Download Application of Artificial Neural Network (ANN) technique for the

Application of Artificial Neural Network (ANN) technique for the
measurement of voltage stability using FACTS Controllers
Abstract: A Methodology is
proposed for the online monitoring and
assessment of voltage stability margins,
using artificial neural networks and FACTS
controllers with a reduced input data set
from the power system. In this methodology,
first the system model is reduced using self
– organized artificial neural networks and an
extended AESOPS algorithm. Then
supervised learning of multilayered artificial
neural networks is carried out on the basis of
this reduced model. Finally, on the basis of
trained network and the reduced set of
system variables, monitoring is carried out
along with the assessment of voltage
stability margins. This methodology is tested
comparatively with a methodology for
monitoring and assessing voltage stability
using a complete input data set. The tests
were carried out on a real power system with
92 buses. The results obtained indicate the
justifiability of using a reduced system
because of the increased efficiency and
accuracy of calculation, both in the learning
stage and in the recall stage of the artificial
neural network.
static voltage collapse the practical
implementation of this method is carried out
while defining an entire series of simple
indicator which show the static voltage
stability margins. Some of this indicators
can be used in an online environment,
however such a simplified approach renders
good results. The dynamic method of
approach is based either on a linearised
dynamic method of the power system and
Eigen value analysis or on a non-linear
dynamic model of the power system and
direct methods. Such as dynamic bifurcation
and numeric transient simulations. The
method based on the dynamic approach give
a much more real picture of voltage stability
margins. However, methods based on the
dynamic approach are time consuming in
terms of computer time for the online
environment. The most attractive means for
solving this problem is found in artificial
neural network (ANNs). These networks can
be trained offline for complicated mapping,
such as the difficult problem of determining
voltage stability margin, and can be used in
an efficient and timely way in the online
environment to monitor and assess the same.
In this paper, an entire series of efficient
methods are proposed which are based on
artificial neural networks (ANNs) and on
various indicators of voltage stability
however all the methods proposed are
mostly based on the complete vector of
system variables, which significantly
diminishes their efficiency.
This paper deals with the
development of a new methodology for
monitoring and assessing voltage stability
margins, it is a methodology based on
artificial neural networks (ANNs) with a
reduced input data set. The motive of this
research is to establish what minimal
necessary input data set is for the precise
I) Introduction: Most of the large
power systems are badly suffered due the
phenomena of voltage instability. The basic
method of preventing such severe system
incidents is timely identification of voltage
instability. In this sense, the development is
made by using suitable computer producers
in online environment for monitoring and
assessing the voltage stability margins. Two
method are adopted to solve the stability
problems I) static method ii) dynamic
method the. This approach of static method
is based on the steady state load flow model
for calculating power flow and assessing
how far the Jacobian matrix is true from the
singularity function, that is, from classis
138
monitoring and assessment of voltage
stability. The methodology proposed is
based on static and dynamic stability
margins, which are determined by eigen
value analysis of the linearised static and
dynamic models of a power system, and also
on the use of multilayered and self
organized artificial neural networks (ANNs).
This methodology is composed, as in the
practice of a learning stage and a recall
stage. In the learning stage, first reduction of
the model of the observed power system is
carried out using self-organized artificial
neural network (ANNs) and an extended
AESOPS algorithm. This reduction is made
for all relevant configurations, while
defining the relevant configurations, all
outages of lines and generators are
considered. As well as the most portable
combinations of double outages. Second, the
training of multilayered feed forward
artificial neural networks (ANNs) is carried
out on this reduced model, by varying the
load power over the range of the nighttime
minimum and daytime peak value. During
the training, mapping is established between
the reduced input set of system variables and
the voltage stability margin. Finally, during
recall or exploitation based on reduced set of
system variables, the monitoring and
assessment of voltage stability margins of
the considered operating regime is
performed. The proposed methodology is
tested with a methodology for monitoring
and assessing voltage stability based on the
complete input set. The results obtained
indicate the justifiability of using a reduced
system in the learning stage through the
increased efficiency and quality of leaning
of the artificial neural networks (ANNs) and
in the exploitation stage through the
increased efficiency and reduction of
measurements required from the system.
(b)
Fig-1. Configuration of artificial neural
network (ANNs) considered
a) UR
b) NT
The proposed methodology for monitoring
and assessment voltage stability is based on
self-organized and multilayered feed
forwarded artificial neural networks
(ANNs). It consists of four stage: 1)
configuration and design procedure. 2)
Stability analysis. 3) Reduction of system
model using a self organized artificial neural
network (ANN) and an extended AESOPS
algorithm, and training the multilayered
artificial neural network (ANN) over such a
reduced model. 4) Exploitation – that is the
monitoring and assessment of voltage
stability margins. In the text that follows the
basic steps in the proposed methodology are
briefly elaborated.
2.1) Configuration and design procedure of
the artificial neural networks (ANNs)
(a): NY
2) Concept of Proposed Methodology:
(b): NR
Fig-2. Configuration of artificial neural
networks (ANNs) considered
(a)
139
Fig-3. Algorithm ‘AY’ based on the
complete input set of system
In the learning stage, four artificial neural
network
(ANN)
configurations
are
considered as in figures 1(a2b), 2(a2b)
i) UR-is a self-organized ANN intended for
the reduction of the vector of system
variables the input to this ANN is the
complete vector of system variables (Y) and
the out put is the reduced vector of system
variables (YR) reduction of the vector of
system variables is carried out for each
relevant topology.
Fig-4. Algorithm ‘AR’ based on the reduced
input set of system variables
ii) NT-is a multilayered ANN for detecting
system topology, the input to this ANN is
the complete vector of system variables, and
the out put is binary identification code of
the topology.
All the artificial neural networks
(ANNs) mention ed were configured with
one input layer, one out put layer, and one
hidden layer each. If the number of hidden
layers is increased to two, the learning time
is increase significantly and there is no
increase in accuracy. The number of neurons
in the hidden layer for all absorbed networks
was designed so that a minimal square error
was obtained. For most of the artificial
neural network observed in this paper, the
number of neurons in the hidden layer is
30% less than the number of neurons in the
input layer. The artificial neural networks
(ANNs) for detecting system topology had a
sigmoid activation function for the neurons
in the hidden layer and for the neurons at the
output layers the activation function was a
pure linear function. The (ANNs) for
evaluating system stability had a tangent
hyperbolic activation function for the
neurons in the hidden layer, and for the
neurons at the out put layer the activation
function is a pure linear function.
iii) NY-is a unified multilayered ANN for
evaluating voltage stability margins for each
considered topology of the power system.
The input to this ANN is the complete
vector of system variables and the
identification code of the topology. The
output is the real part of the critical Eigen
value (Sc) and also the voltage stability
margin M.
iv) NR- is a multilayered ANN for
evaluation of the voltage stability margin for
the considered topology of the power
system. The input to this ANNs is a reduced
vector of system variables and the output is
the real part of the critical Eigen value and
also the voltage stability margin.
By
combining
these
four
configurations two global algorithms for
analyzing stability are obtained.
2.2) Analysis of stability:
The basic stability analysis model is
referred in this paper. In addition, this model
is extended to accept the following
components. Standard types of turbines and
governing systems, basic types of automatic
voltage regulators (AVRs) including the
power system stabilizer (PSS) and finally an
aggregate model of an induction machine
are also included. In this model the fast
transient processes in all machines
stator/network is ignored. The elimination of
these transient leads to standard singular
perturbation form of the description:
AY-an algorithm based on a combination of
‘NT’ and ‘NY’ configurations of the neural
network as in fig-3.
AR-an algorithm based on a configuration of
‘NT’ ‘UR’ and ‘NR’ configurations of
neural network as in fig-4
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x=f(x,y) ………… (1)
0=-g(x,y) …………(2)
Where x & y are the vectors of dynamic
state variables and algebraic system
variables, respectively.
x1=(xM, xAVR, xG) ... (3)
y=(,V,P,Q) ………(4)
Where the meaning of following
symbols is i.e., xM, xAVR, xG – the sub vector
of dynamic state variables of synchronous
and induction machines, the sub vector of
AVR dynamic state variables including the
pss and dynamic state variables of the
turbine and governing system respectively;
,V,P,Q – the sub vector of voltage phase
angles, the sub vector of voltage
magnitudes, the sub vector of injected active
powers and the sub vector injected reactive
powers of all nodes in the observed power
system,
respectively.
Monitor
and
assessment of voltage stability in this paper
are actually based on the algebraic system
variable vector and on various combinations
of its sub vectors-(,V,P,Q).
For analysis of stability, the
algorithm of fig-5 is used. Both the static
and dynamic aspects of voltage stability are
considered. The static aspect is based on the
distance of operating point from the saddle
node bifurcation point which corresponds to
the singularity of the linearised matrix of the
static power system model, while the
dynamic aspect is based on the distance of
the operating point from the critical node
bifurcation point which corresponds to the
null real parts of the pair of complex eigen
values linearised the matrix of the dynamic
power system model.
Fig-5. Flow chart
Stability analysis is basically
differentiated for the ‘AR’ algorithm that is
based on a reduced system model and for the
‘AY’ algorithm, which is based on the
complete system model. Stability analysis
for the ‘AR’ algorithm is carried out in six
steps as in fig-6. The calculation of stability
analysis begins by taking the night minimum
as the starting system load, this load is then
distributed to all loaded buses according to
the specified distribution co-efficient and a
random number generators. In the first step
the observed load level ‘P’ determine the
stationary point (x0,y0). In the second step,
linearisation of the dynamic model is carried
out using the defined stationary point. Then,
elimination is carried out of the algebraic
system variable vector (y0) for dynamic
stability analysis, and then the elimination of
the dynamic state variable vector (x0) for the
analysis of the static stability. In this way,
linear models are obtained for dynamic and
voltage stability. In this way linear models
are obtained for dynamic and static voltage
stability analysis. In the third step system
reduction is carried out first by applying the
self organized ANN from the vector of
dynamic and algebraic variables (x,y) the
reduce dynamic and algebraic vectors (xR,
yR ) are obtained, and then by applying an
141
Fig-6. Measurement of the real part of the critical
Eigen values for the algorithms AY and AR
a) AY-(,V) (----), AR-(,V),(----), simulation
b) AY-(P,Q) (----), AR-(P,Q),(----), simulation
Fig-8. Measurement of the voltage stability margin for
the algorithm AY and AR
a) AY-(,V) (----), AR-(,V),(----), simulation
b) AY-(P,Q) (----), AR-(P,Q),(----), simulation
Fig-7. Measurement of the real part of the critical
Fig-9. Measurement of the voltage stability margin Eigen
values for the algorithms Ay & AR
for the algorithm AY & AR
a) AY-(V,P,Q) (----), AR-(V.P,Q),(----), simulation
a) AY-(V,P,Q) (----), AR-(V,P,Q),(----), simulation
b) AY-(,V,P,Q) (----), AR-(,V,P,Q),(----), simulation b) AY-(,V,P,Q) (----), AR-(,V,P,Q),(----), simulation
142
Extended AESOPS algorithm from the
dynamic and static system matrix E & F, a
reduced system matrices ER & FR are
obtained. In this way reduced dynamic and
static models are obtained. The application
of self organized ANNs for the reducing
vector x & y is described in the next section,
while the application of the extended
AESOPS algorithm for system reduction is
described in the section 4.1. In the fourth
step by applying QR algorithm to the
reduced static and dynamic models the
Eigen values (i, vi) are determined. In the
fifth step the smallest Eigen values (f) are
determined and then the following variables
are recorded: actual load level (p), the
reduced vector of algebraic system variables
(yR) and the smallest critical Eigen value
(f ). In the final step testing is done to see if
the critical load level (Pcrit) has been reached
and if all condition for completing the
calculation have been fulfilled.
When all condition for completing
the calculation have been fulfilled the
voltage stability margin is determined for all
operating points using the following
equation:
i) System reduction using self organize
ANN: The self organized ANN is introduce
in order to reduce the learning time of the
ANN and to reduce the number of system
variables needed for voltage stability margin
monitoring and assessment. According to
this method the problem of reducing the
vector of system variables from the ANN
can be presented as a problem of extracting
characteristics variables and the variables,
which differ from one another. In other
words the algorithm developed for the
reduction of vector system variables groups;
its coordinates, which are sufficiently,
approximate in time, and separate the
dissimilar once. The straight variable, which
is closes to the calculated center of group, is
taken as the characteristic representatives of
the group. Finally by grouping all
characteristics representative of the group in
one vector, the reduced system vector (yR) is
obtained. If only the chosen representative
of each group is observed during the
formation of the system model, the
dimension of the system description can be
decrease. A more detail review of the ANN
algorithm used is to be found in the, section
4.2.
M = Pcrit – p ………(5)
Pcrit
Calculation of the stability analysis
for the AY algorithm is standard and is
different from the described calculation in
that it does not contain step 3 for system
reduction. In this way stability analysis and
determination of Eigen values is done for
the entire system model.
ii) Supervised learning of the ANNs:
After system reduction using the
result of stability analysis on a reduced set
of system variables learning is carried our.
Supervise learning of multilayered ANN is a
process which the correction of synaptic
weights between neurons in proximate
layers to learn carry out. The error back
propagation algorithm with a variable
learning rate is used for these purposes. The
general expression for the correction of the
synaptic weights is well known as the
generalized delta rule.
2.3) Supervised and unsupervised learning
of Artificial Neural Networks (ANNs):
The multilayered and self-organized
artificial neural networks are used in the
proposed methodology. Self-organized
ANNs are used for reduction of the system
model and of the input set for multilayered
artificial neural network learning, while
multilayered ANNs are used for detecting
topology and for monitoring and assessment
of voltage stability.
2.4) Exploitation of artificial neural
network (ANNs):
After selecting required ANN configuration
with the most suitable characteristics, it is
possible to move on to the next stage is
143
called exploitation.
topology is detected.
First,
the
system
The proposed methodology for
stability analysis is based on ANNs with
complete and reduced sets of input data. It
can be tested on high voltage power system
about 92 buses 12 number of generator
buses and 174 lines. In doing so the
following four cases may be considered for
the complete input vector of system
variables.
i)
Vector y=(,V), state variable
system
ii)
Vector
y=(P,Q),
directly
measured system variables
iii)
Vector y=(V,P,Q), directly
measured variables
iv)
Vector y=(,V,P,Q), all system
variables.
Next, and evaluation of the stability
of the power system with the identified
topology on the basis of complete or
reduced vectors of the system variables is
carried out. A schematic over view of the
considered variants of the exploitation
stages shown in fig-3 and fig-4.
Monitoring of voltage stability
margins is performed by introducing
measured and estimated system variable
values into the input of the corresponding
neural network with algorithm ‘AY’ and
‘AR’. Voltage stability assessment is carried
out by testing, if there is an outage of any
single element that is lines or generators or
if any of there probable combinations leads
to a disruption of the voltage stability
margins. For the simulation of the
considered outages two approaches are
followed.
i) Indirect stability assessment (ISA):
The first approach is indirect and is
performed so that for each outage the
evaluation is first done for the vector of
system variables by applying the fastdecoupled load flow calculation. Then on
the basis of the vector of system variables
and the specified identification code for the
topology of the outage in question, the
voltage stability margin of the observed
outage is tested by applying one of the
suggested ANNs.
By combining this input vector by
combining this input vector with the
proposed algorithm or detecting the
topology ‘NT’ and monitoring and
evaluating voltage stability ‘AY’ and ‘AR’
the following configurations/algorithms are
obtained which are tested in this paper:
NT-(,V),
NT-(P,Q),
NT-(V,P,Q),
NT-,V,P,Q),
AY-(,V),
AY-(P,Q),
AY-(V,P,Q),
AY-(,V,P,Q),
AR-(,V),
AR-(P,Q),
AR-(V,P,Q),
AR-(,V,P,Q),
As, such, of special interest are the
algorithms which are based on the vector of
system variables y=(V,P,Q) which are
directly measurable from the system, while
the remaining input vectors require
estimated values of the sub vector of the
voltage phase angles of the buses ().
3.1) System reduction and ANN learning
on a test sample:
The following relevant configurations are
considered in this paper: base configuration,
all single outages of lines and generators,
and most probable combinations of double
outages. For each relevant configuration,
first the reduced set of system variables is
determined using the self-organized neural
network UR. This reduction renders an
average reduction of the system variable
vector by 95%, so that the reduced vector
represents only about 5% of the whole
ii) Direct stability assessment (DSA):
The second approach is direct, and
is performed so that the input of the ANN is
the input vector y = (P,Q) consisting of
actual injected active and reactive powers in
all buses and the identification code of the
outage. In these way voltage stability
margins of the observed outage is directly
evaluated with out the fast-decoupled load
flow calculation.
3) Application of the methodology:
144
vector. In forming these vectors as separate
groups, the following are usually set apart:
i) Voltage phase angles and magnitudes in
buses where low, rated power generators are
connected;
ii) Active and reactive powers in generators
buses;
Table-1. Basic learning
characteristics of the observed ANNs
iii) active and reactive powers in loaded
buses of a single region;
iv) voltages in load buses of single compact
region.
It is interesting to note that it was
actually those buses which would be
assigned as pilot buses, according to the
secondary voltage regulation algorithm
(7,16), which were chosen as characteristic
representatives of the group with the
voltages of load buses. In this way, the
proposed algorithm for reduction of the sys
tem variable set can also be used for
selecting a candidate for the pilot buses of
secondary voltage regulation.
On the basis of the complete and
reduced vectors of system variables,
supervised learning of ANNs was carried
out. The ANNs were trained using either one
network for all relevant configurations
(algorithm AY), or a single network for each
relevant configuration AY), or a single
network for each relevant configuration
(algorithm AR). For each relevant
configuration, the load was varied over the
range of the nighttime minimum to the
daytime peak value, during which 100
different margins were observed. The basic
characteristics are presented in Table-1.
In Table 2, the relationship is
presented comparatively of the time needed
for stability analysis, system reduction and
ANN learning for algorithms AY and AR.
Learning was done on a workstation
ALPHA 4100/300 MHz. From the Table it
can be seen that the AR algorithm is
significantly more efficient in the learning
phase. This efficiency is achieved because
of the shorter time necessary for calculating
the stability analysis for the reduced system
model. The other more significant reduction
in computer time was achieved in
the shorter learning time for several smaller
ANNs that for the one large ANN. However,
the obstacle is the application of algorithm
AR that is necessary for the configuration of
the large number of smaller ANNs. This
number is approximately equivalent to the
number of relevant configurations.
Table-2. Comparative computer
learning time for algorithms AY and AR
145
The test results of the neural
networks for the monitoring and assessment
of the voltage stability margin are shown in
figs. 6-9. In figs. 6 & 7 there is a
comparative presentation of the dependence
of the real part of the critical Eigen values
on the system load level for algorithms AY
and AR. In figs. 8 & 9 there is comparative
presentation of the dependence of the
voltage stability margin on the system load
level for algorithms AY and AR. From Fig.
6a and Fig. 8a, it can be seen that in the
situation where the input set contains only
system state variables (,V), greater
precision is achieved by using algorithm AY
with the whole input set, than by using
algorithm AR with a reduced input set.
3.2) Testing the methodology in
exploitation:
The testing of ANNs on a real
power system was carried out for the all
configurations for topology detection, and
for monitoring and assessing voltage
stability margins. Testing of the trained
ANNs for the detection of system topology
was carried out on an unknown set of 12960
samples (40 each for every topology
observed). In doing so, the following four
ANNs for topology detection were
considered:
NT-(,V), NT-(P,Q), NT-(V,P,Q), NT(,V,P,Q). On the observed test set, all four
networks detected all the considered
topologies with a high degree of accuracy.
Namely, the output neurons of neural
network NT were so designed that they give
a binary code for each topology considered.
The mean square errors calculated from the
real values at the output neurons are given in
table-3.
When the input set consist of input vectors
(P,Q), the precision achieved with reduced
or complete input set, is almost equal with
reference to fig-6(b) and fig-8(b). Finally
from fig-7 (a), 9(a) and fig-9(b). It can b e
seen that when the input set contains enough
data input vectors (V,P,Q) and (,V,P,Q),
greater precision is achieved by using
algorithm AR with a reduced input set then
by using algorithm AY with the entire input
set. In addition algorithm AR is one order of
magnitude faster in exploitation than
algorithm AY in terms of ms, which is
exceptionally important because exploitation
is done in an online environment.
As mention above, when assessing
voltage stability, both the ISA and DSA
approaches were analyzed the computer
time needed for analyzing stability in both
approach are given in table-4. From the table
it can be seen that the direct calculation of
stability using DSA and AR-(P,Q),
decreases computer time remarkably without
significant loss in precision.
Table-3. Mean square errors for output
neurons and for the considered power
system topology:
Table-4. Computer time for assessing
voltage stability using the indirect
approach ISA and the Direct approach
DSA
146
sx=[ARR-B1.D1-1(s)C1] xR ……….. (9)
Where the following symbols are used:
B1=[ARO BR], D1(s)= A RR-SI Bo
Co D
C1= AOR
CR …………………….. (10)
To analyze the dynamic reduce model
equation-9 it is necessary to detetemine its
Eigen values. Determining these Eigen
values is done by an iterative approach (1)
individual or (2) simultaneous.
In this paper the individual method
is used which is based on the following
iterative approach
sK+1=Aii-B1D-1(SK)C1, k=1,2, ….. n (11)
4) Details of Model:
4.1) System Reduction
The basic model for system
reduction is a linearised mathematical model
of power system
x=Ax+By ………….. (1)
0=cx+Dy …………….(2)
It the vector of state x is divided into two
sub vectors xR is the sub vector which
contains the variables of state whose
dynamics are under consideration and x0 is
the sub vector which contains the variables
of the state whose dynamics are not under
consideration the equations above obtain the
following form:
xR= ARRxR+ AROx0+ BR .y ……. (3)
xO= AORxR+ AOOxO+ BO .y …..... (4)
0= CRxR+ COxO + D.y ……… (5)
Where
ARR, ARO, AOR, AOO, correspond to the sub
matrix of matrix A
BR, BO correspond to the sub matrix of
matrix B
CR, CO, correspond to the sub matrix of
matrix C
That is the system of equations 3 – 5 in the
Laplace frequency domain are
sxR= ARRxR+ AROx0+ BR .y ……. (6)
sxO= AORxR+ AOOxO+ BO .y …..... (7)
0= CRxR+ COxO + D.y …………. (8)
Where SK SK+1 is the approximation of the
required Eigen values in the and k and k+1
iteration. For improving the convergence of
these iterative methods product linearisation
B1D-1(SK) is used.
With this iterative method the Eigen
value is determined which predominantly
associated to the ith component of the sub
vectors of xR. By varying the index I=1, …..,
R it is possible to determine all Eigen values
of the reduced model as in equation 9 with
great efficiency.
4.2) Artificial neural network (ANN)
algorithm:
The
goal
of
unsupervised
classification is to group the state variables
of the system which are characteristic by
similar dynamic in the process the cosine of
angle covered by vectors of individual state
variables at all interval is taken as the
measure of similarity between them. If two
arbitrary state variables are observed over a
period of time, for example xi=[x1i, x2i, ….
xmi]T and xj=[x1j, x2j,…. Xmj]T, They will be
similar if they both cover a small angle. In
order the measure there similarity the
following express is used
cosij = xiTxj
(1)
xi x i
If equation (1) is larger than the given value,
represented by the similarity threshold
(cos0), that is if the condition ij < 0 then
In order to observed the dynamics of only
relevant state variables, which are contained
in sub vector xR, it is necessary to eliminate
the sub vector of variables of state x0 from
models equations 6-8, and the vector of
algebraic variable y. After these elimination,
which are realized using the gauss method, a
reduced system model is obtained
147
the vectors xi and xij are similar and belong
to the same group. Keeping the criterion for
defining similar vector as in equation (1) it
is necessary to normalize them before hand
by using the formula
i = xi
xi
I=1,2, …. n
assessment the voltage stability, which is
based on artificial neural networks (ANNs)
with a reduced input system data set. The
input data set is reduced using a selforganized neural network, and then a
multilayered feed forward neural network is
trained using this reduced data set to
monitor and assess the voltage stability
margins. The proposed methodology tested
comparatively with a method for monitoring
and assessing the voltage stability, which is
based on a complete input data set. Tests are
carried out on a power system with 92
buses. The obtained results indicate that,
both in the learning and exploitation stages
the efficiency of the methodology is
significantly increase by reducing the input
data set. More over, when the input data set
is representative enough, the ANNs based
on the reduced data set achieved greater
precision in calculating voltage stability than
did the ANNs, which is based on the
complete input data set. In addition, by
using the reduction of input data, there is a
significant reduction also in the demand for
the measurements within the power system.
(2)
The grouping of similar variables is
achieved in the following way. The arbitrary
group g(g=1, … , G) is observed consisting
of k variables. The arbitrary ith variable xi,
that is its normalized value i, will belong
to this group if it satisfies the condition that
ig < 0 where 0 is the threshold of group
membership, ig is the angle between ith
vector
i and the center of the gth group Cg.
As the new element enters the
group, the center of group moves toward the
position of the variable
formula
i according to the
Cgk+1= Cgk+á Cgk …….. (3)
6) References:
Where
Cgk
and
Cgk+1
are the centers of g
1. Voltage stability and security in bulk
power systems voltage phenomena-voltage
stability
and
security,
engineering
foundation conference, Potosi, Missouri
USA, sep-1988 – by CONCOR DIA, C.
group before and after vector i enters this
group. The coefficient ‘a’ is a factor of
proportionality define by the intuitive
expression ‘a’=1/k+1, where k is the number
of variables making of the group, and the
coefficient Cgk is the correction of the
position of the group center (Cgk=xi- Cgk).
By replacing ‘a’ and Cgk finally we get
2. Neural network and fuzzy system by Bard
Kosco.
Cgk+1=a i+ Cgk(1-a) i.e.,
3. Understanding of FACTS – by narain.
G.Hingorani, Laszlo Gyugyi.
Cgk+1= KCgk+ i ….. (4)
K+1
The calculated group centers of equation 4
serve for the basis for finding the closest
state variables for them, which become the
characteristic representatives of the group.
4. Monitoring and assessment of voltage
stability margins using artificial neural
network with a reduced input set IEEE,
june-1997 by D.Popovic/D.Kukalj and
F.Kulid.
5. Reactive power controller
electronic systems by T.J.E. Millers.
5) Conclusion & Result:
These paper deals with the developments of
new methodology for monitoring and
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