1
Power system small signal stability analysis using
genetic optimization techniques
Zhao Yang Dong
Department of Electrical Engineering
Yuri V. Makarov
David J. Hill
The University of Sydney
NSW 2006, Australia
Abstract
Power system small signal stability analysis aims to explore dierent small signal stability conditions and
controls, namely, 1) exploring the power system security domains and boundaries in the space of power system
parameters of interest, including load ow feasibility, saddle node and Hopf bifurcation ones, 2) nding the
maximum and minimum damping conditions, and 3) determining control actions to provide and increase small
signal stability. These problems are presented in the paper as dierent modications of a general optimization
problem, and each of them has multiple minima and maxima. The usual optimization procedures converge
to a minimum/maximum depending on the initial guesses of variables and numerical methods used. In the
considered problems, all the extreme points are of interest. Additionally, there are diculties with nding the
derivatives of the objective functions with respect to parameters. Numerical computations of derivatives in
traditional optimization procedures are time consuming. In the paper, we propose a new black box genetic
optimization technique for comprehensive small signal stability analysis, which can eectively cope with highly
nonlinear objective functions with multiple minima and maxima and derivatives which can not be expressed
analytically. The optimization result then can be used to provide such important informations as system optimal control decision making, assessment of the maximum network's transmission capacity, etc.
Keywords: Genetic Algorithms; Power System Security; Bifurcations; Stability
1. Introduction
In the open access environment, the power utilities
sometimes are forced to work far away from their predesigned conditions. In this situation, it is necessary
to re-approach the problems related to power system
security, stability and transfer capability [1]. On the
other hand, several recent major system blackouts in
dierent countries and voltage collapses require additional attempts in power system stability area. Several power system oscillatory instability problems occurred recently again put forward attention in the
research area of small signal stability.
In the space of power system parameters, the small
signal stability domain is restricted by complicated
surfaces of dierent kinds. These surfaces can be load
ow feasibility, aperiodic and oscillatory boundaries.
The last two are often referred to as Saddle node and
Hopf bifurcation boundaries. One of the most important tasks is to obtain the system security measure or,
in other words, the adequate stability margin. There
are many denitions for the stability margin [2] -[11].
But the denition based on the distances from the
system's state point to the stability domain boundary in the space of power system controlled parameters seems to be more appropriate [12], [13]. Due
to complexity of the boundary, it is quite dicult to
nd out the critical shortest distance which is used
as stability margin. Additionally, their are normally
several subcritical distances which are close to the
critical one. So the security margin must be assessed
in several critical and subcritical directions.
One of the approaches developed recently is the analytical approach [12], [14] -[17].
Corresponding to the general method proposed in
[18] the optimization problem can be depicted by
Fig. 1. Then in each direction dened by input y,
Input
Φ’
∆y
2. Small Signal Stability Analytical Approaches
In certain practical cases, it is necessary to analysis
the maximum transfer capability in a certain loading direction. For example, the problem can consist
in assessment of the maximum power transfer from a
particular generator to a particular load. To locate
the saddle node and Hopf bifurcations as well as the
load ow feasibility boundary points in a given loading direction within one procedure, the following constrained optimization problem is proposed, see [18],
) max=min
(1)
2
subject to
f (x; y0 + y)
t
~
J (x; y0 + y)l0 ? l0 + !l00
J~t (x; y0 + y)l00 ? l00 ? !l0
li0 ? 1
=
=
=
=
li00 =
0
0
0
0
0
(2)
(3)
(4)
(5)
(6)
where is the real part of an eigenvalue of interest, y0
is the current operation point, and y denes the
distance and direction to security boundaries from y0 ,
f (x; y0 + y) = 0 represents the load ow condition,
and J~ is the system Jacobian with l = l0 + jl00 as
its eigenvector corresponding to the eigenvalue =
+ j!. To consider the load ow constraint, this
constrained optimization problem can be represented
with Lagrange function form as
= 2 + f t (x; y0 + y) ) minx;; (7)
= 2 + 0 ) minx;;
(8)
The problem has many solutions including the load
ow feasibility, saddle node and Hopf bifurcations and
minimum and maximum damping points [18]. The result depends on the initial guesses of variables and selected eigenvalue. The goal is to determine the small
signal stability boundary point which is closest to the
point y0 . To get this point, it is necessary to solve
the problem for dierent eigenvalues and initial values of the loading parameter , which is a very time
consuming task.
Φ
State
Matrix
x
Output
f
Eigenvalues
τ
η
State Variables
Optimization
procedure
Rotation
Fig. 1. System Model Diagram for Small Signal Stability General Method Optimization
with considering of the state variables shown in Fig. 1,
the load ow constraint is computed rst, while the
state variables, which are to be optimized, provide input value to set up the state matrix. The eigenvalue
computation is then based on this matrix. In the
algorithm, the real part of the critical eigenvalue(s)
will be used to form the objective function . The
value of the objective function f , as the output, will
then provide information used to proceed the optimization procedure and the state variables are to be
adjusted accordingly. When the optimization process
converged, all characteristic points along the direction dened by y can be located. To located all
these points in the whole plane, a loop is needed to
rotate the direction and repeat the optimization procedure locating all characteristic points in the whole
plane/space of interest. Generally, this plane/space
is a cut set of the space spanned by all parameters of
interest [19].
In case of exploring the shortest distances, the problem becomes even more complicated. In principle, it
is possible to use the problem represented by equations (1) - (6) to get the critical distance vector. Ideas
similar to those put forward in [12] can be tried. For
example, after obtaining the closest instability point
along the given direction y, it is possible to analyze
the angle between the loading and left eigenvector of
the corresponding matrix [load ow Jacobian or state
matrix] in this point. As this vector shows the normal direction with respect to the stability boundary,
the angle computed can be used to rotate y in the
direction where the distance decrease.
A more ecient way consists in the direct computation of the closest instability point. To get this point,
the following modication of the general optimization
problem (1) -(6) can be used,
jjy ? y0 jj2 ) min
(9)
subject to
f (x; y) = 0
(10)
t
0
0
00
J~ (x; y)l ? l + !l = 0
(11)
t
00
00
0
~
J (x; y)l ? l ? !l = 0
(12)
0l ? 1 = 0
(13)
i
li00 = 0
(14)
Once again, the result depends upon the initial
guesses of variables and selected eigenvalues. The
choice of these parameters is a complicated task. One
can use some practical ideas regarding the most dangerous loading direction and critical eigenvalues [for
example, corresponding to the inter-area oscillatory
modes.]. Nevertheless, it looks quite dicult to get
all of the critical and subcritical distances by this approach.
One of the most confusing diculties in all above
mentioned procedures is computing the small signal
characteristic points in view of breaks in objective
function and constraints. For example, to take account of the reactive power limits, for generators reactive power limits, we must use dierent models in the
constraint set. A sudden change in the model causes
sever problems for the optimization procedures. In
fact, it can lead to an instant instability [12], [20].
To nalize, we should conclude that the analytical
approaches to the small signal stability analysis have
many problems which can be hardly solved by traditional optimization techniques. That is why we are
attempting to apply the genetic optimization procedures.
3. Genetic Algorithms with Sharing
Function Optimization Method
Genetic Algorithms(GAs) [21] are heuristic probabilistic optimization techniques inspired by natural
evolution process. In genetic algorithms, the tness
function is used instead of objective function as in
the traditional optimization procedures. Each concrete value of variables to be optimized is called as
an individual. Then a certain number of individuals
composes the generation. In the process of GAs, individuals with better tness survive and those with
lower tness die o, so to nally locate individual with
the best tness as the nal solution. They are capable
of locating the global optimum of a tness function in
a bounded search domain, provided a sucient population size is given. The GA sharing function method
is able to locate the multiple local maxima as well.
Genetic algorithms have been already eectively
applied in complicated multidimensional optimization problems, which can be hardly solved by traditional optimization methods. Genetic algorithms,
which mimic the natural evolution, usually contain
the following steps. Firstly, produce the initial population; secondly, evaluate tnesses for all individuals
in the current population; thirdly, perform such operations as crossover, reproduction and mutation depending on the existing generation tnesses, and form
a new generation. Then the procedure is repeated till
some termination criterion is met and the optimum
is thus obtained [21].
3.1. Sharing Function Method
To compute multiple maxima of the tness function, the genetic algorithm sharing function method,
[21], [22], can be applied. The method decreases the
tnesses for similar individuals by the \niche count",
m0 (i). For each individual i, the \niche count" is computed as a sum of sharing function values between
the individual and all individuals j in generation, see
Eqn. (16). The similarity of individuals is evaluated
by the distance, d(i; j ) from each other, see Eqn. (15).
The resulting shared tness 0 is changed through
dividing the original tness by the corresponding
niche count, see Eqn. (17), [21] -[23].
d(i; j ) = d(xi ; xj )
n
X
sh[d(i; j )]
m0 (i) =
j =1
i)
0 (i) = Pn (
sh
j =1 [d(i; j )]
(15)
(16)
(17)
The sharing function is dened so that it fullls,
8
<
sh(d) = :
0 sh(d) 1
sh(0) = 1
limd!1 sh(d) = 0
(18)
For example the sharing function can have the form,
sh(d) =
1 ? ( d ) ; if d < 0;
otherwise
(19)
where is a constant, and is the given sharing factor. By doing so, an individual receives its full tness
value if it is the only one in its own niche, otherwise
its shared tness decreases due to the number and
closeness of the neighboring individuals.
In this paper we apply the genetic algorithm with
sharing to small signal stability analysis. This optimization problem is highly non-linear and some
times, even non-dierential-able, which makes it very
dicult to be solved by traditional optimization
methods.
3.2. Black Box System Model
For the analytical approaches, the optimization
problems is highly non-linear and some times, even
non-dierential-able. It is known that the traditional
optimization methods meet serious diculties with
convergence while solving such problems. Besides the
rest, the constraint sets in Eqn. (2) -(6) take account
of only one eigenvalue during the optimization. To
get the stability margin for all eigenvalues of interest, as well as the critical load ow feasibility conditions, it is necessary to vary the initial guesses and
repeat the optimization. Additionally, the functions
in Eqn. (2) -(6) can have breaks due to dierent limitations applied to power system parameters. For example, the generator current limiters may cause sudden changes in the model, and consequently breaks in
the constraint functions. This makes the analytical
optimization problem even more complicated.
In the genetic optimization procedures, those diculties can be overcome by using the black box power
system model given in Fig. 2, described in the sequel.
Unlike the model used in the analytical form of small
Black Box System Model
γ
Load flow
calculation
converged
not converged
State variable
State matrix
Eigenvalue
calculation
formation
calculation
Fitness
Function
Φ
α=0
Fig. 2. Black Box System Model for Optimization
signal stability problems, this black box has control
parameters as inputs, and the tness function as
outputs. Inside the black box, the load ow is computed rst. If it converges, then the state variables
and matrix are computed, and then the eigenvalues of
the state matrix are obtained. Thereafter, the critical
eigenvalue is chosen for analysis. The critical eigenvalue's real part is used to compute a particular value
of the tness function. If the load ow does not not
converge, which means that a load ow solution does
not exist, we put the critical eigenvalue real part to
zero. By such a way, the load ow feasibility points
are treated in the same way as the saddle node and
Hopf bifurcation points. The tness function can
be changed quite exibly depending on the concrete
task to be solved, see the next section for explanation.
To demonstrate the advantages of the black box
model, let's consider the tasks in Abstract. To reveal
all characteristic small signal stability points, such
as maximum loadability, saddle and Hopf bifurcation
etc. along a given ray y0 + y in the space of power
system control parameters, y, the general small stability problem Eqn. (1) -(6) can be used. If the above
problem is solved by traditional optimization methods, the solution obtained depends on initial selection
of the eigenvalue traced, and variables x. Moreover,
even for one eigenvalue selected, it is not possible to
get all the characteristic points in one optimization
procedure. By applying the black box model and GA
techniques all the problem characteristic points can
be found within one optimization procedure. In this
case, the input is the loading parameter, = , and
the tness function is 2 for maximization and 12
for maximization. To compute the function, the load
ow is computed for a given value of . If the load
ow converges, then the state matrix and its eigenvalues are computed, an eigenvalue of interest is selected (for example, the critical eigenvalue with the
minimum real part.), and used to get the tness function. The black box model has only one input and one
output, and is used in the standard GA optimization.
To nd out all the critical distances to the load ow
feasibility and bifurcation boundaries in the problem,
the same black box system model can be used. In
this case, the inputs are y and , and the tness
function is increased when the distance decrease and
the critical eigenvalue real part tends to zero.
It is understood that the shape of power system
small signal stability boundaries can be very complicated, and there exist many niches or in other words,
local maxima/minima. In order to ensure GA to locate the multiple maxima of the tness function, and
to avoid the noise induced by genetic draft, sucient
population size should be considered. However, too
large population size will result in slow convergence.
Techniques for choosing the population size can be
found in [23]. In our test systems, the population
size in the range from 30 to 160 was selected. It has
been discovered that this population size is sucient
to locate the maxima in the space of power system
variables.
3.3. Fitness Function Formulation
1
0.9
(; d) = 1 (d)2 ()
(20)
where is the real part of the system critical eigenvalue, d = jj(y ?y0)jj is the distance from the current
operating point y0. The diagonal matrix scales the
power system parameters of dierent physical nature
and range of variation. The rst multiplier in Eqn.
(20) reects the inuence of distance, and the second
one keeps the point y close to the small signal stability boundary. For example, the following expressions
for 1 and 2 can be exploited, Eqn. (22) and (22),
1 (d) = 1=d
2 () = ek?2
(21)
(22)
where k is the factor dening the range of critical
values of . The second multiplier acts as a lter.
If k is very large, say about 1000 or more, then only
those s which are very close to zero can pass the lter
and survive during the genetic optimization process.
The lter eliminates a large number of negative s,
to force the GA select individuals close to the small
signal stability boundaries. The lter function takes
the shape as shown in Fig. 3
0.8
0.7
Filter Function Values
The tness function plays an important role in genetic algorithms. The optimization result depends
extensively on the tness function's ability to reveal
the inuence of the factors of interests. There are
many ways to convert a practical objective function
into a tness function, [21]. Here, since the small
signal stability properties of the system are of prime
importance, the tness function should be selected to
reect the inuence of the critical eigenvalue and the
critical distance to the small signal stability boundaries. In this paper, we suggest the following general
form of the tness function, see Eqn. (20)
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.01 −0.008 −0.006 −0.004 −0.002
0
0.002
Input variable
0.006
0.008
0.01
Fig. 3. The Filter Function for the Fitness Function
4. The Optimum Operation Direction
As a result of the former parts of the paper, critical and sub-critical distances are computed by GAs.
In a wider meaning, these distances can also be any
most inuential directions of operation in the space of
any power system parameters of interest. For example, they can associated with critical/subcritical distances, minimum damping conditions, Saddle node
or Hopf bifurcations, load ow feasibility boundaries
etc. Upon obtaining these directions, the optimum
operation direction can be dened thereafter. The
approach can be visualize by Fig. 4, where any two or
γ2
V1
3.4. Population Size
The shape of power system small signal stability
boundaries can be very complicated, and there are
many niches existing. In order to ensure GA to locate
the multiple maxima of the tness function, and to
avoid the noise induced by genetic draft, the sucient
population size should be considered. Techniques for
choosing the population size can be found in [23]. In
our test systems, the population size in the range from
30 to 200 was selected. It has been discovered that
this population size is sucient to locate the maxima
in the space of power system variables.
0.004
V3
γ1
O
V2
γ3
Fig. 4. The Cutset of Power System Security Space
more vectors, for example, the critical and subcritical
distance vectors, in the parameter space are located
by GA sharing function method as V~1 and V~i , where i
= 2,...,m are subcritical distance vectors. They form
a cutset in the space and dene a new vector
m
X
V~k = ?( kvi V~i ); i = 1, 2, ..., m
i=1
(23)
where kvi and kvj are weighting factors depending
on the inuence of parameter sensitivity as well as
the dierence between dierent jjVi jjs. For example,
to reveal the inuence of the critical and subcritical
distance toward instability, they can be taken as
kvi / 1=jjVi jj:
Pj
Closest Small signal
stability boundariies
subcritical
solution 4
5. GAs with Classical Optimization
Methods for Optimization
In cases where more detailed information about the
small signal stability boundaries are required, both
genetic algorithm and classical optimization methods should be used provided that the problem can
be handled by classic optimization methods. Since
the stability boundaries are very complicated, we are
only interested in those cut-sets where most inuential stability conditions exist, i.e. areas close to critical and subcritical distances, minimum damping etc.
In such cases, GAs can be used rstly to locate the
approximate optima, then classic method will be applied in the vicinity of these approximate optima to
explore only these critical and subcritical areas in detail. Note that classic method will be used only if
the problem formulation can be handled by it, i.e.
dierentiable, convex etc. As shown in Fig. 5, these
solutions obtained by GA is the initial guessed solution for classic optimization, and the area close to
them is the area where exact optima calculation is
required.
Since GA can locate only approximate optimum,
classic method will have to be used to get the exact optimum depending on the problem accuracy requirement and problem dimension. Note that classic
method will be used only if the problem formulation
can be handled by it, i.e. dierentiable, convex etc.
As shown in Fig. 5, these solutions obtained by GA
Critical
Solution
0
γ2
(25)
Then V~k gives the direction of optimum operation
which enables, at least in the meaning of system parameters involved, safest operation direction. Stability problems of other kinds can be investigated accordingly.
γ1
γ4
(24)
Or in the sense of parameter sensitivity, they can be,
kvi = @Vi =@pi :
is the initial guessed solution for classic optimization, and the area close to them is the area where
exact optima is required. When the most evident
γ3
Pi
subcritical solution 3
subcritical solution 2
Fig. 5. The most evident small signal stability condition points
stability conditions or critical & subcritical distances
have been located, the initial most inuential scope
of these characteristic conditions will be explored by
classic optimization method in the vicinities if necessary and possible.
By combination of these two methods the computation costs will be reduced because un-important conditions will not need to be computed, and subsequent
control activities can be based on the information obtained from the optimization procedures. Similar step
size/direction control algorithms - the multi-rate algorithms can be found in [24].
6. Power System Model for Small Signal Stability Analysis
6.1. Single Machine Innite Bus Model
Analysis
A well known single machine innite bus system is
studied here with GAs and sharing function method
to locate its critical and subcritical distances to instability. The model consists of four dierential equations, which cover both generator and load dynamics
[25], see Fig. 6.
The mathematical model of the system is the following:
m_ = !
(26)
M !_ = ?dm ! + Pm +
+Em ymV sin( ? m ? m ) +
Eo
~
0
yo
(−θ o- π/2)
V δ
Em
ym (−θ m- π/2)
δm
20
~
15
10
Pd+jQd
Fig. 6. The single-machine innite-bus power system model
+Em2 ym sinm
(27)
Kqw _ = ?Kqv2 2 V 2 ? Kq vV +
+E00 y00 V cos( + 00 ) +
+Em ym V cos( ? m + m ) ?
?(y00 cos0 + ym cosm )V 2 ?
?Q0 ? Q1
(28)
2
2
_
k4 V = Kpw Kqv V + (Kpw Kqv ? Kqw Kpv )V +
q
2 + K 2 [?E 0 y 0 V cos( + ? h) ?
+ Kqw
0
pw
0 0
?Em ym V cos( ? m + m ? h) +
+(y00 cos(0 ? h) + ym cos(m ? h))V 2 ] ?
?Kqw (P0 + P1 ) + Kpw (Q0 + Q1 )
(29)
qw
). Paramwhere k4 = TKqw Kpv and h = tan?1 ( KKpw
eters of the system are the following [25]: Kpw = 0:4,
Kpv = 0:3, Kqw = ?0:03, Kqv = ?2:8, Kqv2 = 2:1,
T = 8:5, P0 = 0:6, Q0 = 1:3; P1 and Q1 are taken
zero at the initial operating point.
Network and generator values are: y0 = 20:0, q0 =
?5:0, E0 = 1:0, C = 12:0, y00 = 8:0, 00 = ?12:0,
E00 = 2:5, ym = 5:0, m = ?5:0, Em = 1:0, Pm = 1:0,
M = 0:3, m = 0:05.
All parameters are given in per unit except for angles, which are in degrees. The active and reactive
loads are featured by the following equations:
Pd = P0 + P1 + Kpw + Kpv (V + T V_ ) (30)
Qd = Q0 + Q1 + Kqw + Kqv V + Kqv2 V 2(31)
The system (26) -(29) depends on four state variables , m , !, V . Their values at the initial load ow
point are the following: = 2:75, m = 11:37, ! = 0,
and V = 1:79. Note that the initial point is not a
physical solution as the voltage V is too high as Q1
is zero.
To show the results in the parameter plane, which
have been obtained formerly by the authors in [18],
[26], the boundaries are plotted in solid line in Fig. 7.
In the same gure, critical and subcritical stability
points located by Genetic Algorithm with sharing
function method are plotted in . It is evident that
this method with the black box model can locate the
Reactive Load Power, p.u.
C
5
0
−5
−10
−15
−20
−25
−30
−30
−20
−10
0
10
Active Load Power, p.u.
20
30
Fig. 7. The Closest Stability Boundary Points
desired results. They can be used to nd out optimum operation directions by Eqn. (23) given in
former section.
6.2. Model Analysis for Optimal Control
Direction
This classical power system model composed of 3
machines and 9 buses is taken from [27] as shown in
Fig. 8. Stability studies for a similar system can be
C
2
3
8
2
3
7
9
5
6
A
B
4
1
1
Fig. 8. The 3-machine 9-bus System
found in [28], [29]. The machines of the system are
modeled by using the classical model for machine 1
and two-axis model for machines 2 and 3 - see equations (32) {(37).
j1 !_1 = Tm1 ? E1 Iq1 ? D1 !1
(32)
!1
?Edi0 ? (xqi ? x0i )Iqi
EFDi ? Eqi0 + (xdi ? x0i )Idi
Tmi ? Di !i ? Idi0 Edi0
?Iqi0 Eqi0 ? Edi0 0 Idi ? Eqi0 0 Iqi
_i = !i
i = 2; 3
=
=
=
=
(33)
(34)
(35)
(36)
(37)
Unlike [28] and [29], in our tests we neglected the excitation system dynamics, so our state variables were
the following: , !, Eq0 , and Ed0 . Algebraic variables
were Id , Iq , , and V ; bifurcation parameters were
Pld and Qld . . The notations, parameter values and
description of the system (32) {(37) can be found in
[27].
In the general case of small signal stability analysis,
the load dynamics should be denitely taken into consideration, but nevertheless some stability aspects can
be studied with the constant load model [30]. This 3
machine 9 bus model considers constant load models
only.
In the preliminary examination, eigenvalue minimization approach is applied. The loads Qld at buses
5, 6 and 8 were chosen as varying parameter to dene
the hyper-space within which the security boundaries
lay. The loading was repeatedly tried by GAs with
sharing function till the point where load ow did
not converge where the tness function was dened
as zero so the individuals associated will die o in the
next generation. At each step, eigenvalues of the state
matrix were computed to nd the bifurcation points
and minimum and maximum damping conditions. In
the problem of locating the critical distance to stability boundaries, the entry representing the real part of
system critical eigenvalue, was set to zero instead
of the tness function itself in case the load ow does
not converge.
Initially two most evident solution points were
found in the space dened by load powers at buses
5, 6 and 8. They form two vector, V~1 and V~2 starting from the initial operation point, and thus the
plane was dened as a cut-set in the whole parameter space. The result was shown in Fig. 9, from
these two vector, the optimum operation direction dened as V~3 which ensures maximum loading increase
while maintaining system security can be calculated
as V~3 = ?(k1 V~1 + k2 V~2 ). The load ow and stability properties of the system alone that direction were
checked to verify the transmission capacity. The result of GA optimization was given in table 1. Note in
table 1. the entries 1; 2; 3 till 6 are GA optimization
results indicating the solutions on a comparative base.
Optimal Loading Direction
40
Active Load Power at Bus 8, p.u.
_1
0
_
q0i Edi0
d0 0i E_ qi0
ji !_i
Vec3
30
20
10
0
−10
Vec2
−20
0.34
Vec1
0.52
0.32
0.51
0.3
0.5
0.49
0.28
0.48
0.26
Active Load Power at Bus 6, p.u.
0.47
Active Load Power at Bus 5, p.u.
Fig. 9. The Critical Directions ( 1
2 ) to Small Signal Stability Boundaries and its Corresponding Optimum
~2
~1
V ec
Direction ( 3 ), where 3 = ? ( VV ec
~ 1 + V ec
~ 2 ).
ec
V~
ec ; V ~
ec
V~
ec
No:
1
2
3
4
5
6
V~
ec
k
jj
jj
jj
Table 1
Load direction where fitness function
reveals the most evident
small signal stability points
rad:
(
)
-1.5708
-1.5708
1.5704
1.5708
1.5708
-2.5417
rad:
(
)
-1.5264
-0.7645
-1.0531
0.1141
2.2257
2.3514
p:u:
(
)
-0.3503
-0.5057
-0.4029
3.0745
0.4413
0.6378
jj
Critical
Eignevalue
0.0368 + j0
0.0300 + j0
0.0186 + j0
0.0158 + j0
7.2969 10 04 + j0
-0.0021+ j0
?
, and dene the loading variation as mentioned
before, i.e. Pi = cos cos , Pj = cos sin ,
Pk = sin . These result is from GA with sharing
function method, and the tness function is on closest distance to critical and sub-critical eigenvalues.
However, to make the solution more practical, more
powerful GA algorithms are needed to provide better
solution reliability and faster computation speed.
7. Conclusion
The GAs based approach to small signal stability analysis method using a black box power system
model was addressed in this paper aimed at providing
means to increase transmission capacity while ensuring the system stability. With appropriate system
model the method provided here can locate global as
well as local optimum solutions in the form of critical
and subcritical distances to to the small signal stability boundaries. They can be used to make decision on
control actions. Ideas of dening the optimum operation direction based on critical and subcritical directions or other conditions were put forward here. To
make the techniques more practical and ecient, further research on GAs with sharing function method
and simplied eigenvalue computation algorithms are
required.
Acknowledgment
This work was sponsored by the ERDC/ESAA
Research Program Contract PN2420/94120. Z. Y.
Dong's work was supported by Sydney University Electrical Engineering Postgraduate Scholarship.
The authors would like to thank Mr. Haining Liu of
Tianjin University for his helpful discussion on Genetic Algorithms.
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Zhao Yang Dong (Student Membe IEEE'96) was
born in China, 1971. He received his BSEE degree
as rst class honor in July, 1993. Since 1994, he had
been studying in Tianjin University (Peiyang University), Tianjin, China for his MSEE degree. Now, he is
continuing his study as a postgraduate research student in the Department of Electrical Engineering, the
University of Sydney, Australia. His research interests include power system analysis and control, genetic algorithms, electric machine and drive systems
areas.
Yuri V. Makarov received the M.Sc. in Computer Engineering (1979), and the Ph.D. in Electrical
Networks and Systems (1984) from the St. Petersburg State Technical University (former Leningrad
Polytechnic Institute), Russia. He is an Associate
Professor at Department of Power Systems and Networks at the same University. Now he is conducting
his research work at the Department of Electrical Engineering in the University of Sydney, Australia. His
research interests are mainly in the eld of power system analysis, stability and control with emphasis on
mathematical aspects and numerical methods.
David J. Hill (M IEEE'76, SM IEEE'91, F
IEEE'93) received his B.E. and B.Sc. degrees from
the University of Queensland, Australia in 1972 and
1974 respectively. In 1976 he received his Ph.D. degree in Electrical Engineering from the University
of Newcastle, Australia. He currently occupies the
Chair in Electrical Engineering at the University of
Sydney. Previous appointments include research positions at the University of California, Berkeley and
the Department of Automatic Control, Lund Institute of Technology, Sweden. From 1982 to 1993 he
held various academic positions at the University of
Newcastle. His research interests are mainly in nonlinear systems and control, stability theory and power
system dynamics and security. His resent applied
work consists of various projects in power system stabilization and power plant control carried out in collaboration with utilities in Australia and Sweden.