contingency analysis using complex valued neural network

50

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

51

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.

52

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)

53

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

54

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)

55

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)

56

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.

57

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

58

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

59

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.

60

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.

61

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.

62

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

63

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.

64

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

66

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:

67

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)

68

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.

69

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.

70

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

71

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

72

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.

73

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 - -

74

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.

75

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.

76

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





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


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



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



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



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.

contingency analysis using complex valued neural network

Documents