21

CHAPTER - 2

POWER FLOW ANALYSIS

_______________________________________________________________________________________________________

2.1 INTRODUCTION

Power flow analysis is a basic and necessary tool for any

electrical system under steady state condition to determine the exact

electrical performance. The load flow solution provide the real (kW)

and reactive power (kVAr) losses of the system and voltage magnitudes

and angles at different nodes of the system subject to the regulating

capability of generators, condensers and tap changing of transformers

under load as well as specified net interchange between individual

operating systems. This analysis is essential for the continuous

evaluation of the existing power system and effective planning of

alternatives for system expansion to meet increased load demand in

future. These analyses require the calculation of numerous load flows

for both normal and emergency operating conditions. The load flow

studies are helpful to confirm selected switchgear, transformer, and

cable sizing. These studies should also be used to confirm adequate

voltage profiles during different operating conditions, such as heavily

loaded and lightly loaded system conditions. Load flow studies can be

used to determine the optimum size and location of capacitors for

power factor correction. The results of load flow studies are also

starting points for other system studies.

The distribution power flow involves, first of all, finding all of the

node voltages. From these voltages, it is possible to directly compute

22

currents, power flows, system losses and other steady state quantities.

Some applications, especially in the fields of optimization of power

system, distribution automation (i.e., VAR planning, network

optimization, state estimation, etc.) need repeated fast load flow

solutions. In these applications it is important that the load flow

problem is solved as efficiently as possible.

Das et al. [41] presented a simple method for distribution load

flow solution which solves a simple algebraic expression of voltage

magnitude. S. Ghosh and D. Das [50] proposed a method for solving

radial distribution networks which involves simple algebraic

expression. S. Mok et.al [53] proposed a new approach for power flow

analysis of balanced radial distribution systems. The convergence

characteristics and the effect of voltage dependency are analyzed. A

general load flow for meshed networks with more than one feeding

node is presented by Haque [54]. Ratio flow method based on forward

– backward for complex distribution system is presented in [65]. R.

Ranjan and D. Das proposed a simple algorithm based on circuit

theory using algebraic recursive expression to solve radial distribution

networks [66]. Load flow solution to distribution system is obtained by

using bus injection to branch current matrix and branch current to

bus voltage matrix and a simple multiplication [69]. A Load flow

technique to solve distribution networks based on sequential branch

numbering scheme by considering committed loads is presented [83].

A backward/ forward sweep load flow solution for three phase radial

distribution systems is proposed [86]. A load flow solution including

23

voltage dependent load models based on forward – backward sweep

method is proposed [87]. Comparison of various distribution load flow

algorithms based on forward – backward sweep is presented in [105].

An iterative technique in which loads are simulated as impedances at

each iteration is proposed for radial distribution load flow [124].

In this chapter a backward – forward sweep method is proposed

for solving the load flow problem of a distribution system. The

proposed method is tested by taking 15, 33 and 69 node radial

distribution systems. The mathematical formulation and the

illustration of node identification of the proposed load flow method are

described in the following sections.

2.2 MATHEMATICAL FORMULATION

The load flow of a single source network can be solved

iteratively from two sets of recursive equations. The first set of

equations for calculation of the power flow through the branches

starting from the last branch and proceeding in the backward

direction towards the root node. The other set of equations are for

calculating the voltage magnitude and angle of each node starting

from the root node and proceeding in the forward direction towards

the last node. These recursive equations are derived as follows.

The fig. 2.1 shows the representation of 2 nodes in a

distribution line. Consider a branch ‘j’ is connected between the nodes

‘i’ and ‘i+1’.

24

Fig.2.1. Representation of two nodes in a distribution line

The effective active ( iP ) and reactive ( iQ ) powers that of flowing

through branch ‘j’ from node ‘i' to node ‘i+1’ can be calculated

backwards from the last node and is given as,

'2 '2P +Qi+1 i+1'P = P +ri i+1 j 2Vi+1

(2.1)

'2 '2P + Qi+1 i+1'Q = Q + xi i+1 j 2Vi+1

(2.2)

where 'P = P + Pi+1 i+1 L i+1and 'Q = Q + Qi+1 i+1 Li+1

PLi+1and QLi+1

are loads that are connected at node ‘i+1’.

Pi+1 and Qi+1 are the effective real and reactive power flows from node

‘i+1’.

The voltage magnitude and angle at each node are calculated in

forward direction. Consider a voltage V δi i at node ‘i’ and V δi+1 i+1at

node ‘i+1’, then the current flowing through the branch ‘j’ having an

impedance , z = r + jxj j j connected between ‘i’ and ‘i+1’is given as,

V δ - V δi i i+1 i+1I =j r + jxj j(2.3)

25

andP - j Qi iI =j V - δi i

(2.4)

On equating the equations (2.3) and (2.4), we have

V δ - V δP - jQ i i i+1 i+1i i =V - δ r j + j xi i j

(2.5)

2V - V V (δ - δ ) = (P - jQ )(r + jx )i i i+1 i+1 i i i j j (2.6)

By equating real and imaginary parts on both sides of equation (2.6),

we have

2V V c o s (δ - δ ) = V - (P r + Q x )i i+1 i+1 i i i j i j(2.7)

V V s i n (δ - δ ) = Q r - P xi i+1 i+1 i i j i j (2.8)

Squaring and adding equations (2.7) and (2.8), we get

2 2 2 2(V V ) = [V - (P r + Q x )] + [Q r - P x ]i i+1 i i j i j i j i j (2.9)

2 4 2 2 2 2 2(V V ) = V - 2V (P r + Q x )+ (r + x )(P + Q )i i+1 i i i j i j j j i i(2.10)

1/22 2(P + Q )2 2 2 i iV = V - 2(P r + Q x )+ (r + x )i+1 i i j i j j j 2Vi

(2.11)

and voltage angle, δi+1 can be derived on dividing equations (2.8) and

(2.7)

Q r - P xi j i jt a n (δ - δ ) =i+1 i 2V - (P r + Q x )i i j i j

(2.12)

Q r - P xi j i j-1δ = δ + ta ni+1 i 2V - (P r + Q x )i i j i j

(2.13)

26

The magnitude and the phase angle equations can be used

recursively in a forward direction to find the voltage and angle

respectively of all nodes of radial distribution system.

The real and reactive power losses of branch ‘j’ can be

calculated as,

2 2P + Qi iPloss (j) = r j 2Vi

(2.14)

2 2P + Qi iQ loss (j) = x j 2Vi

(2.15)

The total real and reactive power loss of radial distribution system can

be calculated as,

2 2P + Qn b i iTP L = r j 2j=1 Vi

(2.16)

2 2P + Qnb i iTQL = x j 2j=1 Vi

(2.17)

Initially, a flat voltage profile is assumed at all nodes i.e., 1.0

pu. The branch powers are computed iteratively with the updated

voltages at each node. In the proposed load flow method, powers

summation is done in the backward walk and voltages are calculated

in the forward walk. The maximum difference of voltage magnitudes in

successive iterations is taken as convergence criteria and 0.0001 is

taken as tolerance value. For branch powers calculation the adjacent

nodes and branches of every node are identified with the help of node

identification algorithm which is explained in the following section.

27

2 3 4 5

7

6

8

9

10

11

12

13

15

14

S/S1 2 3 4

5

6

7

9

8

10

11

12

13

14

Fig.2.2 Single line diagram of 15-node radial distribution system

2.3 ILLUSTRATION OF NODE IDENTIFICATION

Consider a 15 node radial distribution system as shown in

fig.2.2 for illustration of node identification. The formation of

various vectors (Adn, Adb, MF and MT) used in sparsity technique

for node identification is given below.

2.3.1 Algorithm for Node Identification

Following algorithm explains the methodology to identify the

nodes and branches connected to a particular node in detail,

which will help in finding the exact load feeding through that

particular node.

Step 1: Read system data

Step 2: Initialize vector MF with 1 & S=0

Step 3: Initialize the count for node i=1

Step 4: Initialize count for branch count j=1

28

Step 5: if (i= = SE [j]) go to step 7 else go to step 6

Step 6: if (i= = RE[j] go to step 8 else go to step 9

Step 7: S=S+1

Adn [S] = RE [j]

Adb [S] = j

Step 8: S=S+1

Adn [S] =SE [j]

Adb [S] =j

Step 9: if (j<nb)

j=j+1 go to step 5 else go to step 10

Step 10: MT[i] =S

MF[i+1] =MT[i] +1

Step 11: if (i<=nd)

i=i+1 go to step 4 else go to step 12

Step 12: Stop

2.4 LOAD FLOW CALCULATION

The forward-backward distribution load flow method is

proposed in this thesis. Initially assume a flat voltage profile i.e.,

set the voltage equal to 1.0 pu at every node. The flow chart for

load flow is shown in fig.2.3. The backward forward load flow

algorithm is given in section 2.4.4. The adjacent branch and node

vectors for 15 node system are tabulated in table 2.1. The MF and

MT vectors are given in table 2.2.

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Table2.1. Adjacent branch & node vectors of fig.2.2

Table2.2. MF and MT vectors of fig.2.2

NodeNo.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

MF[i] 1 2 6 9 13 14 17 18 19 21 22 24 26 27 28

MT[i] 1 5 8 12 13 16 17 18 20 21 23 25 26 27 28

where,

MF[i] =Memory location from

MT[i] =Memory location to for a particular node ‘i’.

where, i=1 to nd

S. No. Node Adn Adb S. No. Node And Adb

1 1 2 1

14

6

2 7

15 7 8

16 8 9

2

2

1 1 17 7 6 8

3 3 2 18 8 6 9

4 6 7 199

2 5

5 9 5 20 10 6

6

3

2 2 21 10 9 6

7 4 3 2211

3 10

8 11 10 23 12 11

9

4

3 3 2412

11 11

10 5 4 25 13 12

11 14 13 26 13 12 12

12 15 14 27 14 4 13

13 5 4 4 28 15 4 14

30

2.4.1 Backward Propagation

The updated effective power flows in each branch are

obtained in the backward propagation computation by considering

the node voltages of previous iteration. It means the voltage values

obtained in the forward path are held constant during the

backward propagation and updated power flows in each branch are

transmitted backward along the feeder using backward path. This

indicates that the backward propagation starts at the extreme end

node and proceeds towards source node. The active and reactive

power flows are calculated using equations (2.1) and (2.2).

2.4.2 Forward Propagation

The purpose of the forward propagation is to calculate the

voltages at each node starting from the feeder source node. The

feeder substation voltage is set at its actual value. During the

forward propagation the effective power in each branch is held

constant to the value obtained in backward walk. The node voltage

magnitudes are calculated using equation (2.11). The voltage angle

is calculated using equation (2.12).

2.4.3 Convergence Criterion

The voltages calculated in the previous and present iterations

are compared. In the successive iterations if the maximum mismatch

between the voltages is less than the specified tolerance i.e., 0.0001,

the solution is said to be converged. Otherwise new effective power

flows in each branch are calculated through backward walk with the

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present computed voltages and then the procedure is repeated until

the solution is converged. The algorithm for load flow solution of radial

distribution system is explained in the following section.

Fig.2.3. Flow chart for Radial Distribution Load Flow

2.4.4 Algorithm for Load Flow Calculation

Step 1: Read distribution system line and load data.

Assume initial node voltages are 1 pu and set ε = 0.0001.

Step 2: Start iteration count, IT =1.

Step 3: Initialize real power loss and reactive power loss vectors to

zero.

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Step 4: Calculate the effective real and reactive power flow in each

branch using equations (2.1) and (2.2).

Step 5: Calculate node voltages, real and reactive power loss of each

branch using equations (2.11), (2.14) and (2.15) respectively.

Step 6: Check for convergence i.e., ΔV <εmax in successive iterations.

If it is converged go to next step otherwise increment iteration number

and go to step 4.

Step 7: Calculate the real and reactive power losses for all branches

and also total real and reactive power loss.

Step 8: Print voltage at each node, the real and reactive power losses

of all branches and total loss.

Step 9: Stop.

2.5 ILLUSTRATIVE EXAMPLES

The effectiveness of the proposed method is illustrated through

three different examples consisting of 15, 33 and 69-node radial

distribution systems.

2.5.1 Example-1

The line and load data of 15-node, 11 kV radial distribution

system [41] shown in fig.2.2 is given in the table A.1. The voltage

magnitudes of the system are given in the table 2.3. The power losses

of the system are given in the table 2.4. The total real and reactive

power losses are 60.35 kW and 58.39 kVAr. The real and reactive

power losses are 4.92% and 4.64% of their respective loads. The

33

minimum voltage is 0.9424 pu at node 13 and the voltage regulation

of the system is 5.76%.

Table2.3. Voltage magnitudes of 15-node system

Node No. Voltage Magnitude (pu) Angle (deg.)

1 1.0000 0

2 0.9713 0.0342

3 0.9547 -0.0599

4 0.9489 -0.0527

5 0.9479 -0.0405

6 0.9618 0.1481

7 0.9611 0.1566

8 0.9593 0.1792

9 0.9652 0.1081

10 0.9635 0.1289

11 0.9478 0.0147

12 0.9437 0.0658

13 0.9424 0.0821

14 0.9466 -0.0243

15 0.9465 -0.0222

Table2.4. Power loss of 15-node radial distribution system

Br.No.

Node Backward ForwardPloss(kW)

Qloss(kVAr)SE RE

P(kW)

Q(kVAr)

P(kW)

Q(kVAr)

1 1 2 1288.2 1308.5 1250.4 1271.67 37.72 36.90

2 2 3 735.48 748.61 724.19 737.57 11.34 14.01

3 3 4 397.24 404.92 395.2 402.53 2.05 2.40

4 4 5 44.16 45.03 44.1 44.99 0.06 0.04

5 2 9 355.27 362.3 350.5 357.41 4.77 4.89

6 9 10 140.39 143.09 140 142.83 0.39 0.27

34

7 2 6 70.11 71.49 70 71.41 0.11 0.08

8 6 7 114.63 116.76 114.16 116.44 0.47 0.32

9 6 8 44.16 45.03 44.1 44.99 0.06 0.04

10 3 11 256.95 261.23 254.77 259.69 2.18 1.54

11 11 12 114.78 116.86 114.17 116.45 0.6 0.41

12 12 13 44.17 45.04 44.1 44.99 0.07 0.05

13 4 14 70.2 71.55 70 71.41 0.2 0.14

14 4 15 140.44 143.12 140 142.83 0.44 0.29

Total loss 60.35 58.39

Fig.2.4 33-node radial distribution system

2.5.2 Example-2

The 33-node, 12.66 kV radial distribution system [20] is shown

in fig.2.4. The line and load data of this system are given in table A.2.

The voltage magnitude at different nodes of this system is given in

table 2.5. Table 2.6 gives the power loss results of the system. The

35

total real and reactive power losses of the system are 201.54 kW and

132.11 kVAr respectively. These are 5.42% and 5.74% of their total

loads. The minimum voltage of the system is 0.9132 pu at node 18.

The maximum voltage regulation of system is 8.68%. Comparison of

load flow results between the proposed method and the existing

method [50] is given in table 2.7. The total real and reactive power

losses are reduced and the minimum voltage is improved in the

proposed method.

Table2.5. Voltage magnitudes of 33-node system

Node No. Voltage Magnitude (pu) Angle (deg.)

1 1.0000 0.0000

2 0.9970 0.0140

3 0.9830 0.0950

4 0.9755 0.1610

5 0.9681 0.2270

6 0.9499 0.1330

7 0.9462 -0.0970

8 0.9414 -0.0610

9 0.9351 -0.1340

10 0.9290 -0.1970

11 0.9283 -0.1900

12 0.9269 -0.1780

13 0.9208 -0.2690

14 0.9185 -0.3470

15 0.9171 -0.3850

16 0.9157 -0.4080

17 0.9137 -0.4850

18 0.9132 -0.4950

19 0.9965 0.0030

36

20 0.9929 -0.0640

21 0.9922 -0.0840

22 0.9916 -0.1040

23 0.9794 0.0640

24 0.9727 -0.0250

25 0.9694 -0.0680

26 0.9478 0.1720

27 0.9452 0.2280

28 0.9337 0.3110

29 0.9254 0.3890

30 0.9220 0.4950

31 0.9178 0.4100

32 0.9169 0.3870

33 0.9166 0.3790

Table2.6. Power loss of 33-node radial distribution system

Br.No.

Node Backward ForwardPloss(kW)

Qloss(kVAr)SE RE

P(kW)

Q(kVAr)

P(kW)

Q(kVAr)

1 1 2 3917.6 2435.2 3905.4 2428.9 12.24 6.33

2 2 3 3444.2 2207.8 3392.4 2181.4 51.79 26.38

3 3 4 2362.8 1684.2 2342.9 1674.0 19.90 10.14

4 4 5 2222.9 1594.0 2204.2 1584.5 18.69 9.52

5 5 6 2144.23 1554.5 2105.98 1521.48 37.12 33.02

6 6 7 1095.24 527.87 1093.32 521.54 1.92 6.33

7 7 8 893.33 421.54 888.49 419.95 4.84 1.59

8 8 9 688.49 319.95 684.31 316.94 4.18 3.00

9 9 10 624.31 296.95 620.75 294.42 3.56 2.52

10 10 11 560.75 274.42 560.2 274.24 0.55 0.18

11 11 12 515.2 244.24 514.32 243.95 0.88 0.29

12 12 13 454.32 208.95 451.66 206.85 2.69 2.09

13 13 14 391.66 171.85 390.93 170.89 0.73 0.96

37

14 14 15 270.9 90.9 270.58 90.58 0.36 0.32

15 15 16 210.58 80.58 210.3 80.37 0.28 0.21

16 16 17 150.3 60.37 150.05 60.04 0.25 0.34

17 17 18 90.05 40.04 90 40 0.05 0.04

18 2 19 361.14 161.08 360.98 160.93 0.16 0.15

19 19 20 270.98 120.93 270.14 120.18 0.83 0.75

20 20 21 180.14 80.18 180.04 80.06 0.10 0.12

21 21 22 90.04 40.06 90 40 0.04 0.06

22 3 23 939.61 457.24 936.43 455.07 3.18 2.17

23 23 24 845.43 404.07 841.28 401.01 4.14 3.06

24 24 25 421.28 201.01 420 200 1.28 1.01

25 6 26 950.75 973.61 948.15 972.28 2.60 1.33

26 26 27 888.15 947.28 884.82 945.59 3.33 1.69

27 27 28 824.82 920.59 813.52 910.62 11.30 6.86

28 28 29 753.52 890.62 745.69 883.8 7.84 5.82

29 29 30 625.69 813.8 621.8 811.82 3.88 1.98

30 30 31 421.8 211.84 420.21 210.26 1.59 1.58

31 31 32 270.22 140.26 270 140.02 0.21 0.24

32 32 33 60.01 40.02 60 40 0.01 0.02

Total loss 201.54 132.11

Table2.7. Comparison of load flow results of 33-node system

Description

Total Loss MinimumVoltage and

it’s nodenumber

Real power(kW)

Reactivepower(kVAr)

Existingmethod [50]

202.67 135.140.9131 at

node 18

Proposedmethod

201.54 132.110.9132 at

node 18

38

2.5.3 Example-3

The fig 2.5 shows a 69-node, 12.66 kV radial distribution

system [17]. The line and load data of this system is given in table A.3.

Table 2.8 gives comparison of the voltage magnitudes (pu) of the

system obtained by the proposed method with the existing method

[50]. The minimum voltage is 0.9094 pu at node 65 and maximum

voltage regulation is 9.06%. The line losses of the system are given in

table 2.9. The total real and reactive power losses of this system are

224.45 kW and 107.14 kVAr respectively. The real and reactive power

losses are 5.91% and 3.78% of their total loads. The load flow results

of the proposed method are compared with the existing method [50] in

table 2.10. The total real and reactive power losses are reduced and

minimum voltage is improved by the proposed method.

Fig.2.5 69-node radial distribution system

39

Table2.8. Voltage magnitudes of 69-node system

NodeNo.

Voltage Magnitude (pu) Angle (deg.)

Proposedmethod

Existing method[50]

Proposedmethod

1 1.0000 1.0000 0.0000

2 1.0000 0.9999 -0.0010

3 0.9999 0.9999 -0.0020

4 0.9998 0.9998 -0.0060

5 0.9990 0.9990 -0.0180

6 0.9901 0.9901 0.0500

7 0.9808 0.9808 0.1220

8 0.9786 0.9786 0.0900

9 0.9775 0.9775 0.0820

10 0.9726 0.9725 0.1660

11 0.9715 0.9714 0.1850

12 0.9684 0.9682 0.2380

13 0.9653 0.9653 0.2850

14 0.9624 0.9624 0.3310

15 0.9596 0.9595 0.3770

16 0.9590 0.9590 0.3860

17 0.9581 0.9581 0.4000

18 0.9580 0.9581 0.4000

19 0.9578 0.9576 0.4090

20 0.9575 0.9573 0.4140

21 0.9569 0.9568 0.4230

22 0.9569 0.9568 0.4230

23 0.9569 0.9568 0.4240

24 0.9566 0.9566 0.4270

25 0.9565 0.9564 0.4300

26 0.9565 0.9564 0.4320

27 0.9563 0.9563 0.4320

40

28 0.9999 0.9999 -0.0030

29 0.9998 0.9999 -0.0050

30 0.9997 0.9997 -0.0030

31 0.9997 0.9997 -0.0030

32 0.9996 0.9996 -0.0009

33 0.9994 0.9994 0.0040

34 0.9992 0.9990 0.0090

35 0.9990 0.9990 0.0100

36 0.9999 0.9999 -0.0030

37 0.9997 0.9998 -0.0090

38 0.9996 0.9996 -0.0120

39 0.9996 0.9995 -0.0120

40 0.9995 0.9995 -0.0130

41 0.9988 0.9988 -0.0240

42 0.9987 0.9986 -0.0280

43 0.9985 0.9985 -0.0290

44 0.9985 0.9985 -0.0290

45 0.9984 0.9984 -0.0310

46 0.9984 0.9984 -0.0310

47 0.9998 0.9998 -0.0080

48 0.9986 0.9985 -0.0530

49 0.9948 0.9947 -0.1920

50 0.9942 0.9942 -0.2110

51 0.9788 0.9785 0.0900

52 0.9788 0.9785 0.0910

53 0.9747 0.9747 0.1030

54 0.9716 0.9714 0.1290

55 0.9669 0.9669 0.1650

56 0.9627 0.9626 0.2000

57 0.9402 0.9401 0.5970

58 0.9291 0.9290 0.8000

41

59 0.9248 0.9248 0.8810

60 0.9197 0.9197 0.9860

61 0.9126 0.9123 1.0550

62 0.9124 0.9121 1.0580

63 0.9118 0.9117 1.0610

64 0.9112 0.9098 1.0790

65 0.9094 0.9092 1.0850

66 0.9715 0.9713 0.1860

67 0.9715 0.9713 0.1860

68 0.9679 0.9679 0.2440

69 0.9679 0.9679 0.2440

Table2.9. Power loss of 69-node radial distribution system

Br.No.

Node Backward ForwardPloss(kW)

Qloss(kVAr)SE RE

P(kW)

Q(kVAr)

P(kW)

Q(kVAr)

1 1 2 4016.3 2785.3 4016.2 2785.1 0.08 0.18

2 2 3 4016.2 2785.1 4016.1 2784.9 0.08 0.18

3 3 4 3748.9 2591.8 3748.7 2591.3 0.20 0.47

4 4 5 2897.9 1980.1 2896 1977.9 1.94 2.27

5 5 6 2896 1977.9 2867.8 1963.5 28.31 14.41

6 6 7 2865.2 1961.3 2836 1946.5 28.41 14.98

7 7 8 2795.6 1916.5 2788.7 1913 6.91 7.10

8 8 9 2669.6 1828 2666.2 1826.2 3.39 2.92

9 9 10 779.5 523.23 774.78 521.67 4.75 1.58

10 10 11 746.78 511.67 745.77 511.33 1.02 0.34

11 11 12 564.77 381.33 562.59 380.61 2.19 0.73

12 12 13 361.56 236.61 360.29 236.18 1.29 0.43

13 13 14 352.28 231.18 351.04 230.77 1.25 0.41

14 14 15 343.04 225.27 341.84 224.8 1.21 0.39

15 15 16 341.84 224.87 341.62 224.8 0.23 0.07

42

16 16 17 296.62 194.8 296.3 194.7. 0.32 0.11

17 17 18 236.3 159.7 236.3 194.7 0.00 0.00

18 18 19 176.3 124.7 176.2 159.7 0.11 0.04

19 19 20 176.2 124.66 176.13 124.66 0.07 0.02

20 20 21 175.13 124.04 175.02 124.64 0.11 0.04

21 21 22 61.02 43.01 61.02 124.01 0.00 0.00

22 22 23 56.02 40.01 56.02 43.01 0.01 0.00

23 23 24 56.02 40.01 56.01 40.01 0.01 0.00

24 24 25 28.01 20 28 20 0.01 0.00

25 25 26 28 20 28 20 0.00 0.00

26 26 27 14 10 14 10 0.00 0.00

27 3 28 81.53 64.01 81.53 64.01 0.00 0.00

28 28 29 55.53 46.01 55.52 46.01 0.00 0.01

29 29 30 29.52 28.01 29.52 28.01 0.01 0.00

30 30 31 29.52 28.01 29.52 28.01 0.00 0.00

31 31 32 29.52 28.01 29.52 28.01 0.01 0.00

32 32 33 29.52 28.01 29.51 28.01 0.01 0.00

33 33 34 15.51 18 15.5 18 0.01 0.00

34 34 35 6 4 6 4 0.00 0.00

35 3 36 185.76 129.16 185.76 129.16 0.00 0.00

36 36 37 159.76 110.61 159.74 110.57 0.02 0.04

37 37 38 133.74 92.02 133.72 92 0.02 0.02

38 38 39 133.72 92 133.72 91.99 0.01 0.01

39 39 40 109.72 74.99 109.72 74.99 0.00 0.00

40 40 41 85.72 57.99 85.67 57.93 0.05 0.06

41 41 42 84.47 56.93 84.45 56.91 0.02 0.02

42 42 43 84.45 56.91 84.45 56.91 0.00 0.00

43 43 44 78.45 52.61 78.45 52.61 0.00 0.00

44 44 45 78.45 52.61 78.44 52.6 0.01 0.01

45 45 46 39.22 26.3 39.22 26.3 0.00 0.00

46 4 47 850.76 611.16 850.73 611.11 0.02 0.06

47 47 48 850.73 611.11 850.15 609.68 0.58 1.43

43

48 48 49 771.15 553.28 769.52 549.28 1.63 4.00

49 49 50 384.82 274.78 384.7 274.5 0.12 0.28

50 8 51 44.1 31 44.1 31 0.00 0.00

51 51 52 3.6 2.7 3.6 2.7 0.00 0.00

52 9 53 1856.8 1281.0 1851 1278.1 5.79 2.95

53 53 54 1846.6 1274.6 1839.9 1271.2 6.73 3.43

54 54 55 1813.5 1252.2 1804.4 1247.5 9.15 4.66

55 55 56 1780 1230.3 1771.2 1225.8 8.81 4.49

56 56 57 1771.2 1225.8 1721.5 1209.2 49.78 16.71

57 57 58 1721.5 1209.2 1697.0 1200.9 24.54 8.23

58 58 59 1697.0 1200.9 1687.5 1197.8 9.52 3.15

59 59 60 1587.5 1125.8 1576.87 1122.6 10.69 3.25

60 60 61 1576.9 1122.6 1562.9 1115.41 14.05 7.16

61 61 62 318.93 227.47 318.82 227.41 0.11 0.06

62 62 63 286.82 204.41 286.68 204.34 0.14 0.07

63 63 64 286.68 204.34 286.02 204.01 0.66 0.34

64 64 65 59.04 42.02 59 42 0.04 0.02

65 11 66 36 26 36 26 0.00 0.00

66 66 67 18 13 18 13 0.00 0.00

67 12 68 56.02 40.01 56 40 0.02 0.01

68 68 69 28 20 28 20 0.00 0.00

Total loss 224.45 107.14

Table2.10. Comparison of load flow results of 69 node system

Description

Total LossMinimum

voltage and it’snode number

Realpower(kW)

Reactivepower(kVAr)

Existing method [50] 224.96 114.15 0.9092 at node 65

Proposed method 224.45 107.14 0.9094 at node 65

44

2.6 CONCLUSIONS

The iterative techniques commonly used in transmission

networks are not suitable for distribution power flow analysis because

of poor convergence characteristics. In this work the distribution

power flow is carried out by the backward and forward propagation

iterative equations. The effective branch powers are calculated in

backward propagation. In forward propagation voltage magnitudes at

each node are calculated. The illustration of node identification is

used for calculating the effective powers of a branch in backward

propagation. It was found that the proposed load flow method is

suitable for fast convergence characteristics.