral of accelerating power input stabilizer [36] (IEEE Type PSS2A). Before beginning PSS tuning, the AVR tuning, needs to be determined [37]. In this case grid requirements for this bus fed static exciter led to a transient gain of 40 per unit, to meet a specific response time and overshoot criteria. A range of reactance values was considered

looking out from the unit into the system. These are given as a total of the generator step up transformer and the Thevenin equivalent looking out from the HV switchyard. To consider the robustness of the parameter choices, a range of impedances is usually chosen to cover normal and contingency cases. In this example 15, 30 and 45% on the unit base were used.

−60

−40

−20

0

20

40

Gai

n(dB

)

PSS Tuning Studies for IEEE TutorialP = 0.9 pu − Q = 0 pu − Xtot = 15/30/45%

Uncompensated (Te/E

tref)+(EXCSIG/Speed)

Xtot=15%Xtot=30%Xtot=45%

10−2

10−1

100

101

102

−300

−200

−100

0

100

200

Pha

se(D

egre

e)Frequency(Hz)

Xtot=15%Xtot=30%Xtot=45%

Fig. 3.3. Uncompensated Transfer Function ∆T/∆ω

The transfer function from speed to torque through the PSS control path is calculated and plotted in Fig 3.3. This transfer function reflects the phase and gain relationship between the component of torque produced by the action of the PSS and the generator speed oscillations, with no PSS phase compensation. We can see from this transfer function that the phase lag in the range of local mode is about 100 degrees, primarily due to the large inductance of the field circuit (T’do). Without compensation, the component of torque produced by the PSS action at the local mode frequency lags the generator rotor speed oscillations by about 100 degrees. This means that this torque component has little, if any, impact on the damping of local mode oscillation, and its influence will be mainly in improving the generator synchronizing torque and increasing local mode frequency. The objective of selecting the PSS phase compensation is to introduce the necessary phase shift in the PSS control path to cause this transfer function to have a phase of nearly zero degrees throughout the range of frequencies of interest, 0.1 to 3-4 Hz where inter-tie and local mode frequencies exist. With a compensated phase of near zero degrees, the component of electrical torque due to PSS action is nearly in phase with speed oscillations, and is almost completely directed to improving the damping of generator oscillations. Also in this and subsequent transfer functions we note the affect of two torsional band-reject filters that were applied at 13.56 and 23.49Hz. The process of PSS tuning actually would begin without application of filters, then determine if filters are required, and then repeat the tuning process for parameter settings with filters, if


- 30 -

required. The torsional filters, if required, for the integral of accelerating power PSS are relatively modest so usually only one iteration in the tuning process is required.

A point to be made here is the slight difference in the transfer function of uncompensated transfer function, and the AVR closed loop, which can be readily measured. This point was mentioned in [31] to [33] and the previous tutorial material. Note the plot on Fig 3.4 compares the two transfer functions and we can see that they agree closely near local mode but differ by as much as 30 degrees in the inter-tie region. In that case if one would assume the AVR closed loop was the target value, the performance goal in the inter-tie region would lead towards overcompensation relative to phase lead. During testing, the measured AVR closed loop transfer function should be compared with the computed AVR closed loop and recognize that is not the one used for tuning. That validates the tuning process and allows confidence in the transfer function that was used in tuning, but which cannot be easily measured.

−60

−40

−20

0

20

40

Gai

n(dB

)

PSS Tuning Studies for IEEE TutorialP = 0.9 pu − Q = 0 pu − Xtot = 15%

Voltage Control Loop Transfer Function (Vt/Vtref) and Uncompensted Open−Loop Transfer Function

10−2

10−1

100

101

102

−300

−200

−100

0

100

200

Pha

se(D

egre

e)

Frequency(Hz)

Vt / VtrefUncompensated Te / Speed

Vt / VtrefUncompensated Te / Speed

Fig. 3.4. Comparison of Uncompensated and AVR Closed Loop

In this example, two PSS lead-lag stages are used to

compensate the uncompensated transfer function. Most stabilizers offer three lead-lag stages for phase compensation (reference the new PSS2B model in 421.5-2005), and some manufacturers offer even more than three stages in their designs. The use of additional lead-lag stages gives the ability to better shape the gain and phase characteristics over a wider range of frequencies to meet performance targets. Older equipment may be limited to two stages for choosing compensation.

A. Phase Compensation The phase compensation is chosen to be close to zero

phase for the range of 0.1 to 3.0Hz. If we leave the phase slightly under-compensated we have both significant positive damping torque contribution, and also positive synchronizing torque contribution. The phase at low frequency is affected by choice of the washout (high pass filter) that is applied to eliminate the steady-state changes

in the input signals on PSS output. The discussion on the deMello Concordia block diagram illustrated how we can view these components are resolved into damping and synchronizing torques. The choice of leads of 0.2 and 0.18 seconds, and lags of 0.035 and 0.04 seconds are shown in Fig 3.5 where the plot also reflects the choice of washout time constant of 5 seconds. Compensated phase is within 30 degrees of target (zero degrees) for most of the range of interest, and no more than 45 degrees of lag at 3 Hz, which is above the local mode frequency (about 1.6 Hz).

10−2

10−1

100

101

102

−300

−250

−200

−150

−100

−50

0

50

100

150

Pha

se (

degr

ees)

Frequency (Hz)

PSS Tuning Studies for IEEE TutorialP = 0.9 pu − Q = 0 pu − Average Transfer Function for Xtot = 15/30/45%

Compensated (Te/E

tref)+(EXCSIG/Speed) − PSS Lead/Lag : 0.18,0.2,0/0.035,0.04,0

Compensated PhaseUncompensated PhasePSS Lead/Lag

Fig. 3.5. Uncompensated, Compensated and PSS Phase Plot

Choice in the phase compensation for adding more lead-

lag stages or changing washout time constant can be made at this point and the effects on phase can be considered. At this point the high frequency gain of the PSS can also be considered. If we are concerned about the application of torsional filters, the characteristics of the lead-lag stages in this frequency range are considered. Generally speaking the torsional frequencies are high enough compared to the local mode that the asymptote in gain which is the product of all the lead time constants divided by the product of all the lag time constants can be used. In this case the high frequency gain is (0.2*0.18)/(0.035*0.04) = 26.5. If two designs are otherwise roughly the same it may be wise to choose the one with lower high frequency gain. A further discussion of the PSS-torsional interaction is in a later part of this section.

B. Root Locus The next step in the process is to consider the root locus

as PSS gain is varied. As shown in Appendix A of [33], the initial direction of local mode eigenvalue migration as the stabilizer gain is increased from zero is determined by the compensated transfer function phase at the local mode frequency. For perfect compensation, i.e., °= 0,Lφ pure positive damping will be obtained and the eigenvalue will move directly into the left half plane with no change in frequency. If phase lag exists, the frequency will increase


- 31 -

in proportion to the amount of damping increase, specifically

LLL σφω ∆−=∆ tan (3.4)

where Lω = local mode frequency (rad/sec)

Lσ = local mode decay rate (sec-1) ∆ = implies change due to stabilizer

For =Lφ -45° frequency will increase at the same rate as damping. For =Lφ -90° a restructuring of (3.4) will show that no change in damping will take place, but frequency will increase. This basic concept is very useful in understanding the root locus.

Root locus plots are shown in Fig. 3.6 for the three different impedances. These plots represent the migration of the local mode eigenvalues as stabilizer gain is increased from zero towards infinity.


Local Mode Locus as a Function of PSS Gain − PSS Lead/Lag : 0.18,0.2,0/0.035,0.04,0

* Kpss

1 0

2 2

3 4

4 6

5 8

6 10

7 15

8 20

Fre

quen

cy (

Hz)

Sigma (1/Seconds)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Xtot = 15%

111111111

1

1

1

222222222

2

2

2

333333333

3

3

3

444444444

4

4

4

555555555

5

5

5

666666666

6

6

6

777777777

7 8888888888

8

Xtot = 30%

111111111

1

1

1

22222222

2

2

2

33333333

3

3

3

44444444

4

4

4

55555555

5

5

5

666666666

6

6

6

777777777

7

7

7

88888888

8

8

8 Xtot = 45%

11111111

1

1

1

22222222

2

2

2

333333333

3

3

3

44444444

4

4

4

55555555

5

5

5

666666666

6

6

6

777777777

7

7

7

888888888

8

8

8

Fig. 3.6. Root Locus of Local Mode for Different System Strengths Although many eigenvalues exist for the total system

only the dominant ones associated with the local mode are shown in this figure. We can see that local mode frequency is increased as the system tie becomes stronger, and significant damping is introduced by the PSS without much frequency change. In this case the frequency increases slightly as PSS gain is increased, illustrating the comments made previously about positive synchronizing torque contribution. Weaker systems have inherently lower damping without PSS, but in all cases the PSS provides significant improvement in damping.

The other aspect of the PSS design comes into the PSS control mode that is becoming dynamically less stable as the local mode dynamic stability is increasing. The plot in Fig 3.7 shows the control mode and also the local mode root locus on the same plot. Here we focus on ensuring the chosen gain allows for adequate gain margin. From a point of view of control system design we need to ensure a minimum of 6dB and typically better 10dB margin (a 3:1

factor between setting and instability point). Here we see the gain of 80 on the PSS will result in instability at about 5.5Hz, well above (18dB) the recommended PSS gain of 10.

If frequency domain tools are not available, the process of PSS tuning is somewhat less rigorous but the damping and oscillation frequency of local mode can be determined from time simulations to get equivalent information as contained in Figs 3.6 and 3.7. Also, that same information can be validated during testing in similar fashion. It is also possible in any system to measure the PSS open loop transfer function (from AVR input to PSS output with PSS not connected) and we have found that the measured gain at the crossover point correlates well with the inverse of the instability gain. The crossover point also provides the instability frequency.


Control Mode Locus as a Function of PSS Gain − PSS Lead/Lag : 0.18,0.2,0/0.035,0.04,0

* Kpss

1 0

2 10

3 20

4 40

5 60

6 80

7 100

Fre

quen

cy (

Hz)

Sigma (1/Seconds)

−25 −20 −15 −10 −5 0 50

1

2

3

4

5

6

7

8

Local Mode

Control M

ode

1111111111111

1

1 2222222222222

2

2

2

3333333333333

3

3

3

444444444444

4

4

4

5555555555555

55

5

666666666666

66

6

6

777777777 777

7 77

7

11111111111

1

1111 22222222222

2

222

2

2

3333333333

3

333

3

3

44444444444

4

444

4

4

55555555555

5

555

5

5

66666666666

6

666

6

6

77777777777

7

777

7

7

111111111

1

1

111111 22222222222

2

222

2

2

33333333333

3

333

3

3

4444444444

4

444

4

4

55555555555

5

555

5

5

6666666666

6

666

6

6

7777777777

7

777

7

7

Fig. 3.7. Control Mode Root Locus as PSS Gain is Increased

Some newer digital based controls have built in test

functions that allow for these frequency responses as well as step tests to be performed and have tools that plot frequency responses directly.

C. Step Test and Fault Simulations After choosing phase compensation and gain, time

simulations are run to verify performance with voltage steps and fault disturbances. Voltage step tests will be used later for performance verification. Cases are normally run for all of the system reactances to check the robustness of the design.

For this example we show the 2% voltage step response in Fig 3.8 for the strong system connection of X=15%. As this is a single machine system the response is local mode and the oscillation without PSS can be seen to be at about 1.6Hz which correlates with the root locus plot in Fig 3.6. The step is applied for 5 seconds and then removed. The oscillation in MW and speed clearly show the response and the damping as provided by the stabilizer.


- 32 -

PSS Tuning Studies for IEEE TutorialP = 0.9 pu − Q = 0 pu − X

tot = 15% − PSS Lead/Lag : 0.18,0.2,0/0.035,0.04,0

Response to a 2% Step in Terminal Voltage Reference − Blue : PSS ON − Red : PSS OFF

0.99

1

1.01

1.02

1.03

1.04

Vt(

pu)

1

1.5

2

2.5

3

3.5

Efd

(pu)

0 2 4 6 8 100.88

0.89

0.9

0.91

0.92

0.93

Pe(

pu)

Time (Seconds)

−0.05

0

0.05

0.1

0.15

0.2

Qe(

pu)

49.96

49.98

50

50.02

50.04

Spe

ed(H

z)

0 2 4 6 8 10−0.01

−0.005

0

0.005

0.01P

SS

Out

put(

pu)

Time (Seconds)

Fig. 3.8 2% Voltage Step Simulation with X=15% - with & without PSS

The fault response in Fig 3.9 shows the same local mode

response following the fault and subsequent line clearing. This case illustrates the principle that the PSS cannot provide effective damping while the PSS output is in limit. During the first two cycles of the local mode swing the PSS is alternating between positive and negative limits, and when the PSS comes out of limit the damping reduces the local mode swing to insignificant levels within 1 swing cycle, similar to what was observed for the step test.

D. PSS – Torsional Interaction For steam turbines with lightly damped low frequency

torsional modes it is possible for the PSS to interact with torsional oscillations [41]. In those cases the calculation of Torsional Interaction Vectors (TIV) are included in the PSS tuning process. There is a criterion for allowable interaction of no more than 10% change in the torsional mode location (vector including damping and frequency) relative to the initial damping of the torsional mode without the PSS. In Fig 3.10 is a plot of the TIV’s for PSS applied to a nuclear plant for mode 1 (near 10Hz), where torsional filters were applied. As part of the dynamic setup, more detailed models of the generator and excitation system are used to accurately represent their behavior at higher frequencies. The torsional dynamics are included in the models using modal form equations for all the sub-synchronous torsional modes. Test results have validated models at dozens of machines studied

PSS Tuning Studies for IEEE TutorialP = 0.9 pu − Q = 0 pu − X

tot(pre−fault) = 15% − X

tot(post−fault) = 20% − PSS Lead/Lag : 0.18,0.2,0/0.035,0.04,0

Response to a 0.1 Second Fault Mid−Way to Infinite Bus Followed by the Loss of Lines − Blue : PSS ON − Red : PSS OFF

0

0.5

1

1.5

Vt(

pu)

0

5

10

15

Efd

(pu)

0 2 4 6 8 10−1

0

1

2

3

Pe(

pu)

Time (Seconds)

−3

−2

−1

0

1

2

Qe(

pu)

49

49.5

50

50.5

51

Spe

ed(H

z)

0 2 4 6 8 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

PS

S O

utpu

t(pu

)

Time (Seconds)

Fig. 3.9 Three Phase Fault Simulation with X=15% - with & without PSS

For each mode 40 cases are studied, for a range of

system impedances and MW and MVAR loading as shown in the bottom of the figure as bar graphs.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

|TIV

|/Sig

ma 0

|TIV|/Sigma0 for Torsional Mode #1 − Less Than 0.1 −> No Interaction − Larger Than 0.1 −> Interaction

ThresholdWithout FiltersWith Filter

Xe

P

0 5 10 15 20 25 30 35 40

Q

Case #

Fig. 3.10. TIV with and without Filters for Mode 1 Response

Note that the interaction increases for stronger systems

and for operation under excited with leading power factor. The TIV calculations are done for all sub-synchronous modes and typically only the lower frequency modes show interaction with the PSS. For any modes showing interaction above the 10% level in TIV/Sigma, filters are specified. Any band reject filter design can be used, and a common form is the bi–quadratic form, as follows:

22

22

22

ssss

nDn

nNn

++++

ωςωωςω (3.5)

In some cases more than two filters are applied to a torsional mode and stagger tuned to give a wider bandwidth. This is common with mode 1 as the modal frequency varies slightly as the unit operating point changes.


- 33 -

E. Use of Modified Lead-lag Compensation For some applications where the performance at both

local mode and inter-area modes are critical, adding phase compensation should be considered. In the following example, an aero-derivative combustion turbine which has very light inertia, H=1.17, and base rating of 60MW is considered. The resultant local mode frequency is about 3.6Hz for the stiff system, and performance at inter-area mode is also important. In this case, it is desired to keep phase compensation in a good region for a 50:1 span in frequency. By using three lead lags, which were available in the equipment, we could set the third lead-lag stage for a net low frequency lag. The lead time constants were adjusted to 0.6, 0.6 and 0.05 seconds, and the lag time constants set to 5.0, 0.015, and 0.01 seconds. In this case we could also retain the washout time constant at 2.0 seconds, to effectively decouple the low frequency power oscillations from affecting the PSS. Figs 3.11 and 3.12 show the plots of phase compensation and local mode root locus.

F. Inter-area Mode Damping So far, we have considered the tuning based on single

machine system and have not validated performance for inter-area modes [42] to [43]. To explicitly evaluate the effect of stabilizers on damping of inter-area modes of oscillation, a four-machine system was simulated having both local and inter-area modes of oscillation.

10−2

10−1

100

101

102

−200

−150

−100

−50

0

50

100

150

200

Pha

se (

degr

ees)

Frequency (Hz)

Aero Turbine Three Lead Lag Example PSS Tuning StudyP = 0.85 pu − Q = 0 pu − Average Transfer Function for Xtot = 15/30/45%

Compensated (Te/E

tref)+(EXCSIG/Speed) − PSS Lead/Lag : 0.6,0.6,0.05/5,0.015,0.01

Compensated PhaseUncompensated PhasePSS Lead/Lag

Fig. 3.11. Phase Compensation for Aero Turbine Example

Aero Turbine Three Lead Lag Example PSS Tuning StudyP = 0.85 pu − Q = 0 pu − Xtot = 15/30/45%

Local Mode Locus as a Function of PSS Gain − PSS Lead/Lag : 0.6,0.6,0.05/5,0.015,0.01

* Kpss

1 0

2 2

3 4

4 6

5 8

6 10

7 15

8 20

Fre

quen

cy (

Hz)

Sigma (1/Seconds)

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 00

0.5

1

1.5

2

2.5

3

3.5

4

Xtot = 15%

1111111

11

1

222222

22

2

2

33333

33

3

3

44444

44

4

4

55555

55

5

5

666666

66

6

6

77777

77

7

7

88888

88

8

8

Xtot = 30%

111111

11

1

22222

22

2

2

333333

33

3

3

444444

44

4

4

555555

55

5

5

66666

66

6

6

77777

77

7

7

888888

88

8

8

Xtot = 45%

11111111

1

1

2222222

2

2

2

333333

3

3

3

444444

4

4

4

5555555

5

5

5

6666666

6

6

6

7777777

7

7

7

8888888

8

8

8

Fig. 3.12. Local Mode Root Locus for Aero Turbine Example

1

2 4

3

PL=5.3 PL=9.5

X=0.5

X=0.07

X=1.2

X=0.6

X=0.025

Pg=0.4

Pg=9.2

Pg=0.9

Pg=4.5

P=0.006

All data on 1000 MVA Base

Fig. 3.13. Four Machine Example System The four-machine example system in Fig 3.13 has

detailed generator models, excitation systems, and integral of accelerating power type PSS which are tuned using three lead-lag stages to maximize damping for both local mode and inter-area modes. For a single machine situation, where local mode is the only consideration, only recovery from the first swing need be considered since the stabilizer will always be acting correctly within its limits to aid damping of the resulting oscillations. In a multi-machine environment, however, more than the first swing may be critical, and the non-linear performance of the stabilizer becomes important. This will be apparent from the large signal performance analysis that follows:

The response of the four-machine system to a three-phase fault on the right side bus (see Fig 3.13) is plotted in Fig. 3.14 for the case of no stabilizer on any machines as compared to all machines having stabilizers. The plot of the speed signals from the simulation clearly shows the improvement that can be obtained in both local mode and inter-area mode damping. The plot in Fig 3.15 shows the root locus that has the local modes and inter-area mode shown graphically in frequency domain.


- 34 -

PSS Tuning Studies for IEEE TutorialMachine Speed Response to a 4−Cycle Fault on Bus 4 − 4−Machine System

Red: All Machines Without PSS − Blue: All Machines With PSS

0 5 10 15 2059

59.5

60

60.5

61

ω1

(Hz)

0 5 10 15 2059

59.5

60

60.5

61

Time (Seconds)

ω2

(Hz)

0 5 10 15 2059

59.5

60

60.5

61

ω3

(Hz)

0 5 10 15 2059

59.5

60

60.5

61

Time (Seconds)

ω4

(Hz)

Fig. 3.14. Plot of Speed Signals from 4 Machine System

The above observations are intended to illustrate a

general relationship between small-signal damping and large-disturbance performance, rather than a comparison of performance between particular input signals. Most new designs use the integral of accelerating power type PSS but many existing plants have speed, frequency and power input PSS designs. Stabilizer output limits and dynamic limiting circuits (third stage switchable washout) [39] also can be part of the tuning process. In the interest of space these concepts will remain documented in the literature.

PSS Tuning Studies for IEEE Tutorial4−Machine Study System

Root Locus as a Function of PSS Gain − PSS Gain on All Machine Varied Simultaneously

* Kpss

1 0

2 10

3 20

4 30

5 40

6 50

Fre

quen

cy (

Hz)

Sigma (1/Seconds)

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.50

0.5

1

1.5

2

2.5

111

111

1

1

1

22222

2

22

22

2

2

222 3333

3

33

33

3

3

333 444444

44

4

44

4

4

444 55555

55

5

5

5

5

5

555 66

6666

6

6

6

6

6

666

Fig. 3.15. Plot of Root Locus from 4 Machine System

IV. SUMMARY AND CONCLUSIONS The objective of applying power system stabilizers is to

extend stability limits on power transfer by enhancing damping of system oscillations via generator excitation control. Lightly damped oscillations can limit power transfer under weak system conditions, associated with either remote generation transmitting power over long distances or relatively weak inter-ties connecting large

areas. Stabilizer performance must therefore be measured in terms of enhancing damping under these weak system conditions. This measure must include not only the small-signal damping contributions to all modes of system oscillation, but the impact upon system performance following large disturbances, when all modes of the system are excited simultaneously. Based upon this measure, it is shown that the most appropriate stabilizer tuning criteria is to provide good damping to local modes of oscillation and also the inter-area modes of oscillation.

The first step before beginning the PSS tuning process is to determine the settings for the AVR control based on the equipment and the grid interconnection requirement that might exist. Once the AVR tuning is determined, we use phase compensation, root locus, and time domain analyses to tune and evaluate the performance of the PSS. Phase compensation lets us select the PSS lead/lag settings to compensate for phase lags introduced by the generator, excitation system, and power system. Root locus analysis is used to select the PSS gain, and to determine the PSS instability gain and ensure adequate gain margin in the PSS control loop. Finally, time domain analysis demonstrates the performance of the PSS in the power system environment. The choice of washout time constants and the use of additional lead-lag stages were illustrated by example. The need to insure adequate margins for low frequency torsional oscillations that might interact with the PSS was also mentioned as impacts the tuning process.

The tuning concepts and performance criteria developed in this chapter, including the relationship of performance to phase compensation characteristics, provides the basis for field tuning procedures. The references [44], [45], [46], [47], [48], [49], [50] and [51] are to be recommended for study and review. Further background and practical application information is in Sections 4, 5, and 6 of this tutorial which are replete with practical experience.


- 35 -

CHAPTER 4 INTEGRAL OF ACCELERATING

POWER TYPE STABILIZERS Roger Bérubé and Les Hajagos

I. INTRODUCTION Despite their relative simplicity, power system stabilizers

may be one of the most misunderstood and misused pieces of generator control equipment. The ability to control synchronous machine angular stability through the excitation system was identified with the advent of high-speed exciters and continuously acting voltage regulators. By the mid-1960’s several authors had reported successful experience with the addition of supplementary feedback to enhance damping of rotor oscillations [52].

The function of a PSS is to add damping to the unit’s characteristic electromechanical oscillations. This is achieved by modulating the generator excitation so as to develop components of electrical torque in phase with rotor speed deviations. The PSS thus contributes to the enhancement of small-signal stability of power systems. Many excellent references are available with guidance on the selection of PSS settings once the required speed signal is provided as an input to the PSS [30], [31], [32], [33], [50], [53].

Early PSS installations were based on a variety of methods to derive an input signal that was proportional to the small speed deviations characteristic of electromechanical oscillations [52], [54], [55]. After years of experimentation the first practical integral-of-accelerating-power based PSS units were placed in service [56], [57], [58]. This design provided numerous advantages over earlier speed-based units and forms the basis for the PSS implementation that is used in most units installed in North America. This design is now a requirement in many Reliability Regions within North America and has been modelled in the IEEE standards as the PSS2A and PSS2B structures [36]. For simplicity, the term PSS2A stabilizer will be used to refer to the integral-of-accelerating power based design in general throughout this paper.

This paper briefly describes some of the earlier structures in order to explain the advantages of the accelerating-power design. This design is then described along with a detailed review of the role of the "ramp-tracking” mechanical filter and the basis for the present structure that is in wide use by many manufacturers.

II. OVERVIEW OF PSS STRUCTURES Shaft speed, electrical power and terminal frequency are

among the commonly used input signals to the PSS. Alternative forms of PSS have been developed using these signals. This section describes the practical considerations that have influenced the development of each type of PSS as well as its advantages and limitations.

A. Speed-Based (Dw) Stabilizer Stabilizers employing a direct measurement of shaft

speed have been used successfully on hydraulic units since the mid-1960s. Reference [52] describes the techniques developed to derive a stabilizing signal from measurement of shaft speed of a hydraulic unit.

In early designs on vertical units, the stabilizer’s input signal was obtained using a transducer consisting of a toothed-wheel and magnetic speed probe supplying a frequency-to-voltage converter. Among the important considerations in the design of equipment for the measurement of speed deviation is the minimization of noise caused by shaft run-out (lateral movement) and other causes [52], [54]. Conventional filters could not remove such low-frequency noise without affecting the electromechanical components that were being measured. Run-out compensation must be inherent to the method of measuring the speed signal. In some early applications, this was achieved by summing the outputs from several pick-ups around the shaft, a technique that was expensive and lacking in long-term reliability.

The original application of speed-based stabilizers to horizontal shaft units (e.g. multi-stage 1800 RPM and 3600 RPM turbo-generators) required a careful consideration of the impact on torsional oscillations. The stabilizer, while damping the rotor oscillations, could reduce the damping of the lower-frequency torsional modes if adequate filtering measures were not taken. In addition to careful pickup placement at a location along the shaft where low-frequency shaft torsionals were at a minimum, electronic filters were also required in the early applications [55].

While stabilizers based on direct measurement of shaft speed have been used on many thermal units, this type of stabilizer has several limitations. The primary disadvantage is the need to use a torsional filter. In attenuating the torsional components of the stabilizing signal, the filter also introduces a phase lag at lower frequencies. This has a destabilizing effect on the "exciter mode," thus imposing a maximum limit on the allowable stabilizer gain [30]. In many cases, this is too restrictive and limits the overall effectiveness of the stabilizer in damping system oscillations. In addition, the stabilizer has to be custom-designed for each type of generating unit depending on its torsional characteristics. The integral-of-accelerating power-based stabilizer, referred to as the Delta-P-Omega (∆Pω) stabilizer throughout this section, was developed to overcome these limitations.

B. Frequency-Based (∆f) Stabilizer Historically terminal frequency was used as the input

signal for PSS applications at many locations in North America. Normally, the terminal frequency signal was used directly. In some cases, terminal voltage and current inputs were combined to generate a signal that


- 36 -

approximates the machine’s rotor speed, often referred to as "compensated” frequency.

One of the advantages of the frequency signal is that it is more sensitive to modes of oscillation between large areas than to modes involving only individual units, including those between units within a power plant. Thus it seems possible to obtain greater damping contributions to these "interarea” modes of oscillation than would be obtainable with the speed input signal [31], [32], [33].

Frequency signals measured at the terminals of thermal units contain torsional components. Hence, it is necessary to filter torsional modes when used with steam turbine units. In this respect frequency-based stabilizers have the same limitations as the speed-based units. Phase shifts in the ac voltage, resulting from changes in power system configuration, produce large frequency transients that are then transferred to the generator’s field voltage and output quantities. In addition, the frequency signal often contains power system noise caused by large industrial loads such as arc furnaces [59].

C. Power-Based (∆P) Stabilizer Due to the simplicity of measuring electrical power and

its relationship to shaft speed, it was considered to be a natural candidate as an input signal to early stabilizers. The equation of motion for the rotor can be written as follows:

( )t

1P Pm e2H

∂∂

∆ω = ∆ − ∆ (4.1)

where H = inertia constant ∆Pm = change in mechanical power input ∆Pe = change in electric power output ∆ω = speed deviation If mechanical power variations are ignored, this equation

implies that a signal proportional to shaft acceleration (i.e. one that leads speed changes by 90°) is available from a scaled measurement of electrical power. This principle was used as the basis for may early stabilizer designs. In combination with both high-pass and low-pass filtering, the stabilizing signal derived in this manner could provide pure damping torque at exactly one electromechanical frequency.

This design suffers from two major disadvantages. First, it cannot be set to provide a pure damping contribution at more than one frequency and therefore for units affected by both local and inter-area modes a compromise is required. The second limitation is that an un-wanted stabilizer output is produced whenever mechanical power changes occur. This severely limits the gain and output limits that can be used with these units. Even modest loading and unloading rates produce large terminal voltage and reactive power variations unless stabilizer gain is severely limited.

Many power-based stabilizers are still in operation although they are rapidly being replaced by units based on the integral-of-accelerating power design.

D. Integral-of-Accelerating Power (∆Pω) Stabilizer The limitations inherent in the other stabilizer designs

led to the development of stabilizers that measure the accelerating power of the generator [56], [57], [36].

MSpeeds Tw1

1 + s Tw1

s Tw2

1 + s Tw2

1

1 + s T6

Powers Tw3

1 + s Tw3

s Tw4

1 + s Tw4

Ks2

1 + s T7

Ks3

+

+

(1+s T8)

(1+s T9)

N

M M

+

-

Ks11 + s T1

1 + s T21 + s T3

1 + s T4

Vstmax

Vstmin

Output

High-Pass Filters

High-Pass Filters

Ramp-Tracking Filter

Stabilizer Gain & Phase Lead Limits

A

B

C D E

F

G H I

Figure 4.1 Accelerating Power PSS Model (PSS2A)


- 37 -

The earliest systems combined an electrical power measurement with a derived mechanical power measurement to produce the required quantity. On hydroelectric units this involved processing a gate position measurement through a simulator that represented turbine and water column dynamics [54]. For thermal units a complex system that measured the contribution of the various turbine sections was necessary [58].

Due to the complexity of the design, and the need for customization at each location, a new method of indirectly deriving the accelerating power was developed. The operation of this design of stabilizer is described in references [56], [57]. The IEEE standard PSS2A model used to represent this design is shown as Fig. 4.1 [36].

The principle of this stabilizer is illustrated by re-writing equation (4.1) in terms of the integral of power.

( )1P P tm e2H

∆ω = ∆ − ∆ ∂∫ (4.2)

The integral of mechanical power is related to shaft

speed and electrical power as follows:

P t 2H P tm e∆ ∂ = ∆ω+ ∆ ∂∫ ∫ (4.3)

The ∆Pω stabilizer makes use of the above relationship

to simulate a signal proportional to the integral of mechanical power change by adding signals proportional to shaft-speed change and integral of electrical power change. On horizontal-shaft units, this signal will contain torsional oscillations unless a filter is used. Because mechanical power changes are relatively slow, the derived integral of mechanical power signal can be conditioned with a low-pass filter to attenuate torsional frequencies.

The overall transfer function for deriving the integral-of-accelerating power signal from shaft speed and electrical power measurements is given by:

Pa t2H

P (s) P (s)e eG(s) (s)2Hs 2Hs

∆ ∂ →∫∆ ∆

− + + ∆ω

(4.4)

where G(s) is the transfer function of the low-pass filter.

The major advantage of a ∆Pω stabilizer is that there is a

greatly reduced requirement for a torsional filters with this design. This alleviates the exciter mode stability problem, thereby permitting a higher stabilizer gain that results in better damping of system oscillations. A conventional end-of-shaft speed measurement or compensated frequency signal can be used with this design.

III. PRACTICAL APPLICATION ISSUES Many excellent papers have been written dealing with

the tuning of PSS [31], [32], [33], [50]. These authors dealt

with the selection of phase compensation, gain and output limit settings and their effect on the overall performance of the PSS. This will not be repeated here. Instead, this section will focus on the derivation of the accelerating-power signal and its use in deriving an equivalent speed signal. Specifically, this section will describe the impact of speed measurement issues and mechanical power variations on the operation of units equipped with this style of PSS and how this has influenced the design of PSS2A stabilizers.

With a large base of installed units, and long history of usage, experience has been acquired with many different vintages of hardware. Early designs suffered from failures due to mechanical components such as speed pickups. Replacement of the measured speed signal with a derived frequency signal has greatly improved reliability at many facilities. The early analog-electronic designs also suffered from reliability problems due to failures of components used to implement the adjustable settings (e.g. switches, potentiometers). Digital designs have eliminated these components and improved reliability and ease of use. Further gains in reliability are achieved when the PSS is implemented as additional software code in a complete digital excitation system, since this eliminates any additional hardware.

A. Signal Mixing Referring to the block diagram of Fig.4.1, the two input

signals to the ∆Pω stabilizer are speed (A) and active power (B). Although the ∆Pω design has many advantages over stabilizers that employ only one of these inputs it is sensitive to the relationship between these two inputs. For optimum performance it is critical that the two signal paths (A-C and B-F) are matched in terms of gain and filter time constants.

The power path employs two high-pass filter stages and an integration to derive the integral-of-electrical power change signal, ∆Pe:

2

e We

W

W3 S2e

W3 7

P sT 1 P2H 1 sT s2H

sT K P1 sT 1 sT

∆ → +

→ + +

∫ (4.5)

The second part of Equation 5 is based on the notation of

Fig.4.1 and the following settings: TW3 = T7 = TW TW4 = 0 (i.e. this block is bypassed) KS2 = TW / (2H) KS3 = 1 In order for the speed signal path to match the power

path it must employ two stages of high-pass filtering as well, and its equivalent filter time constant must be kept as small as possible:

TW1 = TW2 = TW


- 38 -

T6 ≈ 0 With these settings the signal appearing at point D is

proportional to changes in the integral-of-mechanical power, ∆Pm. When re-combined with the ∆Pe signal at point G, the integral-of-accelerating power, ∆Pa, is formed. This signal is then treated as equivalent speed and the phase lead blocks that follow are set to compensate in order to maximize the contribution of the stabilizer to damping torque.

B. Mechanical Power Variations Although the original requirement for the PSS units was

based on a need to provide damping for the local plant modes of oscillation, many new installations and retrofits have been applied to improve damping of inter-area modes of oscillation [50] as is common in western U.S. utilities. In order to be effective at damping these modes of oscillation, the high-pass filters, parameters Tw1 to Tw4 in Fig.4.1, must be set to admit frequencies as low as 0.1 Hz without significant attenuation or the addition of excessive phase lead.

Early attempts at re-tuning PSS for these frequencies identified some side effects related to mechanical power variations on the units. Tests on the original ∆Pω design on thermal units included fast intercept valve closures that produced a step change in power of approximately 5%, followed by a ramp of 0.55%/s [55]. The maximum generator terminal voltage change produced by a PSS configured with short washout time constants was below 2%, for the normal in-service gain. On the first tests of this design on hydraulic units, mechanical power ramp-rates in excess of 10%/s were achieved under gate limit control.

The introduction of long high-pass filter time constants produced excessive terminal voltage and reactive power deviations. In response to this problem, researchers identified the root cause of the variations and modified the designs accordingly.

When mechanical power is changed rapidly, electrical power follows quickly but there is a limited change in the rotor speed. Although this depends on the strength of the system interconnection, the speed changes will always be relatively small and are considered to be negligible in the following analysis.

Referring to Fig.4.1, when electrical power (B) is ramped, the integral-of-electrical power signal (F) will change with a rate and magnitude determined by the selected washout time constants and unit inertia. From this point forward, the signal follows two paths to the output. The lower path is a direct connection to the derivation of the equivalent speed signal at point G. The signal produced at point F also travels through the mechanical power low-pass filter (E) before appearing at the output. Ideally these signals would exactly cancel each other, since the PSS was not intended to produce an output for this condition. With long washouts and high ramp rates, this is

not the case and a large error signal can propagate to the PSS output, thereby changing terminal voltage and reactive power on the unit. This problem forced the selection of low PSS gains or output limits, severely limiting the effectiveness of the PSS.

The transfer function between the power input, PE, and the integral-of-accelerating power signal, PA, (points B and G in Fig.4.1) may be written as follows:

( )W3 S2A

E W3 7

sT KP (s) G(s) 1P (s) 1 sT 1 sT

= − + +

(4.6)

The original design of mechanical power low-pass filter

consisted of a simple multi-pole filter of the form:

M9

1G(s)(1 sT )

=+

(4.7)

which is achieved in the model by setting the following

values: T8 = 0 N = 1 The filter order, M, and time constant, T9, can be

selected to provide adequate attenuation of the lowest torsional frequency for horizontal-shaft applications.

Researchers [60] discovered that they could reduce the sensitivity to mechanical power variations by re-designing the mechanical power low-pass filter to utilize a transfer function of the form:

M

o2

2o o

21 sG(s)

s 2+ s+1

ξ+ ω =ξ

ω ω

(4.8)

Further analysis and tests on actual hardware

implementations confirmed that the complex-pole implementation was not optimal and that the following transfer function could be used to reduce mechanical power effects on the PSS output.

N

M9

8

)sT(1)sT(1G(s)

++= (4.9)

The filter of equation (4.9) is frequently identified as a

"ramp-tracking” filter based on its properties when the coefficients, T8, T9, M and N are selected correctly.

The criteria used to analyze the merits of different mechanical power filter designs are the following:

• Attenuate high-frequency components in the input


- 39 -

signal. • Allow low-frequency mechanical power changes to

pass through with negligible attenuation. • Minimize the PSS output deviation that occurs when

the mechanical power is changing rapidly. Based on torsional frequencies as low as 7 Hz, the first

two criteria dictated the selection of filters with four poles (M=4) and time constants (T9) of 0.08 seconds. These filters were used on numerous large horizontal units but did not meet the third criteria, especially when applied to hydroelectric units with their rapid ramp rates.

To understand the advantages of the "ramp-tracking” filter and the required selection of coefficients it is instructive to compute the accelerating power signal that is generated when mechanical power changes rapidly. For this purpose, the integral-of mechanical power changes are characterized as combinations of the following time-domain inputs:

• step, A*u(t) • ramp, B*t • parabola, C*t2 where t is time in units of seconds and A, B and C are

the magnitudes of the associated components in per unit. The steady-state PA signal for each of these inputs can be

calculated using the final value theorem by evaluating the following:

t A s 0lim p (t) lim (s*Input *(G(s) 1))→∞ →= − (4.10) Appendix A provides details of the evaluation of

equation (4.10) for a conventional low-pass filter (equation 4.7) and the ramp-tracking filter (equation 4.9). The result for each type of input is summarized in Table 4.1.

TABLE 4.1:

STEADY STATE RESPONSE TO POWER VARIATIONS Steady-State Output Input

Low-Pass Ramp-Tracking step input 0 0 ramp input -B*M*T9 0 parabolic input infinite -C*F(M,T9)

The key result in this table is that the ramp-tracking filter

produces a zero steady-state output for a ramp input and a bounded output for a parabolic input. This is only true if the coefficients are selected to satisfy

8 9T M *T= (4.11)

The derivation of the results provided in Table 1,

including the relationship of equation (4.11) is included as Appendix A.

The most commonly used ramp-tracking filter coefficients are N=1 and M=5 since this provides four net poles with the minimum number of numerator and

denominator terms. To obtain 40 dB of attenuation at 7 Hz, the denominator time constants are set to 0.1 s, resulting a numerator time constant of 0.5 s.

With this design, the filtered integral-of-mechanical power signal can track rapid rates-of-change in the measured electrical power signal, greatly reducing the terminal voltage modulation produced by the PSS. Fig.4.2 displays the simulated output of stabilizers equipped with a conventional and ramp-tracking low-pass filter to a power ramp on a hydraulic turbine. Clearly the ramp-tracking filter greatly reduces the PSS output deviation for this condition.

-0.05

0

0.05

0.10

0.15

0.20

0 5 10 15 20

ramp-trackinglow-pass

Time (seconds

PSS

Out

put

(pu)

00.20.40.60.81.01.2

0 5 10 15 20

Activ

e P

ower

(pu)

Figure 4.2 Simulated Ramp Response

Different coefficients and time constants can be used to

improve the tracking of power ramps or to provide greater attenuation of low-frequency torsional components. Increasing the denominator order or the denominator time constant is a viable alternative to introducing notch filters at torsional frequencies since it does not interfere with the selected phase compensation of the resulting accelerating power signal. This will increase the sensitivity of the stabilizer to power changes however this is normally acceptable on large horizontal shaft units with their slow loading rates.

The performance of this filter may also be critical to the behaviour of the unit, in the event of inadvertent islanded operation resulting in large frequency and mechanical power variations.

C. Input Signals Electrical power is readily available as an input. In

analog implementations it can be measured using a three-phase Hall-effect watt transducer or equivalent device that produces an instantaneous output proportional to the generator active power. Selective filtering is required to remove the characteristic harmonics present in the output


- 40 -

measurement. In digital implementations a variety of techniques are available to calculate power from the sampled ac voltage and current measurements. In either case the key is to not add unnecessary filtering and phase lag that will affect the phase compensation in this signal path. This has been achieved with good success in various manufacturers’ implementations for many years.

The original ∆Pω stabilizers employed a physical measurement of shaft speed using magnetic speed pickups as the source. A frequency-to-voltage converter was then used to generate the required direct measurement of speed. This necessitated the use of filtering and as a result, the input speed probe signals had to be relatively high frequency, necessitating multiple probes and toothed wheel or milled slot. Once again careful selection of the filtering was necessary to avoid the introduction of phase lag in this path. In applications where excessive filtering is used, the time constant, T6, can be used in the model of Fig.4.1 to simulate the effect on overall stabilizer performance.

Although there is a long history of speed measurement in excitation control, it introduces several complications to the application of the stabilizer. Since it requires the only moving parts in the entire device, it is the least reliable element of the design. Numerous stabilizers have been temporarily disabled or have failed during operation due to improper gapping of speed measurement probes or failure of physical or electrical connections. On vertical shaft hydraulic units, there was the significant additional complication of dealing with shaft runout. On these units there can be a significant lateral movement of the shaft that varies with load level. Regardless of the location of the pickups, once-per-revolution noise appears at some level. On units with speed in the range of 100 rpm this is very significant since the noise component may coincide with the local mode electromechanical frequency of the unit. Early speed based stabilizers coped with this problem through an ingenious mechanical arrangement that made use of up to 5 speed probes mounted equidistant around the circumference of the shaft to eliminate the runout component [50]. Although this worked and formed the basis for many successful stabilizer installations it was costly due to the need for customization at each location. It was also relatively unreliable due to the requirement to have all probes in operation for the cancellation effect to function properly.

For these reasons, direct speed measurement was gradually phased out in favour of compensated frequency, which can be measured using the same PT and CT inputs that are already available for measurement of electrical power.

D. Compensated Frequency Direct terminal frequency, measured from the generator

PTs, has been used as an input signal in many stabilizers in the past. Its advantages and disadvantages were discussed earlier. It cannot be used directly in a ∆Pω stabilizer

configuration. Referring to the signal nomenclature of Fig.4. 1, it is a requirement that the "speed” signal at point A match the power signal at point B so that the derived integral-of-mechanical power signal at point D represents equation 3 accurately. Any error in the derivation of the signal at point D due to signal mismatch will pass through the filter to point E and will result in an error in the stabilizer output.

The extent to which the electromechanical components appear in terminal frequency is dependent on the component and the system strength. For example inter-machine modes between two units connected together at their low-voltage bus will be completely absent in a frequency signal measured from the generator PTs. Inter-area modes involving large groups of units will be visible in the terminal frequency but local machine modes will be greatly attenuated in for strong system connections.

Based on the above, frequency measurement can only be used if the ac source can simulate a voltage that is coupled directly to shaft position changes. Both the generator terminal voltage and a voltage proportional to the generator's terminal current are used in deriving the "internal voltage”. A voltage behind quadrature axis reactance is used for this purpose:

i t q tE E jX I= + (4.12)

where Xq has been used to denote an impedance proportional to the generator’s quadrature axis impedance.

For steady-state conditions the phasor derived from the synchronous q-axis reactance will be aligned with the quadrature axis is depicted in Fig.4.3.

D-AXIS

ItEt

jXqIt

Ei Q-AXIS

Fig. 4.3 Compensated Phasor

As the rotor moves, the phasor derived in this manner

will maintain its position where the frequency derived from the compensated phasor will contain the desired electromechanical components. Since the rotor is in motion, the compensating reactance should represent the


- 41 -

quadrature reactance that applies to the frequency range of interest. For round-rotor machines this normally requires an impedance value close to the transient quadrature reactance.

Each generator will be somewhat different, and the compensating reactance should be selected based on knowledge of the machine reactances and time constants.

IV. HARDWARE CONSIDERATIONS The hardware should be designed so as to allow setting

of the PSS parameters over a sufficiently wide range. The design should also ensure a high degree of functional reliability and allow sufficient flexibility for maintenance. These requirements are often overlooked, resulting in unreliable and unsatisfactory performance of the PSS, much to the frustration of operators. There have been many instances of operators turning off the PSS because of poor performance resulting from inadequate hardware design and improper selection of control parameters.

The requirement for high reliability and maintainability of PSS and other elements of the excitation system may be in part satisfied by component redundancy. Duplicate voltage regulators and PSS [30], [57] have been used on critical generating units. One voltage regulator with its PSS would be in service at any one time with the other tracking it. In the event of a PSS malfunction, various protective features would initiate transfer to the alternate regulator and PSS. In addition to improving the detection of PSS failures, this feature limits the adverse consequences of such failures. The improved reliability and reduced parts count of newer digital exciters, with built-in PSS, have mitigated the need for such complex systems.

Another feature worth incorporating in a PSS is built-in facility for dynamic tests. This allows routine testing of PSS periodically by station personnel in order to detect latent failures [57]. A convenient way to test the performance of a PSS is to inject a small (1 to 2%) change in the PSS output (AVR terminal voltage reference) signal and monitor the responses of key variables such as generator terminal voltage, field voltage, power output, frequency, and PSS output. Such a test facility is also very useful during PSS commissioning.

V. PSS COMMISSIONING & FIELD VERIFICATION During field commissioning, the actual response of the

generating unit with the PSS is measured and used to verify some of the analytical results. Typical tests performed during commissioning include:

• measurement of the on-line closed-loop excitation system phase compensation requirements (Fig. 4.4)

• step response tests to measure damping improvement at local mode frequencies (Fig. 4.5)

• load-ramping tests to ensure that the PSS does not produce undesirable modulation of the unit’s terminal voltage under normal or emergency

operating conditions (Fig 4.6)

0

20

40

60

80

100

120

0.1 0.2 0.5 1 2 5

stabilizer phase compensationclosed-loop exciter phase lag

lead-lagselection

washout&lag-lead selection

Frequency (Hz)

Pha

se (d

egre

es)

Figure 4.4 Closed-Loop Exciter Phase Compensation

-0.0010-0.0005

00.00050.0010

delta

spe

ed(p

u)

1.02

1.03

1.04

PSS ONPSS OFFTe

rmin

al V

(pu)

-0.010

-0.005

0

0.005

0 1 2 3 4 5

Time (seconds)

PS

S O

utpu

t(p

u)

0.90

0.95

1.00

Act

ive

Pow

er(p

u)

Figure 4.5 Stabilizer On-Line Step Response

As noted in the previous section, the tests usually consist

of injecting small step changes to the voltage regulator terminal voltage reference and monitoring a number of generator variables. If there are discrepancies between computed and measured responses, the models are appropriately modified; if necessary, revised PSS settings are determined and implemented. This "closed loop" design and commissioning process is very effective [61].

Initially, the PSS gain should be increased slowly, with transient testing at each setting. To insure sufficient stability margin, a good practice is to check the performance of the PSS with the gain increased up to twice


- 42 -

the normal in-service setting. The objective is to ensure that the PSS gain is set at a value well below the limit at which either the exciter mode is unstable or there is excessive amplification of input signal noise.

Tests and simulations performed on all types of utility-scale generators, including large and small hydro, large fossil-fired and nuclear units and combustion turbines, have consistently demonstrated that a conventional PSS tuned and tested in this manner, will improve stability for any reasonable operating scenario.

0.20.40.60.81.0

Activ

e P

ower

(pu)

-0.0010-0.0006-0.00020.00020.0006

delta

spe

ed(p

u)

-0.60-0.45-0.30-0.15

00.15

0 5 10 15 20

Time (seconds)

filte

r mec

h po

wer

(pu)

-300-150

0150300450

Fiel

d(V

dc)

-0.010

0.010.020.03

PSS

Out

put

(pu)

-0.050

0.050.100.150.20

Rea

ctiv

e Po

wer

(pu)

Figure 4.6 Fast Load Ramp

VI. APPENDIX - DERIVATION OF FILTER RESPONSES

A. Background The conventional low-pass filter and ramp-tracking filter

are both based on the general form of a filter:

8M

9

(1 sT )G(s)(1 sT )

+=+

(A4.1)

The steady-state response of the output, y, to various

inputs, u, is calculated from the final value theorem.

t s 0lim y(t) lim(s*U(s)*(G(s) 1))→∞ →

= − (A4.2)

B. Conventional Low Pass Filter The conventional low-pass filter is obtained from A.1 by

setting T8 = 0. The denominator of A.1 can be expanded as follows:

M

M i9 i 9

i 0(1 sT ) a (sT )

=

+ =∑ (A4.3)

Some of the coefficients may be written by inspection as

follows: a0 = aM = 1 a1 = aM-1 = M The other coefficients are not critical to the analysis of

the steady-state response. Substituting A.3 into A.1 yields:

Mi

i 9i 0

Mi 1

9 i 9i 1

Mi

i 9i 1

1G(s) 1 1a (sT )

sT a (sT )

1 a (sT )

=

−

=

=

− = −

= −+

∑

∑

∑

(A5.4)

where the fact that a0=1 has been used to reduce the numerator and expand the denominator.

Step input: U(s) = A/s

Mi 1

9 i 9i 1

Mt s 0 ii 9

i 1

sT a (sT )Alim y(t) limss 1 a (sT )

0

−

=

→∞ →

=

= − +

=

∑

∑ (A4.5)

Ramp input: U(s)=B/s2

Mi 1

9 i 9i 1

M2t s 0 ii 9

i 1

Mi 1

9 1 i 9i 2

Ms 0 ii 9

i 1

9

sT a (sT )Blim y(t) limss 1 a (sT )

BT a a (sT )lim

1 a (sT )

B*T *M

−

=

→∞ →

=

−

=

→

=

= − + +

= − +

= −

∑

∑

∑

∑ (A4.6)

Parabolic input: C/s3

tlim y(t)→∞

= ∞

C. Ramp-Tracking Filter


- 43 -

8M

ii 9

i 0M

i8 9 i 9

i 2M

ii 9

i 0

1 sTG(s) 1 1a (sT )

s(T MT ) a (sT )

a (sT )

=

=

=

+− = −

− −=

∑

∑

∑

(A4.7)

Step input: U(s) = A/s (A4.8)

Mi

8 9 i 9i 2

Mt s 0 ii 9

i 0

s(T MT ) a (sT )Alim y(t) limss a (sT )

0

=

→∞ →

=

− − = −

=

∑

∑

Ramp input: U(s) = B/s2 (A4.9)

Mi

8 9 i 9i 2

M2t s 0 ii 9

i 0

8 9

s(T MT ) a (sT )Blim y(t) limss a (sT )

T MT

=

→∞ →

=

− − = −

= −

∑

∑

A4.9 equates to zero as long as T8=M*T9 Ramp input: U(s) = C/s3 (A4.10)

Mi

i 9i 2M3t s 0 i

i 9i 0

Mii 2

2 9 i 9i 3

Ms 0 ii 9

i 1

M 1

9i 0

a (sT )Clim y(t) limss a (sT )

a T a s Tlim C

1 a (sT )

CT i

=

→∞ →

=

−

=

→

=

−

=

= − + = − +

= −

∑

∑

∑

∑

∑

(A4.11)

The reduction is based on the assumption that the

coefficient relationship, T8=M*T9, has been used. In this case the response to a parabolic input will be bounded and will increase with the number of poles and time constant as expected.


- 44 -

CHAPTER 5 FIELD TESTING TECHNIQUES

J. C. Agee and Shawn Patterson

I. INTRODUCTION In this chapter the implementation of techniques for the

adjustment of power system stabilizers at generating stations will be considered. The presentation begins with a review of measurement techniques and instrumentation, continues with a discussion of field testing techniques with examples, and concludes with a brief look at the phenomenon of shaft torsional oscillation.

II. MEASUREMENT TECHNIQUES AND INSTRUMENTATION To assess the performance of the excitation control

system and PSS (power system stabilizer), it is necessary to be able to observe or record several system quantities. These include: terminal voltage deviation, electrical power output deviation, generator field voltage, shaft speed deviation or frequency deviation, and PSS output. While the measurements of generator field voltage and electrical power are not essential for stabilizer setup, they do add insight into the performance of this control system. In addition, the static measurement of torque angle at various loads may be necessary to properly tune certain types of PSS.

There are three types of testing commonly used for dynamic stability assessment: step or impulse response tests, frequency response tests, and system line or load switching tests. For the first two of these, the test or disturbance signal is normally added to the terminal voltage reference. Step or impulse response testing is easy to perform and suitable for model checking, stabilizer commissioning, and routine maintenance performance evaluation.

Frequency response testing provides a much deeper insight into the control system than step response testing and is the best tool for stabilizer tuning. It is, however, more difficult to perform than step response testing, requiring experienced personnel and more expensive test equipment. Switching tests are sometimes used as a final stabilizer check, particularly where inter-area modes of stability are of concern. They are much more difficult to co-ordinate, but allow an excellent final check on the dynamic stability of an area.

A. Signal Transducers and Conditioning Many digital excitation systems have built-in transducers

and data acquisition features that can be used for PSS tuning, but in general, the parameters to be monitored during field testing are not available in a form suitable for direct recording or input to control instrumentation. Frequently the quantities must be transduced, filtered and amplified to obtain a useable signal level proportional to the primary quantity with the appropriate bandwidth and

amplification. Generally, for PSS tuning, all transducers must have at least a 10 Hz bandwidth, be low noise (noise should be less than 5% of the deviations to be recorded), and must faithfully represent small deviations about the steady-state point; however, they do not need to have absolute accuracy over the full range of each quantity.

The required degree of transducer signal to noise depends upon the signal to noise ratio and the sensitivity with which the signal must be observed (i.e. signal deviation to noise). The power station electrical noise environment is a relatively harsh one and hence special care in the shielding and grounding of signal leads is especially important. Depending upon the type, the excitation system itself may provide substantial noise to nearby instrumentation.

B. Terminal Voltage This is an important quantity for the assessment of

performance of both the excitation control system and the PSS. Terminal voltage is readily measurable by means of a simple three phase rectifier bridge transducer. This transducer is quite linear provided diode voltage drop is low and the phases are relatively well balanced. For such a transducer, the main ripple component is at 360 Hz which can readily be filtered to provide a noise free signal with the required bandwidth.

The transducer output without filtering would contain about 6% of 6th harmonic (360 Hz) as well as lower amplitudes of higher frequency components. Small amounts of 2nd harmonic may also be present if the phase voltages are not perfectly balanced. Assume that it is desired to record terminal voltage variations of about 2% of rated and follow accurately frequencies up to 10 Hz with less than 7% amplitude error and 30° phase error at 10 Hz. Assume also that the recorder will reproduce all signals provided to its inputs. It can readily be shown that the desired characteristics can be achieved with a simple second order filter (with breakpoints at 38 Hz). There are several refinements in voltage transducers which provide greater linearity, greater accuracy or reduce filtering requirements. Such refinements, however, are not generally required for the measurement of voltage for the purposes of stabilizer tuning.

C. Electrical Power Generator electrical power is readily measured with the

aid of Hall Watt Transducers (or the equivalent). Such devices when obtained without internal signal conditioning, produce an isolated millivolt output level proportional to the instantaneous power out of the generator with response times of less than 1 ms. Watt transducers permanently installed for metering purposes will normally not fulfill this requirement. The transducer output, for balanced 3 phase generator output contains no inherent noise components. Transducer deficiencies and signal lead pickup can contribute some noise at 60 Hz and even harmonics which are readily filtered by techniques similar to those described


- 45 -

for terminal voltage transducers. Notch rejection filters or a double order low pass filter with break points at 60 Hz or lower will normally be sufficient if good signal shielding practices are followed.

D. Field Voltage Field voltage is already a direct quantity although it may

contain noise components (360 Hz and higher) several times larger than its direct voltage (particularly in thyristor controlled systems). In addition, it is necessary to provide a transducer which will isolate the instrumentation from the high power floating field circuit, while maintaining the desired 10 Hz or higher signal bandwidth. Transducers which will provide an output voltage proportional to the generator field voltage for frequencies up to about 1 kHz while maintaining the required (≈2000 V) level of isolation are commercially available. As field voltage varies widely with input to the voltage regulator, suppression of the steady-state value may not be necessary.

For the measurement of excitation response times on high initial response systems, maximum bandwidth (minimum filtering) should be used, but for stabilizer tuning, a reduced bandwidth will suffice. [62] Techniques similar to those used for terminal voltage will allow the specification of adequate filtering for stabilizer setup. A simple second order low pass filter with break points at about 50 Hz will normally be adequate for the measurement of field voltage of most excitation systems.

E. Generator Speed The measurement of shaft speed is extremely useful for

assessing power system damping (although electrical power can also be used for this purpose). In addition, the PSS transfer function has historically been defined with shaft speed as its input. Thus, measurement of shaft speed has been critical to the development of PSS. However, implementation of dual-input stabilizers using electrical power and “internal” frequency has lessened the importance of shaft speed measurement.

Shaft speed measurement is still important in turbo-generators because the PSS, if improperly designed or adjusted, has the capability of exciting shaft torsional oscillations through excitation control as discussed in Section V. Therefore, in these types of machines it is important to monitor these mechanical modes and ensure that they are not affected by the stabilizer. The first torsional mode is the one most likely to be excited through the stabilizer and is also the one most readily monitored at the ends of the generator shaft. Typical torsional mode profiles are shown in Figure 5.7. The extent to which various torsional modes are observable at any shaft location can be assessed from such profiles.

Speed is the most difficult signal of those described to monitor successfully. This is particularly due to the fact that the speed changes to be observed are of the order of 0.05% of rated and to "see" the torsional components, wide bandwidth is required, (≈50 Hz for a 3600 RPM machine).

Speed should be monitored directly from the turbine-generator shaft. A gear wheel mounted directly on the turbine-generator shaft with stationary electro-magnetic pickups (or the equivalent realized through electro-optics) is normally employed. This produces a signal whose frequency or pulse repetition rate is proportional to the speed of the shaft. (Speed transducers driven by couplings are normally unsatisfactory for observation of the torsionals as they tend to produce additional modes representative of the coupling in addition to, or instead of, those of the shaft system.) The speed measurement technique is usually based upon a conventional tachometric circuit, although other methods have also been used successfully.

In all machines, the generator shaft moves around somewhat in the bearings. This can lead to "noise" components at 60 Hz for a 3600 rpm machine and 30 Hz for an 1800 rpm machine. Such components can be partially removed by the use of either multiple pickups to cancel this motion (diametrically opposed speed sensors have been shown to be quite effective) or sharply tuned notch filters to attenuate them. Slowly rotating hydro generators have noise components below 10 Hz (e.g. 3 Hz for a 180 rpm machine), so this signal is seldom used in stabilizers for hydro applications.

The first (lowest frequency) torsional mode is lightly damped and normally subjected to random excitation by the steam supply system. Hence it can frequently be seen to come and go in a more or less random fashion in a sufficiently sensitive recording. The amplitude of such torsional oscillations under normal operation might typically be of the order of 0.005% of rated speed when monitored at the end of the shaft. They might be higher or much lower depending upon the loading condition and the nature of the system load.

Normally, the filtering used for speed measurement is more complex than that used with other quantities, as dictated by the requirements for high gain, wide bandwidth and high noise attenuation. Because one is interested only in very small changes about the steady-state value, the steady state component can be subtracted, easing the instrumentation requirements. Such direct signal suppression must be extremely stable. Alternatively, high pass filtering can be used in which case the signal is "rolled off" below some low (e.g. 0.01 Hz) cutoff frequency.

The combination of transducing and filtering should then allow the desired signal amplification with a minimum of phase shift and attenuation in a band from about 0.1 Hz or lower to about 50 Hz. For four-pole steam turbine generators, the high frequency end of the band can be further restricted to about 25 Hz as torsional components are proportionately lower in frequency. This allows the possibility of filtering of components at 30 Hz.

To monitor this signal adequately, notch rejection filters at 30 Hz (for 4 pole machines only), 60 Hz and higher harmonics, combined with low pass filters would normally


- 46 -

be required. For example, consider a tachometric circuit which produces 2000 pulses per second at rated speed. Assume 100% of 2 kHz ripple and that to observe the torsionals adequately this must be attenuated to 0.001% of rated. In other words 2 kHz must be attenuated by a factor of 100,000 or 100 db. If a 4th order filter is chosen so that 100 db attenuation occurs at 2 kHz, then a 4th order break point at 110 Hz would be possible. A torsional component at 40 Hz would then experience an attenuation of only 28% or 2.2 db, but would be phase shifted by 80° in the filter. The effects of any notch filters used would have to be added to this.

F. Terminal or Internal Frequency In lieu of shaft speed, terminal or internal frequency is

often used. Internal frequency is defined as the frequency of the internal voltage phasor obtained by adding the voltage drop across the quadrature axis impedance to the generator terminal voltage. The voltage drop across the quadrature axis impedance is determined by multiplying the generator terminal current phasor by the quadrature axis impedance, Xq. This measurement has proven to be easier to obtain and more accurate for salient pole hydro generators than for round rotor machines that are subject to quadrature axis saturation; however, it has been used successfully with all types of machines.

To transduce frequency from either the terminal voltage or internal voltage phasor a method much like the one associated with shaft speed is used. In this case the pulses are generated by zero crossings or other points on the waveform. Care must be taken to filter out the effects of harmonics on the input waveforms. A signal with 10 Hz bandwidth much like the terminal voltage signal is desired.

G. Power System Stabilizer Output This quantity is normally available in a form suitable for

direct observation or recording. It may be necessary to provide appropriate isolation, particularly if the voltage regulator is not grounded. Instrumentation must present a high input impedance so that it does not overload the stabilizer output.

H. Generator Torque Angle Measurement of generator torque angle may be

necessary to properly tune stabilizers that use internal frequency as an input. Typically, only steady-state measurements of angle are required. For these measurements, an index pulse related to a position on the generator shaft, such as a key phasor from a vibration monitoring system, can be compared to the 60Hz terminal voltage waveform to determine the change in relative shaft position as the generator is loaded. This change in position can then be used to calculate the generator torque angle and the effective quadrature-axis impedance. Dynamic measurement of torque angle requires automation of this process using a phase angle transducer. Generators with multiple pole pairs may require more than one pulse per

revolution to produce adequate dynamic torque angle results.

I. Signal Recording Many varieties of signal conditioning, recording, and

data acquisition equipment are suitable for PSS and excitation system testing. Basically, a multichannel recording system with built in signal amplification and a bandwidth of about 100 Hz is required. Increased flexibility is provided by a recording system with high impedance differential inputs, high common mode capability or isolation, internal signal bias, built in amplification and signal conditioning and a wide range of recording speeds. There are a multitude of recorders and data acquisition systems that will meet these requirements. Some excitation systems may have built-in data recording capabilities that can be used for this purpose as well.

III. TESTING TECHNIQUES Once the desired signals are in a form suitable for

observation or measurement with the desired accuracy, gain and bandwidth, the following test procedures can be performed.

A. Step and Impulse Response Testing While various types of perturbations can be considered

for the time domain evaluation of a system, the ones used most often in PSS testing are step and impulse disturbances added to the terminal voltage reference signal. If the desired worst case power system conditions can be set up, they permit quick and simple checks on the stability of the overall system. Without extensive testing or comparative model test results, they do not allow much insight into the way parameters should be modified for improved tuning; however, a series of time domain tests with various PSS adjustments can be useful in verifying model accuracy.

14.85

14.90

14.95

15.00

15.05

15.10

15.15

15.20

15.25

15.30

15.35

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (s)

Term

inal

Vol

tage

, Vt

(kV)

605

610

615

620

625

630

635

640

645

650

655

Out

put P

ower

(MW

)

Vt MW Figure 5.1a Typical system step response with PSS off

Possibly the greatest value of the step or impulse

response is as a simple "finger print" check on the overall performance of the control system. As such it is useful for final commissioning and regular maintenance checks. A


- 47 -

typical step response with the stabilizer out of service and in service is shown in Figures 5.1a and 5.1b

14.90

14.95

15.00

15.05

15.10

15.15

15.20

15.25

15.30

15.35

15.40

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

Time (s)

Term

inal

Vol

tage

, Vt

(kV

)

590

595

600

605

610

615

620

625

630

635

640

Out

put P

ower

(MW

)

Vt MW Figure 5.1b Typical system step response with PSS on

This test is easily performed with the transducers and

data acquisition discussed in section II and a simple circuit and switch (or software tools) to generate the input signal. Results of step response tests can be compared with similar responses from time domain models. This allows validation of the model and hence increased confidence in use of the model for machine or system conditions beyond the scope of field testing. When compared with previously obtained responses it allows validation of the condition of the PSS and excitation system.

B. Frequency Response Testing Frequency response characteristics permit much greater

insight into the small signal operation and tuning of a control system than time responses. There are three frequency response characteristics that are important for the tuning of the PSS:

1. The transfer function relating generator terminal

voltage variation to a signal at the stabilizer output, ∆Et(s) / ∆ PSS(s). This characteristic is useful because ∆Et(s) is in phase with the component of electrical torque produced by exciter action if machine angle were held constant [23]. This transfer function is normally the same as the transfer function relating generator terminal voltage variation to terminal voltage reference changes, ∆Et(s)/∆Et-Ref(s). An example of this function is shown in Figure 5.2.

2. The transfer function (∆ω(s)/∆PSS(s)) describing the

turbine generator shaft speed change resulting from a change in stabilizer output signal, for speed or power based stabilizers. The transfer function can also be deduced from the more readily measurable change in electrical power vs. stabilizer output as shown in Figure 5.3.

3. The overall stabilizer frequency response:

∆PSS(s)/∆ω(s); ∆PSS(s)/∆Pe(s); ∆PSS(s)/∆∫Pacc(s); or equivalent, as shown superimposed on the ∆Pe(s) / ∆

PSS(s) frequency response in Figure 5.4. The combination of characteristics 1 and 3 determine the

ability of the stabilizer to contribute damping at the various machine-system and inter-area modes. The combination of characteristics 2 and 3 (the stabilizer open loop response) allows determination of gain and phase margins of the stabilizer control loop. This technique is shown in [31], [32], [33].

0.1 1.0 10

Gai

n (d

B)

0

- 5

-10

-15

-20

-25

-30

Without PSSWith PSS

Phas

e (d

eg)

0

-50

-100

-150

-200

-250

-3000.1 1.0 10

Frequency (Hz)

Frequency (Hz)

Figure 5.2 Frequency Response of ∆Et(s)/∆Et-Ref(s)

0.1

Pe/VrefFreq/Vref

100

-10-20

-30-40

-50

-60-70

Gai

n (d

B)

0.1 1.0 10

200

100

0

-100

-200

-300

-400

-500

Phas

e (d

eg)

1.0 10Frequency (Hz)

Frequency (Hz)

Figure 5.3 – Frequency Responses of ∆Pe(s) / ∆Et - Ref(s)

40

20

0

-20

-40

-60

Gai

n (d

B)

1.0

Gain Margin

10

180

90

0

-90

-180

-270

-360

Phas

e (d

eg)

1.0 10

Frequency (Hz)

Frequency (Hz)

Speed/VrefPSSSpeed/ Vref + PSS

Figure 5.4 – Overall PSS Frequency Response (Open Loop Responses)


- 48 -

C. Equipment and Techniques for Frequency Domain Analysis

While frequency response characteristics could be measured using a signal generator and recorder, commercially available signal analyzers greatly facilitate such measurements, automate the procedure and allow much higher transfer function resolution with lower levels of disturbance input. These commercially available analyzers include both frequency response analyzers that take sine-wave measurements at discrete frequencies and analyzers that use random noise techniques.

Sine-wave analyzers allow the measurement of the magnitude and phase of the ratio of output to input of a control block at the distinct frequency at which the system is excited. Because correlation techniques are employed, accurate measurements can be obtained even when the measured signal at the test frequency is deeply "buried" in noise.

To determine the transfer characteristic of an arbitrary control system block, the instrument actually measures two characteristics and performs the following calculation:

)()(

)()(

)()(

)(1

2

1

0

0

2

ωω

ωω

ωωω

jVjV

jVjV

jVjV

jG =•= (5.1)

The instrument generates the sinusoidal signal V0 which

drives the control system, generally through addition to the terminal voltage reference signal to obtain the system transfer functions ∆Et(s) / ∆PSS(s) and ∆ω(s) /∆ PSS(s).

With this instrument, care must be exercised in the choice of signal frequency and amplitude, particularly near system resonance. The machine-system resonant “local mode” frequency (0.8-2 Hz) should be approached with caution.

Due to possible excitation of torsional modes, tests

should not be conducted at frequencies higher than 7 Hz on turbogenerators without special precautions which are beyond the scope of this work.

With random-noise type analyzers, the distinct frequency

excitation V0(jω) is replaced by a random noise input, generated either by the instrument or by a suitable external source. The source does not have to be synchronized with the measurement system. Using the ratio of the cross-power to auto-power spectra, the system obtains information about the transfer function over the chosen frequency range and displays or records the appropriate magnitude and phase characteristics. Averaging is provided to allow increased resolution and the use of a coherence function allows an indication of the quality of the measurement.

As with sine-wave analyzers, useful results can be obtained even when the response to the test signal is smaller than the uncorrelated system noise. Random noise analyzers can obtain the frequency response over the entire spectrum simultaneously, thereby reducing the testing time.

In addition, because of the random frequency distribution, there is a lower probability of over exciting sharply tuned resonances.

Most random-noise analyzers have features that allow them to be used in a manner similar to sine-wave analyzers by replacing the random noise input signal with a sinusoidal input signal.

D. General Comments It is not always convenient or expedient to perform

frequency response tests over the desired range of possible machine and system conditions. If the field test results can be used to confirm computer models, the models can then be used to investigate the desired range of system conditions. When the frequency response characteristics have been measured, the stabilizer phase lead characteristics can be adjusted as described in other sections.

After the stabilizer phase compensation has been suitably adjusted, the stabilizer frequency response can be checked. The combination of the appropriate machine/system transfer function and the stabilizer transfer function provides the open loop characteristic which can be used for the determination of control loop stability.

IV. ON SITE TUNING AND STABILITY ASSESSMENT

A. The Excitation System The generator excitation system is normally designed to

meet a host of cost and performance criteria described by the purchaser. It is then set up to meet specified steady-state regulation requirements and to respond at the rate and with a forcing capability specified for the desired transient stability enhancement through excitation control. The voltage regulator should be adequately damped on open circuit. Typical open circuit voltage responses to step change in voltage reference are shown in Figure 5.5.

1.025

1.020

1.015

1.010

1.005

1.0000 1 2 3 4 5

Time (seconds)

Term

inal

Vol

tage

(pu)

Figure 5.5 – Off-line voltage step responses

After the excitation system has been properly adjusted, the power system stabilizer should be able to contribute sufficient damping to prevent oscillatory instability over the range of possible system configurations and conditions envisaged for any machine. In other words, the stabilizer should have the effect of moving the small-signal stability


- 49 -

limit beyond other power system stability limits. High speed excitation systems, while contributing to

better transient stability and allowing larger operating angles than slower excitation systems, can unfortunately produce an additional negative component of damping. Fortunately, a well-tuned PSS can overcome this negative damping and provide substantial additional positive damping for both inter-area and local machine-system modes. While providing the required damping, action of the PSS should not detract significantly from the high-speed excitation system contribution to improvement of system transient stability.

Slower excitation systems do not contribute as much to the improvement of transient stability, but at the same time do not contribute significantly to negative damping of the local mode. With slow systems it is difficult to add much damping to the machine-system mode via a power system stabilizer, but in turn little is required. Depending upon the system, it still may be possible to provide a significant contribution to inter-area mode stability.

With modern excitation systems, high steady-state gain (≈200) or integral gain (in many digital systems) is normally employed. Depending upon power system requirements and design philosophy, transient gain reduction may also be used. Systems without transient gain reduction can contribute more to transient stability, but at the expense of an additional negative contribution to local mode damping. However, this can be overcome by using slightly higher stabilizer gains. For this reason, care should be exercised in the comparison of stabilizer gains for systems with and without transient gain reduction. A factor of approximately the value of the transient gain reduction separates the two types of systems for an equivalent contribution to system damping (when the stabilizing signal is injected before the TGR circuit) [30]. Also, gain for inter-area modes and the machine-system local mode will be different for these two types of systems. In either case, it should be possible to tune the stabilizer to entirely remove the power system oscillatory instability.

B. Tuning Criteria The most important PSS tuning criterion is that, after

choosing and setting up the excitation system, the power system stabilizer should move any oscillatory stability limitations beyond all other power system limitations. This criterion should hold for all possible machine and system operating conditions. In some power systems, providing damping of inter-area modes may be of primary importance. If so, adequate gain and phase relationships must be maintained for all known oscillatory modes. Many times these modes are not visible during field testing, so computer models and analytical techniques must be used to verify proper tuning.

The degree of difficulty experienced in providing damping for all oscillatory modes is dependent on the performance required from the excitation system. If

excitation system response requirements are low, it will be relatively easy to provide a satisfactory stabilizer - in fact it may not be necessary to add damping at all for the local machine/system mode of oscillation. For inter-area mode stability, the unit must provide an appropriate damping contribution relative to its share of generation in the area.

C. PSS Testing The first critical step in PSS testing is measuring the

transfer function from the PSS output to terminal voltage as shown in Figure 5.2. This is normally performed with the unit on-line at light load, so the effect of rotor swings on the response is minimal. Then, the PSS compensation transfer function is selected and documented as in Figure 5.4. [30] Finally, PSS gain is chosen and a benchmark step response is obtained as in Figure 5.1.

V. SHAFT TORSIONAL OSCILLATION Consider first a simple model of a spring and mass in

torsion as shown in Figure 5.6. The equation for rotational motion of the mass can be written as:

02

2

=∆++ TKdtdJ θθ

(5.2)

where J is the inertia K is the spring constant of the shaft ∆T is an external torque input. If the external torque is a constant or zero (∆T=0), the

equation is one of an undamped system which would oscillate continuously if set in motion. Assume now that the external torque input ∆T can be manipulated, say as a function of the speed of the mass as shown in Fig 5.6.

Figure 5.6 - Simplified torsional oscillation model

In a turbine-generator, torque changes could be produced

in this manner through the field of the generator, if shaft speed were used as an input to the excitation control.

If this feedback path constitutes a straight gain G, equation 5.2 becomes:

02

2

=++ θθθ KdtdG

dtdJ (5.3)

Θ

G

∆ω

∆T


- 50 -

in which case damping of the torsional oscillation is produced as a function of the amplitude of G. If, however, the transducer produces not just gain but phase lag of greater than 90° at the oscillating frequency, a negative damping component of torque is produced and oscillation will build spontaneously.

In the frequency domain a transfer function of the form

)1)(1()()()(

BsAsG

ssTsG

++′

=∆∆

=ω

(5.4)

could produce the necessary phase lag to provide a negative component to undamp this mode.

The rotating components of a large steam turbine form a more complex torsional system with many shaft sections. The torsional components are all very lightly damped with decay time constants measured in seconds. The frequencies of the torsionals are functions of turbine generator configuration and design. Some large units may have first torsional modes as low as 7 Hz, while other units may have a highest torsional mode near 55 Hz.

The rotating system is connected to the outside world through the rotating air gap torque. The strength of this tie can be represented by a spring connected to a fixed reference. The steady state rotation of the shaft is now effectively removed from the problem. A dashpot in parallel with this spring represents machine-system damping. The total mass, spring and dashpot define the machine-system mode of oscillation to be damped by the PSS.

The objective of the PSS is to enhance this machine-system damping by generating a torque in phase with speed changes at the frequency of this oscillation. If the signal is generated directly from shaft speed measurement, it is difficult to find a place on the shaft where the transducer responds only to the mode in which the shaft acts as a rigid body oscillating against the power system. If speed is measured at the generator end of the shaft and if the speed to torque transfer function constitutes strictly a gain and no phase lag, damping would be provided for both the system mode and the first torsional mode. Analysis of damping of this first torsional mode is directly related to the simple modal analysis in the first part of this section.

Unfortunately, the phase lags through the stabilizer and generator (Equation 5.4) are sufficient to cause instability of the first torsional mode when the stabilizer is adjusted for appropriate damping of the system mode. This possibility of torsional excitation through the power system stabilizer and excitation system has been well documented. [55][41]

Placing the speed sensor at the node of the first torsional mode would make this mode invisible to the control system, but this location would normally not be equipped for speed measurement and might be inaccessible. Even if speed could be measured at this node, other torsional signal components would still be detected by the transducer, and

it would be necessary to ensure that these torsional modes could not be excited.

In practice, a combination of selective speed location and/or sharply tuned rejection filters at the torsional frequencies is employed to preclude torsional excitation. Because the lowest torsional frequency is only about a decade away from frequencies at which stabilization is required, the torsional filter does add some small undesirable phase lag to the stabilizing loop, and has the effect of limiting the maximum usable stabilizer gain.

In a stabilizing system which uses terminal frequency as an input, torsional frequencies are attenuated somewhat compared to a stabilizer which uses an end of shaft speed signal [31], [32], [33]. In stabilizers using electrical or accelerating power as input, the torsionals are inherently highly attenuated [31], [32], [33], [58].

Figure.5.7 Typical Shaft System And Torsional Mode Shapes


- 51 -

CHAPTER 6 APPLICATION CONSIDERATIONS

Murray Coultes

I. INTRODUCTION Around the year 2000 two significant changes occurred

in the field of excitation systems. The most significant was the use of digital processors rather than circuit components to implement the voltage regulator, limiter and PSS functions. The second was widespread installation of small gas turbine generators in utility rather than industrial applications.

II. UNIDIRECTIONAL EXCITERS Small gas turbine generators are usually equipped with

rotating main exciters and low-power (less than 10A) pilot exciter/voltage regulators. These pilot exciters are often equipped with half-controlled full-wave bridges. These bridges can only produce positive voltages, so no negative field forcing is possible. Although this may be acceptable for isolated or industrial applications, it is highly undesirable in a utility application when a PSS is also used. During large disturbances, in which the PSS signal can exceed 0.05 pu of the terminal voltage reference, the result is a generator overvoltage.

Fig.6.1 is a simulation of an external fault. The first part of the record is for a full-controlled bridge; the second part is a half-controlled bridge. The half-controlled bridge produces a generator terminal voltage increase of over 10%. The settings for PSS on these systems must consider this effect. Typically, the output limits of the PSS are set at +/-0.05 pu, instead of the recommended +/-0.1 pu.

III. DIGITAL EXCITERS When voltage regulators and power system stabilizers

were implemented with operational amplifiers, speed and resolution were seldom issues in the bandwidth that they were required to handle. Such is not the case with digital types.

A. Processor Cycle Time The rate at which the PSS algorithm is recalculated

limits the minimum time constant that can be implemented - the faster the execution time, the shorter the time constant that can be used. Fig.6.2 compares the nominal phase lead compensation that was set with what was measured on a voltage regulator with a 5 ms cycle time.

( )( )( )( )s.s.

s.s.0101010115011501

++++ (6.1)

-60

-40

-20

0

20

40

60

80

Exci

ter

Fiel

d V

olta

ge, V

R

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Gen

erat

or F

ield

Vol

tage

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Gen

erat

or T

erm

inal

Vol

tage

0.40.50.60.70.80.91.01.11.21.3

Pow

er

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 1 2 3 4 5 6 7 8

Time (s)

PSS

Figure 6.1 Performance With Bi-Directional And Uni-Directional Exciters

The calculation of phase lead settings for a digital exciter

must account for the phase error at higher frequencies if the compensation is to be correct.

Most digital voltage regulators utilize two or more different cycle times for different parts of the algorithm. It is important that the PSS be in the fastest loop so that the short time constants are realized with as much fidelity as possible.

B. Integer Or Floating Point Arithmetic Because of the large dynamic range that a PSS must

handle correctly, floating point arithmetic is preferred. If integer is used it is important to check that both large and small signals are reproduced with sufficient accuracy.

C. Passwords And Security The computer industry is accustomed to using passwords

to protect software from unauthorized tampering. Unfortunately this mindset has carried over into digital exciters and power system stabilizers. While there is some merit in this security, it is a dangerous practice.

The commissioning of a PSS is a good example. A common method is to calculate and apply all of the settings except the gain, and then raise the gain slowly toward the final setting. On circuit component types, this was usually done via a knob; it was easy to turn the gain up slowly and also quick and easy to turn it down if the system started to


- 52 -

become unstable. On digital systems, there are two problems. First, it is

not difficult to enter a gain that is a factor of 10 higher than was intended. Second, it is inherently slower to set the gain back to zero quickly since it has to be entered via keystrokes rather than turning a knob. If a password also has to be entered before a gain change can be made, it is a recipe for disaster.

If a customer insists on password protection, it should only have to be entered once at the beginning of the commissioning procedure, never in the midst of it. A preferable solution is no password at all.

IV. MINIMUM EXCITATION LIMITERS Minimum excitation limiters are usually one of two

types: takeover, or additive. The takeover versions effectively disable the voltage

regulator and PSS functions while they are active. The additive ones inject a "raise” signal into the voltage regulator summing junction.

If a PSS is part of the excitation system, the additive type is preferred since it will permit the PSS to continue to function during potentially unstable situations when the minimum excitation limiter is active.

1.00

10.00

100.00

1000.00

0.10 1.00 10.00Frequency (Hz)

Mag

nitu

de (V

/V)

0

20

40

60

80

100

120

140

Phas

e (d

egre

es)

MeasuredNominal

Magnitude

Phase

Figure 6.2 Phase Error Caused By Limited Processor Speed


- 53 -

CHAPTER 7 FUTURE DIRECTIONS IN PSS DESIGN

Om Malik, José Taborda, Robert Grondin, Innocent Kamwa, Gilles Trudel

I. INTRODUCTION Power systems are non-linear and operate over a wide

range. For example, the gain of the plant increases with generator load. Also, the phase lag of the plant increases as the ac system becomes stronger. A power system stabilizer (PSS) is employed on an electric generating unit to improve its damping and the stability of the power system.

Design of the conventional power system stabilizer (CPSS), done off-line, is primarily based on the linear control theory using a model of the power system linearized at one operating point. Such a CPSS can provide optimal performance only for the specific parameters used in the design. To further improve the performance and stability of the power system, various other approaches using the linear quadratic optimal control, H-infinity, variable structure, rule based, and artificial intelligence (AI) techniques [31], [32], [33], [63] - [69] have been proposed in the literature to design a fixed parameter PSS. One common feature of all fixed parameter controllers is that the design is done off-line.

Due to the non-linear characteristics, wide operating conditions and unpredictability of perturbations in a power system, the fixed parameter PSS generally cannot maintain the same quality of performance under all conditions of operation.

Adaptive control can be described as the changing of controller parameters on-line based on the changes in system operating conditions. Whenever an adaptive controller detects changes in system operating conditions, it responds by determining a new set of control parameters. The adaptive control theory provides a possible way to solve many of the problems associated with the control of non-linear time-varying systems, such as power systems.

A number of examples of development and successful implementation of adaptive PSSs (APSSs) based on analytical and artificial intelligence techniques, that are capable of being employed in the power systems in the future, are described.

II. ANALYTICAL ADAPTIVE CONTROL BASED APSS The common procedure in process control is to compare

the actual measured values of the output with the desired values and the difference, the control error, is fed as input to the process via a regulator and an actuator. Various criteria are available for the computation of the control.

Using this technique, the desired control law is obtained as:

)]Tt(u),t(y),t([f)t(u s −θ= (7.1)

where: θs(t) is the system parameter vector y(t) is the output vector [y(t), y(t-T) …]′ U(t-T) is the control vector [u(t-T) u(t-2T) …]′ ′ denotes the transpose, T is the sampling period f[.] denotes function If the parameter vector is known, control to meet specific

performance criterion can be computed directly. However, the dynamics of a complex non-linear system vary with time depending upon the operating conditions, disturbances, etc.

An adaptive controller has the ability to modify its behavior depending on the performance of the closed-loop system. The basic functions of the adaptive controller may be described as:

• Identification of unknown parameters, or measurement of a performance index,

• Decision of the control strategy, • On-line modification of the controller parameters. Depending on how these functions are synthesized,

different types of adaptive controllers are obtained. Various adaptive control techniques have been proposed for excitation control since the mid 1970s. A brief review of the adaptive control techniques from the excitation control aspect is presented in this section.

Two distinct approaches – direct adaptive control and indirect adaptive control – can be used to control a plant adaptively. In the direct control, the parameters of the controller are directly adjusted to reduce some norm of the output error. In the indirect control, the parameters of the plant are estimated as the elements of a vector at any instant k, and the parameters vector of the controller is adapted based on the estimated plant vector.

A. Direct Adaptive Control A very common form of direct adaptive control is the

Model Reference Adaptive Control (MRAC). The objective of an MRAC system is to update the controller parameters such that the closed-loop system maintains a performance specified by a reference model. It requires a suitable model, an adaptive mechanism and a controller.

The structure of a MRAC system is shown in Fig.7.1. In MRAC, the actual system performance is measured against a desired closed-loop performance specified by a reference model that is driven by the same input as the controlled system. The objective is to minimize the error, the difference between the actual system output and the reference model output. The "Adaptation Mechanism” block in Fig.7.1 is used to update the parameters of the controller. Various methods are available to minimize the error function.


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Fiure.7.1 MRAC Structure Reference model must be selected such that the actual

system is capable of matching its performance characteristics. The ideal reference model is ‘1’. However, when this is not achievable due to system limitations, delay of inputs will occur even for a first order model. This might cause the response of the reference model to be substantially different from that of the actual system and will be interpreted as system fault even under normal conditions.

The most important feature in ensuring the success of MRAC is the selection of a proper reference model and its parameters. The selected parameters must be such that the system is capable of following the reference model output and that the control signal remains within the physical control limits.

A systematic method to determine a proper reference model for the plant is described in [70]. In this approach time domain performance of the controlled system with an analytical pole-shift APSS [71] was studied by simulation studies under various operating conditions and disturbances. The best closed-loop pole locations that do not violate the control constraints were obtained. A set of average zeros and closed-loop pole locations of the poles obtained from these studies was used as the parameters of a discrete third order reference model.

Application of an APSS based on the MRAC principle is shown in Fig.7.2. An FLC with self-learning capability is used to adapt the system performance to track the reference model. Two inputs, generator speed deviation and its derivative, and the supplementary control output, each have seven membership functions. The FLC uses the Mamdani-type fuzzy PD rule base [72]. Updating the center points of the controller input membership functions, i.e. the weights of the fuzzy controller, using the steepest descent algorithm provides it with a self-learning capability. It can thus adapt the system performance to track the reference model.

Figure 7.2 System Configuration

Results of a number of studies show that this APSS

provides good damping over a wide operating range and improves the performance of the system. An illustrative example showing the system response to a three phase to ground fault with the self-learning MRAC based FLC and a fixed center FLC is given in Fig.7.3.

Figure 7.3 Three-Phase To Ground Fault At The Middle Of One

Transmission Line And Successful Reclosure With APSS (MRAFC) And Fixed FLC PSS (P= 0.95 pu, 0.9 PF Lag)

B. Indirect Adaptive Control A general configuration of the indirect adaptive control

as a self-tuning controller is shown in Fig.7.4. At each sampling instant, the input and output of the generating unit are sampled and a plant model is obtained by some on-line identification algorithm to represent the dynamic behavior of the generating unit at that instant in time. It is expected that the model obtained at each sampling instant can track the system operating conditions.

u uc y _ + yr

Reference model

Adaptation Mechanism e

System Controller

Field

Transmission Lines

Vt Vg

AVR

& Exciter

Generating unit

yr + - − y yc

u

FLC Z-1-

+ Reference Model

e ∆e


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Figure 7.4 Block Diagram Of A Self-Tuning Controller

The required control signal is computed based on the

identified model. Various control techniques can be used to compute the control. All control algorithms assume that the identified model is the true mathematical description of the controlled system.

In the analytical approach to the design of an adaptive controller, sampled data design techniques are used to compute the control. The indirect adaptive control procedure involves:

-Selection of a sampling frequency, fs, about ten times the normal frequency of oscillation to be damped.

-Updating of the system model parameters (coefficients of system transfer function in the z-domain) each sampling interval T(= 1/fs) using an identification technique suitable for real-time application. A number of identification routines, in recursive form, e.g. recursive least squares (RLS), recursive extended least squares (RELS), etc. can be used to determine the transfer function of the controlled plant in the discrete domain.

-Use the updated estimates of the parameters to compute the control output based on the control strategy chosen. Various control strategies, among them optimal, minimum variance, pole-zero assignment, pole assignment, pole shift, etc. have been proposed.

C. System Model The generating unit is described by a discrete ARMAX

model of the form:

)t(e)t(u)z(B)t(y)z(A += −− 11 (7.2) where A(z-1) and B(z-1), polynomials in the delay

operator z-1, are of the form: a

a

nn

ii zazazazA −−−− +++++= .........1)( 1

11 (7.3)

b

b

nn

ii zbzbzb)z(B −−−− ++++= LL1

11 (7.4)

ba n n ≥ the variables y(t) and u(t) are the system output and

system input, respectively, and e(t) is assumed to be a sequence of independent random variables with zero mean.

D. System Parameter Estimation The control is computed based on the identified model

parameters, ai and bi. Thus to compute the control appropriate to the varying conditions the system parameters have to be estimated on-line. The correctness of the identification determines the preciseness of the identified model that tries to reflect the true system. For a time-varying system the tracking ability of the identification method is very desirable.

An on-line estimate of the system parameters is obtained by providing in the regulator a mathematical model having a desired structure describing the actual process. Such a model may be expressed as:

)]t(,[g)t(y m ξθ= (7.5)

where: )t(y is the predicted (estimated) value of the

system output. θm is the model parameter vector, and ξ(t) is the information known at the time of

prediction. The model parameter vector may either be constant, θm,

or be a function of time, θm(t). For the model to track the system dynamics, i.e. tune itself to the system, its parameters must be updated continuously at an interval that is consistent with the time constants of the system.

Several methods can be used to obtain an estimate for the model parameter vector, θm(t) [73]. A commonly used technique of achieving a continuous tracking of the system behavior is the RLS parameter estimation technique. It minimizes the square of the error between the actual system output and the model output, and the estimated parameter vector, )t(ˆ

mθ , is given by:

)]t(),t(P),Tt(ˆ[h)t(ˆmm ξ−θ=θ (7.6)

where P(t) is the covariance matrix of the error of

estimates. In general terms it contains the entire history of the process.

To enhance the ability of the identifier to track the operating conditions of the actual system, a forgetting factor is used to discount the importance of the older data. It can be chosen as a constant or a variable. A variable forgetting factor, employed to improve the tracking ability especially under large disturbances, is calculated on-line every sampling interval [74].

III. INDIRECT ADAPTIVE CONTROL STRATEGIES Four control strategies that need explicit identification

are described below.

A. Linear Quadratic Control In the linear quadratic (LQ) control algorithm the


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objective is to minimize a performance index [75]. The performance is chosen so that the system output error is minimized with respect to the system input. The LQ controller has the advantage that it will always result in a stable closed-loop system provided that the parameter estimates are exact. However, the achievement of this characteristic imposes heavy computational burden because it requires the solution of a matrix Ricaati equation. Also, this controller is designed in the state space form and a common identification technique estimates the system parameters in the input/output form. Thus an observer is required to convert the system parameters into a canonical form.

B. Minimum Variance (MV) Control In this control strategy, the objective is to minimize the

variance of the output [76]. Output error at the next sampling instant for zero control is predicted first. The control that will drive this predicted error to zero is then computed. Although this control strategy has nice properties, it has characteristics that make it difficult to use for excitation control.

The closed-loop system will be unstable if the dynamics of the sampled system are non-minimum phase, i.e. the system has a zero on or outside the unit circle in the z-domain. In this strategy, the controller poles are obtained directly from the identified system zeros. This might cause an unstable control computation if identified zeros are not cancelled exactly with the system zeros. When the cancellation of large parameter errors is not possible within one sample due to the limits on the control signal, the MV controller will produce an oscillatory response. The excitation signal is band limited and the use of MV controller will result in excessive control and a poor control action. These problems associated with the MV controller can be avoided by using a pole-zero or pole assigned controller.

C. Pole-Zero And Pole Assigned Control In the pole-zero assignment (PZA) controller the poles

and zeros in the closed-loop are pre-specified by the designer [77]. Whereas, in the MV case all the poles are shifted towards the center of the unit circle, poles and zeros in the PZA case are shifted to locations that produce the desired closed-loop characteristics. This permits a trade-off between performance and control effort. Although this controller does not suffer from the problems of non-minimum phase and band limited output associated with the MV controller, the designer has to know the system characteristics to achieve the desired characteristics. In this respect, this algorithm can be compared to MRAC.

Pre-selection of the locations of poles and zeros is difficult for non-deterministic case and their poor choice may lead to unstable control computations.

In the pole-assigned (PA) controller only poles, instead of both poles and zeros, are assigned [78]. Otherwise, it is exactly the same as the PZA controller.

D. Pole Shift Control The pole-shift (PS) controller is in essence the PA

controller but the closed-loop poles are obtained by shifting the open-loop poles radially towards the center of the unit circle in the z-domain. Shifting the poles towards the center is directly related to increased damping. This approach has the advantage of producing a stable controller. Detailed description of the PS control algorithm and its application as an adaptive PSS is given in the next section.

IV. PS CONTROL BASED ADAPTIVE PSS Extensive amount of work has been done to develop and

implement an APSS based on the pole-shift strategy. Such a PSS can adjust its parameters on-line according to the environment in which it works and can provide good damping over a wide range of operating conditions of the power system.

A. Self-Adjusting Pole-Shift Control Strategy In the pole-shift control strategy, in closed-loop (with

PSS) the poles of the controlled system are shifted from their open-loop (without PSS) locations towards the center in the z-plane by a factor less than one. This factor, called the ‘pole shifting factor’, is varied on-line to always produce maximum damping contribution without exceeding the control limits. To determine the desired control, such a system may be modeled by a linear low order discrete model with time-varying parameters.

The parameters of the system model of a given structure, estimated as in section g above, are used in the control algorithm to compute the updated control. A block-diagram of the regulator is shown in Fig.7.4. Because the control is based on the estimated model parameter vector,

)t(ˆmθ , equation (7.1) now becomes:

)]Tt(U),t(y),t(ˆ[f)t(u m −θ= (7.7) For the system modeled by equation (7.2), assume that

the feed-back loop has the form (c.f. Fig.7.5)

)z(F)z(G

)t(y)t(u

1

1

−

−−= (7.8)


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Figure 7.5 Closed-Loop System Block Diagram

Figure 7.6 Pole-Shifting Process

From (7.2) and (7.8) the closed loop characteristic

polynomial T(z -1 ) can be derived as

)T(z ))G(zB(z ) )F(z A(z -1-1-1-1-1 =+ (7.9) Unlike the pole-assignment algorithm in which T(z –1)

is prescribed [78], the pole-shift algorithm makes T(z –1) take the form of A(z –1) but the pole locations are shifted by a factor α, i.e.

)z(A)T(z -1-1 α= (7.10)

In the pole-shift algorithm, α, a scalar, is the only

parameter to be determined and its value reflects the stability of the closed-loop system. Supposing λ is the absolute value of the largest characteristic root of A(z –1), then α.λ is the largest characteristic root of T(z –1). To guarantee the stability of the closed-loop system, α ought to satisfy the following inequality (stability constraint):

λ>α<

λ− 11 (7.11)

The pole-shift process is presented schematically in

Fig.7.6. It can be seen that once T(z -1) is specified, F(z -1) and G(z –1) can be determined by (7.9), and thus the control signal u(t) can be calculated from (7.8).

To consider the time domain performance of the controlled system, a performance index J is formed to

measure the difference between the predicted system output, ŷ(t+1), and its reference, yr(t+1):

2

r 1)](ty - 1)(tyE[ J ++=∧

(7.12) E is the expectation operator. Ŷ(t+1) is determined by

system parameter polynomials A(z –1), B(z –1) and past y(t) and u(t) signal sequences. Considering that u(t) is a function of the pole-shifting factor α, the performance index J becomes

1)]+( ,α ),( ),( ),(Β ),(Α[ = −−

α tytytuzzfJmin r11 (7.13)

The pole-shifting factor α is the only unknown variable

in (7.13) and thus can be determined by minimizing J. Constraints: When minimizing J(t+1, α), it should be noted that α will

be subject to the following constraints: 1. The stabilizer must keep the closed-loop system

stable. It implies that all roots of the closed-loop characteristic polynomial A(z –1) must lie within the unit circle in the z-plane (c.f. equation 7.11).

2. The control limit should be taken into account in the stabilizer design to avoid servo saturation or equipment damage. The optimal solution of α should also satisfy the following inequality (control constraint):

maxu u(t, min u ≥)α≤ (7.14)

Pole-patterns of T(z –1) for a 50 ms three-phase to ground

fault at the middle of one line of a double circuit transmission line connecting a generator to a constant voltage bus, Fig.7.7, are shown in Figs. 7.8 and 7.9. The pole-pattern before the application of control is shown in Fig.7.8. Since two poles map outside the unit circle, the closed-loop system is in an unstable state. The pole-pattern after the pole-shift control is applied is shown in Fig.7.9. Since all the poles lie within the unit circle, the closed-loop system is stable. It shows that the pole-shift control assures the stability of the closed-loop system and also optimizes the performance given by (7.13).


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Figure 7.7 Power System With Adaptive PSS

Figure 7.8 Pole Pattern For T(z –1) With Pole-Shift Control

Figure 7.9 Pole Pattern For T(z –1) Without Pole-Shift Control

V. PERFORMANCE STUDIES WITH POLE-SHIFTING CONTROL PSS

Performance of the self-tuning adaptive controller based on the pole-shifting control algorithm has been investigated by conducting simulation studies on single machine [71], [74] and multi-machine system [79], on a single machine

[80] and on a multi-machine physical model [81] in the laboratory and on a 400MW thermal machine under fully loaded conditions connected to the system [82].

The single machine power system consists of a synchronous generator connected to a constant voltage bus through two transmission lines (Fig.7.6). A non-linear seventh order model is used to simulate the dynamic behavior of this system. The differential equations used to simulate the synchronous generator and the parameters used in simulation studies are given in [71], [74]. The generator has an IEEE Standard 421.5, Type ST1A AVR and Exciter. An IEEE Standard 421.5, PSS1A Type CPSS [36] is used for comparative studies.

The system output is sampled at the rate of 20 Hz for parameter identification and control computation. Studies performed with various sampling rates show that the performance is practically the same for a sampling rate in the range of 20-100 Hz. Sampling frequencies above 100 Hz are of no practical benefit and the performance deteriorates for sampling rate under 20 Hz. A sampling rate of 20 Hz is chosen to make sure that there is enough time available for updating the parameters and control computation. In most studies, deviation of electrical power output is used as the input to the PSS. The control output is limited to 0.1 pu.

Results of a simulation study to demonstrate the effect of the APSS on the transient stability margin are shown in Table VII. With the single machine infinite bus system initially operating at 0.95 pu power, 0.9 pf lag, a three phase to ground fault was applied near the sending end of one transmission line. It can be seen from Table VII that the APSS provides the largest maximum clearance time.

TABLE VII TRANSIENT STABILITY MARGIN RESULTS

Without PSS With CPSS With APSS

Maximum Clearing Time

120 ms 150 ms 165 ms

This APSS was implemented on a microprocessor and

tested in real-time on a physical model of a single-machine infinite bus system. With the system operating at a stable operating point, the APSS was applied and the torque reference increased gradually to the level, P = 1.307 pu, pf = 0.95 lead, vt = 0.950 pu. At this load, the system was still stable with the APSS.

At 5s, Fig.7.10, the APSS was replaced by the CPSS. After the switch over, the system began to oscillate and diverge, which means that the CPSS is unable to keep the system stable at this load level. At about 25s, the APSS was switched back to control the unstable system and the system came under control very quickly as shown in Fig.7.10. This test demonstrates that the ASPSS can provide a larger dynamic stability margin than the CPSS. Also, more power can be transmitted with the help of the APSS if an overload operation is necessary under certain circumstances.


- 59 -

Figure 7.10 Dynamic Stability Improvement By The APSS

VI. ARTIFICIAL INTELLIGENCE BASED PSS

A. Adaptive PSS With NN Predictor And NN Controller Identification of the power plant model using an on-line

recursive identification technique is a computationally extensive task. Neural networks (NNs) offer the alternative of a model-free method. An adaptive neural network based controller using indirect adaptive control method has been developed. It combines the advantages of neural networks with the good performance of the adaptive control. This controller employs the learning ability of the neural networks in the adaptation process and is trained each sampling period.

The controller consists of two sub-networks as shown in Fig.7.11. One network is an adaptive neuro-identifier (ANI) that identifies the power plant in terms of its internal weights and predicts the dynamic characteristics of the plant. It is based on the inputs and outputs of the plant and does not need the states of the plant. The second sub-network is an adaptive neuro-controller (ANC) that provides the necessary control action to damp the oscillations of the power plant.

The success of the control algorithm depends upon the accuracy of the identifier in predicting the dynamic behaviour of the plant. The ANI and ANC are initially trained off-line over a wide range of operating conditions and a wide spectrum of possible disturbances. After the off-line training stage, the controller is hooked up in the system. Further training of the ANI and ANC is done on-line every sampling period. On-line training enables the controller to track the plant variations as they occur and to provide control signal accordingly.

Employing a feed-forward multi-layer network in each of the two sub-networks, an adaptive neural network based PSS has been built [83]. The two networks are trained further in each sampling period using an on-line version of the back-propagation algorithm. The errors used to train the ANI and ANC are both scalar and the learning is done only once in each sampling period for each of the two sub-networks. This simplifies the training algorithm in terms of the computation time.

System performance in response to a three-phase to ground fault on one circuit of a double-circuit transmission line in a five machine interconnected network is shown in

Fig.7.12. Details of the five-machine system are given in [84]. The adaptive neural network based PSSs were installed on two generators and CPSSs were installed on the other three generators. It can be seen that both the local mode and the inter-area mode oscillations are damped effectively.

Figure 7.11 Controller Structure For Single-Machine Study

Figure 7.12 System Response With NAPS Installed On Generators G1

And G3 And CPSS On G2 , G4 ,G5

B. Adaptive Network Based Fuzzy Logic Controller The characteristics of fuzzy logic and neural networks

complement each other in respect of their pros and cons. That offers the possibility of using a hybrid neuro-fuzzy approach in the form of an adaptive network based Fuzzy Logic controller (FLC) whereby it is possible to take advantage of the positive features of both Fuzzy Logic and neural networks. Such a system can automatically find an appropriate set of rules and membership functions [85].

C. Architecture In the neuro-fuzzy controller, the system is implemented

in the framework of a network architecture. Considering the functional form of the fuzzy logic controller, Fig.7.13, it becomes apparent that the FLC can be represented as a five layer feed-forward network, in which each layer


- 60 -

corresponds to one specific function with the node functions in each layer being of the same type. With this network representation of the fuzzy logic system, it is straightforward to apply the back-propagation or a similar method to adjust the parameters of the membership functions and inference rules.

Figure 7.13 Basic Structure Of Fuzzy Logic Controller

In this network, the links between the nodes from one

layer to the next layer only indicate the direction of flow of signals and part or all of the nodes contain the adjustable parameters. These parameters are specified by the learning algorithm and should be updated according to the given training data and a gradient-based learning procedure to achieve a desired input/output mapping. It can be used as an identifier for non-linear dynamic systems or as a non-linear controller with adjustable parameters.

D. Training And Performance Because the neuro-fuzzy controller has the property of

learning, fuzzy rules and membership functions of the controller can be tuned automatically by the learning algorithm. Learning is based on the error in the controller output. Thus it is necessary to know the error that can be evaluated by comparing the output of the neuro-fuzzy controller and a desired controller.

To train this controller as an adaptive network based fuzzy PSS (ANF PSS), training data was obtained from a self-optimizing pole-shifting APSS. Training was performed over a wide range of operating conditions of the generating unit including various types of disturbances. Based on earlier experience, seven linguistic variables for each input variable were used to get the desired performance.

Extensive simulation [86] and experimental studies with the ANF PSS show that it can provide good performance over a wide operating range and can significantly improve the dynamic performance of the system over that with a fixed parameter CPSS.

E. Self-Learning ANF PSS In the above case the ANF PSS was trained by data

obtained from a desired controller. However, in a general situation, the desired controller may not be available. In that case, the neuro-fuzzy controller can be trained using a self-learning approach [87].

In the self-learning approach two neuro-fuzzy systems

are used in a manner similar to Fig.7.11, one acting as the controller and the other acting as the predictor. The plant identifier can compute the derivative of the plant’s output with respect to the plant’s input by means of the back-propagation process illustrated by the line passing through the forward identifier and continuing back through the neuro-fuzzy controller that uses it to learn the control rule.

The self-learning ANF PSS was initially trained off-line on a power system simulation model over a wide range of operating conditions and disturbances. Electric power deviation and its integral were used as the input to the stabilizer. The ANF PSS, with the parameters, membership functions and inference rules obtained from the off-line training procedure, was implemented on a DSP mounted on a PC and its performance was evaluated on a physical model of a power system in the laboratory. A digital CPSS was also implemented in the same environment on the DSP board for comparative studies.

Out of the various tests, results for a 0.25 pu step decrease in the input torque reference applied at 1s and removed at 9s with the generator operating at 0.9 pu power, 0.85 pf lag and 1.10 pu Vt are shown in Fig.7.14. The ANF PSS provides a consistently good performance for either of the two disturbances.

Simulation studies on a single machine connected to a constant voltage bus and on a multi-machine power system [80] and experimental studies on a physical model of a power system have demonstrated the effectiveness of the ANN PSS in improving the performance of a power system over a wide operating range and a broad spectrum of disturbances.

Figure 7.14. Comparison Of ANF PSS and CPSS Responses to a

0.25 p.u. Step Torque Disturbance (P= 0.9 p.u., 0.85 PF Lag)

F. Neuro-Fuzzy Controller Architecture Optimization Adaptive fuzzy systems offer a potential solution to the

knowledge elicitation problem. The controller structure, expressed in terms of the number of membership functions and the number of inference rules, is usually derived by trial and error. The number of inference rules has to be determined from the standpoint of overall learning capability and generalization capability.

The above problem can be resolved by employing a


- 61 -

genetic algorithm to determine the structure of the adaptive fuzzy controller. By employing both genetic algorithm and adaptive fuzzy controller, the inference rules parameters can be tuned and the number of membership functions can be optimized at the same time.

VII. AMALGAMATED ANALYTICAL AND AI BASED PSS

A. Adaptive PSS With NN Identifier And Pole-Shift Control

A self-tuning APSS described above can improve the dynamic performance of the synchronous generator by allowing the parameters of the PSS to adjust as the operating conditions change. However, proper care needs to be taken in the design of the RLS algorithm for identification to make it stable, especially under large disturbances.

It is possible to make the identification more robust by using a NN for identifying the system model parameters. An analytical technique, such as the pole-shift control, can be retained to compute the control signal. One approach, using a radial basis function (RBF) network for model parameter identification, is described below. The PSS shown in Fig.7.4 now consists of an ANN identifier and the pole-shifting control algorithm described above.

The RBF network, Fig.7.15, is used to identify the system model parameters, ai, bi, (7.13) and (7.14). The network consists of three layers: the input, hidden and output layers. The input vector is:

)]Tt(u),Tt(u),Tt(u),Tt(P

),Tt(P),Tt(P[)t(V

e

ee

323

2

−−−−∆

−∆−∆= (7.15)

Figure 7.15 Radial Basis Function Network Model

Each of the six input variables is assigned to an

individual node in the input layer and passes directly to the hidden layer without weights. The hidden nodes, called the RBF centers, calculate the Euclidean distance between the centers and the network input vector. The result is passed through a widely used Gaussian function characterized by a response which has a maximum value of 1 when the distance between the input vector and the center is zero. Thus a radial basis neuron acts as a detector which produces 1 whenever the input vector is identical to the

center (active neuron). The other neurons with centers quite different from the input vector will have outputs near zero (non-active neurons).

The connections between the hidden neurons and the output node are linear weighted sums as described by the equation:

∑=

σ

−−θ′=

nh

i

icp.expy

12

2

(7.16)

where ci, σ, θ′ and nh are the centers, widths, weights

and the number of hidden layer neurons, respectively. To make the proposed RBF identifier faster for on-line

applications, the hidden layer is created as a competitive layer wherein the center closest to the input vector becomes the winner and all the other non-active centers are deactivated. Also, the scalar weights are modified as a vector θ′ whose size equals the size of the input vector. The weight vector is given by:

[ ]'''''' bbbaaa)t( 321321=θ′ (7.17)

Linearizing the output of the RBF, y(t) = f [y(t-1), u(t-

1)], by Taylor series expansion at each sampling instant, a one-to-one relationship between the weight vector θ′ and the system model parameters )t(ˆ

mθ (equation 7.6) can be obtained. These parameters are then used in computing the control signal.

The RBF identifier was first trained off-line to choose appropriate centers using data collected at a number of operating points for various disturbances. The n-means clustering algorithm used for training yielded 15 centers for the RBF model. After the off-line training, the weights (system parameters) were updated on-line to obtain the appropriate control signal using the pole-shifting controller. A 100ms sampling period was chosen for digital implementation.

Results of an experimental study for a 0.10 pu decrease in torque reference applied at 10s and removed at 20s, with the generator operating at 0.6 pu power, 0.92 pf lead and Vt of 0.99 pu are shown in Fig.7.16. It can be seen that the APSS can provide a well-damped response.


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Figure 7.16 ∆Pe Response For 0.1 p.u. Input Torque Reference Step

Change With APSS

B. Adaptive PSS with Fuzzy Logic Identifier and Pole-Shift controller

Takagi-Sugeno (TS) fuzzy systems have been successfully employed in the design of stabilization control of non-linear systems.

A non-linear plant can be represented by a set of linear models interpolated by membership functions of a TS fuzzy model. Although the TS system identifier is a NARMAX model, at each sample an average linear discrete ARMA model can be determined to identify the controlled plant according to the current active rules. This ARMA model can be used to determine the control signal by the pole-shifting control strategy. Using this approach, a self-tuning adaptive controller has been developed and applied as a PSS [88].

The proposed single-input single-output TS model used for the identification of dynamic systems is composed of fuzzy rules , the consequent part of which provides the rule output at time k based on the past inputs and past outputs with fuzzy sets designed in universe of discourse. The consequent part of the rule then identifies the parameters of a desired order discrete model of the plant. Two parallel on-line learning procedures, one each for the identification of premise and consequent parameters, are used to track the plant in real-time [89].

In the proposed TS system for generating unit identification, two input signals, the past control input, u(k-1), and the past generator speed output, y(k-1), are used to identify a 3rd. order model of the plant. The output at sample k is the estimated generator speed output, ŷ(k). The TS system is trained by using the steepest descent algorithm for the premise parameters and RLS algorithm for the consequent parameters using the error of the system output and the estimated TS output. Initially a set of three equally spaced membership functions, over the normalized universe, are used for the inputs of the system.

The response of the system with the TS system based identifier and pole-shift controller based APSS has been studied for various disturbances at different operating conditions. One illustrative result for a three phase to ground fault is shown in Fig.7.17.

Figure 7.17 Three Phase To Ground Fault At The Middle Of One

Transmission Line And Successful Reclosure (P=0.95 pu, 0.9 PF Lag)

C. Adaptive PSS With RLS Identifier And Fuzzy Logic Control

Fuzzy logic controllers (FLCs) have attracted considerable attention as candidates for novel computational systems because of the advantages they offer over the conventional computational systems. They have been successfully applied to the control of non-linear dynamic systems, especially in the field of adaptive control, by making use of on-line training.

A self-learning adaptive fuzzy logic controller has been developed. Only the inputs and outputs of the plant are measured and there is no need to determine the states of the plant. Using on-line training by the steepest descent method and the identified system model, the adaptive FLC is able to track the plant variations as they occur and compute the control.

In the proposed controller, a discrete model of the plant is first identified using the RLS parameter identification method. This allows a continuous tracking of the system behavior.

The control learning is based on the prediction of the identified model. The identified model output is used as input to the Mamdani–type PD controller [72]. The center points of the controller inputs are updated [89] by treating them exactly the same as the weights of an NN and by using the steepest descent algorithm with chain rule.

The proposed adaptive FLC has been applied as an adaptive fuzzy PSS (AFPSS) [90]. For the AFPSS, the generating unit is identified as a 3rd. order model. The controller has two input signals, the generator speed deviation and its derivative, with an initial set of seven equally spaced membership functions over the normalized universe of discourse. The output, the supplementary control signal, also having seven membership functions, is added to the AVR summing junction. A number of simulation studies have been performed for various disturbances at different operating conditions. An illustrative result for a 0.05 pu increase in torque and return to initial condition, shown in Fig.7.18, demonstrates the performance of this AFPSS.


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0 1 2 3 4 5 6 70.62

0.63

0.64

0.65

0.66

0.67

0.68

0.69

time [s]

Pow

er A

ngle

[rad

]

APSSCPSSNo PSS

Figure 7.18 Response To A 0.05 p.u. Step Increase In Torque And Return

To Initial Condition (P= 0.95 p.u., 0.9 PF Lag)

VIII. MULTI BAND PSS The multi band PSS consists of robust classical 3-band

pass parallel filter structure to enable optimum tuning for all oscillation modes. The design has been included in the IEEE standard 421.5 2005 with designation PSS4B and it already been installed as a commercial product by Hydro Québec and ABB.

The simplified structure of the MB-PSS is represented in Figure 7.19. It uses the calculated shaft speed deviation obtained from two transducers. The stabilizing network consists of three sets of band-pass filters for three different frequency ranges. The filters are tuned for low frequency, intermediate frequency and high frequency.

To build the stabilizing signal, the MB-PSS uses the calculated speed deviation and active power. These quantities are calculated from the primary electrical machine quantities (stator voltages and currents). Figure 7.20, shows the representation of the speed deviation transducers that are required to feed the three band structure used as lead-lag compensation as shown in Figure 7.21.

Two tunable notch filters are provided for applications in turbo generating units and are mainly intended for selective rejection of the first and second torsion oscillation modes.

As shown in figure 7.21, each band-pass filter is formed by the two branches of symmetrical lead-lag filters that contain three lead-lag elements and three individual gain factors. This simple and robust filter structure provides a remarkable tuning flexibility.

The low band is taking care of very slow oscillating phenomena such as common modes found on isolated system. The intermediate band is used for inter-area modes usually found in the range of 0.2 to 0.8 Hz. The high band is dealing with inter-machine/plant and local modes.

VH MIN

VSTMIN

VSTMAX

VIMIN

VLMIN

VH MAX

VST

∆ω LI

∆ωH

∆ω SpeedTransducers

KIVIMAX

VLMAX

FI

FH

KH

FL

KL

Σ+

+

+

VH MIN

VSTMIN

VSTMAX

VIMIN

VLMIN

VH MAX

VST

∆ω LI

∆ωH

∆ω SpeedTransducers

KIVIMAX

VLMAX

FI

FH

KH

FL

KL

ΣΣ+

+

+

Figure 7.19 Multiband PSS Algorithm Structure

Figure 7.20 Multiband Speed Deviation Transducers

+ +

+

∆ω∆ω∆ω∆ω LI

∆ω∆ω∆ω∆ω H

VHmax

VHmin

VHmax

VHmin

VLmax

VLmin

VLmax

VLmin

ΣΣΣΣ

KL

KL2KL17 + sTL7

1+ sTL8

1+ sTL2

KL11 + sTL1KL11+ sTL3 +

-ΣΣΣΣ

1+ sTL4

1+ sTL51+ sTL6

1+ sTL9

1+ sTL10

1+ sTL11

1+ sTL12

KL

KL2KL17 + sTL7

1+ sTL8

1+ sTL2

KL11 + sTL1KL11+ sTL3 +

-ΣΣΣΣ

1+ sTL4

1+ sTL51+ sTL6

1+ sTL9

1+ sTL10

1+ sTL11

1+ sTL12

VImax

VImin

VImax

VImin

VSTmax

VSTmin

VSTmax

VSTmin

KI

KI2KI17 + sTI7

1+ sTI8

1+ sTI2

KI11 + sTI1KI11+ sTI3 +

-ΣΣΣΣ

1+ sTI4

1+ sTI51+ sTI6

1+ sTI9

1+ sTI10

1+ sTI11

1+ sTI12

KI

KI2KI17 + sTI7

1+ sTI8

1+ sTI2

KI11 + sTI1KI11+ sTI3 +

-ΣΣΣΣ

1+ sTI4

1+ sTI51+ sTI6

1+ sTI9

1+ sTI10

1+ sTI11

1+ sTI12

KH

KH2KL17 + sTH7

1+ sTH8

1+ sTH2

KH11 + sTH1KH11+ sTH3 +

-ΣΣΣΣ

1+ sTH4

1+ sTH51+ sTH6

1+ sTH9

1+ sTH10

1+ sTH11

1+ sTH12

KH

KH2KL17 + sTH7

1+ sTH8

1+ sTH2


-ΣΣΣΣ

1+ sTH4

1+ sTH51+ sTH6

1+ sTH9

1+ sTH10

1+ sTH11

1+ sTH12

Figure 7.21 Multiband PSS according to IEEE 421.5 2005 PSS4B

The lead-lag filter branches can be adjusted in many

different ways upon the specific application needs. Figure 7.22 shows in a detailed view of the high frequency band pass filter.

∆ω∆ω∆ω∆ω H

KH

KH2KL17 + sTH7

1+ sTH8

1+ sTH2


-ΣΣΣΣ

1+ sTH4

1+ sTH51+ sTH6

1+ sTH9

1+ sTH10

1+ sTH11

1+ sTH12

KH

KH2KL17 + sTH7

1+ sTH8

1+ sTH2


-ΣΣΣΣ

1+ sTH4

1+ sTH51+ sTH6

1+ sTH9

1+ sTH10

1+ sTH11

1+ sTH12


- 64 -

Figure 7.22 The high frequency band differential filter Each band pass filter branch has an independent output

limitation for the individual influence restriction according to the needs stabilizing in the three different frequency ranges. The resulting stabilizing signal is formed by the summation of all three filters and the maximum and minimum amplitudes of stabilizing signal can be limited as well by individual and adjustable maximum and minimum adjustable limitation parameters.

A. Tuning Methodology Each branch of a differential filter was designed to

provide similar flexibility to a conventional PSS. Taking the figure 7.21 as base and concentrating just in the high frequency filter branch, it has two lead-lag blocks and an hybrid block that either can be used as washing-out when KH11= K17=0 or for lead-lagging when KH11=KH17=1. With such an arrangement, one can use a single branch (the positive or negative) to tune the PSS similar to the usual PSS.

As published in [92], [93] there are different ways to tune the parameters of MB-PSS, however, the simplest and most natural tuning strategy of the differential band filters is the symmetrical approach as presented in [92]. With this approach, it is possible to set the PSS with only two high-level parameters and the whole lead-lag compensation circuit is defined by six parameters. They are the three filter central frequencies FL, FI, FH and gains KL, KI, KH.

The time constants and gains are derived from simple equations as shown in [92] and on annex H21 of [36].

B. Application Experience Beside the several laborious theoretical investigations,

Hydro Québec also is carrying out extensive field testing of the MB-PSS equipment. The obtained field results were also compared with other conventional PSS technologies and published in [92] and [93]. The example of figure 7.23 shows 50MVAR step in closed loop (applied to one out of sixteen generators) in the La Grande 2 (LG2) hydro power station.

Figure 7.23a is showing the simulation results and figure 7.23b the results obtained during field testing. The blue and red curves represent the signals (active power deviation, PSS output signal and speed deviation) obtained with two different gain levels of MB-PSS (high and low gain). The black curves correspond to the response to the original conventional PSS installed prior to the inclusion of MB-PSS. This practical case illustrates the superior performance of MB-PSS when compared with the original PSS equipment.

For the same LG2 HPS, Figure 7.24 shows a MW step response for which both global (0.05Hz) and inter area (0.5Hz) oscillation modes are excited.

Figure 7.23a MVAr step response in LG2 HPS-simulation

Figure 7.23b MVAr step response in LG2 HPS-field test

The system response for the new MB-PSS (red) and

original PSS (blue) are being compared. Interestingly, it may be observed that the MB-PSS response to the global mode is such that it significantly contributes to a reduction of the plant frequency deviation by decreasing its generator output voltage in accordance with the decreasing speed. On the other hand, the original PSS, with its inherent transfer function limitations, can not be properly tuned at this mode and reacts in a reverse way. The resulting frequency deviation is 0.12 Hz instead of 0.02 Hz with the MB-PSS.

0.5 1 1.5 2 2.5

−10

0

10

MW

De

via

tion

+50MVARS STEP

0.5 1 1.5 2 2.5

−2

−1

0P

SS

Sig

na

l(%

)

PSS4B−Normal Gain (Test #7d16)PSS4B−High Gain(Test #17d16)Existing PSS (Test #5d16)

0.5 1 1.5 2 2.5

−0.04

−0.02

0

0.02

Sp

ee

d D

evi

atio

n(H

z)

Time in seconds

0.5 1 1.5 2 2.5

−10

0

10

MW

De

via

tion

0.5 1 1.5 2 2.5

−2

−1

0

PS

S S

ign

al(%

)

0.5 1 1.5 2 2.5

−0.04

−0.02

0

0.02

Sp

ee

d

De

via

tion

(Hz)

Time in seconds

PSS4B(Normal Gain)PSS4B(High Gain)Existing PSS


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0 5 1 0 1 5- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0

MW

Dev

iatio

n

M W S TE P - B b lu e: A c tu a l L G 2 P S S ; Re d : M B -P S S ( No r m al G ain )

M B - P S S in c los e d -lo o p ( Tes t #1 0 d1 6 )C on ven ti o na l P S S in c lo s e d- lo op ( Tes t #4 d 1 6)

0 5 1 0 1 5

- 2

- 1 .5

- 1

- 0 .5

0

0 .5

1

PSS

Sign

al(%

)

0 5 1 0 1 5- 0 .1 2

- 0 .1

- 0 .0 8

- 0 .0 6

- 0 .0 4

- 0 .0 2

0

Spee

d D

evia

tion(

Hz)

0 5 1 0 1 5

- 6 0

- 4 0

- 2 0

0

2 0

MVA

R D

evia

tion

s e c

Figure 7.24 MW step response test in closed loop at LG2

Figure 7.24 shows the results of a 1% AVR setpoint

pulse done in the unit 3 of Outardes HPS. The tests were carried out at power level of 200MW. Figure 7.23a shows the system response without PSS and figure 7.23b with MB-PSS. Also in this case a significant damping improvement has been achieved.

Tests have also been carried out in Gentilly Nuclear Power Station in order to verify the influence of torsional modes on MB-PSS signal. Figure 7.26 shows for an open loop test the raw speed signal including torsional oscillation modes, the filtered speed signal after the notch filters and the MB-PSS output signal compared with the conventional PSS (values multiplied by 10). From these measurements, it is possible to verify how effective the notch filters are suppressing the turbo-machine 9.95 Hz and 17.82 Hz main torsional modes while not affecting the MB-PSS.

Figure 7.25a 1% setpoint pulse test with MB-PSS out of

service

Fig 7.25b 1% setpoint pulse test with MB-PSS in service

Fig 7.26 Measurement of torsional modes on PSS signal

C. Multiband PSS Conclusions After years of theoretical investigations [91] - [93],

equipment development and extensive field testing, the MB-PSS is proving its efficacy and flexibility for all kind of electromechanical oscillation modes. Due to its robust and flexible structure it can be tuned in order to attend all specific power system needs on damping enhancement.

The notch filters are able to suppress any kind of undesired influence of torsional modes on turbo units and provides a very selective stabilizing signal in the frequency range of interest.

Since 2005 the MB-PSS is included in the IEEE 421.5 2005 and named as PSS-4B and most commercial power system analysis software already includes the PSS-4B model.

Due to its properties and characteristics, the MB-PSS

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

- 0 .0 5

0

0 .05

0 .1

0 .15

0 .2

Dev

iatio

n (H

z)

G e n ti lly 2 e ve nt r e c o r d e d o n J uly 1 8 , 2 0 0 0 1 2 : 1 8 d ur in g o p en - lo o p te s tin g o f th e M B - P S S

R a w S p e e d E s tim a te s

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

0

0 .05

0 .1

0 .15D

evia

tion

(Hz)

T un e d -N o tc h F i lte r ed S p e e df n 1= 9 .9 5 Hz a nd f n2 = 17 .8 2 H z

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

- 0 .0 1

0

0 .01

0 .02

0 .03

0 .04

0 .05

0 .06

PSS

(p.u

.)

T im e (s e c )

N o m ina l G a in M B - P S S1 0 x C o n ve n tion a l P S S


- 66 -

will certainly play in important role on the world wide on going restructuring of power systems and energy markets.

IX. CONCLUDING REMARKS Power system stabilizers based on the control algorithms

described above have been studied extensively in simulation. They have also been implemented and tested in real-time on physical models in the laboratory with very encouraging results. The pole-shifting control algorithm based adaptive PSS has also been tested on a multi-machine physical model [81], on a 400 MW thermal machine under fully loaded conditions connected to the system [82], and is now in regular service in a hydro power station after extensive testing in the field [97]. These studies have shown clearly the advantages of the advanced control techniques and intelligent systems.

Very satisfactory adaptive controllers can be developed and implemented using a number of approaches, i.e. purely analytical, purely AI techniques or by amalgamating the analytical and AI approaches. Which approach to use depends upon the expertise of the designer and the developer of the controller, and the confidence that they or the client have in a particular technology.


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CHAPTER 8 REFERENCES

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[2] Robert A. Gabel, Richard A. Roberts, "Signals And Linear Systems," John Wiley & Sons.

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CHAPTER 9 BIOGRAPHIES

J. C. Agee received his BSEE degree from Rose-Hulman Institute of Technology in 1979. Upon graduation, he joined the Bureau of Reclamation as a power systems engineer. He is currently employed in the Hydroelectric Research and Technical Services Group as a technical specialist in the fields of

governor control, excitation control, and power system stability. His research interests include feedback control systems, digital control, and power system stability analysis. He is a senior member of IEEE, past chair of the Excitation Systems Subcommittee, and is currently chair of the IEEE Energy Development & Power Generation Committee.

Roger Beaulieu graduated from the University of Waterloo in Ontario Canada, with a B.A.Sc, in 1967. After a 26-year career at Ontario Hydro, in the areas of power system protection, stabilizers and power system modelling and testing, he retired to the life of an adviser to utilities and electrical

equipment manufacturers. He is a senior engineer with Goldfinch Power Engineering.

Roger Bérubé received his Bachelor and Masters of Engineering degrees from McGill University in Montreal Canada 1981 and 1982 respectively. He worked for Ontario Hydro between 1982 and 2000 in various roles involving the design, testing and simulation of generator control systems.

Since 2000 he has been working as Senior Engineer with Kestrel Power Engineering Ltd. a leading firm in the area of generator control consulting and regulator compliance testing. He is a member of the IEEE Power Engineering Society and a participant in the Excitation Systems Sub-Committee and associated Working Groups.

Dr. George E. Boukarim received the B.S., M.E., and Ph.D. degrees in electric power engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1987, 1988, and 1998, respectively. From 1988 to 1994 and from 1998 to the present he was with the Power Systems Energy Consulting

(PSEC) Department of the General Electric Company, Schenectady, NY, working in the area of power system

dynamics and control. He also worked for the Transmission Technology Institute, ABB Power T&D Company, Raleigh, NC, for a short period from 1997 to 1998. His interests include robust multivariable control, and power system dynamics and control.

Murray Coultes has an electrical engineering degree from the University of Western Ontario and an MBA from the University of Toronto. He worked at Ontario Hydro Research on the early development of speed-based power system stabilizers and frequency response testing for generator models.

He then moved to Ontario Hydro Operations and retired in 1993 as the System Security Manager. Since then, he has been a senior engineer at Goldfinch Power Engineering where he works on excitation system testing and modeling and commissioning power system stabilizers.

Robert Grondin received his B.Sc.A. in Electrical Engineering from University of Sherbrooke, Canada in 1976 and his M.Sc.from INRS Energie, Varennes, Canada in 1979. He then joined Hydro-Québec research institute, IREQ. Currently a Senior Research Engineer in the Power

System Analysis, Operation and Control Department, he is leading research activities in the field of power system dynamics and defense plans. Member of IEEE Power Engineering society and of CIGRÉ, he is also a registered professional engineer in province of Québec, Canada

Professor Arjun Godhwani is an emeritus professor of Electrical Engineering at Southern Illinois University Edwardsville. He has been a member of IEEE Excitation Systems Subcommittee for over 10 years. He actively consults in the field of excitation systems.

Les Hajagos received his B.A.Sc. in 1985 and his M.A.Sc. in 1987 from the University of Toronto. Since 1988 he has worked mainly in the analysis, design, testing and modeling of generator, turbine and power system control equipment and power system

loads, first at Ontario Hydro, and since 2000 as one of the principals at Kestrel Power Engineering. He is a registered Professional Engineer in the Province of Ontario and an active member of the IEEE Power Engineering Society as chair of the Generator Model Validation and Excitation System Modeling task forces.


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Dr. Innocent Kamwa received a PhD in electrical engineering from Laval University, Québec, Canada, 1988, after graduating in 1984 at the same university. Since then, he has been with the Hydro-Québec Research Institute, where he is at present a Principal Researcher with interests broadly in bulk system dynamic

performance. Since 1990, he has held an associate professor position in Electrical Engineering at Laval University. A member of CIGRÉ, Dr. Kamwa is a recipient of the 1998 and 2003 IEEE PES Prize Paper Awards and is currently serving on the System Dynamic Performance Committee AdCom. He is also the acting Standard Coordinator of the PES Electric Machinery Committee.

Professor Om P. Malik has done pioneering work in the development of adaptive and artificial intelligence based controllers for application in electric power systems over the past thirty years. After extensive testing in the laboratory and in actual power systems, these controllers are now employed on large generating units.

Professor Malik graduated in 1952 from Delhi Polytechnic, India. After working for nine years in electric utilities in India, he returned to academia and obtained a Master’s Degree from Roorkee University, India in 1962, a Ph.D. from London University and a DIC from the Imperial College, London in 1965.

He was teaching and doing research in Canada from 1966 to 1997 and continues to do research as Professor Emeritus at the University of Calgary, Canada.

Professor Malik is a Fellow of the Engineering Institute of Canada, Canadian Academy of Engineering, Institution of Electrical Engineers, World Innovation Foundation and a Life Fellow of IEEE. He has received many awards from IEEE, EIC and APPEGA and the University of Calgary. He is a member of the Association of Professional Engineers, Geologists and Geophysicists of Alberta and Professional Engineers Ontario. Professor Malik is also actively involved in IFAC and is currently Chair of the IFAC Technical Committee on Power Plants and Power Systems Control.

Dr. Alexander Murdoch is received the B.S.E.E. degree from Worcester Polytechnic Institute, Worcester, MA, in 1970, and the M.S.E.E. and Ph.D. degrees from Purdue University, West Lafayette, IN, in 1972 and 1975, respectively. Currently, he is a Consulting Engineer with General

Electric, working in Energy Consulting, Schenectady, NY. He joined General Electric in 1975 and has worked primarily at Energy Consulting but also for Drive System, Salem, VA. His areas of interest include rotating machine modeling, excitation system design and testing, and advanced control theory. Dr. Murdoch is a member of the Excitation System Subcommittee in the IEEE.

Shawn Patterson received his BS and MS degrees in Electrical Engineering from the University of Colorado in 1985 and 1995. He works for the Bureau of Reclamation specializing in power system stability, computer modeling, and technical studies. He is

a registered Professional Engineer in the state of Colorado and an active member of the WECC Modeling and Validation Work Group and IEEE PES working groups.

José Taborda was born in Brazil 1961. He received the degree on electrical engineering from the University of São Paulo in 1985. Since 1985 he has been working for ABB where he occupied several positions starting from development,

commissioning, design, product management, technical sales support and electrical studies. His speciality is control systems and power electronics applied to electrical machines. José Taborda is IEEE member and currently working as senior system consultant of Excitation System Group in ABB Switzerland Ltd.

Robert Thornton-Jones received his BEng. Degree in Electrical and Electronic Engineering in 1986 from the University of Bradford, United Kingdom, after which he joined Brush Electrical Machines Ltd. He specializes in generator excitation systems and power management systems for

industrial power generation projects providing load shedding, automatic generator dispatch, power and power factor control. Robert is a Chartered Engineer in the United Kingdom, a member of the IET, a member of the IEEE Power Engineering Society and a participant in the Excitation Systems Sub-Committee and associated Working Groups.


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Gilles Trudel received his B.Sc.A. (1978) and M.Eng (1986) in Electrical Engineering from École Polytechnique Université de Montréal, Canada. In 1978, he joined Hydro-Québec, where he was first involved in the design of control and protection for substations. Eight years later, he moved to the

System Planning department where he is now involved in high-voltage network planning and design of special protection systems.Trudel is a member of the IEEE Power Engineering Society and a registered professional engineer in the province of Québec.

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