Chapter 12 Small-Signal Stability

8/17/2019 Chapter 12 Small-Signal Stability

1/128

small Signal Stability

Chapter provided a general introduction to the power system stability

problem, including a discussion of the basic concepts, classification, and definitions

of related terms. We will now consider in detail the various categories of system

stability, beginning with this chapter on small-signal stability. Knowledge of the

characteristics and modelling of individual system components as presented in

Chapters to 11 should be helpful in this regard.

Small-signal stability, as defined in Chapter

2

is the ability of the power

system to maintain synchronism when subjected to small disturbances. In this context,

a disturbance is considered to be small if the equations that describe the resulting

response of the system may be linearized for the purpose of analysis. Instability that

may result can be of two forms: (i) steady increase in generator rotor angle due to

lack of synchronizing torque, or (ii) rotor oscillations of increasing amplitude due to

lack of sufficient damping torque. In today s practical power systems, the small-signal

stability problem is usually one of insufficient damping of system oscillations. Small-

signal analysis using linear techniques provides valuable information about the

inherent dynamic characteristics of the power system and assists in its design.

This chapter reviews fundamental aspects of stability of dynamic systems,

presents analytical techniques useful in the study of small-signal stability, illustrates

the characteristics of small-signal stability problems, and identifies factors influencing

them.


2/128

7

Small Signal Stabi l i ty Chap 1

12 1 FUNDA MENTA L CONCEPTS OF STABILITY

OF

DYN AM IC SYSTEMS

12 1 State Space Representation

The behaviour of a dynamic system such as a power system may be described

by a set of n first order nonlinear ordinary differential equations of the following

form:

where

n

is the order of the system and r is the number of inputs. This can be written

in the following form by using vector-matrix notation:

where

The column vector is referred to as the state vector and its entries

x

as

state

variables. The column vector

u

is the vector of inputs to the system. These are the

external signals that influence the performance of the system. Time is denoted

by

t

and the derivative of a state variable with respect to time is denoted by 1

f

the

derivatives of the state variables are not explicit functions of time the system is said

to be autonomous. In this case Equation

12 2

simplifies to

We are often interested in output variables which can be observed on the

system. These may be expressed in terms of the state variables and the input variables

in the following form:

where


3/128

Set 12 1 Fundame nta l Concepts o f Stab i l ity o f D ynam ic Systems

701

The column vector is the vector of outputs, and is a vector of nonlinear functions

,,lating state and input variables to output variables.

he concept of state

The concept of state is fundamental to the state-space approach. The state of

a

system represents the minimum amount of information about the system at any

instant in time

to

that is necessary so that its future behaviour can be determined

reference to the input before

to.

Any set of n linearly independent system variables may be used to describe the

state of the system. These are referred to as the

state variables;

they form a minimal

set

of

dynamic variables that, along with the inputs to the system, provide a complete

description of the system behaviour. Any other system variables may be determined

from a knowledge of the state.

The state variables may be physical quantities in a system such as angle, speed,

voltage, or they may be abstract mathematical variables associated with the differential

equations describing the dynamics of the system. The choice of the state variables is

not unique. This does not mean that the state of the system at any time is not unique;

only that the means of representing the state information is not unique. Any set of

state variables we may choose will provide the same information about the system.

f we overspecify the system by defining too many state variables, not all of them will

be independent.

The system state may be represented in an n-dimensional Euclidean space

called the

state space.

When we select a different set of state variables to describe the

system, we are in effect choosing a different coordinate system.

Whenever the system is not in equilibrium or whenever the input is non-zero,

the system state will change with time. The set of points traced by the system state

in the state space as the system moves is called the

state trajectory

Equilibrium or singular) points

The equilibrium points are those points where all the derivatives

21 d2

n

are simultaneously zero; they define the points on the trajectory with zero velocity.

The system is accordingly at rest since all the variables are constant and unvarying

with time.

The equilibrium or singular point must therefore satisfy the equation

where xo is the state vector at the equilibrium point.

If the functions f; i=1,2,

...

n) in Equation 12.3 are linear, then the system is

linear. A linear system has only one equilibrium state if the system matrix is non-

singular). For a nonlinear system there may be more than one equilibrium point.


4/128

Small Signal Stability

Chap

The singular points are truly characteristic of the behaviour of the dynamic

system, and therefore we can draw conclusions about stability from their nature

12 1 2

Stability

of

a Dynamic System

The stability of a linear system is entirely independent of the input,

and

state of a stable system with zero input will always return to the origin of the state

space, independent of the finite initial state.

In contrast, the stability of a nonlinear system depends on the type

and

magnitude of input, and the initial state. These factors have to be taken into account

in defining the stability of a nonlinear system.

In control system theory, it is common practice to classify the stability of

nonlinear system into the following categories, depending on the region of state space

in which the state vector ranges:

Local stability or stability in the small

Finite stability

Global stability or stability in the large

Local stability

The system is said to be loc lly st ble about an equilibrium point if, when

subjected to small perturbation, it remains within a small region surrounding the

equilibrium point.

If as

t

increases, the system returns to the original state, it is said to

be

symptotic lly st ble in the small.

It should be noted that the general definition of local stability does not require

that the state return to the original state and, therefore, includes small limit cycles. In

practice, we are normally interested in asymptotic stability.

Local stability (i.e., stability under small disturbance) conditions can be studied

by linearizing the nonlinear system equations about the equilibrium point in question.

This is illustrated in the next section.

inite stability

If the state of a system remains within a finite region R, it is said to be stable

within R. If, further, the state of the system returns to the original equilibrium point

from any point within R it is asymptotically stable within the finite region R.

Global stability

The system is said to be glob lly st ble if

R

includes the entire finite space.


5/128

Set-

12 1

Fundamenta l Concep ts of Stab i l i t y o f Dynamic Systems 703

2 1.3

inearization

We now describe the procedure for linearizing Equation 12.3. Let xo be the

initial state vector and uo the input vector corresponding to the equilibrium point

about

which the small-signal performance is to be investigated. Since xo and

u,

satisfy

Equation 12.3, we have

Let us perturb the system from the above state, by letting

where the prefix denotes

a

small deviation.

The new state must satisfy Equation 12.3. Hence,

AS the perturbations are assumed to be small, the nonlinear functions f(x,u) can be

expressed in terms of Taylor s series expansion. With terms involving second and

higher order powers of

x

and

u

neglected, we may write

Since

o

f xo u,) we obtain

with i=1,2, .. n In a like manner, from Equation 12.4, we have


6/128


7/128


8/128

Sm all Signal

Stability Chap. 1

12 1 4 nalysis of Stability

Lyapunov s first method [ I ]

The stability in the sma of a nonlinear system is given by the roots of the

characteristic equation of the system of first approximations, i.e., by the eigenvalues

of A:

(i)

When the eigenvalues have negative real parts, the original system is

asymptotically stable.

(ii)

When at least one of the eigenvalues has a positive real part, the original

system is unstable.

(iii)

When the eigenvalues have real parts equal to zero, it is not possible on the

basis of the first approximation to say anything in the general.

The stability in the large may be studied by explicit solution of the nonlinear

differential equations using digital or analog computers.

A method that does not require explicit solution of system differential

equations is the direct method of Lyapunov.

Lyapunov s second method, or the direct method

The second method attempts to determine stability directly by using suitable

functions which are defined in the state space. The sign of the Lyapunov function and

the sign of its time derivative

with respect to the system state equations are

considered.

The equilibrium of Equation 12.3 is

stable

if there exists a positive definite

function

V x , ,: ... n ) such that its total derivative

v

with respect to Equation 12.3

is not positive.

The equilibrium of Equation 12.3 is

asymptotically stable

if there is a positive

definite function V xl

, , n)

such that its total derivative with respect to Equation

12.3 is negative definite.

The system is stable in that region in which

v

is negative semidefinite, and

asymptotically stable if v is negative definite.2

The stability in the large of power systems is the subject of the next chapter.

This chapter is concerned with the stability in the small of power systems,

nd

this

is given by the eigenvalues of

A

As illustrated in the following section, the natural

A

function is called

deflnite

in a domain

D

of state space if it has the same sign

for

ll

x

withinD and vanishes for

x=O

For example, V x,, x,, x3) =x: + +x: is positive definite.

A function is called semidefinite in a domain

D

of the state space if it has the same sign

or is zero for all

x

withinD For example, V x, 3)

=

x, -x212 +x: is positive semi-definite

since

it

is zero for x, =x, x,

O


9/128

Set

1

2.2

Eigenproper ties of the State M atr ix

707

of system response are related to the eigenvalues. Analysis of the

,igenproperties of A provides valuable information regarding the stability

characteristics of the system.

It is worth recalling that the matrix is the Jacobian matrix whose elements

are given by the partial derivatives af i lx evaluated at the equilibrium point about

Ghich the small disturbance is being analyzed. This matrix is commonly referred to

the

state matrix

or the

plant matrix.

The term plant originates from the area of

process control and is entrenched in control engineering vocabulary. It represents that

art of the system which is to be controlled.

12 2EIGENPROPERTIES OF THE S TA TE M A TR IX

12 2 1 Eigenvalues

The eigenvalues of a matrix are given by the values of the scalar parameter h

for which there exist non-trivial solutions (i.e., other than =0) to the equation

A +

= A

(12.16)

where

A

is an nxn matrix (real for a physical system such as a power system)

is

n

nx 1 vector

To find the eigenvalues, Equation 12.16 may be written in the form

For non-trivial solution

det A-AI) =

Expansion of the determinant gives the characteristic equation. The solutions of

h=A

h .. h,, are eigenvalues of A

The eigenvalues may be real or complex. If

A

is real, complex eigenvalues

always occur in conjugate pairs.

Similar matrices have identical eigenvalues. It can also be readily shown that

a matrix and its transpose have the same eigenvalues.

12 2 2 Eigenvectors

For any eigenvalue hi, the n-column vector 4 which satisfies Equation 12.16

is called the right eigenvector of A associated with the eigenvalue hi herefore, we

have


10/128

A+i pi

The eigenvector bi has the form

Small-Signal Stabi l i ty

Chap 12

i=ly2,...yn 12.19)

Since Equation 12.17 is homogeneous,

k9i

where is a scalar) is also a solution.

Thus, the eigenvectors are determined only to within a scalar multiplier.

Similarly, the n-row vector q i hich satisfies

is called the lefr eigenvector associated with the eigenvalue hi

The left and right eigenvectors corresponding to different eigenvalues are

orthogonal. In other words, if hi s not equal to A

However, in the case of eigenvectors corresponding to the same eigenvalue,

4 i 12.22)

where i s a non-zero constant.

Since, as noted above, the eigenvectors are determined only to within a scalar

multiplier, it is common practice to normalize these vectors so that

Vibi 12.23)

I 2 2 3Modal Matrices

In order to express the eigenproperties of A succinctly, it is convenient to

introduce the following matrices:

diagonal matrix, with the eigenvalues Al, A2,

... n

as diagonal elements


11/128

Set

12.2 Eigenproperties o f th e Sta te M atr ix 709

~~~h of the above matrices is

nxn.

In terms of these matrices, Equations 12.19 and

12 23 may be expanded as follows.

A@ @ A 12.27)

~tfollows from Equation 12.27

@ l ~ @

A

12 2 4 Free Mo tion of

a

ynamic System

Referring to the state equation 12.9, we see that the free motion with zero

input) is given by

A x AAx 12.30)

A set of equations of the above form,

derived porn physical considerations

is often not the best means of analytical studies of motion. The problem is that the

rate of change of each state variable is a linear combination of all the state variables.

As the result of cross-coupling between the states, it is difficult to isolate those

parameters that influence the motion in a significant way.

In order to eliminate the cross-coupling between the state variables, consider

a new state vector z related to the original state vector Ax by the transformation

where is the modal matrix of A defined by Equation 12.24. Substituting the above

expression for Ax in the state equation 12.30), we have

The new state equation can be written as

n

view of Equation 12.29, the above equation becomes

The

important difference between Equations 12.34 and

12 30

is that A is a diagonal

matrix whereas

A

in general, is non-diagonal.


12/128


13/128

set

12 2

Eigenproperties of the State Matrix

In other words, the time response of the ith state variable is given by

The

above equation gives the expression for the free motion time response of the

system in terms of the eigenvalues, and left and right eigenvectors.

Thus, the free (or initial condition) response is given by a linear combination

o n

dynamic modes corresponding to the n eigenvalues of the state matrix.

The scalar product i q x 0) represents the magnitude of the excitation of

the ith mode resulting from the initial conditions.

If the initial conditions lie along the jth eigenvector, the scalar products

y.~x O)

or all i j are identically zero. Therefore, only the jth mode is excited.

If the vector representing the initial condition is not an eigenvector, it can be

represented by

a

linear combination of the n eigenvectors. The response of the system

will be the sum of the responses. If a component along an eigenvector of the initial

conditions is zero, the corresponding mode will not be excited (see Example

12.1

for

an

illustration).

igenvalue and stability

The time dependent characteristic of a mode corresponding to an eigenvalue

hi

is given by

e .

Therefore, the stability of the system is determined by the

eigenvalues as follows:

(a)

A real eigenvalue corresponds to a non-oscillatory mode. A negative real

eige~ivalue epresents a decaying mode. The larger its magnitude, the faster the

decay. positive real eigenvalue represents aperiodic instability.

The values of c s and the eigenvectors associated with real eigenvalues are

also real.

b)

Complex eigenvalues occur in conjugate pairs, and each pair corresponds to an

oscillatory mode.

The associated c s and eigenvectors will have appropriate complex values so

as to make the entries of x(t) real at every instant of time. For example,

has the form


14/128


15/128


16/128

714 Small Signal Stabi l i ty

Chap

Cases I), 3) and 5) ensure local stability, with 1) and 3) being asymptotically

stable.

12.2.5 Mode Shape Sensitivity and Participation Factor

a) Mode shape and eigenvectors

In the previous section, we discussed the system response in terms of

the

state

vectors x and z which are related to each other as follows:

and

The variables Axl, Ax,, ... Axn are the original state variables chosen to represent the

dynamic performance of the system. The variables z,,z,,

...

z, are the transformed state

variables such that each variable is associated with only one mode. In other words,

the transformed variables

z

are directly related to the modes.

From Equation 12.47A we see that the right eigenvector gives the mode shape,

i.e., the relative activity of the state variables when a particular mode is excited. or

example, the degree of activity of the state variable xk in the ith mode is given by the

element ki of the right eigenvector gi

The magnitudes of the elements of i ive the extents of the activities of the

n state variables in the ith mode, and the angles of the elements give phase

displacements of the state variables with regard to the mode.

As seen from Equation

12.47B, the left eigenvector q identifies which

combination of the original state variables displays only the ith mode. Thus

the

kth

element of the right eigenvector

i

easures the activity of the variable

xk

in the ith

mode, and the Ath element of the left eigenvector q eighs the contribution of this

activity to the ith mode.

b)

Eigenvalue sensitivity

Let us now examine the sensitivity of eigenvalues to the elements of the state

matrix. Consider Equation 12.19 which defines the eigenvalues and eigenvectors:


17/128

see 12 2 Eigenproperties of the State Matrix

7

5

Differentiating with respect to a the element of in kth row and jth column) yields

prernultiplying by @ i and noting that

qii

=

1

and I

A

Ii

=0 we see that the

above equation simplifies to

~ l llements of

dAlaa

are zero, except for the element in the kth row and jth column

which is equal to 1 Hence,

Thus the sensitivity of the eigenvalue

i

to the element a,, of the state matrix is equal

to the product of the left eigenvector element

v k

nd the right eigenvector element

4 j i

(c) Participation factor

One problem in using right and left eigenvectors individually for identifying

the relationship between the states and the modes is that the elements of the

eigenvectors are dependent on units and scaling associated with the state variables. As

a solution to this problem, a-matrix called the participation

matrix

P),

which

combines the right and left eigenvectors as follows is proposed in reference 2 as a

measure of the association between the state variables and the modes.

with


18/128

71 Small-Signal Stability Chap 1

where

ki

= the element on the kth row and ith column of the modal matrix

= kth entry of the right eigenvector i

yik

the element on the ith row and kth column of the modal matrix

rp

= kth entry of the left eigenvector q

The element

pki= kiyik

s termed the participation factor

[2]

t is a measure

of the relative participation of the kth state variable in the ith mode, and vice versa.

Since ki measures the activity of xk in the ith mode and

yikw ighs

the

contribution of this activity to the mode, the product pk measures the net

participation. The effect of multiplying the elements of the left and right eigenvectors

is also to make pkidimensionless i.e., independent of the choice of units).

In view of the eigenvector normalization, the sum of the participation factors

n n

associated with any mode xp,) or with any state variable

Cp ,

is equal to 1.

i l k 1

From Equation 12.48, we see that the participation factor pk s actually equal

to the sensitivity of the eigenvalue hi to the diagonal element akkof the state matrix

As we will see in a number of examples in this chapter, the participation

factors are generally indicative of the relative participations of the respective states

in the corresponding modes.

12 2 6

ontrollability and Observability

In Section 12.1.3 the system response in the presence of input was given as

Equations 12.8 and 12.9 and is repeated here for reference.

Expressing them in terms of the transformed variables z defined by Equation 12.31

yields


19/128

set

12.2 Eigenproper ties of the State M atr ix 7 1 7

The state equations in the normal form (decoupled) may therefore be written as

Ay

=

C z

DAu

~~ferr ingo Equation 12.51, if the ith row of matrix

B

is zero, the inputs have no

effect on the ith mode. In such a case, the ith mode is said to be uncontrollable.

From Equation 12.52, we see that the ith column of the matrix

C

determines

or not the variable z contributes to the formation of the outputs. If the

column is zero, then the corresponding mode is unobservable. This explains why

some poorly damped modes are sometimes not detected by observing the transient

,response of a few monitored quantities.

The nxr matrix B = 8 - I

B is referred to as the mode controllability matrix, and

the mxn matrix

C

=C@as the mode observability matrix.

By inspecting B and

C

we can classify modes into controllable and

observable; controllable and unobservable; uncontrollable and observable;

uncontrollable and unobservable.

12.2.7 The oncept of omplex Frequency

Consider the damped sinusoid

The

unit

of

s radians per second and that of 8 is radians. The dimensionless unit

neper (Np) is commonly used for a t in honour of the mathematician John Napier

(1550-1

617) who invented logarithms. Thus the unit of is neper per second (Npls).

For circuits in which the excitations and forced functions are damped

sinusoids, such as that given by Equation 12.55, we can use phasor representations

of damped sinusoids. This will work as well as the phasors of (undamped) sinusoids

normally used in ac circuit analysis because the properties of sinusoids that make the

phasors possible are shared by damped sinusoids. That is, the sum or difference of

two

or more damped sinusoids is a damped sinusoid and the derivative or indefinite

This is referred to as Kalman s canonical structure theorem, since it was first proposed

y R.E. Kalman in 1960.


20/128

7 8 Sm al l Signal Stabi l i ty Chap.

2

integral of a damped sinusoid is also a damped sinusoid. In all these cases, v and

8 may change; and o are fixed.

Analogous to the form of phasor notation used for sinusoids, in the

case

of

damped sinusoids, we may write

v

=

V m e u t c o s o t + B )

= R e [ v m Ot e

I

- ~[v, je 0 +ja t ]

With

s

o jo, we have

where

is the phasor (V L8) and is the same for both the undamped and damped

sinusoids. Obviously, we may treat the damped sinusoids the same way

we do

undamped sinusoids by using

s

instead of j o .

Since

s

is a complex number, it is referred to as complex frequency,

and

V( )

is called a generalized phasor.

All concepts such as impedance, admittance, Thevenin s and Norton s

theorems, superposition, etc., carry over to the damped sinusoidal case.

It follows that, in the s-domain, the phasor current I(s) and voltage V(s),

associated with a two-terminal network are related by

where Z(s) is the generalized impedance.

Similarly, input and output relations of dynamic devices can be expressed as

In the factored form,

The numbers z, z2, ... are called the zeros because they are values of s for which

G(s) becomes zero. The numbers p , p2 .. p, are called the poles of G(s). The values


21/128

set

12 2

i genpropert ies o f the S ta te M at r ix

71

9

of

and zeros, along with a and b uniquely determine the system transfer

functionG s).Poles and zeros are useful in considering frequency domain properties

dynamic systems.

12 2 8

Relationship between igenproperties and Transfer Functions

The state-space representation is concerned not only with input and output

properties of the system but also with its complete internal behaviour. In contrast, the

rnsfer function representation specifies only the inputloutput behaviour. Hence, one

can

make an arbitrary selection of state variables when a plant is specified only by

a transfer function. On the other hand, if a state-space representation of a system is

known

the transfer function is uniquely defined. In this sense, the state-space

is a more complete description of the system; it is ideally suited for the

analysis of multi-variable multi-input and multi-output systems.

For small-signal stability analysis of power systems, we primarily depend on

the eigenvalue analysis of the system state matrix. However, for control design we are

interested in

an

open-loop transfer function between specific variables. To see how

this is related to the state matrix and to the eigenproperties, let us consider the transfer

function between the variables

y

and u. From Equations 12.8 and 12.9, we may write

where is the state matrix, x is the state vector, Au is a single input, y is a single

output, is a row vector and is a column vector. We assume that y is not a direct

function of i.e., D=O).

The required transfer function is

This has the general form

If D s) and N s) can be factored, we may write


22/128

720 Sm all-Signal Sta bil i ty Chap.

12

As discussed in Section 12.2.7, the n values of s, namely, p1,p2, ..P,, which make the

denominator polynomial D s) zero are the poles of G s). The 1 values of s, namely,

z,, z,, ... zl, are the zeros of G s).

Now, G s) can be expanded in partial fractions as

and i s known as the residue of G s) at pole pi

To express the transfer function in terms of the eigenvalues and eigenvectors,

we express the state variables x in terms of the transformed variables z defined by

Equation 12.31. Following the procedure used in Section 12.2.4, Equations 12.57 nd

12.58 may be written in terms of the transformed variables as

nd

Hence,

Since

A

is a diagonal matrix, we may write

where

Ri =

c @ q i b

We see that the poles of G s) are given by the eigenvalues of A Equation 12.67 gives

the residues in terms of the eigenvectors. The zeros of G s) are given by the solution

of


23/128

Set

12.2

igenpropert ies of t he S ta te Ma t r i x

In this exam ple we will study a second-order linear system. S uch a system is easy to

analyze and is helpful in understanding the behaviour of higher-order systems. The

performance of high-order systems is often viewed in terms of a dominant set of

second-order poles or eigenvalues. Therefore a thorough understand ing of the

characteristics of a second-order system is essential before w e study complex system s.

Figure E12.1 show s the fam iliar RL circuit which represents a second-order system .

Study the eigenproperties of the state matrix of the system and examine its modal

characteristics.

igure E12 1

Solution

The differential equation relating v to v is

This may be written in the standard form

where

o

I @

=

undamped natural frequency

= ~/2)/mdamping ratio


24/128

Small Signal Stability Chap

In order to develop the state-space representation we define the following state input

and output variables:

Using the above quantities Equation E12 2 can be expressed in t a m s of

two

first-

order equations:

In matrix form

The output variable is given by

These have the standard state-space form:

=

Ax bu

y

=

cx du

The eigenvalues of A are given by

Hence


25/128

12 2

igenproperties

of

th State Matrix

Solving for the eigenvalues, we have

The right eigenvectors are given by

A-AI)@ =

Therefore,

This may be rewritten as

If we attempt to solve the above equations for l i and 2i, we realize that they are not

independent. As discussed earlier, this is true in general; for an nth order system, the

equation A- AI)

0

gives only n 1 independent equations for the n components of

eigenvectors. One com ponent of the eigenvector may be fixed arbitrarily and then the

other components can be determined from the n-1 independent equations. It shodd ,

however,

be noted that the eigenvectors themselves are linearly independent if

eigenvalues are distinct.

For the second-order system, we can fix

l i=l

nd determine 2i, from one of the two

relationships in Equation E12.10, for each eigenvalue.

The eigenvector corresponding to 3L is

and the eigenvector corresponding to

3L

is


26/128

7 4 Small Signal Stabi l i ty Chap.

1

The nature of the system response depends almost entirely on the damping rati

The value of

o

as the effect of simply adjusting the time scale.

C;

If < is greater than 1 both eigenvalues are real and negative; if

5

is equal to

1 both

eigenvalues are equal to

-a ;

and if is less than 1 eigenvalues are complex

conjugates given as

The location of the eigenvalues in the complex plane with respect to < and n s

indicated in Figure E12 2

Damping angle 0

=

cos lr

Figure

E12 2

We will first examine the singularities of the second-order system and discuss the

shape of the state trajectories near the singularity. We will then discuss in detail the

case where both eigenvalues are real and negative with

h

greater than A] but with

A

and

A2

not far different.

The state equations in the normal form are given by

Hence


27/128


28/128

Small-Signal Stabi l i ty Chap.

transformation

x z

If the input

v

is zero, and if the initial cond itions are such tha t xl , x2) is on one of

the eigenvectors, the state vector will remain in the same direction but will vary in

magnitude by the factor

e l'

or

eh*

as the case may be.

9

If the vector representing the initial condition is not an eigenvector, it can e

represented by a linear combination of the two eigenvectors. The response of the

circuit will be the sum of the two responses. As time increases, the component

in the

direction of the eigenvector

42

becomes less significant because e h t decays faster

than

eal'.

Thus the trajectories always approach the origin along the 4, direction

unless the co mpo nent of this eigenvector was initially zero. If the eig env ecto r~ re not

real, such a sim ple physical interpretation of eigenvectors is not possible.

A

and

1 1 2 1 >

A real and

negative;

k higenvector 4,

\\ slow decay)

~ i ~ e n v e c t o r,

fast decay)

igure E12 4

12 2 9

omputation of Eigenvalues

In the above example we computed the eigenvalues by solving the

characteristic equation of the system. This was possible because we were analyzing

a simple second-order system. For higher-order systems with eigenvalues of widely

differing magnitudes this approach fails. The method that has been widely used

for

the computation of eigenvalues of real non-symmetrical matrices is the

R

transformation method originally developed by J.G.F. Francis [3] The method is

numerically stable robust and converges rapidly. It is used in a number of very good

general purpose commercial codes and has been successfully used for analyzing small-

signal stability of power systems with several hundred states. The right eigenvectors

may be computed by using the inverse iteration technique. A good description of

the

QR

transformation and inverse iteration methods may be found in reference

4.


29/128

Set.

12 3

Single Machine

Infinite

Bus Sys tem 727

For large systems involving several thousand states, the QR method cannot be

for computing the eigenvalues. The reason for this and a description of special

for eigenvalue analysis of very large systems are presented in Section 12.8.

12 3 SMALL SIGNAL STABILITY OF A SINGLE MACHINE

INFINITE BUS SYSTEM

In this section we will study the small-signal performance of a single machine

connected to a large system through transmission lines. general system

configuration is shown in Figure 12.3(a). Analysis of systems having such simple

configurations is extremely useful in understanding basic effects and concepts. After

we develop an appreciation for the physical aspects of the phenomena and gain

experience with the analytical techniques, using simple low-order systems, we will be

in a better position to deal with large complex systems.

Large

system I

(a) General configuration

(b) Equivalent system

Figure

12 3

Single machine connected to a large system

through transmission lines

Infinite bus

For the purpose of analysis, the system of Figure 12.3(a) may be reduced to

the form of Figure 12.3(b) by using Thtvenin s equivalent of the transmission

network external to the machine and the adjacent transmission. Because of the relative

size of the system to which the machine is supplying power, dynamics associated with

the machine will cause virtually no change in the voltage

and

frequency of Thevenin s

Zeq

=

R

jXE


30/128

Small Signal Stability Chap. 1 ~

voltage EB. Such a voltage source of constant voltage and constant frequency

is

referred to as an infinit bus.

For any given system condition, the magnitude of the infinite bus voltage

remains constant when the machine is perturbed. However, as the steady-state system

conditions change, the magnitude of EBmay change, representing a changed operating

condition of the external network.

@

In what follows we will analyze the small-signal stability of the system of

Figure 12.3 b) with the synchronous machine represented by models of varying

degrees of detail. We will begin with the classical model and gradually increase the

model detail by accounting for the effects of the *dynamics of the field circuit,

excitation system, and amortisseurs. In each case, we will develop the expressions for

the elements of the state matrix as explicit functions of system parameters. This will

help make clear the effects of various factors associated with a synchronous machine

on system stability. In addition to the state-space representation and modal analysis,

we will use the block diagram representation and torque-angle relationships to analyze

the system-stability characteristics. The block diagram approach was first used by

Heffron and Phillips [5] and later by deMello and Concordia

[6]

to analyze the small-

signal stability of synchronous machines. While this approach is not suited for a

detailed study of large systems, it is useful in gaining a physical insight into the

effects of field circuit dynamics and in establishing the basis for methods of

enhancing stability through excitation control.

12 3 1

Generator Represented y the lassical M od el

With the generator represented by the classical model see Section 5.3.1 and

all resistances neglected, the system representation is as shown in Figure

12.4.

Here

E

is the voltage behind

Xj

ts magnitude is assumed to remain constant

at the pre-disturbance value. Let 6 be the angle by which

E

leads the infinite -bus

voltage

EB.

As the rotor oscillates during a disturbance, 6 changes.

With

E

as reference phasor,

igure

12 4


31/128

set 12 3

Single-Machine Inf in i te us Sys tem

The

complex power behind

j

s given by

With stator resistance neglected, the air-gap power (P,) is equal to the terminal power

(p) . In per unit, the air-gap torque is equal to the air-gap power see Section 5 .1.2).

Hence,

Linearizing about an initial operating condition represented by 6=a0yields

The equations of motion Equations.3.209 and 3.2

10

of Chapter 3) in per unit

are

where

Am

is the per unit speed deviation, 6 is rotor angle1 in electrical radians,

w

is the base rotor electrical speed in radians per second, and

p

is the differential

operator l t with time in seconds.

Linearizing Equation 12.73

and

substituting for

AT,

given by Equation 12.72,

we obtain

where

K

is the synchronizing torque coefficient given by

As discussed in Section 5.3.1, for a classical generator model, the angle of

E

with

respect to a synchronously rotating reference phasor can be used as a measure of the rotor

angle. Here we have chosen EB as the reference, and the rotor angle

6

is measured as the angle

by which E leads EB


32/128

Small-Signal Stabi l i ty Chap

12

Linearizing Equation 12.74 we have

p A 6 w 0 A q

Writing Equations 12.75 and 12.77 in the vector-matrix form we obtain

This is of the form x =Ax+bu. The elements of the state matrix A are seen to

be

dependent on the system parameters

KD H,

X nd the initial operating condition

represented by the values of E and Fo The block diagram representation shown in

Figure 12.5 can be used to describe the small-signal performance.

From the block diagram of Figure 12.5 we have

Rearranging we get

KD

s 2 A 6 ) + - s A ~ ) + - o , A ~ ) T,

H H H

Therefore the characteristic equation is given by

This is of the general form


33/128

Set

12 3

Single M achine Inf in i te Bus Sys tem

Ks

=

synchronizing torque coefficient in pu torquehad

K

= damping torque coefficient in pu torque/pu speed deviation

H = inertia constant in MW-s/MVA

Am, = speed deviation in pu = m, -mo)/wo

A6

=

rotor angle deviation in elec. rad

= Laplace operator

w, = rated speed in elec. rad/s

= nfo

=

377

for a 6 Hz system

Synchronizing torque

igure

12 5

Block diagram of a single-machine infinite

bus system with classical generator model

component

Therefore the undamped natural frequency is

and the damping ratio is

Ks

A8

ATe

AT

Amr

0

Hs

Damping torque

K

component


34/128

73 Small Signal Stab i l i ty Chap.

12

As the synchronizing torque coefficient

Ks

increases the natural frequency

increases

and the damping ratio decreases.

An

increase in damping torque coefficient K

increases the damping ratio whereas an increase in inertia constant decreases both

and 6 .

Example

12 2

Figure E12.5 shows the system representation applicable to a thermal generating

station consisting of four 555 MVA, 24 kV, 60

Hz

units.

HT

Infinite

bus

Figure E12 5

/

/

/

/

5

/

The network reactances shown in the figure are in per unit on 2220 MV A, 24 kV base

(referred to the LT side of the step-up transformer). Resistances are assumed to be

negligible.

j0.5

CCT

j0.93

C C T 2

The objective of this example is to an'alyze the small-signal stability characteristics

of the system about the steady-state operating condition following the loss of circuit

2. The postfault system condition in per unit on the 2220 MVA, 24 kV base is as

follows:

P =

0.9

=

0.3 (overexcited)

E = 1

OL36

EB =

0.99510

The generators are to be modelled as a single equivalent generator represented by the

classical model with the following parameters expressed in per unit on 2220

MVA

24 kV base:

(a)

Write the linearized state equations of the system. Determine the eigenvalues,

damped frequency of oscillation in Hz, damping ratio and undamped natural

frequency for each of the following values of damping coefficient (in pu

torquelpu speed):

(b)

For the case with KD=lO.O, find the left and right eigenvectors, and

participation m atrix. Determine the time response if at

t =O

A6=5 and A@=O.


35/128

12.3

Single Machine Infinite Bus System

Solution

a)

Figure E12.6 shows the circuit model representing the postfault steady-state

operating condition with all parameters expressed in per unit on 2220 MVA base.

igure E12 6

With t as reference phasor, the generator stator current is given by

The voltage behind the transient reactance is

The angle by which

E

leads

EB

is

The total system reactance is

The corresponding synchronizing torque coefficient, from Equation 12.76, is


36/128

Small-Signal S tabi l i ty Chap 1

inearized system equations are

The eigenvalues of the state matrix are given by

or

12

. 1 4 3 ~ ~ ~40.79

=

0

This is of the form

2

h 2 + 2 [ o n h + o n 0

with

o = m 6.387

radls

= 1.0165 z

= 0 .1 43K d 2~ 6 .3 87 ) 0 .01 12KD

The eigenvalues are

The damped frequency is

The following are the required results for different values of KD


37/128

Set.

12 3 Single Machine Infinite Bus System 735

b) The right eigenvectors are given by

KD

Eigenvalues

Damped frequency od

Damping ratio

Undamped natural frequency on

For the given system, with KD=lO, he above equation becomes

For =-0.7 14+j6.35, he corresponding equations are

The above equations are not linearly independent. A s discussed in Exam ple

12.1,

one

of the eigenvectors corresponding to an eigenvalue has t o be set arbitrarily. Therefore,

let

10

0.714g6.36

1.0101

Hz

-0.1 12

1.0165 z

0

096.39

1.0165 Hz

0

1.0165

z

412 =

1 0

then

@

-0.0019+j0.0168

Similarly, eigenvectors corresponding to =

-0.7 4 -j6.35

are

412

= 1.0

@

-0.0019 -j0.0168

The right eigenvector modal matrix is

10

-0.71496.35

1.0101

Hz

0.1 12

1.0165

Hz


38/128

Small Signal

Stability

Chap

he

left

eigenvectors normalized

so

that i i

=

1.0 are given by

he participation matrix

is

he time response is given by

With A6

=5

=0.0873 r d and Ao =O at t=O we have


39/128

et. 12 3

Sing le M achine In f in i te

us System

The time response of speed deviation is

Similarly, the time response of rotor angle deviation is

A6 t)

=

0.088e

-0.714

cos 6.35

t

0.112)

rad

This is

a

second-order system with an oscillatory mode of response having a damped

frequency of 6.35 radls or 1.0101 Hz. The oscillations decay with a time constant of

110.714 s. This corresponds to a damping ratio of 0.1 12. As this is a rotor angle

mode,

Am

and

A6

participate in it equally.

12.3.2 Effects of Synchronous Machine Field Circuit Dynamics

We now consider the system performance including the effect of field flux

variations. The arnortisseur effects will be neglected and the field voltage will be

assumed constant manual excitation control).

In what follows, we will develop the state-space model of the system by first

reducing the synchronous machine equations to an appropriate form and then

combining them with the network equations. We will express time in seconds, angles

in electrical radians, and all other variables in per unit.

ynchronous machine equations

As in the case of the classical generator model, the acceleration equations are

where oo=2xf0 elec. radls. In this case, the rotor angle 6 is the angle in elec. rad) by

which the q-axis leads the reference

EB.

As shown in Figure 12.6, the rotor angle

is the sum of the internal angle

i

see Section 3.6.3) and the angle by which

E

leads

4

e need a convenient means of identifying the rotor position with respect to an

appropriate reference and keeping track

of

it as the rotor oscillates. As discussed in


40/128

Small Signal Stabi l i ty

Chap.

2

q-axis

\my

//

d-axis

igure 12 6

Chapter 3 (Section 3.6), the q-axis offers this convenience when the dynamics of rotor

circuits are represented in the machine model. The choice of

EB

as the reference for

measuring rotor angle is convenient from the viewpoint of solution of network

equations.

The per unit synchronous machine equations were summarized in Section 3.4.9

and the simplifications essential for large-scale stability studies were discussed in

Section 5.1. From Equation

5.10

with time in seconds instead of per unit, the field

circuit dynamic equation is

where Efd is the exciter output voltage defined in Section 8.6.1. Equations 12.83 to

12.85 describe the dynamics of the synchronous machine with Am,,

6

and y as the

state variables. However, the derivatives of these state variables appear in these

equations as functions of T and d, which are neither state variables nor input

variables. In order to develop the complete system equations in the state-space form,

we need to express

if

and T in terms of the state variables as determined by the

machine flux linkage equations and network equations.

With amortisseurs neglected, the equivalent circuits relating the machine

flux

linkages and currents are as shown in Figure 12.7.


41/128

12 3

Single Machine Inf in i te Bus Sy stem

Figure

12 7

The stator and rotor flux linkages are given by

L

i

=

@ad f f

In the above equations y dand v qre the air-gap mutual) flux linkages, and

and

La,,

are the saturated values of the mutual inductances.

From Equation

12.88

the field current m y be expressed as

The d-axis mutual flux linkage can be written in terms of

yf

and

i

as follows:


42/128

740

where

Small-Signal Stabi l i ty

Chap 1

Since there are no rotor circuits considered in the q-axis the mutual

flux

linkage is

given by

q q

The air-gap torque is

With py terms and speed variations neglected as discussed in Section

5.1

the stator

voltage equations are

As a first step we have expressed i and T in terms of yl- id i vad nd yaq

Sd

In addition ed and e have been expressed in terms of these variables and will be used

in conjunction with the network equations to provide expressions for id and iq in terms

of the state variables.

-.

The advantages of using w and yl as intermediate variables in the

elimination process will be more apparent when we account for the effects

of

amortisseur circuits in Section 12.6.

etwork equations

Since there is only one machine the machine as well as network equations

c n

be expressed in terms of one reference frame i.e. the d-q reference frame of the

machine. Referring to Figure 12.6 the machine terminal and infinite bus voltages in

terms of the

d

and q components are


43/128

Set 12.3

Single Machine Inf in i te Bus Sy stem

The network constraint equation for the system of Figure 12.3 b) is

E =

E , + R , + ~ x , ) ~ ,

ed+jeq = EBd jEBq) RE jXE ) id jiq

Resolving into and

q

components gives

ed

=

RE d-x E q+EBd

eq

=

REiq+XEid+EBq

where

Using

Equations 12.94

nd

12.95 to eliminate ed, eq in Equations 12.99 and 12.100,

and using the expressions for v dnd yr g given by Equations 12.90 and 12.92, we

obtain the following expressions for

id

and

iq

in terms of the state variables

yfd

and

:


44/128

74

where

Sma ll Signal Stab i l i ty Chap.

The reactances X and

XAs

are saturated values. In per unit they are equal to

the

corresponding inductances.

Equations

12.103 and 12.104 together with Equations 12.89 12.90

and

12 92

can be used to eliminate qdand

T

from the differential equations 12.83 to 12.85 and

express them in terms of the state variables. These equations are nonlinear and

have

to be linearized for small-signal analysis.

inearized system equations

Expressing Equations 12.1 03 and 12.1 04 in terms of perturbed values we may

write

where

By

linearizing Equations

12.90

and

12.92

and substituting in them the above

expressions for

Aid

and

Ai

we get


45/128

set. 12.3

Single Machine Infinite

Bus

System

Linearizing Equation 12.89 and substituting for Ayrad from Equation 12.109 gives

The linearized form of Equation 12.93 is

Substituting for Aid Ai d y a d and A y from Equations 12.106 to 12.110, we obtain

AT =

KlA8

12.1 12)

where

y

linearizing Equations 12.83 to 12.85

nd

substituting the expressions for

A

nd

AT

given by Equations 12.111 and 12.112, we obtain the system equations in the

desired final form:


46/128

Small-Signal Stability Chap

where

and

AT

and

AEfd

depend on prime-mover

and

excitation controls. With constant

mechanical input torque,

AT

=O; with constant exciter output voltage,

AEfd=O.

t

is interesting to compare the above state-space equations with those derived

in Section 12 3 1 by assuming the classical generator model which is equivalent to

assuming Rfd=0, Ra

O

and X, =Xi .

The mutual inductances Lads and Laps in the above equations are saturated

values. The method of accounting for saturation for small-signal analysis is described

below.

Representation of saturation in small signal studies

Since we are expressing small-signal performance in terms of perturbed values

of flux linkages

and currents, a distinction has to be made between total saturation


47/128

~ e c12.3


Bnd

incremental saturation.

Total saturation is associated with total values of flux linkages and currents.

The method of accounting for total saturation was discussed in Section 3.8.

Incremental saturation is associated with perturbed values of

flux

linkages and

,urent~. Therefore, the incremental slope of the saturation curve is used in computing

th incremental saturation as shown in Figure 12.8.

Denoting the incremental saturation factor

we have

Lads

incr)

-

Ksd incr) L?m

Based on the definitions of A , and

y

in Section 3.8.2, we can show that

B

atAsat B m t d m - * ~ ~ )

similar treatment applies to q-axis saturation.

For computing the initial values of system variables denoted by subscript o),

total saturation is used. For relating the perturbed values, i.e., in Equations 12.105,

12.108, 12.1 13, 12.1 14, and 12.1 16, the incremental saturation factor is used.

The method of computing the initial steady-state values of machine parameters

was described in Section 3.6.5.

Slope represents saturated value of

ad

relating total values of v and

\

Slope represents saturated

value of

ad

relating incremental

values of v and

I

I I

I

jd

or mmf

Figure 12 8

Distinction between incremental and total saturation


48/128

Small Signal Stability Chap l

Summary of procedure for formulating the state matrix

a)

The following steady-state operating conditions, machine parameters nd

network parameters are given:

+

p t

Qt

Et

RE

XE

Ld Lq Ra Lfd Rfd Asat sat

WT

Alternatively EB may be specified instead of Qt or El

b)

The first step is to compute the initial steady-state values of system variables:

It, power factor angle

Total saturation factors Ksd and Ksq see Section 3.8

xds

=

Lds = KsdLadu+L~

Xqs = 9s = Ksq Lagu Lz

ItXqscosQ I

Ra

sin@

6, =

tan-

Et+It

Ra

OSQ+Zt qs

sin@

EBdO

= tan- -1

EBq

c) The next step is to compute incremental saturation factors

nd the

corresponding saturated values of La,,, Lags, LLds, and then

R , XQ T ~ from Equation

12.105

m l , m 2,

1, n

from Equation

12.108

4 , 2

fiom Equations

12.113

and

12.114


49/128

set. 12.3

single-~achinenfinite Bus System

4

Finally compute the elements of matrix fiom Equation 12.1

16

~ ~ o c k

i gr m represent tion

Figure 12.9 shows the block diagram representation of the small-signal

rformance of the system. In this representation the dynamic characteristics of the

are expressed in terms of the so-called

K

constants

[5 ]

The basis for the block

diagram and the expressions for the associated constants are developed below.

Field circuit

Figure 12 9

Block diagram representation with constant Efd

From Equation 12.1 12 we may express the change in air-gap torque as a

function of A6 and Ayfd as follows:

AT, =

KlA6

+

K2A fd

where

K , = AT,/AF with constant yfd

K2 = ATe/Ayfdwith constant rotor angle

The expressions for Kl and K2 are given by Equations 12 113 and 12.114

The component of torque given by K,A6 is in phase with A6 and hence

represents a synchronizing torque component.

The component of torque resulting fiom variations in field flux linkage is

given by

K2Ay

The variation of vfds determined by the field circuit dynamic equation:


50/128

7 8

Small Signal Stabi l i ty

Chap

1

By grouping terms involving

Ayfd

and rearranging we get

where

Equation 12.119 with s replacing p accounts for the field circuit block in Figure

12.9.

Expression for the K constants in the expanded form

We have expressed the

K

constants in terms of the elements of matrix

A

In

the literature [5 6] hey are usually expressed explicitly in terms of the various system

parameters as summarized below.

The constant

K l

was expressed in Equation

12.113

as

4 = nl( ado+Laqsido)

- m l ( a q o + L ~ i q O )

From Equation 12.95 the first term in parentheses in the above expression for

Kl

may

be written as

where Eqo is the predisturbance value of the voltage behind R, jX,. The second term

in parentheses in the expression for K may be written as

@aqo

= -L

i

+L

aqs q0 q0

Substituting for

nl

m

fi-om Equation

12 108

and for the terms given by Equations

12.121 and 12.122 in the expression for K yields


51/128

Set

12 3

Single M achine Inf in i te

Bus

Sys tem

similarly, the expanded form of the expression for the constant K, is

rom

Equations 12.91, 12.108, and 12.1 16, we may write

a

= o0

-

a h

Ld fd

(La Lfd) ( ah f d )

Substitution of the above in the expression for K3 and j given by Equation 12.120

yields

where

T , is

the saturated value

o T ,

Similarly, from ~~uations2.91, 12.108

and


52/128

Small Signal Sta bil i ty Chap.

12

12.1 1

6

we may write

Substitution of the above in the expression for K4 given by Equation 12.120 yields

If the effect of saturation is neglected, this simplifies to

If the elements of matrix A are available, the constants may be computed directly

from them. The expanded forms are derived here to illustrate the form of expressions

used in the literature. An advantage of these expanded forms is that the dependence

of the

constants on the various system parameters is more readily apparent. A

disadvantage, however, is that some inconsistencies appear in representing saturation

effects.

In the literature, Ei= LadILld)~fd is often used as a state variable instead of

v l

see Section 5.2). The effect of this is to remove the Lad/ Lad+Lfd) erm from the

expressions for K2 and K3. The product K2K3 would, pqwever, remain the same.

Effect of field lux inkage variation on system stability

We see from the block diagram of Figure 12.9 that, with constant field voltage

Mfd=O), the field flux variations are caused only by feedback of A through the

coefficient K4. This represents the demagnetizing effect of the armature reaction.

The change in air-gap torque due to field flux variations caused by rotor angle

changes is given by

s ue to

A d

1+ST,

The constants K2, K3, and K4 are usually positive. The contribution of Avfd to

synchronizing and damping torque components depends on the oscillating frequency

as discussed below.

a)

In the steady state and at very low oscillating frequencies s=jo - 0):


53/128

set 12 3

Single M achine Inf in i te Bus Sys tem

ATe

due

to q jd = KZK3K4A6

The field

flux

variation due to A6 feedback (i.e., due to armature reaction)

introduces a negative synchronizing torque component. The system becomes

monotonically unstable when this exceedsK1A6 The steady-state stability limit

is reached when

b)

At oscillating frequencies much higher than 1/T3:

K K K

ATe

=

4 ~ 6

j 0 T 3

K K K

4 i ~ 6

Thus, the component of air-gap torque due to Ayfd is 90 ahead of A6 or in

phase with do Hence, Avfd results in a positive damping torque component.

c)

At typical machine oscillating frequencies of about 1 Hz

27-c

radls), Avfd

results in a positive damping torque component and a negative synchronizing

torque component. The net effect is to reduce slightly the synchronizing torque

component and increase the damping torque component.

Figure 12 10 Positive damping torque and negative

synchronizing torque due to KzAvfd


54/128

Small Signal Stability

Chap 1

Special situations with

-K4

negative:

The coefficient K4 is normally positive. As long as it is positive the effect

of

field flux variation due to armature reaction

Aw

with oonstant E ) is to introduce

f

a positive damping torque component. However there Can be situations where K~ is

negative. From the expression given by Equation 12.128, K4 is negative

when

a

XE+Xq)sin60- R,+RE)cos60s negative. This is the situation when a hydraulic

generator without damper windings is operating at light load and is connected bv

line of relatively high resistance to reactance ratio to a large system. This typiof

situation was reported in reference 7.

Also K4 can be negative when a machine is connected to a large local load

supplied partly by the generator and partly by the remote large system [8]. Under such

conditions the torques produced by induced currents in the field due to armature

reaction have components out of phase with Am and produce negative damping.

Example 12 3

In this example we analyze the small-signal stability of the system of Figure E12.5

considered in Example 12.2) including the effects of the generator field circuit

dynamics. The parameters of each of the four generators of the plant in per unit on

its rating are as follows:

The above parameters are unsaturated values.

The effect of saturation is to

be

represented by assum ing that and axes have similar saturation characteristics with

A

= 0.03 1 B

=

6.93 wT = 0.8

The effects of the amortisseurs may be neglected. The excitation system is on manual

control constant

E:/d

and transmission circuit 2 is out of service.

a) If the plant output in per unit on 2220 MVA, 24 kV base is

P

=

0.9

Q

= 0.3 overexcited)

E = 1.0

compute the following:

i)

The elements of the state matrix A representing the small-signal

performance of the system.

ii) The constants

K

to

K4

and

T3

associated with the block diagram

representation of Figure 12.9.


55/128

12 3

S i n g l e - M a c h i n e I n f i n i t e Bus S y s t e m 753

iii)

Eigenvalues of A and the corresponding eigenvectors and participation

matrix; frequency and damping ratio of the oscillatory mode.

iv) Steady-state synchronizing torque coefficient; damping and

synchronizing torque coefficients at the rotor oscillating frequency.

b)

Determine the limiting value of

P

within

k0.025

pu) and the corresponding

value of the rotor angle 6 beyond which the system is unstable, with

0 Saturation effects neglected

ii) Saturation effects included

Assume that

Q=P/3

as

P

varies and

El=l.O.

Comment on the mode of

instability and the effect of representing saturation.

Solution

The four units of the plant m ay be represented by a single generator whose parameters

on 2220 MVA base are the sam e as those of each unit on its rating. The circuit model

of the system in per unit on

2220

MVA base is shown in Figure

E12.7.

igure

E12 7

The generators of this example have the same characteristics as the generator

considered in examples of Chapters 3 and 4 except for LI.

The per unit fundamental parameters elements of the d and q-axis equivalent

circuits) of the equivalent generator following the procedure used in Example 4.1 are

Lad 1.65 L 1.60

L

0.16

R 0.003

Rfd 0.0006 Lfd 0.153

a) i) The initial steady-state values of the system variables are computed by using

the procedure summarized earlier in this section.


56/128

75 Sma l l Signal Stabi l i ty Chap

From Equation 12.108,

From Equation 12.1 16,

ii) From Equations 12.113, 12.1 14, and 12.120 , the constan ts of the block d iagram

of Figure 12.9 are

iii)

Eigenvalues computed by using a standard routine based on the R

transformation method are

A h = -0.1 1kj6.41

ad=1.02 Hz,


57/128

Set

12.3 S i n g l e - M a c h i n e I n f i n i t e B u s S y s t e m 755

From the participation m atrix, we see that

Am

and A6 have a high participation in the

oscillatory mo de corresponding to eigenvalues

hl

and k2); he field flux linkage has

a high participation in the non-oscillatory mode, represented by the eigenvalue h3

iv) The steady-state synchronizing torque coefficient due to Ayfd is

The total steady-state synchronizing torque coefficient is

Ks K , K2K3K4

0.7643-0.3963 0.3679

pu torquelrad

From the block diagram of Figure 12.9

AS(s) due to A fd 1+ST, s2T:

Therefore, AT due to Ayfdis

From the eigenvalues, the com plex frequency of rotor oscillation is -0.1 +j6.41. Since

the real component is much smaller than the imaginary component, we can compute

Ks and

K

at the oscillation frequency by setting s=j6.41 without loss of much

accuracy.

- K ~ K 3 K 4 -0.3963

KS(A fdfd)

1-s2T: 1 - ~ 6 . 4 1 ~ 2 . 3 6 5 ) ~

-0.00172 pu torquelrad

1.53

pu torquelpu

spee

change


58/128

Smal l S igna l Stab i l i ty

Chap.

1

The effect of field flux variation i.e., armature reaction) is thus to reduce the

synchronizing torque slightly and to add a damping torque component.

The net synchronizing torque component is

The only source of damping is due to field flux variation. Hence, the net damping

torque coefficient is

KD KD AJrfd) 1 53

pu torquelpu speed ch nge

From Equation 12.81 he undamped natural frequency is

and from Equation 12.82, the damping ratio is

The above values of w and gree with those computed from the eigenvalues.

b) The stability limit is determined by increasing with Q= P /3 and E =1.0 pu EB

is allowed to take appropriate values so as to satisfy the network equations).

The

results with and without saturation effects are as follows.

i) With saturation effects:

The limiting within .025 pu) and the corresponding system conditions in per

unit

are

The corresponding

K

constants are


59/128

see 12.3


Hence, the steady-state synchronizing torque coefficient is

Ks = K 1 K2K3K4 = -0.00 14

pu torquelrad

The eigenvalues of the system state matrix are

A,, h,

= -0.226kj4.95

a, =

0.79 Hz

=

0.046)

h

=

+0.00142

The above represents conditions just past the stability limit. The system instability s

due to lack of synchronizing torque. This is reflected in the real eigenvalue becoming

slightly positive, representing a mode of instability through non-oscillatory mode.

ii) Without saturation effects:

The limiting value of and the corresponding system conditions in this case are

The

K

constants are

The steady-state synchronizing torque coefficient is

Ks K1 K2K3K4

=

0.0001 pu torquelrad

The eigenvalues are

The system is on the verge of instability. The limiting rotor angle

6

is very close to

90'.

With constant Efd and negligible saliency, the limiting rotor angle w ill be equal

to

90'

if the values of

Ld

and

Laq

used to com pute the initial operating condition are

the same as the values used to relate incremental flux linkages and currents.

In case i), when we represented saturation, we made a distinction between total

saturation and incremental saturation. Hence, the limiting rotor angle was about

102 ,

significantly higher than 90'. r


60/128


12 4 EFFECTS OF EXC IT TION SYS TEM

In this section we will extend the state-space model and the block diagram

developed in the previous section to include the excitation system. We will then

examine the effect of the excitation system on the sma@-signal stability performance

of the single-machine infinite bus system under consideration.

The input control signal to the excitation system is normally the generator

terminal voltage E . In the generator model we implemented in the previous section

t is not a state variable. Therefore

E

has to be expressed in terms of the stat;

variables A m A , and

A ~ J ~

In Section 3.6.2 we showed that t may be expressed in complex form:

Hence

Applying a small perturbation we may write

By neglecting second-order terms involving perturbed values the above equation

reduces to

Therefore

edo

e

AE,

=

- ~ e ~ + ~ e

t

t

4

In terms of the perturbed values Equations 12.94 and 12.95 may be written as

Ae, = -RaAid+LlAi,-A ,,

Ae, = Ra A iq 4A i Aqad

Use of Equations 12.106 12.107 12.109 and 12.110 to eliminate

Aid Ai Ayad

and

Ayr,,

from the above equations in terms of the state variables and substitution

of

the


61/128

set

12 4

f fects o f xc i ta t ion System

resulting expressions for Aed and Ae in Equation 12.131 yield

For the purpose of illustration and examination of the influence on small-signal

stability, we will consider the excitation system model shown in Figure 12.11. It is

representative of thyristor excitation systems classified as type ST 1A in Chapter

8.

The model shown in Figure 12.11, however, has been simplified to include only those

elements that are considered necessary for representing a specific system. A high

exciter gain, without transient gain reduction or derivative feedback, is used.

Parameter

TR

represents the terminal voltage transducer time constant.

I erminal voltage

transdilcer Exciter

igure 12 1

Thyristor excitation system with AVR

The only nonlinearity associated with the model is that due to the ceiling on

the exciter output voltage represented by

FMM

and

Fm

For small-disturbance

studies, these limits are ignored as we are interested in a linearized model about an

operating point such that Efd is within the limits. Limiters and protective circuits

UEL, OXL, VIHz) are not modelled as they do not affect small-signal stability.

From block

1

of Figure 12.1 1, using perturbed values, we have

Hence


62/128

76 Small Signal

Stability Chap

Substituting for AE, from Equation 12.132 we get

From block

2

of Figure 12.1 1

Efd =

KA

re

n

terms of perturbed values we have

Efd

= KA

Avl)

The field circuit dynamic equation developed in the previous section with the

effect

of excitation system included becomes

where

The expressions for a a32and

remain unchanged and are given by Equation

12.1 16.

Since we have a first-order model for the exciter the order of the overall

system is increased by 1 he new state variable added is

Av,.

From Equation 12.13

5

where

and

K

and

g

re given by Equations 12.133 and 12.134.


63/128

Set

12 4

Ef fec ts o f Exc i ta t ion Sys tem

Since pAw and pA6 are not directly affected by the exciter

14 a =

he complete state-space model for the power system including the excitation system

of Figure 12.11 has the following form:

With constant mechanical torque input

Block di gr m including the excit tion system

Figure 12.12 shows the block diagram obtained by extending the diagram of

Figure 12.9 to include the voltage transducer and AVR/exciter blocks. The

Voltage transducer

K

Figure

12 12 Block diagram representation with exciter and AVR

vref

Exciter

K

1 sT3

Field circuit

Kl

I

AvL

A

K2

1

Hs+ KD


64/128

76

Small Signal Stability Chap.

12

representation is applicable to any type of exciter, with Gex s) representing the

transfQ

function of the VR and exciter. For a thyristor exciter,

G, s)

=

K

The terminal voltage error signal, which forms the input to the voltage

transducer

block, is given by Equation 12.132:

The coefficient K6 is always positive, whereas

Kg

an be either positive or negative

depending on the operating condition and the external network impedance

The value of K5 has a significant bearing on the influence of the VR on the damping

of system oscillations as illustrated below.

ffect of AVR on synchronizing and damping torque components

With automatic voltage regulator action, the field flux variations are caused by

the field voltage variations, in addition to the armature reaction. From the block

diagram of Figure 12.12, we see that

By grouping terms involving

Ay

and rearranging,

The change in air-gap torque due to change in field flux linkage is

As noted before, the constants K2, K3, K4, and K are usually positive; however,

K

may take either positive or negative values. The effect of the

VR

on damping and

synchronizing torque components is therefore primarily influenced by K and Gex s).

We will illustrate this by considering a specific case with parameters as follows:


65/128

~ e c 2 4

Effects of Excitation System

763

hi^ represents a system with a thyristor exciter and system conditions such that

K

is negative.

a

steady-state synchronizing torque coefficient:

From Equations 12.143 and 12.144, with s =jw =0, AT, due to Avfd is

.

Hence, the synchronizing torque coefficient due to Avfd is

We see that the effect of the AVR is to increase the synchronizing torque component

at steady state. With KA

O

i.e., constant EP), K q AI

=

-0.9. When KA

=

15, the AVR

tii

compensates exactly for the demagnetizing effect- of the armature reaction. With

K~

=2 9

K s ~ q f d )0 529 and the total synchronizing torque coefficient is

Here, we considered a case with K5 negative. With a positive K5 the AVR would have

an effect opposite to the above; that is, the effect of the AVR would be to reduce the

steady-state synchronizing torque component.

Although we have considered a thyristor exciter in our example, the above

observations apply to any type of exciter with a steady-state exciter1AVR gain equal

to KA.

b)

Damping and synchronizing torque components at the rotor oscillation

frequency:

Substitution of the numerical values applicable to the specific case under

consideration in Equation

12 143

yields


66/128


From Equation 12.144,

We will assume that the rotor oscillation frequency is 10 rad/s 1.6 Hz .

With

s=ja

=jlO,

With

K

=-0.12 and KA=200,

Thus the effect of the AVR is to increase the synchronizing torque component

and

decrease the damping torque component, when

K

is negative.

The net synchronizing torque coefficient

s

Ks

=

~

+ K s ~ u d

1.591 0.2804

=

1.8714

pu

torquelrad

The damping torque component due to Aty- is

K ~ ~ I V l d )

-0.3255 UA6

Since

A o , =sM/o =jaAGlo,,


67/128

Set 12 4

Ef fec ts o f E xc i ta tion Sys tem

with

=10 radls, the damping torque coefficient is

K

D A q f d )

=

12.27 pu torquelpu

speed

change

the absence of any other source of damping, the total

KD

=

KDcA6,.

It is readily apparent that, with K5 positive, the synchronizing and damping

torque components due to

Ayfd

would be opposite to the above.

For the system under consideration, Table 12.1 summarizes the effect of the

AVR

on Ks and KD at =10 radls for different values of KA.

able

12 1

With KA=O,Ay- is entirely due to armature reaction. The effect of the AVR

is to decrease KD for all positive values of KA. The net damping is minimum most

negative) for KA=200,and is zero for KA=m For low values of KA, the effect of the

AVR is to decrease Ks very slightly, the net Ks being minimum at KA of about 46. As

K

is increased beyond this value, Ksincreases steadily. For infinite value of KA, he

torque due to Ayfd is in phase with A6, and hence has no damping component.

We are normally interested in the performances of excitation systems with

moderate or high responses. For such excitation systems, we can make the following

general observations regarding the effects of the AVR:

KA

0.0

10.0

15.0

25.0

50.0

100.0

200.0

400.0

1000.0

Infinity

With K5 positive the effect of the AVR is to introduce a negative

synchronizing torque and a positive damping torque component.

The constant K5 is positive for low values of external system reactance and

low generator outputs.

K f ~ q f d )

0.0025

0.0079

0.0093

0.0098

0.0029

0.0782

0.2804

0.4874

0.5847

0.6000

The reduction in Ks due to AVR action in such cases is usually of no

particular concern, because K, is so high that the net Ks is significantly greater

s =K I K S C A ~ ,

1.5885

1.5831

1.5817

1.5812

1.5939

1.6692

1.8714

2.0784

2.1757

2.1910

K ~ ~ q f d )

1.772

0.614

0.024

1.166

4.090

8.866

12.272

9.722

4.448

0.000


68/128

766 Small-Signal Stability Chap 2

than zero.

With Kg negative, the AVR action introduces a positive synchronizing to,

ue

component and a negative damping torque component. This effect is

pronounced as the exciter response increases.

For high values of external system reactance and high generator outputs

~

is

negative. In practice, the situation where KQis negative are commonl

encountered. For such cases, a high response exciter is beneficial in increasing

synchronizing torque. However, in so doing it introduces negative damping

We thus have conflicting requirements with regard to exciter response.

one

possible recourse is to strike a compromise and set the exciter response so that

it results in sufficient synchronizing and damping torque components for the

expected range of system-operating conditions. This may not always be

possible. It may be necessary to use a high-response exciter to provide

the

required synchronizing torque and transient stability performance. With a very

high external system reactance, even with low exciter response the net

damping torque coefficient may be negative.

An

effective way to meet the conflicting exciter performance requirements with

regard to system stability is to provide a power system stabilizer as described in the

following section.

12 5 POWER

SYSTEM

ST BILIZER

The basic function of apower system stabilizer PSS) is to add damping to the

generator rotor oscillations by controlling its excitation using auxiliary stabilizing

signal s). To provide damping, the stabilizer must produce

a

component of electrical

torque in phase with the rotor speed deviations.

The theoretical basis for a PSS may be illustrated with the aid of the block

diagram shown in Figure 12.13. This is an extension of the block diagram of Figure

12.12 and includes the effect of a PSS.

Since the purpose of a PSS is to introduce a damping torque comp

Documents

Chapter 12 Small-Signal Stability