CHAPTER 3 NEURAL NETWORK FOR REAL TIME POWER FLOW ANALYSIS...

38

CHAPTER 3

NEURAL NETWORK FOR REAL TIME

POWER FLOW ANALYSIS

3.1 INTRODUCTION

The power flow analysis assumes importance for various real time power

systems applications. The conventional methods used to solve these real time

problems are iterative techniques and takes longer time for computation. Numerous

methods are proposed to overcome the drawbacks of conventional methods. All

these methods are computationally rigorous, time consuming. Neural Network based

methods trained from input/output data provide an alternate solution for on-line

power flow analysis. The nonlinear mapping capability of NN model trained from

input/output data is well proven in the literature [27,64] and avoids the direct use of

complex mathematical model. They are computationally less rigorous and give

faster estimation in real time.

The major issues in NN based power flow analysis are; the NN model

used for power flow analysis should be accurate, simpler in design, structurally

compact and computationally less complex to ensure faster execution time in real

time implementation. This in turn depends on the type of neural architectures and

learning algorithms. Different neural architectures and neural learning algorithms

are proposed in the literature. The popular neural architectures used for function

approximation are Feed-forward architecture and Cascade architecture [27,55,56].

The popular neural learning algorithms are Backpropagation with Momentum,

Variable Learning Rate and Levenberg Marquardt [57-60,65,66]. This chapter

investigates the suitability of neural architecture and learning algorithm for on-line

power flow analysis.

39

The NN models are built from input-output data. Using the collected data

three types of neural architectures and neural learning algorithm for on-line power

flow analysis are built. The performance of the obtained NN models for power flow

analysis is compared in terms of estimation accuracy, structural compactness and

computational complexity which assume importance for faster analysis of power

flow problem in real time. The most compact neural network model for power flow

analysis is identified.

3.2 TRAINING OF NN MODELS FOR POWER FLOW ANALYSIS

The three architectures are trained with chosen learning algorithms. To

design SLFF-NN, single neuron is added in the hidden layer at a time till the target

MSE is reached. In MLFF-NN and cascade-NN, the choice of number of layers and

number of neurons in each layer is decided by trial and error. The design of MLFF-

NN and cascade-NN is more of an art than a science. Therefore, in this research

work, MLFF-NN and cascade-NN with two hidden layers is designed by trial and

error method. For ease in design, equal number of neurons is chosen in each hidden

layers. The initial parameter value for various algorithms is presented in the Table 3.1.

Table 3.1 Initial parameter value for various algorithms

SL.No Algorithm Learning Rate )

Momentum )

Constant Value (µ)

1. BPM 0.01 0.9 - 2.

VLR

Initial =0.01

Ratio to increase learning rate ( ) = 1.05

Ratio to decrease learning rate ( ) = 0.7

0.9 -

3.

LM - -

Initial µ =0.001

µ decrease factor = 0.1

µ increase factor = 10

The weights and biases are initialized using Nguyen Window algorithm.

40

The results are extensively validated using four test cases. The test cases

considered for investigation are IEEE30 bus test system, IEEE57 bus test system,

IEEE118 bus test system and practical Indian Practical 76 bus test system.

For IEEE30, IEEE57 and IEEE118 bus test systems, around 1000 data

sets are obtained using Newton Raphson method through MATLAB simulation by

varying both the real and reactive load power demand (45%-145%) with the

increment of 0.1%. 750 data are used for training and 250 data are used for testing.

In IEEE30 bus test system, totally there are 24 PQ buses, 5 PV buses and

1 slack bus. The total numbers of outputs are 53. The target MSE is chosen as 1×10-10.

In IEEE57 bus test system, totally there are 50 PQ buses, 6 PV buses and

1 slack bus. The total numbers of outputs are 106. The target MSE is chosen as

1×10-10.

In IEEE118 bus test system, totally there are 64 PQ buses, 53 PV buses

and 1 slack bus. The total numbers of outputs are 181. The target MSE is chosen as

1×10-10.

For Indian Practical 76 bus test system, around 900 data sets are obtained

using Newton Raphson method through MATLAB simulation by varying both the

real and reactive power load demand (45%-135%) the increment of 0.1%. 675 data

are used for training and 225 data are used for testing. In this test case, totally there

are 63 PQ buses, 12 PV buses and 1 slack bus. The total numbers of outputs are 76.

The target MSE is chosen as 1×10-7.

For training the neural network, the inputs to the NN model is chosen as

real power load demand (PD) for all the cases. The real power load demand is

presented in equation (3.1).

1

n

D ii

P p (3.1)

where Pi is the real power at ith load bus, i=1,2,…n.

41

The outputs from the NN model are voltage magnitude (|V|) of PQ buses

and voltage angles ( ) of both PQ and PV buses. The block diagram of NN model

for voltage magnitude and angle estimation with inputs and outputs is shown in

Figure 3.1.

Figure 3.1 NN based voltage magnitude and angle estimator

The performance have been compared with all the three NN architectures

(SLFF-NN, MLFF-NN & CC-NN) which are trained with same input/output data,

same learning algorithms, and same target MSE. This is repeated for different

learning algorithms (BPM, VLR & LM). The tan-sigmoid function is chosen for

hidden layer neurons and pure-linear function is chosen for output layer neurons.

The flow chart for training the NN architecture for power flow analysis is shown in

Figure 3.2.

For IEEE30 bus test system, training Performance and MSE convergence

graph of SLFF-NN using BPM Algorithm is shown in Figure 3.3 and Figure 3.4

respectively. For IEEE30 bus test system, Training Performance and MSE

convergence graph of SLFF-NN using VLR Algorithm is shown in Figure 3.5 and

Figure 3.6 respectively. For IEEE30 bus test system, Training Performance and

MSE convergence graph of SLFF-NN using LM Algorithm is shown in Figure 3.7

and Figure 3.8 respectively.

42

Figure 3.2 Training process of NN for power flow analysis

Choose the NN

architecture and

learning

algorithm

Choose the

number of layers

and number of

neurons per layer

Train the NN using

chosen learning

algorithm using

the LFA training

data

Is Target

MSE met

Stop

Change the

number of

layers and

number of

neurons per

Start

Initialize

weights, biases

and set the

target MSE

No

Yes

43

Figure 3.3 Training performance of SLFF-NN using BPM algorithm for IEEE30 bus test system

Figure 3.4 MSE convergence graph of SLFF-NN using BPM algorithm for IEEE30 bus test system

44

Figure 3.5 Training Performance of SLFF-NN using VLR Algorithm for IEEE30 bus test system

Figure 3.6 MSE Convergence graph of SLFF-NN using VLR Algorithm for IEEE30 bus test system

45

Figure 3.7 Training Performance of SLFF-NN using LM Algorithm for IEEE30 bus test system

Figure 3.8 MSE convergence graph of SLFF-NN using LM algorithm for IEEE30 bus test system

46

The training MSE achieved for all the NN models for power flow analysis for all the test cases is tabulated in Table 3.2. From the Table 3.2, it is observed that all the three NN architectures trained with LM algorithm performed well as compared to BPM, VLR trained NN architectures. Thus LM trained NN models shows better performance as compared to other models. Hence, it is concluded that LM algorithm is most suitable for offline training.

Table 3.2 Performance comparison of NN models trained for same accuracy for various test cases

Test Case Architecture NN Model for Power Flow

Analysis Algorithm MSE Achieved

IEE

E 3

0 bu

s tes

t Sys

tem

Single Layer Feedforward(SLFF) 1-8-53 BPM 4.54 10-2 VLR 1.3 10-4 LM 1 10-10

Multi Layer Feedforward (MLFF) 1-8-8-53

BPM 2.70 10-2 VLR 7.49 10-5 LM 1 10-10

Cascade (CC) 1-2(h)-2(h)-53 BPM 2.87 10-2 VLR 6.41 10-6 LM 1 10-10

IEE

E 5

7 bu

s tes

t Sys

tem

Single Layer Feedforward (SLFF) 1-8-106 BPM 0.195 VLR 1.96 10-4 LM 1 10-10


BPM 7.42 10-2 VLR 1.03 10-4 LM 1 10-10

Cascade (CC) 1-2(h)-2(h)-106 BPM 0.124 VLR 8.87 10-5 LM 1 10-10

IEE

E 1

18 b

us te

st S

yste

m

Single Layer Feedforward (SLFF) 1-13-181 BPM 0.854 VLR 1.79 10-3 LM 1 10-10


BPM 0.935 VLR 9.02 10-5 LM 1 10-10

Cascade (CC) 1-3(h)-3(h)-181

BPM 0.151 VLR 1.54 10-4 LM 1 10-10

Indi

an P

ract

ical

76

bus

test

Sys

tem

Single Layer Feedforward (SLFF) 1-8-138

BPM 0.220 VLR 1.65 10-3 LM 1 10-7


BPM 0.115 VLR 2.61 10-4 LM 1 10-7

Cascade (CC) 1-2(h)-2(h)-138

BPM 0.438 VLR 3.81 10-4 LM 1 10-7

h – Hidden layers

47

3.3 TESTING OF NN MODELS FOR POWER FLOW ANALYSIS IN

TERMS OF ACCURACY

To determine the most suitable architecture, the performance of

LM-trained NN models using three NN architectures is compared in terms of

accuracy, structural compactness and computational complexity. Firstly, the

performance of LM trained SLFF-NN, MLFF-NN and CC-NN models is

compared in terms of accuracy. The off-line LM trained three NN models are

tested for on-line estimation of voltage magnitudes and angles.

Case(i) IEEE30 bus test system

The sample results for voltage magnitude of PQ bus no. 25 and angle

of PQ bus no. 30 estimated using SLFF-NN, MLFF-NN and CC-NN for the

IEEE30 bus test system is presented in Figure 3.9, Figure 3.10 and Figure 3.11

respectively. The test MSE of voltage magnitude for all the PQ buses found all

the three architectures namely SLFF-NN, MLFF-NN and CC-NN is consolidated

and presented in Table 3.3. The test MSE of voltage angle for all the PV & PQ

buses using all the three architectures is consolidated and presented in Table 3.4.

The Performance Comparison of LM Trained NN Models designed using

SLFF-NN, MLFF-NN and CC-NN in terms of average test MSE is shown in

Table 3.5

48

(a)

(b)

Figure 3.9 Voltage magnitude and angle estimated using SLFF-NN for IEEE30 bus test system (a) Voltage magnitude for bus no. 25 (b) Voltage angle for bus no. 30

49

(a)

(b)

Figure 3.10 Voltage magnitude and angle estimated using MLFF-NN for IEEE30 bus test system (a) Voltage magnitude for bus no. 25 (b) Voltage angle for no. bus 30

50

(a)

(b)

Figure 3.11 Voltage magnitude and angle estimated using CC-NN for IEEE30 bus test System (a) Voltage magnitude for bus no. 25 (b) Voltage angle for bus no. 30

51

Table 3.3 MSE of voltage magnitude for various NN architectures-IEEE30 bus test system

SL.NO BUS NO ERROR_ SLFF ERROR_

MLFF ERROR_ CC

1 3 4.4402x10-10 5.4846x10-10 4.5666x10-10

2 4 4.8563x10-10 5.4682x10-10 5.1806x10-10

3 6 2.6037x10-10 2.8072x10-10 2.7104x10-10

4 7 2.5937x10-10 2.6291x10-10 2.6602x10-10

5 9 6.5968x10-10 6.7845x10-10 6.6718x10-10

6 10 1.7747x10-10 1.7918x-09 1.7767x10-09

7 12 7.4694x10-10 7.7645x10-10 7.6372x10-10

8 14 9.5004x10-10 9.4996x10-09 9.4993x10-10

9 15 1.9255x10-09 1.9360x10-09 1.9752x10-09

10 16 1.5474x10-09 1.5613x10-09 1.6041x10-09

11 17 2.0260x10-09 2.0337x10-09 2.0642x10-09

12 18 2.7691x10-09 2.7645x10-09 2.7824x10-09

13 19 3.0659x10-09 3.0562x10-09 3.1228x10-09

14 20 2.7758x10-09 2.7728x10-09 2.7768x10-09

15 21 2.6745x10-09 2.6751x10-09 2.7130x10-09

16 22 2.6381x10-09 2.6408x10-09 2.6660x10-09

17 23 2.9013x10-09 2.9004x10-09 2.9243x10-09

18 24 3.6915x10-09 3.6875x10-09 3.7052x10-09

19 25 3.1329x10-09 3.1830x10-09 3.1569x10-09

20 26 4.8843x10-09 4.9124x10-09 5.0368x10-09

21 27 2.1299x10-09 2.2221x10-09 2.2141x10-09

22 28 3.2687x10-10 3.4557x10-10 3.3725x10-10

23 29 3.8767x10-09 3.9711x10-09 3.9405x10-09

24 30 5.1179x10-09 5.2093x10-09 5.2096x10-09

52

Table 3.4 MSE of voltage angle for various NN architectures-IEEE30 bus test system

SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC

25 2 2.6163x10-05 2.5590x10-05 2.5948 x10-05

26 3 3.9664x10-05 3.8845x10-05 3.9356 x10-05

27 4 6.1496x10-05 6.0190x10-05 6.0985 x10-05

28 5 1.4053x10-04 1.3763x10-04 1.3925 x10-04

29 6 8.6909x10-05 8.5043x10-05 8.6152 x10-05

30 7 1.1519x10-04 1.1275x10-04 1.1421 x10-04

31 8 1.0270x10-04 1.0047x10-04 1.0191 x10-04

32 9 1.3407x10-04 1.3117x10-04 1.3317 x10-04

33 10 1.6407x10-04 1.6048x10-04 1.6312 x10-04

34 11 1.3407x10-04 1.3117x10-04 1.3309 x10-04

35 12 1.5065x10-04 1.4736x10-04 1.4977x10-04

36 13 1.5065x10-04 1.4736x10-04 1.4994x10-04

37 14 1.6810x10-04 1.6443x10-04 1.6748x10-04

38 15 1.6938x10-04 1.6568x10-04 1.6867x10-04

39 16 1.6106x10-04 1.5754x10-04 1.6020x10-04

40 17 1.6768x10-04 1.6401x10-04 1.6712x10-04

41 18 1.8247x10-04 1.7848x10-04 1.8165x10-04

42 19 1.8626x10-04 1.8219x10-04 1.8519x10-04

43 20 1.8174x10-04 1.7777x10-04 1.8092x10-04

44 21 1.7310x10-04 1.6931x10-04 1.7213x10-04

45 22 1.7269x10-04 1.6891x10-04 1.7192x10-04

46 23 1.7677x10-04 1.7290x10-04 1.7622x10-04

47 24 1.7933x10-04 1.7541x10-04 1.7848x10-04

48 25 1.7544x10-04 1.7160x10-04 1.7497x10-04

49 26 1.8515x10-04 1.8110x10-04 1.8457x10-04

50 27 1.6725x10-04 1.6360x10-04 1.6628x10-04

51 28 9.6072x10-05 9.4014x10-05 9.5235 x10-04

52 29 1.9492x10-04 1.9068x10-04 1.9462x10-04

53 30 2.1708x10-04 2.1243x10-04 2.1698x10-04

53

Table 3.5 Performance comparison of LM trained NN Models designed using SLFF-NN, MLFF-NN and CC-NN in terms of average test MSE for IEEE30 bus test system

Test Case Average Test MSE

for SLFF-NN Average Test MSE

for MLFF-NN Average Test MSE

for CC-NN

IEEE 30 bus test system

8.2183×10-5 8.0437×10-5 8.1784×10-5

The average test MSE for all IEEE30 bus test system is consolidated and

presented in Table 3.5. From the above investigation, it is understood that the

voltage magnitude and angle obtained from all the LM trained NN models using all

the three NN architectures is found to closely match with the voltage magnitude and

angle estimated using conventional method. Thus all the three LM trained NN

models performed equally well.

Case(ii) IEEE57 bus test system

The sample results for voltage magnitude of PQ bus no. 15 and angle of

PV bus no. 12 estimated using SLFF-NN, MLFF-NN and CC-NN for the IEEE57

bus test system is presented in Figure 3.12, Figure 3.13 and Figure 3.14 respectively.

The test MSE of voltage magnitude for all the PQ buses using all the three

architectures namely SLFF-NN, MLFF-NN and CC-NN is consolidated and

presented in Table 3.6. The test MSE of voltage angle for all the PV & PQ buses

using all the three architectures is consolidated and presented in Table 3.7.

54

(a)

(b)

Figure 3.12 Voltage magnitude and angle estimated using SLFF-NN for IEEE57 bus test System (a) Voltage Magnitude for bus no. 15 (b) Voltage angle for bus no. 12

55

(a)

(b)

Figure 3.13 Voltage magnitude and angle estimated using MLFF-NN for IEEE57 bus test system (a) Voltage magnitude for bus 15 no. (b) Voltage angle for bus no. 12

56

(a)

(b)

Figure 3.14 Voltage magnitude and angle estimated using CC-NN for IEEE57 bus test system (a) Voltage magnitude for bus no. 15 (b) Voltage angle for bus no. 12

57



1 4 7.9171x10-11 6.3593x10-09 8.2636x10-11

2 5 6.3531x10-11 3.2110x10-09 1.0758x10-10

3 7 2.0504x10-10 8.8919x10-10 2.0540x10-10

4 10 4.1744x10-10 2.0971x10-09 4.4653x10-10

5 11 7.2983x10-10 2.4049x10-08 7.4438x10-10

6 13 1.1570x10-09 4.9427x10-08 1.1798x10-09

7 14 2.2324x10-09 9.1525x10-08 2.2647x10-09

8 15 1.4229x10-09 1.0157x10-07 1.5123x10-09

9 16 1.5275x10-09 1.1973x10-07 1.5812x10-09

10 17 2.2266x10-09 1.5712x10-07 2.3138x10-09

11 18 1.4035x10-09 1.6074x10-08 1.4107x10-09

12 19 5.3150x10-09 4.2471x10-08 5.3349x10-09

13 20 7.1812x10-09 6.5305x10-08 7.2408x10-09

14 21 8.4362x10-09 1.2446x10-07 8.4740x10-09

15 22 8.5747x10-09 1.3367x10-07 8.6308x10-09

16 23 8.9016x10-09 1.3359x10-07 9.0720x10-09

17 24 1.2077x10-08 1.1718x10-07 1.2135x10-08

18 25 2.9816x10-08 2.2785x10-07 3.0022x10-08

19 26 9.4686x10-09 9.7404x10-08 9.4972x10-09

20 27 3.9937x10-09 2.4719x10-08 4.0426x10-09

21 28 2.1098x10-09 9.1088x10-09 2.1125x10-09

22 29 1.0871x10-09 3.2942x10-09 1.0916x10-09

23 30 3.6451x10-08 2.6306x10-07 3.6827x10-08

24 31 4.5680x10-08 3.2368x10-07 4.6018x10-08

25 32 3.6226x10-08 3.0593x10-07 3.6667x10-08

26 33 3.7106x10-08 3.0948x10-07 3.7344x10-08

58

Table 3.6 (Continued)


27 34 1.9785x10-08 1.7768x10-07 1.9923x10-08

28 35 1.7337x10-08 1.6458x10-07 1.7514x10-08

29 36 1.4650x10-08 1.5380x10-07 1.4835x10-08

30 37 1.2740x10-08 1.4873x10-07 1.2807x10-08

31 38 7.9854x10-09 1.3577x10-07 8.0427x10-09

32 39 1.3019x10-08 1.4776x10-07 1.3125x10-08

33 40 1.4858x10-08 1.5107x10-07 1.4991x10-08

34 41 5.0816x10-09 5.5489x10-08 5.1183x10-09

35 42 1.0889x10-08 7.8821x10-08 1.0962x10-08

36 43 1.7096x10-09 3.3943x10-08 1.7213x10-09

37 44 6.4029x10-09 1.5428x10-07 6.4873x10-09

38 45 2.2349x10-09 1.5969x10-07 2.2816x10-09

39 46 3.8077x10-09 1.2722x10-07 3.9023x10-09

40 47 6.0193x10-09 1.2751x10-07 6.0616x10-09

41 48 6.5687x10-09 1.2555x10-07 6.7084x10-09

42 49 5.5014x10-09 9.0261x10-08 5.5910x10-09

43 50 5.6430x10-09 4.8428x10-08 5.7016x10-09

44 51 1.3666x10-09 6.2307x10-09 1.3692x10-09

45 52 3.6901x10-09 1.7895x10-09 3.6905x10-09

46 53 4.9735x10-09 1.4814x10-09 4.9764x10-09

47 54 1.9295x10-09 5.2140x10-10 1.9334x10-09

48 55 1.1492x10-10 4.7514x10-11 1.3748x10-10

49 56 1.2385x10-08 1.0234x10-07 1.2447x10-08

50 57 1.4158x10-08 1.2000x10-07 1.4226x10-08

59



51 2 1.7618x10-05 1.1538x10-05 1.7654x10-05

52 3 2.9994x10-04 4.9683x10-05 3.0026x10-04

53 4 4.5205x10-04 4.3681x10-05 4.5263x10-04

54 5 7.3162x10-04 6.1742x10-05 7.3216x10-04

55 6 8.7253x10-04 7.6791x10-05 8.7333x10-04

56 7 0.0010 1.0584x10-04 0.0010

57 8 0.0011 1.2677x10-04 0.0011

58 9 0.0010 9.5262x10-05 0.0010

59 10 8.8970x10-04 7.6730x10-05 8.9065x10-04

60 11 8.1267x10-04 7.3194x10-05 8.1353x10-04

61 12 7.6464x10-04 6.6962x10-05 7.6548x10-04

62 13 6.3496x10-04 5.7158x10-05 6.3570x10-04

63 14 5.0717x10-04 4.7236x10-05 5.0777x10-04

64 15 3.0778x10-04 2.7053x10-05 3.0812x10-04

65 16 4.3673x10-04 4.3390x10-05 4.3736x10-04

66 17 1.2848x10-04 1.1843x10-05 1.2861x10-04

67 18 6.1246x10-04 5.2033x10-05 6.1318x10-04

68 19 6.9273x10-04 5.8938x10-05 6.9345x10-04

69 20 7.1470x10-04 6.4375x10-05 7.1545x10-04

70 21 7.2094x10-04 6.3839x10-05 7.2187x10-04

71 22 7.2104x10-04 6.5677x10-05 7.2172x10-04

72 23 7.3247x10-04 6.6556x10-05 7.3350x10-04

73 24 9.0108x10-04 7.9472x10-05 9.0214x10-04

74 25 0.0012 2.4749x10-04 0.0012

75 26 9.0880x10-04 8.4468x10-05 9.0958x10-04

76 27 0.0011 9.7153x10-05 0.0011

77 28 0.0011 1.0306x10-04 0.0011

78 29 0.0011 1.0607x10-04 0.0011

79 30 0.0013 3.1166x10-04 0.0013

60



80 31 0.0013 4.2607x10-04 0.0013

81 32 0.0012 2.9503x10-04 0.0012

82 33 0.0012 3.0027x10-04 0.0012

83 34 8.0316x10-04 7.0727x10-05 8.0414x10-04

84 35 7.8886x10-04 6.9665x10-05 7.8966x10-04

85 36 7.7416x10-04 6.8776x10-05 7.7512x10-04

86 37 7.5776x10-04 6.7985x10-05 7.5886x10-04

87 38 6.9934x10-04 6.4391x10-05 7.0022x10-04

88 39 7.6328x10-04 6.8216x10-05 7.6436x10-04

89 40 7.8309x10-04 6.8908x10-05 7.8407x10-04

90 41 9.8153x10-04 8.2372x10-05 9.8286x10-04

91 42 0.0010 8.7775x10-05 0.0010

92 43 8.5916x10-04 7.5301x10-05 8.6014x10-04

93 44 6.1327x10-04 6.0597x10-05 6.1396x10-04

94 45 4.3191x10-04 4.1906x10-05 4.3234x10-04

95 46 5.9100x10-04 5.4717x10-05 5.9164x10-04

96 47 6.7293x10-04 6.1079x10-05 6.7363x10-04

97 48 6.8853x10-04 6.2363x10-05 6.8938x10-04

98 49 7.4833x10-04 6.4803x10-05 7.4931x10-04

99 50 8.4195x10-04 7.1111x10-05 8.4302x10-04

100 51 9.2015x10-04 7.6379x10-05 9.2145x10-04

101 52 0.0012 1.1245x10-04 0.0012

102 53 0.0012 1.1388x10-04 0.0012

103 54 0.0011 1. 1325x10-04 0.0011

104 55 0.0011 1.0556x10-04 0.0011

105 56 0.0010 8.6193x10-05 0.0010

106 57 0.0010 8.8643x10-05 0.0010

61

Table 3.8 Performance comparison of LM trained NN models designed using SLFF-NN, MLFF-NN and CC-NN in terms of average test MSE for IEEE57 bus test system

Test Case Average Test MSE for

SLFF-NN Average Test MSE

for MLFF-NN Average Test

MSE for CC-NN

IEEE 57 bus test system

4.3307×10-4 4.9513×10-5 4.3364×10-4

The average test MSE for all IEEE57 bus test system is consolidated and

presented in Table 3.8. From the above investigation, it is understood that the





Case(iii) IEEE118 bus test system


PQ bus no. 118 estimated using SLFF-NN, MLFF-NN and CC-NN for the IEEE118

bus test system is presented in Figure 3.14, Figure 3.15 and Figure 3.16 respectively.

The test MSE of voltage magnitude for all the PQ buses using all the three

architectures namely SLFF-NN, MLFF-NN and CC-NN is consolidated and



62

(a)

(b)

Figure 3.15 Voltage magnitude and angle estimated using SLFF-NN for IEEE118 bus test System (a) Voltage magnitude for bus no. 114 (b) Voltage angle for bus no. 118

63

(a)

(b)

Figure 3.16 Voltage magnitude and angle estimated using MLFF-NN for IEEE118 bus test system (a) Voltage magnitude for bus no. 114 (b) Voltage angle for bus no. 118

64

(a)

(b)

Figure 3.17 Voltage magnitude and angle estimated using CC-NN for IEEE118 bus test system (a) Voltage magnitude for bus no. 114 (b) Voltage angle for bus no. 118

65


SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 1 2 1.8967x10-11 4.6929x10-08 3.0254x10-11 2 3 1.7462x10-11 3.9560x10-08 3.5823x10-11 3 5 1.1182x10-13 1.4717x10-08 2.4227x10-11 4 7 3.2426x10-13 1.2269x10-08 1.2263x10-12 5 9 4.8909x10-12 2.0082x10-08 9.3067x10-12 6 11 3.4776x10-11 1.1168x10-08 5.9718x10-11 7 13 2.3965x10-10 2.8766x10-09 2.1912x10-10 8 14 4.4631x10-12 3.3215x10-09 9.6680x10-12 9 16 8.7137x10-11 1.5957x10-08 1.6214x10-10

10 17 2.2379x10-11 2.4764x10-08 4.4301x10-11 11 20 5.2852x10-10 4.4334x10-08 5.8313x10-10 12 21 1.1401x10-09 2.9187x10-08 1.2203x10-09 13 22 1.1650x10-09 8.5437x10-09 1.2074x10-09 14 23 1.8577x10-10 2.0481x10-08 2.0078x10-10 15 28 2.8944x10-11 6.4487x10-08 4.4211x10-11 16 29 1.1454x10-11 1.4763x10-08 2.1916x10-11 17 30 4.0512x10-10 2.9074x10-08 3.9055x10-10 18 33 1.9442x10-10 2.6646x10-08 2.6881x10-10 19 35 5.7621x10-12 3.9442x10-08 3.6367x10-11 20 37 8.1473x10-11 8.0503x10-09 1.3586x10-10 21 38 2.7630x10-09 1.1930x10-07 2.7553x10-09 22 39 6.2239x10-11 5.6895x10-09 5.6120x10-11 23 41 4.1738x10-11 2.3447x10-08 3.3275x10-11 24 43 1.7054x10-09 3.3267x10-08 1.4710x10-09 25 44 3.7304x10-09 4.9810x10-08 3.6464x10-09 26 45 2.4811x10-09 3.4694x10-08 2.6422x10-09 27 47 2.8439x10-09 1.2834x10-07 2.4918x10-09 28 48 2.7905x10-11 1.0903x10-08 6.9501x10-11 29 50 6.3417x10-11 1.1790x10-08 1.2528x10-10 30 51 4.2944x10-10 4.4290x10-09 4.4967x10-10 31 52 6.1790x10-10 4.5079x10-08 7.9484x10-10 32 53 3.8119x10-10 1.5596x10-08 4.9962x10-10

66


SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 33 57 5.9558x10-11 3.8970x10-09 1.0445x10-10 34 58 2.0431x10-10 3.6252x10-08 2.9019x10-10 35 60 2.9904x10-12 6.2508x10-08 1.7657x10-11 36 63 1.2652x10-10 6.8335x10-08 1.5949x10-10 37 64 8.5506x10-11 5.2970x10-09 1.5878x10-10 38 67 4.0110x10-11 5.0277x10-09 1.2786x10-10 39 68 1.3983x10-10 2.0834x10-08 1.7189x10-10 40 71 2.7577x10-11 1.3005x10-08 1.9681x10-11 41 75 7.4550x10-10 6.9339x10-08 6.6488x10-10 42 78 4.3217x10-11 8.5745x10-09 5.1072x10-11 43 79 9.3584x10-11 2.2071x10-09 1.7098x10-10 44 81 1.2565x10-10 1.7250x10-08 1.0509x10-10 45 82 7.6354x10-10 4.1434x10-08 7.3409x10-10 46 83 4.2532x10-10 5.5095x10-08 3.5179x10-10 47 84 8.9447x10-11 2.4859x10-08 1.8758x10-10 48 86 8.1713x10-11 7.3455x10-09 7.9094x10-11 49 88 1.7270x10-11 4.6982x10-08 3.1205x10-11 50 93 1.2299x10-10 2.6380x10-08 2.2244x10-10 51 94 3.1832x10-10 3.0485x10-08 3.1012x10-10 52 95 8.3073x10-10 4.2487x10-08 8.5728x10-10 53 96 6.2787x10-10 4.0716x10-08 7.0742x10-10 54 97 4.1014x10-10 6.3143x10-08 4.5423x10-10 55 98 3.4101x10-10 3.0263x10-08 4.3989x10-10 56 101 1.0205x10-10 2.6530x10-09 2.4682x10-10 57 102 1.8224x10-11 8.2632x10-08 5.9509x10-11 58 106 1.0823x10-10 2.1627x10-08 1.5163x10-10 59 108 6.9287x10-12 8.6653x10-09 1.5417x10-11 60 109 8.7714x10-12 1.5283x10-08 1.4949x10-11 61 114 1.5526x10-11 1.2623x10-07 3.9529x10-11 62 115 1.7953x10-11 5.1524x10-09 6.3957x10-11 63 114 1.8045x10-10 7.1371x10-08 1.9654x10-10 64 118 4.2166x10-10 1.0128x10-08 3.5812x10-10

67


SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 65 1 9.0x10-3 4.5x10-3 8.8x10-3 66 2 8.8x10-3 4.5x10-3 8.6x10-3 67 3 8.8x10-3 4.5x10-3 8.7x10-3 68 4 8.4x10-3 4.1x10-3 8.2x10-3 69 5 8.3x10-3 4.0x10-3 8.2x10-3 70 6 8.6x10-3 4.2x10-3 8.5x10-3 71 7 8.7x10-3 4.8x10-3 8.5x10-3 72 8 7.8x10-3 3.8x10-3 7.7x10-3 73 9 7.8x10-3 3.8x10-3 7.7x10-3 74 10 7.8x10-3 3.8x10-3 7.7x10-3 75 11 8.6x10-3 4.2x10-3 8.4x10-3 76 12 8.7x10-3 4.3x10-3 8.5x10-3 77 13 8.5x10-3 4.2x10-3 8.3x10-3 78 14 8.4x10-3 4.3x10-3 8.3x10-3 79 15 7.6x10-3 3.6x10-3 7.4x10-3 80 16 8.3x10-3 4.2x10-3 8.1x10-3 81 17 7.3x10-3 3.7x10-3 7.2x10-3 82 18 7.5x10-3 3.7x10-3 7.4x10-3 83 19 7.4x10-3 3.6x10-3 7.3x10-3 84 20 7.2x10-3 3.4x10-3 7.0x10-3 85 21 6.8x10-3 3.2x10-3 6.7x10-3 86 22 6.4x10-3 3.3x10-3 6.3x10-3 87 23 5.6x10-3 2.7x10-3 5.5x10-3 88 24 4.4x10-3 2.1x10-3 4.3x10-3 89 25 6.3x10-3 3.0x10-3 6.2x10-3 90 26 6.5x10-3 3.2x10-3 6.3x10-3 91 27 7.2x10-3 3.5x10-3 7.1x10-3 92 28 7.4x10-3 3.5x10-3 7.2x10-3 93 29 7.5x10-3 4.0x10-3 7.4x10-3 94 30 6.8x10-3 3.3x10-3 6.7x10-3 95 31 7.5x10-3 3.6x10-3 7.3x10-3 96 32 7.1x10-3 3.4x10-3 7.0x10-3 97 33 6.8x10-3 3.2x10-3 6.6x10-3 98 34 5.9x10-3 2.8x10-3 5.7x10-3 99 35 5.9x10-3 2.8x10-3 5.8x10-3 100 36 5.9x10-3 3.0x10-3 5.8x10-3 101 37 5.8x10-3 2.8x10-3 5.6x10-3 102 38 5.2x10-3 2.5x10-3 5.1x10-3

68


SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 103 39 5.8x10-3 2.7x10-3 5.6x10-3 104 40 5.6x10-3 2.7x10-3 5.5x10-3 105 41 5.5x10-3 2.6x10-3 5.4x10-3 106 42 4.8x10-3 2.3x10-3 4.7x10-3 107 43 5.1x10-3 2.4x10-3 4.9x10-3 108 44 3.7x10-3 1.8x10-3 3.7x10-3 109 45 3.2x10-3 6.0x10-3 3.2x10-3 110 46 2.8x10-3 1.5x10-3 2.7x10-3 111 47 2.1x10-3 1.1x10-3 2.1x10-3 112 48 2.7x10-3 1.3x10-3 2.6x10-3 113 49 2.6x10-3 1.3x10-3 2.6x10-3 114 50 2.8x10-3 1.4x10-3 2.7x10-3 115 51 3.1x10-3 1.4x10-3 3.0x10-3 116 52 3.2x10-3 1.6x10-3 3.1x10-3 117 53 3.3x10-3 1.7x10-3 3.3x10-3 118 54 3.3x10-3 1.6x10-3 3.3x10-3 119 55 3.3x10-3 1.9x10-3 3.3x10-3 120 56 3.3x10-3 1.6x10-3 3.3x10-3 121 57 3.1x10-3 1.5x10-3 3.0x10-3 122 58 3.2x10-3 1.5x10-3 3.1x10-3 123 59 2.9x10-3 1.4x10-3 2.9x10-3 124 60 2.5x10-3 1.2x10-3 2.5x10-3 125 61 2.5x10-3 1.2x10-3 2.4x10-3 126 62 2.5x10-3 1.4 x10-3 2.5x10-3 127 63 2.5x10-3 1.2 x10-3 2.4x10-3 128 64 2.2x10-3 1.1 x10-3 2.2x10-3 129 65 1.7x10-3 7.8x10-4 1.7x10-3 130 66 2.2x10-3 1.0 x10-3 2.1x10-3 131 67 2.4x10-3 1.2 x10-3 2.3x10-3 132 68 9.4x10-4 4.6x10-4 9.3x10-4 133 70 7.8x10-4 6.1x10-4 7.5x10-4 134 71 9.9x10-4 7.2x10-4 9.8x10-4 135 72 2.4x10-3 1.5 x10-3 2.4x10-3 136 73 1.0x10-3 7.2x10-4 9.9x10-4 137 74 7.5x10-4 4.9x10-4 7.2x10-4 138 75 6.4x10-4 4.4x10-4 6.2x10-4 139 76 1.0x10-3 5.9x10-4 1.0x10-3 140 77 1.3x10-3 6.6x10-4 1.2x10-3 141 78 1.3x10-3 7.0x10-4 1.3x10-3

69


SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 142 79 1.4x10-3 7.6x10-4 1.4x10-3 143 80 1.6x10-3 7.8x10-4 1.6x10-3 144 81 1.2x10-3 5.7x10-4 1.1x10-3 145 82 2.3x10-3 1.2x10-3 2.3x10-3 146 83 2.7x10-3 1.3x10-3 2.6x10-3 147 84 3.2x10-3 1.6x10-3 3.1x10-3 148 85 3.5x10-3 1.9x10-3 3.4x10-3 149 86 3.6x10-3 2.2x10-3 3.5x10-3 150 87 3.6x10-3 1.9x10-3 3.5x10-3 151 88 3.9x10-3 2.2x10-3 3.8x10-3 152 89 4.0x10-3 2.1x10-3 3.9x10-3 153 90 4.4x10-3 2.3x10-3 4.3x10-3 154 91 4.2x10-3 2.2x10-3 4.1x10-3 155 92 3.9x10-3 2.4x10-3 3.8x10-3 156 93 3.5x10-3 1.9x10-3 3.5x10-3 157 94 3.3x10-3 1.6x10-3 3.2x10-3 158 95 3.0x10-3 1.7x10-3 2.9x10-3 159 96 2.5x10-3 1.4x10-3 2.5x10-3 160 97 2.1x10-3 1.1x10-3 2.0x10-3 161 98 2.3x10-3 1.1x10-3 2.3x10-3 162 99 3.1x10-3 1.5x10-3 3.0x10-3 163 100 3.6x10-3 1.9x10-3 3.6x10-3 164 101 3.8x10-3 2.0x10-3 3.7x10-3 165 102 3.9x10-3 2.0x10-3 3.8x10-3 166 103 4.1x10-3 2.2x10-3 4.0x10-3 167 104 4.4x10-3 2.4x10-3 4.3x10-3 168 105 4.5x10-3 2.6x10-3 4.4x10-3 169 106 4.5x10-3 2.4x10-3 4.4x10-3 170 107 4.8x10-3 3.2x10-3 4.7x10-3 171 108 4.7x10-3 2.7x10-3 4.6x10-3 172 109 4.8x10-3 3.2x10-3 4.7x10-3 173 110 5.0x10-3 2.7x10-3 4.8x10-3 174 111 5.0x10-3 3.6x10-3 4.9x10-3 175 112 5.3x10-3 3.1x10-3 5.2x10-3 176 113 7.3x10-3 3.5x10-3 7.2x10-3 177 114 7.2x10-3 3.4x10-3 7.1x10-3 178 115 7.3x10-3 3.9x10-3 7.1x10-3 179 116 9.6x10-4 4.5x10-4 9.4x10-4 180 117 8.9x10-3 4.6x10-3 8.7x10-3 181 118 8.4x10-4 4.9x10-4 8.1x10-4

70

Table 3.11 Performance comparison of LM trained NN Models designed using SLFF-NN, MLFF-NN and CC-NN in terms of average test MSE for IEEE118 bus test system

Test Case Average Test MSE for

SLFF-NN

Average Test MSE

for MLFF-NN

Average Test

MSE for CC-NN

IEEE 118 bus

test system 2.90×10-4 8.45×10-5 2.90×10-4

The average test MSE for all IEEE118 bus test system is consolidated

and presented in Table 3.11. From the above investigation, it is understood that the





Case(iv) Indian practical 76 bus test system


PQ bus no. 40 estimated using SLFF-NN, MLFF-NN and CC-NN for the Indian

practical 76 bus test system is presented in Figure 3.18, Figure 3.19 and Figure 3.20

respectively. The test MSE of voltage magnitude for all the PQ buses found all the

three architectures namely SLFF-NN, MLFF-NN and CC-NN is consolidated and



71

(a)

(b)

Figure 3.18 Voltage magnitude and angle estimated using SLFF-NN for indian practical 76 test System (a) Voltage magnitude for bus no. 54 (b) Voltage ANGLE for bus no. 40

72

(a)

(b)

Figure 3.19 Voltage magnitude and angle estimated using MLFF-NN for Indian practical 76 Test System (a) Voltage magnitude for bus no. 54 (b) Voltage angle for bus no. 40

73

(a)

(b)

Figure 3.20 Voltage magnitude and angle estimated using CC-NN for Indian practical 76 test system (a) Voltage magnitude for bus no. 54 (b) Voltage angle for bus no. 40

74

Table 3.12 MSE of voltage magnitude for various NN architectures- Indian practical 76 bus test system

SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 1 14 5.4283x10-08 2.5216x10-07 7.7225x10-08 2 15 2.6499x10-09 5.0411x10-09 8.1746x10-09 3 16 3.6683x10-09 2.3419x10-08 1.2767x10-08 4 17 2.3373x10-08 6.4392x10-08 2.8612x10-08 5 18 2.7667x10-08 5.2529x10-08 2.7826x10-08 6 19 2.0081x10-08 7.7087x10-08 1.8788x10-08 7 20 1.7404x10-08 3.3589x10-08 1.9020x10-08 8 21 2.7024x10-08 9.5974x10-08 2.6015x10-08 9 22 3.4650x10-08 6.5397x10-08 3.9771x10-08

10 23 2.5620x10-08 8.9083x10-08 2.5681x10-08 11 24 6.6742x10-08 2.7735x10-07 7.8931x10-08 12 25 2.7958x10-08 5.1521x10-08 2.9308x10-08 13 26 8.5759x10-09 1.6514x10-08 1.4173x10-08 14 27 1.9506x10-08 5.7199x10-08 3.1931x10-08 15 28 3.6567x10-08 2.4681x10-07 4.4433x10-08 16 29 7.6991x10-09 1.9244x10-08 9.9542x10-09 17 30 5.4522x10-08 2.8978x10-07 6.2632x10-08 18 31 2.0736x10-08 7.3395x10-08 2.9055x10-08 19 32 1.7145x10-08 2.7975x10-08 2.0979x10-08 20 33 1.0262x10-08 1.6688x10-08 1.4618x10-08 21 34 1.9660x10-08 4.6433x10-08 4.1234x10-08 22 35 2.1787x10-08 3.8492x10-08 2.3930x10-08 23 36 4.5097x10-09 8.1707x10-08 5.5492x10-09 24 37 1.0973x10-08 1.9591x10-08 1.3611x10-08 25 38 7.5319x10-08 2.8337x10-07 8.1088x10-08 26 39 9.7486x10-09 4.2157x10-08 1.2161x10-08 27 40 3.2335x10-08 1.5623x10-07 5.5007x10-08 28 41 1.8174x10-08 3.4722x10-08 1.8719x10-08 29 42 1.6188x10-08 1.0045x10-07 2.4907x10-08 30 43 1.6754x10-08 4.0706x10-08 1.8863x10-08 31 44 9.4560x10-09 2.3043x10-08 1.3251x10-08 32 45 1.1743x10-08 4.1410x10-08 2.1955x10-08 33 46 4.4891x10-08 2.6817x10-07 5.5265x10-08 34 47 4.5910x10-08 1.9921x10-07 5.4117x10-08

75


SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 35 48 7.1516x10-08 2.6268x10-07 7.4597x10-08 36 49 1.6527x10-08 3.5892x10-08 5.8802x10-08 37 50 1.1951x10-08 2.7119x10-08 2.6400x10-08 38 51 3.6500x10-08 2.1260x10-07 4.5909x10-08 39 52 2.0950x10-08 4.2450x10-08 3.2863x10-08 40 53 2.3713x10-08 5.4226x10-08 2.3061x10-08 41 54 2.4377x10-08 8.4562x10-08 2.4338x10-08 42 55 4.6544x10-08 2.7519x10-07 5.8393x10-08 43 56 1.6154x10-08 4.4912x10-08 1.8280x10-08 44 57 1.5490x10-08 3.0262x10-08 2.4878x10-08 45 58 4.8213x10-08 3.1936x10-07 6.1907x10-08 46 59 5.3114x10-08 3.0469x10-07 6.2006x10-08 47 60 2.3230x10-08 4.2438x10-08 2.4809x10-08 48 61 2.1506x10-08 3.6106x10-08 2.1443x10-08 49 62 1.0940x10-08 6.5528x10-08 1.3359x10-08 50 63 3.0012x10-08 1.2966x10-07 3.9653x10-08 51 64 1.3655x10-08 3.9754x10-08 2.4105x10-08 52 65 1.0305x10-08 6.2213x10-08 1.2444x10-08 53 66 3.6697x10-08 1.4143x10-07 5.3503x10-08 54 67 1.6456x10-08 5.2981x10-08 2.0704x10-08 55 68 1.6538x10-08 5.9186x10-08 4.2636x10-08 56 69 2.0542x10-08 3.6402x10-08 4.0493x10-08 57 70 1.3264x10-08 4.0122x10-08 1.4074x10-08 58 71 7.6289x10-09 1.4613x10-08 8.6534x10-09 59 72 8.2444x10-09 2.3118x10-08 3.5931x10-08 60 73 4.2325x10-09 4.1635x10-08 5.1988x10-09 61 74 9.1292x10-09 1.9369x10-08 1.3819x10-08 62 75 5.2619x10-08 3.2477x10-07 6.1718x10-08 63 76 3.5895x10-08 2.7008x10-07 4.8767x10-08

76

Table 3.13 MSE of voltage angle for various NN architectures- Indian practical 76 bus test system

SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 64 2 0.0067 0.0055 0.0053 65 3 0.0147 0.0116 0.0114 66 4 0.0150 0.0118 0.0116 67 5 0.0144 0.0113 0.0112 68 6 0.0185 0.0148 0.0142 69 7 0.0163 0.0130 0.0125 70 8 0.0043 0.0035 0.0034 71 9 0.0175 0.0138 0.0135 72 10 0.0183 0.0147 0.0140 73 11 0.0145 0.0115 0.0111 74 12 0.0181 0.0144 0.0139 75 13 0.0087 0.0070 0.0068 76 14 0.0060 0.0049 0.0047 77 15 0.0181 0.0145 0.0139 78 16 0.0143 0.0115 0.0110 79 17 0.0184 0.0146 0.0141 80 18 0.0182 0.0145 0.0139 81 19 0.0176 0.0138 0.0134 82 20 0.0176 0.0141 0.0135 83 21 0.0169 0.0133 0.0130 84 22 0.0146 0.0115 0.0112 85 23 0.0178 0.0141 0.0136 86 24 0.0089 0.0072 0.0070 87 25 0.0173 0.0136 0.0133 88 26 0.0163 0.0129 0.0125 89 27 0.0162 0.0128 0.0125 90 28 0.0044 0.0035 0.0034 91 29 0.0186 0.0148 0.0143 92 30 0.0052 0.0042 0.0041 93 31 0.0137 0.0108 0.0106 94 32 0.0160 0.0126 0.0123 95 33 0.0164 0.0130 0.0126 96 34 0.0159 0.0125 0.0122 97 35 0.0178 0.0142 0.0137 98 36 0.0185 0.0150 0.0142 99 37 0.0151 0.0119 0.0116 100 38 0.0101 0.0082 0.0079 101 39 0.0189 0.0151 0.0145

77


SL.NO BUS NO ERROR_ SLFF ERROR_ MLFF ERROR_ CC 102 40 0.0088 0.0072 0.0069 103 41 0.0169 0.0136 0.0130 104 42 0.0156 0.0125 0.0121 105 43 0.0165 0.0131 0.0127 106 44 0.0163 0.0128 0.0126 107 45 0.0175 0.0139 0.0134 108 46 0.0041 0.0033 0.0032 109 47 0.0066 0.0054 0.0052 110 48 0.0067 0.0054 0.0053 111 49 0.0159 0.0125 0.0122 112 50 0.0185 0.0147 0.0142 113 51 0.0044 0.0035 0.0034 114 52 0.0161 0.0128 0.0125 115 53 0.0164 0.0130 0.0126 116 54 0.0174 0.0137 0.0133 117 55 0.0047 0.0038 0.0037 118 56 0.0156 0.0123 0.0120 119 57 0.0183 0.0145 0.0140 120 58 0.0032 0.0026 0.0025 121 59 0.0048 0.0039 0.0038 122 60 0.0165 0.0130 0.0127 123 61 0.0162 0.0128 0.0125 124 62 0.0145 0.0115 0.0112 125 63 0.0069 0.0056 0.0054 126 64 0.0177 0.0141 0.0136 127 65 0.0149 0.0117 0.0115 128 66 0.0135 0.0109 0.0105 129 67 0.0161 0.0126 0.0123 130 68 0.0175 0.0140 0.0134 131 69 0.0162 0.0127 0.0124 132 70 0.0176 0.0141 0.0135 133 71 0.0186 0.0148 0.0142 134 72 0.0150 0.0121 0.0116 135 73 0.0185 0.0150 0.0141 136 74 0.0163 0.0129 0.0125 137 75 0. 0060 0.0049 0.0047 138 76 0.0044 0.0035 0.0034

78

Table 3.14 Performance comparison of LM Trained NN models designed using SLFF-NN, MLFF-NN and CC-NN in terms of average test MSE for Indian practical 76 bus test system

Test Case Average Test MSE for SLFF-NN

Average Test MSE for MLFF-NN

Average Test MSE for CC-NN

Indian Practical 76 bus test System

2.48×10-4 2.59×10-4 8.86×10-5

The average test MSE for all Indian practical 76 bus test system is

consolidated and presented in Table 3.14. From the above investigation, it is

understood that the voltage magnitude and angle obtained from all the LM trained

NN models using all the three NN architectures is found to closely match with the

voltage magnitude and angle estimated using conventional method. Thus all the

three LM trained NN models performed equally well.

The average test MSE for all the four test cases is consolidated and

presented in Table 3.15. For all the four test cases LM trained SLFF-NN, MLFF-NN

and CC-NN architectures performed equally well.

Table 3.15 Performance comparison of LM Trained NN models designed using SLFF-NN, MLFF-NN and CC-NN in terms of average test MSE for various test cases

Test Case Average Test MSE for SLFF-NN

Average Test MSE for MLFF-NN

Average Test MSE for CC-NN

IEEE 30 bus test system 8.2183×10-5 8.0437×10-5 8.1784×10-5



Indian Practical 76 bus test System 2.48×10-4 2.59×10-4 8.86×10-5

79

3.4 TESTING OF NN ARCHITECTURES FOR POWER FLOW

ANALYSIS IN TERMS OF STRUCTURAL COMPACTNESS AND

COMPUTATIONAL COMPLEXITY

The structural compactness and computational complexity assumes

importance in real time implementation. This is the motivation to compare all the

NN models in terms of structural compactness and computational complexity. For

the desired accuracy, the number of hidden neurons is used as an index to measure

the structural compactness of model. The neural network architecture with lesser

number of hidden neurons is found to be compact and gives ease in real time

implementation of the on-line power flow analysis. The number of parameters and

nonlinear function extraction in the network indicates its computational complexity.

Each parameter warrants some mathematical operations. The number of parameters

for FF-NN and CC-NN can be calculated using equation (2.23) and equation (2.24)

respectively.

For various test cases, the parameters, neurons and computations required

by the SLFF-NN, MLFF-NN and CC-NN models are tabulated in Table 3.16. From

the Table 3.16, it is seen that for IEEE30 bus test system, SLFF-NN and MLFF-NN

requires 8 and 16 hidden neurons respectively, where as CC-NN model requires

much lesser number of hidden neurons 4 as compared to SLFF-NN and MLFF-NN

models. For IEEE57 bus test system, SLFF-NN and MLFF-NN requires 8 and 18

hidden neurons respectively, where as CC-NN model requires much lesser number

of hidden neurons 4 as compared to SLFF-NN and MLFF-NN models. For IEEE118

bus test system, SLFF-NN and MLFF-NN requires 13 and 14 hidden neurons

respectively, where as CC-NN model requires much lesser number of hidden

neurons 6 as compared to SLFF-NN and MLFF-NN models. For practical Indian

Practical 76 bus test system, SLFF-NN and MLFF-NN requires 8 and 18 hidden

neurons respectively, where as CC-NN model requires much lesser number of

hidden neurons 4 as compared to SLFF-NN and MLFF-NN models.

80

Table 3.16 Performance comparison of LM trained NN models for power flow analysis in terms of structural compactness and Computational complexity

Test

Cas

e

NN

Arc

hite

ctur

es

NN

Mod

els

No.

of H

idde

n N

euro

ns

No.

of P

aram

eter

s

No.

of A

dditi

ons

No.

of M

ultip

licat

ions

No.

of

Tan-

sigm

oids

Exec

utio

n T

ime

in

mill

iseco

nds

IEEE

30

bus t

est

Syst

em SLFF 1-8-53 8 493 432 432 8 3256

MLFF 1-8-8-53 16 565 496 496 16 5776 CC 1-2(h)-2(h)-53 4 330 273 273 4 1742

IEEE

57

bus t

est

Syst

em SLFF 1-8-106 8 970 856 856 8 4104

MLFF 1-9-9-106 18 1168 1044 1044 18 7470 CC 1-2(h)-2(h)-106 4 648 538 538 4 2272

IEEE

118

bu

s tes

t Sy

stem

SLFF 1-13-181 13 2560 2366 2366 13 8619 MLFF 1-14-14-181 28 2953 2744 2744 28 13860

CC 1-3(h)-3(h)-181 6 1469 1282 1282 6 4358

Indi

an

Prac

tical

76

bus t

est

Syst

em SLFF 1-8-138 8 1258 1112 1112 8 4616

MLFF 1-9-9-138 18 1488 1332 1332 18 8046

CC 1-2(h)-2(h)-53 4 840 698 698 4 2592

On Intel(R) Core(TM) i5 32-bits processor with the clock frequency of

2.53GHz, 4 GB RAM, the execution time required to compute voltage magnitude

and angle using SLFF-NN, MLFF-NN and CC-NN are computed and presented in

table 3.16.From the Table 3.16, it is seen that for IEEE30 bus test system, SLFF-NN

and MLFF-NN requires 3256ms and 5776 ms respectively, where as CC-NN model

requires much lesser execution time of 1742ms as compared to SLFF-NN and

MLFF-NN models. For IEEE57 bus test system, SLFF-NN and MLFF-NN requires

SLFF-NN and MLFF-NN requires 4104ms and 7470 ms respectively, where as CC-

NN model requires much lesser execution time of 2272 ms as compared to SLFF-

NN and MLFF-NN models. For IEEE118 bus test system, SLFF-NN and MLFF-NN

81

requires SLFF-NN and MLFF-NN requires 8619ms and 13860ms respectively,

where as CC-NN model requires much lesser execution time of 4358ms as

compared to SLFF-NN and MLFF-NN models. For Indian practical 76 bus test

system, SLFF-NN and MLFF-NN requires SLFF-NN and MLFF-NN requires

4616ms and 8046ms respectively, where as CC-NN model requires much lesser

execution time of 2592ms as compared to SLFF-NN and MLFF-NN models.

From the results obtained, it is found that for all the four test cases, the

execution time required for CC-NN is lesser as compared to SLFF-NN AND MLFF-

NN architectures.

Hence CC-NN model results in structurally compact model as compared

to SLFF-NN and MLFF-NN model. The total number of parameters and

computations time required for CC-NN is found to be lesser as compared to SLFF-

NN and MLFF-NN. Hence, CC-NN model is of lesser complexity as compared to

SLFF-NN and MLFF-NN model. As the nonlinear function is the most complex part

of the computation it can be observed that the computational complexity is

proportional to the total number of hidden neurons.

3.5 SUMMARY

Thus, from the above simulation result, it can be summarized that CC-

NN architecture provides the required accuracy, structurally compact,

computationally less complex model for real time power flow analysis. Hence,

cascade architecture trained with Levenberg Marquardt algorithm is identified to be

most suitable NN model for on-line Power flow analysis.

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CHAPTER 3 NEURAL NETWORK FOR REAL TIME POWER FLOW ANALYSIS...

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