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CHAPTER 3
CONTINGENCY ANALYSIS USING COMPLEX
VALUED NEURAL NETWORK
3.1 INTRODUCTION
Contingency analysis is one of the most important tasks
encountered by the planning and operation engineers of bulk power
system. Power system engineers use contingency analysis to examine the
performance of the system and to assess the need for new transmission
expansions due to load increase or generation expansions. In power
system operation contingency analysis assists engineers to operate at a
secured operating point where equipment are loaded within their safe
limits and power is delivered to customers with acceptable quality
standards. Real time implementation of power system analysis and
security monitoring is still a challenging task for the operators.
In general the state of the system is determined on the basis
of ability to meet the expected demand under all levels of contingencies.
The objective of contingency analysis is to find voltage violations or line
overloads under such contingencies and to initiate proper measures that
are required to alleviate these violations. Exhaustive load flow
calculations are involved in ascertaining these contingencies and
determining the remedial actions. The necessity for such tool is
increasingly critical due to the emerging complexity of power systems
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that results from network expansions and the fact that the power
systems are pushed to operate at their limits due to financial and
environmental constraints.
Voltage stability is defined as the ability of a power system to
maintain steadily acceptable bus voltage at each node under normal
operating conditions, after load variation following a change in system
configuration or when the system is subjected to contingencies like line
outage or generator outage. Single or multiple contingencies cause
voltage violations which are known as voltage contingencies. The line
outages may lead to the most severe violations in line flow which
necessitates the line over load alleviation of the network.
The different methods used for analyzing these contingencies are
based on full AC load flow analysis or reduced load flow or sensitivity
factors. But these methods need large computational time and are not
suitable for on line applications in large power systems. It is difficult to
implement on line contingency analysis using conventional methods
because of the conflict between the faster solution and the accuracy of
the solution. In this chapter application of real valued neural networks
for contingency analysis of a 6 bus system is discussed. A
computationally efficient method using complex valued neural network is
proposed for contingency analysis of 6 bus and 30 bus system.
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3.2 METHODS OF CONTINGENCY ANALYSIS
There are various methods used for contingency analysis purpose.
Methods based on AC power flow calculations are considered to be
deterministic methods which are accurate compared to DC power flow
methods. In deterministic methods line outages are simulated by actual
removal of lines instead of modeling. AC power flow methods are
accurate but they are computationally expensive and excessively
demanding of computational time. Because contingency analysis is the
only tool for detecting possible overloading conditions requiring the study
by the power system planner computational speed and ease of detection
are paramount considerations. A brief description of these methods is
given below.
3.2.1 DC LOAD FLOW METHOD OF CONTINGENCY ANALYSIS
This method is based on DC power flow equation to simulate single
or multiple contingencies. These equations are N-1 in number, where N is
the number of buses. In this method the line resistances are neglected,
only real power flows are modeled ignoring the reactive power flows. This
results in a linear model of the network to facilitate performing multiple
contingency outages using the principle of super position.
Each transmission line is represented by its susceptance Bij.
Impedance jxrZ (3.1)
Inverse of impedance jBGY (3.2)
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022
xr
rG (3.3)
xxr
xB
122
(3.4)
In this method only the real part of the power flow equations are
considered, that is the effect of reactive power Q is neglected and all the
bus voltages are assumed to be 1 p.u. the matrix B' is computed on the
basis that all the resistances are zero from equation 3.5
ij
ijikx
BB1
(3.5)
Where xij is the reactance of the line connecting buses i and j.
The angles and real powers are solved by iterating Equation 3.6.
PB 1
(3.6)
3.2.2 Z-MATRIX METHOD OF CONTINGENCY ANALYSIS
This method makes use of bus impedance matrix associated with
both base case system and the system modified by either line removals
or additions [22]. Z-matrix of a system can be obtained by inverting the
bus admittance matrix or it can be constructed by using available
algorithms. The fundamental approach to contingency analysis using z-
matrix method is to inject a fictitious current in to one of the buses
associated with the element to be removed, of such value that the
current flow through the element equals the base case flow; all the other
bus currents are set equal to zero. In effect, this procedure creates
throughout the system a current flow pattern that will change in the
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same manner as the current flow pattern in the AC load flow solution
when the element in question is removed.
This method is more accurate compared to DC load flow method
and the results are comparable to those obtained using AC power flow.
3.2.3 VOLTAGE STABILITY INDEX (L-INDEX) COMPUTATION
This method is also based on load flow analysis and used for
determining the voltage collapse proximity. The value of L-index ranges
from 0 (no load condition) to 1 (Voltage collapse). The bus with highest L-
index value will be the most vulnerable bus in the system. Calculation of
L- index for a power system is briefly disused below.
Consider an N bus system with number of generators Ng
The voltages and currents are represented by equation
L
G
LLLG
GLGG
L
G
V
V
YY
YY
I
I (3.7)
Where IG, IL and VG, VL represent the complex current and voltage
vectors at generator and load nodes.
[YGG], [YGL], [YLL] and [YLG] are corresponding partitioned portion of
the network admittance matrix.
Rearranging the equations
G
L
GGGL
LGLL
G
L
V
I
YK
FZ
I
V (3.8)
Where
LGLLLG YYF1
(3.9)
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The L-index of the jth node is given by
gN
j
jiij
j
ijijV
VFL
1
1 (3.10)
The value of Fji are obtained from matrix FLG from equation 3.9.
3.2.4 DECOUPLED LOAD FLOW
The Newton Raphson power flow equations in Jacobian form are
represented by the following equation.
V
VLJ
NH
Q
P (3.11)
Where ;k
i
ik
PH
;k
k
i
m VV
PN
;k
i
km
QJ
;k
k
i
km VV
QL
(3.12)
H, N, J and L are the sub-matrices of the Jacobian represented by
Equation 3.12.
In a power system there is a strong dependence between injected
real powers and bus voltage angles and between the injective reactive
power and bus voltage magnitudes that is strong couplings between P
and δ variables and between Q and |V|. The coupling between Q and δ
and P and |V| is weak. Therefore the matrices N and J can be set to
zero. Resulting linear equations reduce to
kkk HP )1()1( (3.13)
||
||)1()1(
V
VLQ kk (3.14)
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the elements of H and L are given in Equation 3.12.
3.2.5 FAST DECOUPLED LOAD FLOW
In decoupled method the elements of Jacobian matrix H and L
have to be calculated at each iteration which requires considerable
computational time. For simplifying these calculations the following
approximations described by equations 3.13, 3.14 and 3.15 are done to
make these matrices constant
cos δik = 1 (3.15)
Gik sinδik << Bik (3.16)
Qi << Bik|Vi|2 (3.17)
Introducing these approximations the H and L matrices are
obtained from equations 3.18 and 3.19.
Hik = Lik = -|Vi||Vk|Bik i≠k (3.18)
Hii = Lii = |Vi|2Bii (3.19)
Dividing each equation by Vi and setting |Vj| = 1, fast decoupled
load flow (FDLF) equations are given by equation 3.20 and 3.21.
BV
P
|| (3.20)
VBV
Q
|| (3.21)
Where elements of B' and B'' are obtained taking the negative of
appropriate elements of B.
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3.3 APPLICATION OF REAL VALUED AND COMPLEX VALUED
NEURAL NETWORKS APPROACHES TO LOAD FLOW ANALYSIS
The importance of load flow studies is discussed in section 1.1.2.
In this section an IEEE-6 bus test system is used to demonstrate the
conventional and complex valued neural network approach to load flow
analysis by taking a simple case of determining the voltage magnitudes
and angles. It is assumed that the demands at the load buses are
dynamic.
3.3.1 CONVENTIONAL NEURAL NETWORK APPROACH
In recent years there has been a confluence of ideas and
methodologies from several different disciplinary areas to give rise to an
extremely interesting research area called artificial neural net research or
connectionist net research. The concept of Artificial Neural Networks
(ANN) is one of the greatest developments of this century. These networks
resemble the functioning of human brain like intelligent guessing and
pattern recognition. ANNs use large number of interconnected,
concurrently operating elemental processors to process the information
in a collective manner. Neurons are the basic building blocks and the
input output relationship solely depends on the interconnection of the
nodes and layers. Artificial neural networks are best suitable for
nonlinear function approximation, estimation and prediction.
A conventional neural network uses real numbers as inputs,
weights and thresholds. To solve a power system problem with complex
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numbers, each complex input has to be split in to two parts (real part
and imaginary part) to present as input to the network. Therefore the
number of input and output nodes or neurons increases. To evaluate the
performance of this network a 7 X 15 X 6 neural network (including bias)
has been developed. The three layered neural network has the following
architecture.
Input layer: It has 6 neurons representing real and reactive
powers at the three load buses.
Hidden layer: It consists of 15 neurons. Hidden layer neurons are
selected by trial and error method.
Output layer: It has 6 neurons. Voltage magnitudes and angles at
each load bus are taken as outputs.
3.3.2 BACK PROPAGATION ALGORITHM
Back propagation algorithm [Annexure –IV] is the most popular
algorithm and it can be applied to networks with any number of layers.
The summary of back propagation algorithm is
1. Initialize the weights of hidden layer V and output layer W to small
random values.
2. Choose error > 0 and learning constant η < 1.
3. Present the input pattern Z to input layer and compute the output.
Response of hidden layer
ZVfy t
jj , for j = 1, 2, ……J
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Response of output layer
yWfO t
kk , for k = 1, 2,……K
4. Compute the error
EodE kk 2
2
1, for k = 1, 2,………K
5. Compute the error signal
Error signal terms for the output layer are
212
1kkkok ood , for k = 1,2,…….K
Error signal terms of the hidden layer are
K
k
kjokjyj wy1
212
1 , for j = 1, 2,…….J
6. Adjust the output layer weights
jokkjkj yww , for k =1,2,……K and
j = 1,2,……J
7. Adjust the hidden layer weights
iyjjiji zvv , for j = 1,2,…….J and
i = 1,2,…….I
8. Present the next input pattern and go to step 2 otherwise go to
step 9
9. Training cycle is completed.
For E < Emax terminate the training
If E > Emax initiate new training cycle and go to step 2.
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3.3.3 RESULTS AND DISCUSSIONS OF CONVENTIONAL NEURAL
NETWORK APPROACH
An IEEE-6 bus test system [63] is used to demonstrate the
conventional neural network approach to load flow analysis. The single
line diagram of the 6 bus system is shown in Fig. 3.1. A simple case of
determining the voltage magnitudes and angles is considered. It is
assumed that the demands at the load buses are dynamic. Back
propagation algorithm is used for training the network.
slack
1 2 3
4
5
6
Fig. 3.1 Six bus system
There are 3-generator buses namely bus 1, bus 2, bus 3 with
specified real powers and voltages. The other 3 buses are load buses i.e.
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bus 4, bus 5 and bus 6 with specified real and reactive powers. The
training samples required for the neural network are generated by Fast-
Decoupled method.
0 500 1000 1500 2000 2500 3000 3500 4000 4500
0
0.5
1
1.5
2
2.5
3
3.5
4
Iterations
Tota
l square
d e
rror
Fig. 3.2 Error convergence graph of real ANN
The above Fig. 3.2 represents the variation of error with number of
iterations. The number of training samples used is 14 and the least
mean squared error of 0.08 is obtained after 5000 iterations.
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Table 3.1: ANN Test data in p.u for Load Flow
S.No. P4 Q4 P5 Q5
P6
Q6
1 0.75 0.75 0.75 0.75 0.75 0.75
2 0.8 0.8 0.8 0.8 0.8 0.8
3 0.9 0.9 0.9 0.9 0.9 0.9
4 0.6 0.6 0.6 0.6 0.6 0.6
5 0.65 0.65 0.65 0.65 0.65 0.65
6 1 1 1 1 1 1
After the training phase the network is tested using the patterns which
shown in Table 3.2.
Table 3.2: Network outputs for the test data
S. No. |V4| Δ4 |V5| δ5 |V6| δ6
1 0.9774 -0.0855 0.9730 -0.1069 0.9921 -0.1225
2 0.9731 -0.0972 0.9672 -0.1215 0.9925 -0.1408
3 0.9620 -0.1209 0.9518 -0.1502 0.9934 -0.1770
4 0.9866 -0.0544 0.9848 -0.0646 0.9904 -0.0699
5 0.9841 -0.0638 0.9817 -0.0782 0.9910 -0.0867
6 0.9472 -0.1438 0.9302 -0.1771 0.9941 -0.2114
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Table 3.3: Actual outputs from load flow analysis
S. No. |V4| Δ4 |V5| δ5 |V6| δ6
1 0.9843 -0.0842 0.9788 -0.1060 0.9993 -0.1213
2 0.9792 -0.0955 0.9721 -0.1202 0.9942 -0.1392
3 0.9685 -0.1186 0.9581 -0.1494 0.9835 -0.1761
4 0.9991 -0.0518 0.9982 -0.0650 1.0144 -0.0697
5 0.9943 -0.0624 0.9919 -0.0784 1.0094 -0.0866
6 0.9573 -0.1428 0.9432 -0.1800 0.9926 -0.2147
It can be observed that the network is able to extrapolate to give
the output for the input data which is outside the training data. The
results are shown in Table 3.2 and Table 3.3. The neural network output
shown in Table 3.2, when compared to that of conventional load flow
method shown in Table 3.3 is within the acceptable limits with excellent
generalization capability.
When the real valued neural network is used for complete load flow
analysis including real and reactive power flows with 7 input and 24
output neurons the network error attained a steady-state value of 0.1
after 5000 iterations and it is prone to local minima as shown in Fig. 3.3.
the training time is 67.2 seconds.
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0 0.5 1 1.5 2
x 104
0
2
4
6
8
10
12
Iterations
Mean s
quare
d e
rror
Fig. 3.3 Local minima in Error convergence
It is observed that the results obtained were inferior to that of
conventional load flow method. Comparison between real ANN, Complex
valued ANN and conventional load flow using FDLF method is shown in
Table 3.5.
3.4 COMPLEX VALUED NEURAL NETWORK APPROACH
In the previous chapter the importance and characteristics of
complex valued neural networks has been discussed. ANNs have been
widely used to provide solution for some of the power system problems
such as energy management, optimal power flow, load forecasting,
65
however most of these applications were developed using real numbers.
In electrical power system applications such as load flow analysis,
contingency evaluation complex numbers are involved extensively.
Although conventional ANNs are able to deal with complex numbers by
treating real and imaginary parts independently their implementation is
not so satisfactory and the proposed complex valued neural network is
superior due to the following reasons.
The size of the neural network will be smaller as the input, weights
and outputs are all complex numbers.
It will be shown that complex valued neural network is less prone
to local minima.
It has an improved ability to evaluate the cases which do not fall
within the training zone.
The training sequence may be shorter than that required by the
conventional multi layer perceptron.
3.4.1 COMPLEX BACK PROPAGATION ALGORITHM
Back propagation learning algorithm [54-58] is the most common
neural network approach as it gives excellent empirical results. It is
simple to implement, faster than other approaches, well tested and easy
to mold in to very specific and efficient algorithms.
Complex Back propagation algorithm is a complex valued version
of probabilistic gradient decent method. In a complex back propagation
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model all inputs, weights, threshold and output patterns are complex
numbers. Derivation of complex back propagation algorithm is given in
Annexure-III and the flow chart is given in Annexure-IV.
The algorithm is stated as follows:
Weight initialization
All the weights of hidden layer and output layer are set to small
random complex numbers. The thresholds are set to -1-1i.
Calculation of activation
The instance which is complex in nature presented to the network
is the activation level of the input.
The activation level of jth neuron Oj of hidden and output unit is
determined by
jijij xWFO (3.22)
where ]Im[]Re[ jijiji WiWW is the complex weight from an input
unit xi and θj the complex valued threshold. The activation
function F is a sigmoid function as shown in Eq. 3.23:
aeaF
1
1)( (3.23)
Weight Training
The weights are adjusted in the following way during the training
period:
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1. Start at the output layer units and work backward to the
hidden layers recursively.
jijiji WtWtW )()1( (3.24)
Where Wji(t) = weight from unit i to unit j at tth iteration
∆Wji = weight adjustment.
2. The weights are modified by computing the weight change by
jjji OW ˆ (3.25)
Where η is learning rate (0 < η < 1)
δj is error gradient of unit j
Ô is the complex conjugate of output O
3. The following equations are used to find the error gradient:
- For output units the real and imaginary parts are:
]Re[]Re[]Re[1]Re[ jjjjj OTOO
]Im[]Im[]Im[1]Im[ jjjj OTOOi (3.26)
Where Re [ ] and Im [ ] are the real and imaginary parts, Tj is
the target output and Oj is the actual output of the unit j
For the hidden units the real and imaginary parts are:
k
kjkjjj WOHH ]Re[]Re[1]Re[]Re[1]Re[ ]
2
k
kjkjj WOHHi ]Im[]Im[1]Im[]Im[1]Im[ ]
2 (3.27)
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Where δk the error gradient of kth unit obtained from
equation 3.24
3.4.2 RESULTS AND DISCUSSIONS WITH COMPLEX VALUED
NEURAL NETWORK
To demonstrate the complex valued neural network method load
flow analysis of 6 bus system used in section 3.3, shown in Fig. 3.1 is
considered for a detailed load flow analysis. The proposed network not
only estimates the bus voltages but also the line flows in complex form.
The neural network structure has only 3 inputs with 14 outputs.
The number of hidden neurons is taken as 30 by trial and error method.
Random function is used to obtain the initial complex weights with a
small value near zero. Normalizing the inputs or outputs is not required
as all the values are represented in p.u. The proposed network is trained
using 14 samples using complex back propagation algorithm explained
in section 3.4.2. The mean squared error has attained a value of
0.0091after 20000 iterations as shown in Fig. 3.4. To train the network
the time taken is 190.11seconds.
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0 0.5 1 1.5 2
x 104
0
1
2
3
4
5
6
7
8
9
Iterations
Mean s
quare
d e
rror
Fig. 3.4 Error convergence graph of Complex ANN
Comparing the Fig. 3.3 and Fig. 3.4 it can be noted that the real
valued neural network error has got itself into a minimum after number
of iterations where as the complex valued neural network error has
converged to a very small value by improving itself during the learning
process. Complex ANN took longer time for learning when compared with
real ANN but the training error is very small therefore it is quite easy for
the complex ANN to predict the data which is out of training range.
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After training phase the network can be used to estimate the
voltages at various buses and the power flows in complex form. It is
assumed that the real and reactive power demands at the three load
buses are variable inputs to the network and the outputs are complex
voltages and line flows.
Different test patterns falling within the range of training patterns
and outside the limits are used to test the ability of the network and
these are included in Table 3.4. The comparison between real ANN and
complex ANN output for one set of load pattern is included in Table 3.5.
Table 3.4: Test data for real and complex ANNS
Loads Test sample 1
Test sample 2
Test sample 3
Test sample 4
Test sample 5
P4+jQ4 0.75+j0.75 0.78+j0.78 0.82+j0.82 0.85+j0.85 0.75+j0.75
P5+jQ5 0.75+j0.75 0.78+j0.78 0.82+j0.82 0.85+j0.85 0.85+j0.85
P6+jQ6 0.75+j0.75 0.78+j0.78 0.82+j0.82 0.85+j0.85 0.9+j0.9
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Table 3.5: Comparison of Real and Complex neural networks for a
test pattern of 0.78+0.78i at all the three buses
Voltages and Line flows
FDLF Real ANN Complex ANN
V4 0.9769-j0.0891 0.9834-j0.0687i 0.9743-j0.0941i
V5 0.9686-j0.1140 0.9882-j0.0109 0.9654-j0.1172
V6 0.9873-j0.1311 0.9967-j0.1121 0.9829-j0.1304
1-2 0.3840-j0.1940 0.3016-j0.1012 0.4256-j0.2023
1-4 0.5300+j0.2280 0.4535+j0.2012 0.5466+j0.2303
1-5 0.4350+j0.1360 0.2829+j0.0525 0.4526+j0.1395
2-3 0.0550-j0.1270 0.0256-j0.1445 0.0696-j0.1284
2-4 0.3360+j0.5430 0.2847+j0.49 0.3528+j0.556
2-5 0.1690+j0.1860 0.1438+j0.1791 0.1677+j0.1902
2-6 0.3080+j0.1530 0.3126+j0.1396 0.3228+j0.1548
3-5 0.1910+j0.2750 0.1895+j0.2530 0.2020+j0.2809
3-6 0.4630+j0.6900 0.4472+j5881 0.4608+j0.7069
4-5 0.0520-j0.0480 -0.0447-j0.0218 0.0446-j0.0477
5-6 0.0300-j0.1080 0.0501-j0.1105 0.0480-j0.1097
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After the training phase the two neural networks are tested for
their ability to generalize the power flow problem. Table 3.5 can be
referred in order to compare the capability of prediction of two ANNs. As
the error has reached to a steady state value the real ANN stopped
learning. There is a mismatch between the actual and predicted value of
power flow in line 4-5, where as the results obtained from the complex
ANN are acceptable and it can be said that this ANN behaves better when
compared to real ANN.
3.5 CONTINGENCY ANALYSIS USING ARTIFICIAL NEURAL
NETWORKS
Contingency analysis is used to detect the line flow and voltage
violations following a set of contingencies. It comprises of simulating a
set of contingencies such as line and/or equipment outages to observe
the behavior of the system. The operational problems are detected after
evaluating the post contingency scenario in order to detect operational
problems. After identifying the potential problems a detailed analysis is
performed this requires exhaustive load flow analysis.
The application of ANNs in solving power system problems include
many areas such as power system security assessment, contingency
analysis, load forecasting. Recently artificial neural networks have been
proposed as tools for contingency analysis and ranking.
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3.5.1 CONTINGENCY ANALYSIS OF 6- BUS SYSTEM USING REAL
VALUED NEURAL NETWORKS
Six-bus system shown in Fig. 3.1 is considered for the purpose of
contingency analysis. It has three generator buses, three load buses and
11 lines. The Neural network consists of three layers, i.e. the input layer,
hidden layer and output layer. The number of inputs to the network is
chosen as five, the hidden neurons considered are twenty and the output
has twelve neurons. The network is required to be trained for estimating
the voltages of all six buses for the single line outage. Error back
propagation algorithm is used for this purpose. Bus numbers and real
power demands have been considered as five inputs, which are the line
numbers where the exact line outage occurred. The other input is a bias.
As the system considered for this problem has 11 lines and the data for
training patterns are obtained for different line outages by changing the
loads within a range of 50% to 150%.
Table 3.6: Neural network inputs for Contingency analysis
S. No. Lost Line S. No. Lost Line
1 1 – 2 7 2 – 6
2 1 – 4 8 3 – 5
3 1 – 5 9 3 – 6
4 2 – 3 10 4 – 5
5 2 – 4 11 5 – 6
6 2 – 5 - -
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Table 3.7: Desired outputs with base case load
S.
No V2 V3 V4 V5 V6
1 1.0493–j0.0868 1.0672-j0.0918 0.9863-j0.0704 0.9657-j0.0849 0.9964-j0.1151
2 1.0420-j0.0735 1.0648-j0.0842 0.9457-j0.1012 0.9685-j0.0854 0.9937-j0.1075
3 1.0520-j0.0640 1.0581-j0.0901 0.9972-j0.0652 0.9384-j0.1074 0.9871-j0.1121
4 1.0437+0.0653 1.0747+j0.0893 0.9816-j0.0704 0.9661-j0.8645 0.9988-j0.1086
5 1.0964+j0.0625 1.0920-j0.0804 0.8912-j0.0878 0.9674-j0.0865 1.0214-j0.1025
6 1.0630-j0.0666 1.0667-j0.0836 0.9912-j0.0713 0.9454-j0.0892 0.9961-j0.1059
7 1.6300-j0.0570 1.0530-j0.0928 0.9956-j0.0640 0.9630-j0.0880 0.9669-j0.1288
8 1.0509-j0.0697 1.1055-j0.0740 0.9802-j0.0740 0.9379-j0.0901 1.0135-j0.1032
9 1.0451-j0.0717 1.1643-j0.0624 0.9380-j0.0745 0.9668-j0.0836 0.9025-j0.1225
10 1.0485-j0.0682 1.0661-j0.0831 0.9884-j0.0698 0.9624-j0.0885 0.9950-j0.1060
11 1.0485-j0.0699 1.0709-j0.0857 0.9835-j0.0723 0.9583-j0.0811 1.0057-j0.1117
In Table 3.6 the lost lines or line outages are given in an order
which is presented as inputs to the neural network. The base case
voltages are given for contingencies 1 – 11.
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Fig. 3.5 Error versus number of iterations
3.5.2 RESULTS AND DISCUSSIONS
The above Fig. 3.5 represents the variation in error with number of
iterations. The error decreases rapidly at beginning and the convergence
becomes slow, as the iteration count is incremented as shown in Fig.3.5.
Mean squared error at the end of 1400 iterations is 0.1 as the input
feature selected has simple numbers representing the bus numbers
between which the outage lines are connected.
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Table 3.8: Desired output with load 0.75+0.75i
Bus voltages Line 1-4 Line 1-5 Line 2-3
V2 1.037+j0.162 1.041-j0.137 1.04-j0.079
V3 1.056-j0.175 1.056-j0.175 1.065-j0.105
V4 0.923-j0.178 0.973-j0.123 0.981-j0.082
V5 0.958-j0.164 0.932-j0.183 0.973-j0.106
V6 0.979-j0.193 0.975-j0.189 0.991-j0.125
The voltages V2 – V6 at the five buses shown in Table 3.8 are
obtained from full AC power flow method. A complex load of 0.75+0.75i
p.u is assumed at the three load buses. The various line outages
considered are line 1-4, line 1-5 and line 2-3 and the post contingency
voltages are as shown.
Table 3.9: Neural Network outputs with load 0.75+0.75i
Bus voltages Line 1-4 Line 1-5 Line 2-3
V2 1.031+j0.156 1.044-j0.136 1.03-j0.076
V3 1.050-j0.176 1.051-j0.175 1.063-j0.111
V4 0.919-j0.172 0.969-j0.123 0.978-j0.083
V5 0.959-j0.161 0.934-j0.180 0.963-j0.110
V6 0.974-j0.196 0.973-j0.185 0.990-j0.121
77
After training the neural network for finding out the post contingency
voltages at different buses it is tested at a load of 0.75+j0.75 at all the
three load buses. The actual output of the neural network is as shown in
Table 3.9. The result shows a good agreement between the deterministic
and neural network methods.
The drawback of the real valued neural network is the number of
neurons and size of the network increases with increase in size of the
power system. The other problem with this method is local minima.
3.6 CONTINGENCY ANALYSIS USING COMPLEX VALUED NEURAL
NETWORK
To reduce the computational time in performing the analysis
artificial neural network approach can be used as discussed in section
3.4 but this approach cannot be used for larger power system due to the
structural disadvantages discussed in the previous section.
3.6.1 PROBLEM FORMULATION
In this section the proposed complex valued neural network
approach for detailed contingency evaluation is applied to IEEE 6 bus
system and IEEE 30 bus system.
Fig. 3.6 shows the architecture of the proposed neural network for
computing post contingency status of the power system. The diagonal
elements of the Y bus matrix and powers in complex form at the load
buses are taken as inputs to the network. As the line outage in a power
system reflects in the diagonal elements of Y bus it is selected as a input
78
feature in addition to the varying load demand at the load buses to train
the network. G+jB represent the admittance element and PD+jQD
represent the load demand.
Fig. 3.6 Block diagram of network
There are two phases for any neural network approach so that the
trained network should be able to give correct output for any kind of
input which is known as generalization. The different phases of training
and validation are shown in Fig. 3.7 and 3.8.
Complex
Valued
Neural
Network
G11+jB11
PD1+jQD1
Gnn+jBnn
PDm+jQDm
Bus Voltages
Line flows
79
Fig. 3.7: Training phase (Off-line process)
During the offline process different contingencies and load
scenarios are presented to the neural network. The required training
patterns of inputs and desired outputs are generated using full AC power
flow method. Complex back propagation is used as the learning
algorithm in this phase. To improve the predefined performance measure
the weights are incrementally adjusted during the learning process. Data
normalization is required to convert the input or output factor in to a
unit vector pointing in the same direction. This makes all the elements to
lie within the range 0 to 1 or -1 to +1. In this problem normalization is
only applied to the output vectors as it is easy to implement by using a
simple sigmoidal activation function.
Present Various
Operating
Scenarios
Perform Full Ac Power Flow
Generate the Training Patterns
Train the Complex
ANN
Obtain the
Generalization Model
80
Fig. 3.8 Validation phase (On-Line process)
The above Fig. 3.8 represents the validation phase in which the trained
network is tested by using unseen patterns. The real time data is
presented as input to estimate the output accurately. Once the neural
network is trained it takes a very small computational time when
compared to the conventional method of solution.
3.7 IMPLEMENTATION OF COMPLEX ANN FOR CONTINGENCY
ANALYSIS OF 6-BUS AND 30 BUS SYSTEMS
From the formulations discussed in the previous sections the
complex ANN approach to solve contingency analysis problem is
presented here. The structure of the neural network is as follow:
1. Input Vector: For 6 bus system the input comprises of upper or
lower triangle elements of the symmetric bus admittance matrix
and loads in complex form at the three load buses.
2. Hidden layer: After trial and error method the number of hidden
neurons is selected as 80.
Power
System
Real Time Data
Trained Network
Contingency
Evaluation
81
3. Output layer: There are 25 output neurons representing bus
voltages and line flows.
3.7.1TRAINING PROCESS
The proposed complex neural network is trained using the complex
back propagation algorithm explained in section 3.4.2. The training and
test patterns are generated using a standard AC load flow method. A
total of 48 training patterns generated with different line outages and
load patterns varying from 70% to 120% of base case values. A learning
constant of very small value 0.02 is used to train the network. The
admittance matrix elements and load patterns map the post contingency
bus voltages and line-flows. This mapping is more suitable for the power
system on-line contingency analysis to provide the operator a list of
critical contingencies and also to assist the engineers to operate the
system at a secured operating point. The most severe violations in line
flow may occur due to line outages therefore an immediate need arises to
alleviate the line over loads and to ensure that the power is supplied to
the consumer with acceptable quality standards.
3.7.2 RESULTS AND DISCUSSIONS
A 6-bus and 30 bus system is considered in this work to
demonstrate the performance of the complex valued neural network
approach to contingency analysis.
82
3.7.3 RESULTS AND DISCUSSIONS OF IEEE 6-BUS SYSTEM
This system has 3 generator buses, 3 load buses and 11 lines.
After training the network various test line outages simulated on the
system to map the output.
Table 3.10: various Contingencies cases
Case scenario Case Scenario
1 Base case 7 Line 6 out
2 Line 1 out 8 Line 7 out
3 Line 2 out 9 Line 8 out
4 Line 3 out 10 Line 9 out
5 Line 4 out 11 Line 10 out
6 Line 5 out 12 Line 11 out
Table 3.10 represents the 11 contingency cases and one base case.
The states of the system i.e. bus voltages and line flows in complex form
for various scenarios are computed using the complex neural network.
These values are represented graphically in Fig. 3.9, Fig. 3.10 (a) and Fig.
3.10(b)
83
0 1 2 3 4 5 6 7 8 9 1011120.5
0.6
0.7
0.8
0.9
Contingency
Real
0 1 2 3 4 5 6 7 8 9 101112-0.2
-0.15
-0.1
-0.05
0
Contingency
Imagin
ery
0 1 2 3 4 5 6 7 8 9 1011120.5
0.6
0.7
0.8
0.9
Contingency
Real
0 1 2 3 4 5 6 7 8 9 101112-0.2
-0.15
-0.1
-0.05
0
Contingency
Imagin
ery
0 1 2 3 4 5 6 7 8 9 101112
0.6
0.8
1
Contingency
Real
0 1 2 3 4 5 6 7 8 9 10 11 12-0.2
-0.15
-0.1
-0.05
0
Contingency
Imagin
ery
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
Voltage at Bus 4
Voltage at Bus 6
Voltage at Bus 5
Fig. 3.9 Comparison of complex bus voltages
84
0 1 2 3 4 5 6 7 8 9 101112-0.5
0
0.5
1
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 101112-0.4
-0.3
-0.2
-0.1
0
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 1011120
0.2
0.4
0.6
0.8
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 101112-0.5
0
0.5
1
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 101112-0.2
0
0.2
0.4
0.6
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 101112-0.1
0
0.1
0.2
0.3
Contingency
Reactive P
ow
er
CVNN
NR
CVNN
NR
CVNN
NR
NR
CVNN
CVNN
NR
CVNN
NR
Line Flow 1-2
Line Flow 1-4
Line Flow 1-5
Fig. 3.10 (a) Comparison of real and reactive power flows
85
.
0 1 2 3 4 5 6 7 8 9 10 11 12-0.6
-0.4
-0.2
0
0.2
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 120
0.1
0.2
0.3
0.4
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 101112-0.3
-0.2
-0.1
0
0.1
0.2
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.2
-0.15
-0.1
-0.05
0
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.2
0
0.2
0.4
0.6
0.8
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Contingency
Reactive P
ow
er
NR
CVNN
NR
CVNN
CVNN
NR
CVNN
NR
NR
CVNN
CVNN
NR
Line Flow (2-1)
Line Flow (2-3)
Line Flow (2-4)
Fig. 3.10 (b) Comparison of real and reactive power flows
86
0 1 2 3 4 5 6 7 8 9 10 11 12-0.1
0
0.1
0.2
0.3
0.4
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 120
0.1
0.2
0.3
0.4
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 120
0.2
0.4
0.6
0.8
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-1
-0.5
0
0.5
1
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.8
-0.6
-0.4
-0.2
0
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.8
-0.6
-0.4
-0.2
0
Contingency
Reactive P
ow
er
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
Line Flow 3-5
Line Flow 3-6
Line Flow 4-1
Fig. 3.10 (c) Comparison of real and reactive power flows
87
0 1 2 3 4 5 6 7 8 9101112-1
-0.5
0
0.5
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9101112-1
-0.5
0
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9101112-0.2
0
0.2
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9101112-0.2
0
0.2
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9101112-1
-0.5
0
0.5
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9101112-0.5
0
0.5
Contingency
Reactive P
ow
er CVNN
NR
CVNN
NR
CVNN
NRCVNN
NR
CVNN
NR
CVNN
NR
Line Flow 4-2
Line Flow 4-5
Line Flow 5-1
Line Flow 4-2
Fig. 3.10 (d) Comparison of real and reactive power flows
88
0 1 2 3 4 5 6 7 8 9 10 11 12-0.4
-0.2
0
0.2
0.4
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.3
-0.2
-0.1
0
0.1
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.4
-0.2
0
0.2
0.4
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.4
-0.3
-0.2
-0.1
0
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.2
-0.1
0
0.1
0.2
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.1
-0.05
0
0.05
0.1
Contingency
Reactive P
ow
er
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
Line Flow 5-2
Line Flow 5-3
Line Flow 5-4
Fig. 3.10 (e) Comparison of real and reactive power flows
89
0 1 2 3 4 5 6 7 8 9 10 11 12-0.1
0
0.1
0.2
0.3
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.2
-0.1
0
0.1
0.2
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.6
-0.4
-0.2
0
0.2
0.4
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.8
-0.6
-0.4
-0.2
0
Contingency
Reactive P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-1
-0.5
0
0.5
Contingency
Real P
ow
er
0 1 2 3 4 5 6 7 8 9 10 11 12-0.8
-0.6
-0.4
-0.2
0
0.2
Contingency
Reactive P
ow
er
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
NR
CVNN
Line Flow ( 5-6)
Line Flow ( 6-2)
Line Flow ( 6-3)
Fig. 3.10 (f) Comparison of real and reactive power flows
90
Fig. 3.10 (g) Comparison of real and reactive power flows
The variation in voltages at bus 4, bus 5 and bus 6 are shown in
Fig. 3.6. The proposed complex valued neural network (CVNN) produced
results that are within the accuracy limits when compared with those
values obtained from Newton Raphson method.
In Fig. 3.10 (a) to Fig. 3.10 (g) various line flows are graphically
represented with respect to the contingencies. It is observed that changes
in line flows and bus voltages are matching exactly. In case of
contingency 3 there is a drop in bus voltage at bus 4 and an appreciable
real power flow change in line 1-2, line 1-4 and line 1-5 and the opposite
effect in other lines. It is observed that over 95% of the cases are
predicted with reasonable accuracy.
91
Table 3.11: Bus Voltages
Outage Condition
BUS VOLTAGE (p.u)
Bus - 4 Bus - 5 Bus - 6
Base case NR CVNN
0.9835 - j0.0775 0.9770 - j0.0917 0.9861 - j0.1040
0.9677 - j0.0805 0.9792 - j0.0969 0.9977 - j0.1117
Line 1
0.9763 - j0.1134 0.9676 - j0.1297 0.9770 - j0.1546
0.9748 - j0.1190 0.9744 - j0.1369 0.9905 - j0.1639
Line 2
0.9914 - j0.1655 0.9876 - j0.1494 0.9928 - j0.1731
0.9197 - j0.1738 0.9642 - j0.1573 0.9861 - j0.1834
Line 3
0.9764 - j0.1073 0.9658 - j0.1618 0.9747 - j0.1656
0.9811 - j0.1081 0.9454 - j0.1611 0.9853 - j0.1647
Line 4
0.9759 - j0.0731 0.9790 - j0.0923 0.9870 - j0.1078
0.9864 - j0.0721 0.9807 - j0.0919 0.9980 - j0.1065
Line 5
0.8929 - j0.0888 0.9539 - j0.0779 0.9774 - j0.0803
0.8876 - j0.0887 0.9668 - j0.0783 0.9977 - j0.0812
Line 6
0.9778 - j0.0698 0.9701 - j0.0953 0.9845 - j0.1048
0.9835 - j0.0705 0.9581 - j0.0969 0.9943 - j0.1065
Line 7
0.9893 - j0.0710 0.9833 - j0.1021 0.9922 - j0.1389
0.9855 - j0.0707 0.9757 - j0.1023 0.9749 - j0.1405
Line 8
0.9900 - j0.0692 0.9838 - j0.0906 0.9918 - j0.0903
0.9815 - j0.0704 0.9496 - j0.0911 0.9949 - j0.0906
Line 9
0.9620 - j0.0765 0.9377 - j0.0849 0.8991 - j0.1246
0.9820 - j0.0762 0.9532 - j0.0852 0.8811 - j0.1253
Line 10
0.9829 - j0.0687 0.9752 - j0.0929 0.9850 - j0.1055
0.9846 - j0.0689 0.9775 - j0.0940 0.9983 - j0.1065
Line 11
0.9842 - j0.0693 0.9764 - j0.0869 0.9854 - j0.1106
0.9853 - j0.0726 0.9740 - j0.0872 1.0010 - j0.1076
92
Table 3.12 (a): Real and Reactive power flows
Line flows
Base Case Line -1 out
CVNN NR CVNN NR
Line flow 1-2 0.3266 - 0.1655i 0.3240 - 0.1700i -0.0017 - 0.0006i 0
Line flow 1-4 0.4530 + 0.2459i 0.4880 + 0.2390i 0.6418 + 0.1882i 0.6810 + 0.2030i
Line flow 1-5 0.3459 + 0.1382i 0.3780 + 0.1140i 0.4855 + 0.0954i 0.5130 + 0.0940i
Line flow 2-1 -0.3129 + 0.1521i 0.3130 + 0.1490i 0.0003 + 0.0006i 0
Line flow 2-3 0.0545 - 0.1142i 0.0270 - 0.1220i -0.0005 - 0.1093i -0.0080 - 0.1150i
Line flow 2-4 0.3343 + 0.5315i 0.3740 + 0.5330i 0.1792 + 0.5596i 0.1940 + 0.6070i
Line flow 2-5 0.1839 + 0.1890i 0.1510 + 0.1600i 0.1109 + 0.1835i 0.0910 + 0.1780i
Line flow 2-6 0.2456 + 0.1562i 0.2600 + 0.1260i 0.2126 + 0.1397i 0.2230 + 0.1370i
Line flow 3-2 -0.0543 + 0.2281i 0.0270 + 0.5700i 0.0005 + 0.0495i 0.0080 + 0.0500i
Line flow 3-5 0.1875 + 0.2448i 0.1890 + 0.2380i 0.1464 + 0.2460i 0.1510 + 0.2550i
Line flow 3-6 0.4643 + 0.1739i 0.4380 - 0.6100i 0.4523 + 0.6511i 0.4400 + 0.6090i
Line flow 4-1 -0.4409 - 0.2164i -0.4740 - 0.2240i -0.6208 - 0.1456i -0.6580 - 0.1510i
Line flow 4-2 -0.3716 - 0.5155i -0.3540 - 0.5140i -0.1814 - 0.5468i -0.1750 - 0.5890i
Line flow 4-5 0.0190 - 0.0516i 0.0280 - 0.0610i 0.0527 - 0.0581i 0.0330 - 0.0600i
Line flow 5-1 -0.3507 - 0.1454i -0.3660 - 0.1310i -0.4617 - 0.0880i -0.4930 - 0.0800i
Line flow 5-2 -0.1741 - 0.2122i -0.1460 - 0.1860i -0.0991 - 0.2124i -0.0860 - 0.2070i
Line flow 5-3 -0.1685 - 0.2689i -0.1780 - 0.2670i -0.1314 - 0.2757i -0.1410 - 0.2840i
Line flow 5-4 -0.0139 - 0.0303i 0.0280 - 0.0150i 0.0332 - 0.0267i 0.0330 - 0.0170i
Line flow 5-6 0.0153 - 0.0903i -0.0180 - 0.1010i -0.0014 - 0.0989i -0.0530 - 0.1120i
Line flow 6-2 -0.1917 - 0.1935i -0.2550 - 0.1620i -0.1769 - 0.1801i -0.2180 - 0.1760i
Line flow 6-3 -0.3912 - 0.5855i -0.4280 - 0.5810i -0.4066 - 0.5678i -0.4300 - 0.5800i
Line flow 6-5 -0.0147 + 0.0478i -0.0180 +0.0430i 0.0032 + 0.0570i 0.0520 + 0.0560i
93
Table 3.12 (b): Real and Reactive power flows
Line flows
Line - 2 Out Line - 3 out
CVNN NR CVNN NR
Line flow 1-2 0.6322 - 0.2846i 0.6820 - 0.3020i 0.5001 - 0.2414i 0.5010 - 0.2390i
Line flow 1-4 0.0061 + 0.0085i 0 0.6181 - 0.1994i 0.6190 - 0.1860i
Line flow 1-5 0.5548 + 0.1138i 0.5890 + 0.1100i -0.0271 - 0.0047i 0
Line flow 2-1 0.5562 + 0.3034i 0.5610 + 0.3060i -0.4760 + 0.2527i -0.4740 + 0.2490i
Line flow 2-3 0.0134 - 0.1180i 0.0110 - 0.1190i 0.1084 - 0.1392i 0.1030 - 0.1360i
Line flow 2-4 0.6711 + 0.6858i 0.6840 + 0.6930i 0.2461 + 0.5301i 0.2570 + 0.5240i
Line flow 2-5 0.1376 + 0.1908i 0.1220 + 0.1840i 0.2883 + 0.2127i 0.2720 + 0.2110i
Line flow 2-6 0.2333 + 0.1417i 0.2430 + 0.1360i 0.3375 + 0.1250i 0.3420 + 0.1280i
Line flow 3-2 0.0135 + 0.0402i 0.0110 + 0.0540i -0.1074 + 0.0757i -0.1020 + 0.0730i
Line flow 3-5 0.1668 + 0.2591i 0.1730 + 0.2640i 0.2745 + 0.3064i 0.2740 + 0.3030i
Line flow 3-6 0.4461 + 0.6717i 0.4380 + 0.6190i 0.4316 + 0.7196i 0.4280 + 0.6660i
Line flow 4-1 -0.0059 - 0.0045i 0 -0.5996 - 0.1461i -0.6000 - 0.1500i
Line flow 4-2 -0.6642 - 0.6197i -0.6400 - 0.6260i -0.2680 - 0.5188i -0.2410 - 0.5130i
Line flow 4-5 -0.0546 - 0.0767i -0.0600 - 0.0740i 0.1472 - 0.0350i 0.1410 - 0.0370i
Line flow 5-1 -0.5107 - 0.0877i -0.5180 - 0.0830i 0.0228 + 0.0041i 0
Line flow 5-2 -0.1308 - 0.2163i -0.1170 - 0.2100i -0.2738 - 0.2189i -0.2610 - 0.2170i
Line flow 5-3 0.0205 - 0.2852i 0.0161 - 0.2910i -0.2545 - 0.3169i -0.2550 - 0.3130i
Line flow 5-4 0.0597 - 0.0024i 0.0610 + 0.0020i -0.1395 - 0.0306i -0.1370 - 0.0310i
Line flow 5-6 0.0373 - 0.1121i 0.0350 - 0.1180i -0.0440 - 0.1393i -0.0480 - 0.1400i
Line flow 6-2 -0.2089 - 0.1791i -0.2380 - 0.1720i -0.2990 - 0.1512i -0.3330 - 0.1550i
Line flow 6-3 -0.4112 - 0.5881i -0.4280 - 0.5890i -0.3781 - 0.6360i -0.4170 - 0.6320i
Line flow 6-5 0.0302 + 0.0636i 0.0340 + 0.0620i 0.0520 + 0.0872i 0.0490 + 0.0870i
94
Table 3.12 (c): Real and Reactive power flows
Line flows
Line - 4 out Line - 5 Out
CVNN NR CVNN NR
Line flow 1-2 0.2900-j0.1471 0.2840-j0.1530 0.1345-j0.0870 0.1370-j0.0880
Line flow 1-4 0.4421+j0.2051 0.4350+j0.2010 0.6438+j0.6666 0.6390+j0.6690
Line flow 1-5 0.3453+j0.1107 0.3600+j0.1120 0.3212+j0.1738 0.3280+j0.1710
Line flow 2-1 -0.2826+j0.1222 -0.2750+j0.1260 -0.1333+j0.0486 -0.1350+j0.0480
Line flow 2-3 0.0023-j0.0022 0 0.0821-j0.1338 0.0800-j0.1320
Line flow 2-4 0.3308+j0.4622 0.3350+j0.4590 -0.0052+j0.0019 0
Line flow 2-5 0.1686+j0.1452 0.1620+j0.1520 0.2441+j0.1848 0.2380+j0.1830
Line flow 2-6 0.2806+j0.1203 0.2780+j0.1200 0.3138+j0.1270 0.3170+j0.1240
Line flow 3-2 -0.0023+j0.0111 0 -0.0811+j0.0670 -0.0790+j0.0680
Line flow 3-5 0.1827+j0.2459 0.1810+j0.2360 0.2495+j0.2683 0.2480+j0.2700
Line flow 3-6 0.4179+j0.6806 0.4190+j0.6120 0.4346+j0.6678 0.4310+j0.6420
Line flow 4-1 -0.4306-j0.1980 -0.4240-j0.2000 -0.6021-j0.5450 -0.5980-j0.5460
Line flow 4-2 -0.3194-j0.4515 -0.3200-j0.4490 -0.0052-j0.0017 0
Line flow 4-5 0.0508-j0.0549 0.0440-j0.0510 0.1071-j0.1550 0.1020-j0.1540
Line flow 5-1 -0.3331-j0.1306 -0.3490-j0.1330 -0.3126-j0.1937 -0.3180-j0.1920
Line flow 5-2 -0.1626-j0.1713 -0.1560-j0.1780 0.2252-j0.1983 0.2290-j0.1970
Line flow 5-3 -0.1744-j0.2767 -0.1700-j0.2660 0.2326-j0.2858 0.2320-j0.2880
Line flow 5-4 0.0388-j0.0254 0.0440-j0.0260 0.1083+j0.0945 0.1080+j0.0970
Line flow 5-6 0.0208-j0.0979 0.0200-j0.0970 -0.0252-j0.1205 -0.0290-j0.1200
Line flow 6-2 0.3012-j0.1546 0.2720-j0.1550 -0.2848-j0.1573 -0.3100-j0.1540
Line flow 6-3 0.4517-j0.5963 0.4090-j0.5840i -0.3940-j0.6146 -0.4210-j0.6100
Line flow 6-5 0.0221-j0.0374 0.0190-j0.0390i 0.0315+j0.0689 0.0300+j0.0650
95
Table 3.12 (d): Real and Reactive power flows
Line flows
Line - 6 out Line - 7 out
CVNN NR CVNN NR
Line flow 1-2 0.2418-j0.1468 0.2630-j0.1440 0.2623-j0.1389 0.2600-j0.1430
Line flow 1-4 0.4272+j0.2200 0.4310+j0.2200 0.4320+j0.2004 0.4290+j0.2070
Line flow 1-5 0.3898+j0.1831 0.3970+j0.1840 0.3933+j0.1266 0.3990+j0.1220
Line flow 2-1 -0.2373+j0.1170 -0.2550+j0.1150 -0.2568+j0.1086 -0.2530+j0.1140
Line flow 2-3 0.0729-j0.1228 0.0710-j0.1300 0.1699-j0.1428 0.1660-j0.1460
Line flow 2-4 0.3601+j0.4713 0.3760+j0.4760 0.3682+j0.4445 0.3700+j0.4510
Line flow 2-5 0.0037-j0.0094 0 0.2217+j0.1567 0.2170+j0.1530
Line flow 2-6 0.2981+j0.1354 0.3080+j0.1340 -0.0018+j0.0043 0
Line flow 3-2 -0.0720+j0.0641 -0.0700+j0.0660 -0.1689+j0.0615 -0.1650+j0.0880
Line flow 3-5 0.2412+j0.3128 0.2400+j0.3030 0.1557+j0.2573 0.1540+j0.2690
Line flow 3-6 0.4295+j0.7274 0.4310+j0.6570 0.6129+j0.8194 0.6110+j0.7900
Line flow 4-1 -0.4163-j0.2178 -0.4200-j0.2180 -0.4209-j0.2054 -0.4180-j0.2060
Line flow 4-2 0.3417-j0.4577 0.3590-j0.4620 -0.3522-j0.4345 -0.3540-j0.4400
Line flow 4-5 0.0749-j0.0220 0.0790-j0.0200 0.0750-j0.0494 0.0720-j0.0540
Line flow 5-1 -0.3887-j0.1896 -0.3820-j0.1890 -0.3797-j0.1396 -0.3860-j0.1340
Line flow 5-2 -0.0054+j0.0094 0 -0.2139-j0.1762 -0.2100-j0.1730
Line flow 5-3 -0.2267-j0.3278 -0.2220-j0.3170 -0.1437-j0.2829 -0.1420-j0.2960
Line flow 5-4 -0.0662-j0.0538 -0.0770-j0.0530 -0.0738-j0.0267 -0.0710-j0.0210
Line flow 5-6 -0.0135-j0.1356 -0.0190-j0.1410 0.1132-j0.0694 0.1090-j0.0760
Line flow 6-2 -0.2655-j0.1659 -0.3000-j0.1640 0.0161-j0.0071 0
Line flow 6-3 -0.3874-j0.6162 -0.4200-j0.6230 -0.5860-j0.6999 -0.5930-j0.7220
Line flow 6-5 0.0209+j0.0805 0.0200+j0.0870 -0.1041-j0.0204 -0.1070-j0.0220
96
Table 3.12 (e): Real and Reactive power flows
Line flows
Line - 8 Out Line - 9 out
CVNN NR CVNN NR
Line flow 1-2 0.2533-0.1500 0.2680-j0.1460 0.3160-j0.1664 0.3220-j0.1690
Line flow 1-4 0.4377+j0.2290 0.4320+j0.2270 0.4642+j0.2218 0.4600+j0.2180
Line flow 1-5 0.3847+j0.2179 0.3850+j0.2160 0.3564+j0.2110 0.3630+j0.2080
Line flow 2-1 -0.2466+j0.1211 -0.2600+j0.1180 -0.3049+j0.1452 -0.3100+j0.1470
Line flow 2-3 -0.0462-j0.1070 -0.0510-j0.1070 -0.2084-j0.0710 -0.2130-j0.0700
Line flow 2-4 0.3606+j0.4860 0.3720+j0.4910 0.3114+j0.5077 0.3180+j0.5080
Line flow 2-5 0.2074+j0.2405 0.2090+j0.2470 0.1527+j0.2556 0.1500+j0.2530
Line flow 2-6 0.2182+j0.1576 0.2290+j0.1620 0.5486+j0.6324 0.5550+j0.6310
Line flow 3-2 0.0461+j0.0371 0.0510+j0.0410 0.2104+j0.0086 0.2150+j0.0130
Line flow 3-5 0.0014+j0.0032 0 0.3849+j0.2613 0.3850+j0.2650
Line flow 3-6 0.5525+j0.6725 0.5490+j0.6460 0.0055+j0.0213 0
Line flow 4-1 -0.4266-j0.2261 -0.4210-j0.2230 -0.4520-j0.2117 -0.4480-j0.2100
Line flow 4-2 -0.3609-j0.4719 -0.3550-j0.4770 -0.3078-j0.4946 -0.3020-j0.4950
Line flow 4-5 0.0717-j0.0010 0.076 0.0508+j0.0041 0.0500+j0.0050
Line flow 5-1 -0.3775-j0.2208 -0.3700-j0.2190 -0.3468-j0.2193 -0.3490-j0.2170
Line flow 5-2 -0.2058-j0.2484 -0.1990-j0.2560 -0.1464-j0.2682 -0.1410-j0.2660
Line flow 5-3 -0.0052-j0.0029 0 -0.3624-j0.2597 -0.3600-j0.2640
Line flow 5-4 -0.0673-j0.0723 -0.0740-j0.0720 -0.0461-j0.0808 -0.0490-j0.0790
Line flow 5-6 -0.0535-j0.1564 -0.0570-j0.1530 0.2002+j0.1266 0.1990+j0.1260
Line flow 6-2 -0.1866-j0.1930 -0.2230-j0.1990 -0.4870-j0.5460 -0.5080-j0.5440
Line flow 6-3 -0.4982-j0.6130 -0.5360-j0.6030 0.0208+j0.0006 0
Line flow 6-5 0.0663+j0.1011 0.0590+j0.1020 -0.1862-j0.1535 -0.1920-j0.1560
97
Table 3.12 (f): Real and Reactive power flows
Line flows
Line - 10 out Line - 11 Out
CVNN NR CVNN NR
Line flow 1-2 0.3304-j0.1540 0.2910-j0.1560 0.3081-j0.1675 0.2920-j0.1560
Line flow 1-4 0.4403+j0.2169 0.4210+j0.2150 0.4487+j0.2014 0.4380+j0.2070
Line flow 1-5 0.3287+j0.1187 0.3000+j0.1220 0.3497+j0.1307 0.3510+j0.1400
Line flow 2-1 -0.3183+j0.1281 -0.2810+j0.1300 -0.2990+j0.1425 -0.2830+j0.1310
Line flow 2-3 0.0358-j0.1221 0.0370-j0.1240 0.0269-j0.1318 0.0330-j0.1230
Line flow 2-4 0.3352+j0.4875 0.3050+j0.4970 0.3284+j0.4828 0.3290+j0.4730
Line flow 2-5 0.1549+j0.1579 0.1680+j0.1620 0.1629+j0.1696 0.1510+j0.1820
Line flow 2-6 0.2924+j0.1264 0.2710+j0.1250 0.2819+j0.0945 0.2690+j0.1100
Line flow 3-2 -0.0356+j0.0500 -0.0370+j0.0590 -0.0267+j0.0934 -0.0330+j0.0580i
Line flow 3-5 0.2002+j0.2576 0.2000+j0.2420 0.1867+j0.2903 0.1870+j0.2660
Line flow 3-6 0.4389+j0.6373 0.4370+j0.6150 0.4439+j0.6118 0.4460+j0.5800
Line flow 4-1 -0.4285-j0.2165 -0.4110-j0.2140 -0.4371-j0.2004 -0.4270-j0.2040
Line flow 4-2 -0.2558-j0.4774 -0.2890-j0.4860 -0.3226-j0.4718 -0.3130-j0.4630
Line flow 4-5 0.0257-j0.0012 0 0.0196-j0.0342 0.0410-j0.0330
Line flow 5-1 -0.3408-j0.1398 -0.3590-j0.1410 -0.3579-j0.1512 -0.3400-j0.1600
Line flow 5-2 -0.1419-j0.1838 -0.1620-j0.1870 -0.1552-j0.1943 -0.1450-j0.2060
Line flow 5-3 -0.1784-j0.2858 -0.1880-j0.2690 -0.1758-j0.3180 -0.1740-j0.2910
Line flow 5-4 -0.0339+j0.0002 0 -0.0298-j0.0364 -0.0410-j0.0440
Line flow 5-6 0.0010-j0.0898 0.0090-j0.1030 0.0064-j0.0057 0
Line flow 6-2 -0.3000-j0.1635 -0.2650-j0.1600 -0.2928-j0.1240 -0.2640-j0.1460
Line flow 6-3 -0.4514-j0.5720 -0.4260-j0.5860 -0.4588-j0.6022 -0.4360-j0.5540
Line flow 6-5 -0.0018+j0.0402 -0.0090+j0.0460 -0.0093-j0.0066 0
98
The result obtained from the proposed method and the comparison
with conventional AC power flow (Newton-Raphson) method is shown in
tables 3.11, 3.12 (a) – 3.12(f). It is observed that in case of a particular
line outage the line flow on that line is zero but the proposed method
gave negligible non-zero value. It is also observed that line 2 outage has
much effect on real and reactive line flows; in some cases post
contingency values are nearly doubled.
3.7.4 CONTINGENCY ANALYSIS OF 30-BUS SYSTEM
In section 3.7.3 contingency analysis of a small power system is
discussed. In a large power system there are hundreds or thousands of
buses therefore contingency analysis and ranking becomes more
arduous. In a large power system widespread studies are required to be
carried out for various contingencies, their combinations, load and
generation margins. Therefore optimum number of training patterns
should be selected to cover all the boundary conditions. In a large scale
power system, the system is divided in to subsystems according to
physical regions and each one can be considered separately and a sub-
problem approach can be preferred for analysis purpose.
In this section a detailed simulation analysis conducted on a 30
bus system is presented using the proposed approach. IEEE 30 bus
system has 6 generator buses, 24 load buses and the number of
transmission lines is 41 as shown in Fig. 3.11 [7][121]. These details are
provided in Annexure -1.
99
1
2
34
56 78
9
10
11
12 13
14
15
16 17
18
19
20
2122
23
2425
26
27
28
29 30
Fig. 3.11 IEEE 30-Bus system
100
3.7.5 STRUCTURE OF THE NEURAL NETWORK
The selection of optimum input neurons plays an important role as
the size of the power system to be analyzed increases. The fact is as the
size of the input vector increases the number of basis functions required
to approximate the given function rises exponentially which may lead to
poor performance of the system. To simplify the solution the diagonal
elements of the Y bus matrix are selected in addition to the load patterns
as shown in Fig. 3.6. A total of 120 training patterns are generated to
train the network for mapping the output voltages at all the buses except
the slack bus.
3.7.6 RESULTS AND DISCUSSIONS OF 30 BUS SYSTEM
To test the capability of the complex valued neural network method
various test patterns comprising of single line outages and load patterns
are used. Trial and error method is followed for selecting the values of
complex weights and learning constant. The magnitude of error for the
initial weights near unity was very large and there was no convergence.
Hence the weights are initialized by multiplying with 10-8 and after
weight up gradation during learning process their values are adjusted to
a value near unity.
101
0 5 10 15 20 25 300.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
Bus Number
Bus V
oltage (
Real P
art
)
Fig. 3.12 Real part of Bus voltages (Post contingency)
102
0 5 10 15 20 25 30-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
Bus Number
Bus V
oltage (
Imagin
ary
Part
)
Fig. 3.13 Imaginary part of Bus voltages (Post contingency)
Fig. 3.12 and Fig. 3.13 represent the bus voltages for all single line
outages at a load 110% of base case total load with constant power
factor.
103
Table 3.13: Bus Voltages Line-1outage
Bus number Actual output Desired output
2 1.0001 – j0.0168 1.0299 - j0.0173
3 1.0050 – j0.0328 1.0066 - j0.0331
4 1.0001 – j0.0390 1.0080 - j0.0390
5 1.0081 – j0.0414 1.0140 - j0.0415
6 0.9991 – j0.0469 1.0030 - j0.0468
7 0.9948 – j0.0536 0.9982 - j0.0534
8 0.9869 – j0.0544 0.9912 - j0.0540
9 0.9998 – j0.0568 1.0014 - j0.0559
10 0.9995 – j0.0621 1.0006 - j0.0606
11 0.9988 – j0.0564 1.0014 - j0.0559
12 1.0110 – j0.0385 1.0248 - j0.0384
13 1.0351 + j0.0112 1.0499 + j0.0112
14 1.0141 – j0.0518 1.0165 - j0.0514
15 1.0001 – j0.0530 1.0202 - j0.0526
16 1.0000 – j0.0537 1.0070 - j0.0527
17 0.9951 – j0.0635 0.9969 - j0.0617
18 0.9973 – j0.0692 0.9998 - j0.0674
19 0.9826 – j0.0758 0.9915 - j0.0730
20 0.9905 – j0.0731 0.9928 - j0.0709
21 0.9995 – j0.0614 1.0034 - j0.0594
22 1.0054 – j0.0585 1.0084 - j0.0569
23 1.0421 – j0.0453 1.0489 - j0.0477
24 1.0125 – j0.0553 1.0177 - j0.0548
25 1.0301 – j0.0410 1.0321 - j0.0425
26 1.0102 – j0.0476 1.0146 - j0.0491
27 1.0281 – j0.0244 1.0496 - j0.0302
28 1.0000 – j0.0484 1.0071 - j0.0484
29 1.0145 – j0.0490 1.0294 - j0.0508
30 1.0169 – j0.0662 1.0175 - j0.0649
104
Table 3.14: Line 4 outage
Bus number Actual output Desired output
2 1.0200 – j0.0239 1.0297 - j0.0261
3 0.9999 – j0.0324 1.0003 - j0.0050
4 1.0000 – j0.0472 1.0086 - j0.0599
5 1.0000 – j0.0508 1.0140 - j0.0546
6 0.9990 – j0.0580 1.0028 - j0.0647
7 0.9947 – j0.0650 0.9980 - j0.0693
8 0.9867 – j0.0655 0.9910 - j0.0717
9 0.9987 – j0.0673 1.0007 - j0.0741
10 0.9984 – j0.0716 0.9996 - j0.0791
11 0.9986 – j0.0658 1.0007 - j0.0741
12 1.0300 – j0.0480 1.0242 - j0.0585
13 1.0400 + j0.0009 1.0500 - j0.0094
14 1.0100 – j0.0613 1.0155 - j0.0712
15 1.0150 – j0.0626 1.0192 - j0.0722
16 0.9999 – j0.0634 1.0061 - j0.0720
17 0.9950 – j0.0728 0.9959 - j0.0803
18 0.9972 – j0.0784 0.9986 - j0.0864
19 0.9885 – j0.0848 0.9902 - j0.0917
20 0.9904 – j0.0824 0.9916 - j0.0894
21 0.9994 – j0.0701 1.0022 - j0.0778
22 1.0003 – j0.0682 1.0072 - j0.0754
23 1.0000 – j0.0567 1.0478 - j0.0674
24 1.0101 – j0.0652 1.0165 - j0.0736
25 1.0200 – j0.0527 1.0312 - j0.0613
26 1.0201 – j0.0591 1.0135 - j0.0676
27 1.0290 – j0.0400 1.0488 - j0.0491
28 0.9999 – j0.0597 1.0069 - j0.0664
29 1.0190 – j0.0593 1.0283 - j0.0694
30 1.0140 – j0.0737 1.0162 - j0.0832
105
Table 3.15: Line 8 outage
Bus number Actual output Desired output
2 1.0241 – j0.0154 1.0299 - j0.0156
3 1.0000 – j0.0328 1.0025 - j0.0379
4 1.0001 – j0.0372 1.0031 - j0.0448
5 1.0001 – j0.0200 1.0340 - j0.0167
6 0.9991 – j0.0495 0.9958 - j0.0550
7 0.9747 – j0.0582 0.9797 - j0.0694
8 0.9868 – j0.0521 0.9841 - j0.0622
9 0.9987 – j0.0546 0.9977 - j0.0638
10 0.9985 – j0.0601 0.9986 - j0.0685
11 0.9987 – j0.0543 0.9977 - j0.0638
12 1.0181 – j0.0465 1.0233 - j0.0455
13 1.0401 + j0.0132 1.0500 + j0.0039
14 1.0001 – j0.0498 1.0151 - j0.0586
15 1.0171 – j0.0509 1.0189 - j0.0601
16 1.0000 – j0.0566 1.0052 - j0.0601
17 0.9951 – j0.0614 0.9950 - j0.0694
18 0.9973 – j0.0672 0.9981 - j0.0750
19 0.9886 – j0.0737 0.9897 - j0.0806
20 0.9905 – j0.0710 0.9910 - j0.0785
21 0.9994 – j0.0595 1.0026 - j0.0678
22 1.0004 – j0.0564 1.0079 - j0.0655
23 1.0321 – j0.0428 1.0485 - j0.0561
24 1.0122 – j0.0532 1.0172 - j0.0633
25 1.0231 – j0.0486 1.0317 - j0.0513
26 1.0101 – j0.0452 1.0141 - j0.0577
27 1.0371 – j0.0212 1.0493 - j0.0392
28 1.0000 – j0.0460 1.0007 - j0.0569
29 1.0091 – j0.0470 1.0289 - j0.0596
30 1.0091 – j0.0646 1.0169 - j0.0736
106
Table 3.16: Line 15 outage
Bus number Actual output Desired output
2 1.0001 – j0.0171 1.0299 - j0.0173
3 1.0000 – j0.0327 1.0038 - j0.0323
4 1.0001 – j0.0392 1.0047 - j0.0381
5 1.0001 – j0.0417 1.0130 - j0.0412
6 0.9991 – j0.0472 1.0009 - j0.0460
7 0.9948 – j0.0540 0.9966 - j0.0528
8 0.9869 – j0.0548 0.9892 - j0.0533
9 0.9988 – j0.0571 1.0009 - j0.0547
10 0.9985 – j0.0624 1.0008 - j0.0592
11 0.9988 – j0.0567 1.0009 - j0.0547
12 1.0001 – j0.0388 1.0300 - j0.0385
13 1.0201 + j0.0108 1.0499 + j0.0111
14 1.0211 – j0.0521 1.0206 - j0.0512
15 1.0205 – j0.0533 1.0234 - j0.0517
16 1.0000 – j0.0540 1.0100 - j0.0521
17 0.9951 – j0.0638 0.9980 - j0.0605
18 0.9993 – j0.0695 1.0020 - j0.0663
19 0.9886 – j0.0760 0.9931 - j0.0718
20 0.9905 – j0.0734 0.9941 - j0.0696
21 0.9995 – j0.0616 1.0036 - j0.0578
22 1.0004 – j0.0588 1.0085 - j0.0554
23 1.0331 – j0.0456 1.0490 - j0.0453
24 1.0142 – j0.0556 1.0178 - j0.0531
25 1.0261 – j0.0414 1.0322 - j0.0412
26 1.0112 – j0.0480 1.0146 - j0.0478
27 1.0351 – j0.0249 1.0496 - j0.0291
28 1.0000 – j0.0488 1.0054 - j0.0477
29 1.0241 – j0.0493 1.0295 - j0.0497
30 1.0131 – j0.0664 1.0176 - j0.0638
107
Table 3.17: Line 24 outage
Bus number Actual output Desired output
2 1.0001 – j0.0165 1.0299 - j0.0173
3 1.0000 – j0.0328 1.0065 - j0.0335
4 1.0061 – j0.0385 1.0079 - j0.0395
5 1.0211 – j0.0409 1.0141 - j0.0414
6 0.9991 – j0.0463 1.0031 - j0.0466
7 0.9947 – j0.0531 0.9984 - j0.0533
8 0.9868 – j0.0539 0.9914 - j0.0539
9 0.9987 – j0.0563 1.0023 - j0.0532
10 0.9985 – j0.0617 1.0019 - j0.0567
11 0.9987 – j0.0559 1.0023 - j0.0532
12 1.0101 – j0.0380 1.0237 - j0.0442
13 1.0001 + j0.0116 1.0500 + j0.0053
14 1.0001 – j0.0514 1.0141 - j0.0592
15 1.0091 – j0.0525 1.0159 - j0.0618
16 1.0000 – j0.0532 1.0070 - j0.0543
17 0.9951 – j0.0630 0.9979 - j0.0594
18 0.9973 – j0.0687 0.9854 - j0.0868
19 0.9886 – j0.0753 0.9715 - j0.0985
20 0.9905 – j0.0726 0.9701 - j0.0998
21 0.9994 – j0.0609 1.0039 - j0.0559
22 1.0064 – j0.0580 1.0086 - j0.0536
23 1.0322 – j0.0447 1.0486 - j0.0547
24 1.0152 – j0.0548 1.0173 - j0.0557
25 1.0001 – j0.0405 1.0320 - j0.0431
26 1.0031 – j0.0471 1.0144 - j0.0496
27 1.0261 – j0.0237 1.0496 - j0.0305
28 1.0000 – j0.0478 1.0073 - j0.0483
29 1.0177 – j0.0485 1.0294 - j0.0511
30 1.0100 – j0.0658 1.0175 - 0.0652
108
The post contingency voltages at all the 29 buses are given in Table
3.13 – Table 3.17. The different contingencies considered are outage of
line 1, line 4, line 8, line 15 and line 24. The newly developed method is
able to determine the voltages correctly with a relative error below 2%.
These results can be used to rank the contingencies using the voltage
stability index or L-Index explained in section 3.2.3 and it can be shown
that the results will match exactly with those obtained from Full AC
power flow analysis.
Table 3.18: L-Index and Ranking
Outage L-index (NR) Rank L-index (CVNN) Rank
Line 1-2 0.3012 1 0.3001 1
Line 1-3 0.1811 4 0.1880 4
Line 4-6 0.1545 9 0.1556 9
Line 4-12 0.2045 2 0.2019 2
Line 6-7 0.1602 8 0.1623 8
Line 9-10 0.1912 3 0.1956 3
Line 10-20 0.1772 5 0.1758 5
Line 10-21 0.1366 10 0.1362 10
Line 19-20 0.1634 7 0.1652 7
Line 28-27 0.1674 6 0.1669 6
109
3.7.7 COMPARISON OF CONTINGENCY RANKING
These results can be used to rank the contingencies using the
voltage stability index or L-Index explained in section 3.2.3 and it can be
shown that the results will match exactly with those obtained from Full
AC power flow analysis. The results for first ten ranks are shown in Table
3.18. To obtain the L-index the post contingency voltages are converted
to polar form.
3.8 CONCLUSIONS
The analysis of contingencies, particularly line outages is
important to utilities in both operation and planning, if the potential
outage of a line or generator would result in overload of another line,
then the system is said to be vulnerable, a condition which should be
quickly detected for possible corrective rescheduling actions in operation,
or for system redesign in planning. In this chapter a method of
contingency analysis using complex valued neural network is proposed
to estimate the post contingency state of the power system. Initially a
real valued neural network method is demonstrated with contingency
number and single load pattern as input the network. Secondly detailed
analysis is described using CVNN approach to a 6 bus system, voltage
contingency analysis of IEEE 30 bus system and the results obtained are
encouraging with an immense advantage of avoiding repeated load flow
solutions for each of a list of potential component failures. The inherent
flexibility of complex valued neural network ensures the increase or
110
decrease in the input output variables without changing the computer
program itself.
In power system operation Contingency evaluation plays vital roles
in two realms one is security monitoring and the other is in online
control during emergencies where the viability of the system needs to be
judged for a specific contingency in progress. These evaluations are done
under extreme time pressure. In some instances of actual viability crisis
the action must be taken within two seconds after the emergency. The
computational time required by the trained network to evaluate each
contingency is 0.00269 seconds for a 6 bus system and 0.0065 seconds
for a 30 bus system which is very much useful for the control center
operators to initiate control action during emergencies. It is observed
that application of CVNN for a large power system requires more inputs
with large amount of training data, requiring considerable amount of
computational time for off line training process. From the present study
it is clear that the complex valued artificial neural network can be
applied to the problem of contingency analysis which can very well
replace the conventional practices performed currently in a power
system.