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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.
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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
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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.
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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.
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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
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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
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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
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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
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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.
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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
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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:
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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:
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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
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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
:
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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
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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:
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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
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~ e c12.3
Single Machine Infinite Bus System
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
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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
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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:
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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
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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
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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):
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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
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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.
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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.
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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,
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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
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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
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see 12.3
Single Machine Infinite Bus System
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
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Small Signal Stability Chap
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
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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
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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.
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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
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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:
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~ 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
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Small Signal Stability Chap
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,,
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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
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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