Chapter - 2

Distribution SystemPower Flow Analysis

18

CHAPTER - 2

Radial Distribution System Load Flow

2.1 Introduction

Load flow is an important tool [66] for analyzing electrical power system network

performance. Load flow is used at the time of solving optimization problem. So, load flow

should be fast and efficient. The distribution network is radial in nature having high R/X

value which makes it ill conditioned. The conventional Gauss Seidel (GS) and Newton

Raphson (NR) method does not converge for the distribution networks.

As discussed in chapter –1(section - 1.5.1), several limitations exist in radial load flow

techniques presently reported in the literature such as complicated bus numbering schemes,

convergence related problems, and the inability to handle modifications to existing

distribution system in a straight forward manner. This motivated the development of a radial

distribution load flow solution method.

Most of the conventional load flow methods consider power demands as specified

constant values. This should not be assumed because in distribution system bus voltages are

not controlled. Loads are specified by constant power, current or impedance requirements.

There are several load flow methods based on backward – forward sweep technique.

These method basically use -

1. Current summation methods

2. Power summation methods

19

Current summation method is more convenient and faster than the Power summation

method because it uses only ‘V’ and ‘I’ instead of ‘P’ and ‘Q’.

Basically, load flow is used for the determination of the distribution system operating

point at steady state under given conditions of loads. First of all, all bus voltages are

calculated. From these bus voltages, it is possible to directly calculate currents, active &

reactive power flows, active & reactive system losses and other steady state quantities.

In this chapter, a backward – forward sweep method is presented for solving the load

flow problem of a distribution system. The mathematical formulation of the above method is

explained in Section 2.2. In this Section, derivation of voltage, angle, active and reactive

power losses, voltage deviation and voltage stability index from single line diagram of

distribution system are discussed. The steps of load flow algorithm calculation are presented

in Section 2.3. In Section 2.4, the effectiveness of the proposed method is tested with 33 and

69 bus radial distribution systems and the results are compared with the existing methods. In

Section 2.5, conclusions of the proposed method are presented.

2.2 Mathematical Formulation

2.2.1 Assumption

In this section, circuit model of radial distribution system (RDS) is presented. It is

assumed that the three phase radial distribution system is balanced and can be represented by

an equivalent single – line diagram. The line shunt capacitance at distribution voltage level is

negligibly small and, hence neglected. The voltage magnitude and phase angle of the source

should to be specified. Also, the complex values of load demands at each bus along the feeder

should be given. Initially, a flat voltage profile is assumed at all buses i.e., 1.0 pu. The Fig. 2.1

shows the single line diagram of n-bus RDS.

Fig. 2.1: Single - line diagram of n-bus RDS

20

2.2.2 Bus indexing (numbering) scheme

Generally, most load flow formulations are the set of equations and unknowns,

associated with individual buses. So the formulations are arranged by appropriate bus

indexing. But in radial distribution system, the equations and unknowns can be reduced such

that the equations and unknowns become corresponding to the laterals. So lateral indexing is

needed for radial distribution system load flow.

A radial distribution system can be structured as a main feeder with laterals and these

laterals may also have sub- laterals, which themselves may have sub- laterals, etc. Each bus is

assigned an index ( l ,m , n ) , where l , m and n corresponds to level of lateral, lateral index

and bus index respectively. The reverse breadth-first (RBF) ordering of the laterals, found by

sorting the lateral indices in descending order, first by level, then by lateral index. The RBF

ordering is typically used for backward sweep type operations. If the laterals are sorted in

ascending order, the result is a breadth-first (BF) ordering, typically used for forward sweep

type operations.

2.2.3 Backward and Forward sweep based load flow algorithm

Iterations of backward and forward sweep based load flow algorithm:

1. First the bus and lateral indexing are done.

2. Then RBF ordering of end nodes are done according to the indexing.

3. All the end nodes voltages are initialized for the three phases (considering the

nominal voltage as base voltage).

4. According to the RBF ordering the backward sweep for the first iteration starts.

5. In backward sweep –

(a) The end node of lowest RBF order is considered.

(b) As the node voltage is known, so the current injecting at this node by

loads, shunt capacitors and DGs can be calculated.

(c) Then the current injection at the node is calculated by applying KCL at

current nod. As this is the end node, no current should be added by the

incoming downstream branch.

21

(d) Then by the initialized voltage and the total current injection at the

current node, using the update formulae, voltage at the next node and the

current injection at the next node through the branch between current

node and next node are calculated.

(e) Go to 5(b) and follow the same process until the branch off node of

current sub lateral or lateral has reached.

(f) The calculation of voltage and current at the current sub lateral or lateral

are stopped and the RBF order is incremented by one and goes to 5 (b)

until source node has reached.

(g) If the node is source node, and RBF order value becomes maximum, it

means that the backward sweep for first iteration ends.

(h) So from backward sweep the branch off current of laterals and sub

laterals are stored for computing the node voltages in forward sweep.

6. In forward sweep -

(a) The forward sweep starts with the specified substation secondary voltage,

and the current injected by the substation to the network which was stored

as branch off current of the main feeder during the backward sweep.

(b) It is therefore easy to calculate the voltage of the downstream node and

the current flowing through the downstream branches of the main feeder

one by one using the update formulae (and the branch off current stored

during backward sweep and the branch off voltages stored during the

calculation of the previous main feeders or laterals is used except for

main feeder calculation) until the end node of that main feeder or laterals

or sub laterals reaches.

(c) During this, voltages at the points from where the laterals are branching

off from this main feeder or laterals or sub laterals are stored as branch

off voltages of corresponding laterals.

(d) Now if the end node of that main feeder or the lateral or the sub lateral

reaches, RBF order is decreased by one and go to 6 (b) until the

calculation of the lateral of RBF order one reaches.

22

7. After completion of the backward and forward sweep for the first iteration, end

node voltages are updated.

8. This new end node voltage is compared with the previous initialized end node

voltages (only for first iteration) or the end node voltages of the previous iteration.

9. If these compared end node voltages values are less than a small error value, it

means that the load flow has converged otherwise go to 5 and do the backward

and forward sweep repeatedly considering the new end node voltages.

2.2.4 Load Flow Calculation

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 by starting from the ending buses and moving in the backward direction towards the

source bus (substation bus). The other set of equations are for calculating the voltage

magnitude and angle of each node starting from the slack bus (substation bus) and moving in

the forward direction towards the ending bus. These recursive equations are derived as follows.

The fig. 2.1 shows the single – line diagram of n-bus radial distribution system.

Consider a branch ‘ ij ’ is connected between the buses ‘ i’ and ‘ j ’.

2.2.4.1 Backward sweep

The updated effective power flows in each branch are obtained in the backward sweep

computation by considering the bus voltages of previous iteration. It means the voltage values

obtained in the forward path are held constant during the backward sweep and updated power

flows in each branch are transmitted backward along the feeder using backward path. This

indicates that the backward sweep starts at the extreme end bus and proceeds towards source

bus (substation bus). The active and reactive power flows are calculated in backward direction.

The effective active (iP ) and reactive (

iQ ) powers that of flowing through branch ‘ ij ’

from node ‘ i ’ to node ‘ j ’ can be calculated backwards from the ending bus and is given as,

( )' 2' 2

2i

j j

j L j ij

j

QPRP P P

V

+= + + ⋅ (2.1)

( )' 2' 2

2

j j

iji L jj

j

QPQQ Q X

V

+= + + ⋅ (2.2)

23

Where 'Ljj j PP P += and

'

Ljj jQ QQ +=

LjP and LjQ are loads that are connected at bus ‘ j ’

jP and jQ are the effective real and reactive power flows from bus ‘ j ’.

2.2.4.2 Forward sweep

The purpose of the forward sweep is to calculate the voltages at each bus starting from the

feeder source bus (substation bus). 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 voltage magnitude and angle at each bus are calculated in forward direction.

Consider a voltage iV iδ at bus ‘ i ’ and jV jδ at bus ‘ j ’, then the current flowing

through the branch ‘ ij ’ having an impedance, ij i ji j

R j XZ += connected between ‘ i ’

and ‘ j ’ is given as,

( )i ji j

i j

i j i j

VVI

jR X

δ δ−=

+

(2.3)

And

( )i j

i i

i i

jQPI

V δ

−=

−

(2.4)

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

( ) ( )i i i jji

i j i ji i

VVjQP

jV R X

δ δ

δ

−−=

+−

(2.5)

( ) ( )( )2

i i i jj ii j i jijXjQV P RV V δ δ− = − +−

(2.6)

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

( ) ( )2

cos ii j i ij i i j i jV V R Q XV Pδ δ− = − + (2.7)

( )sini j i i i j i i jj

Q R P XVV δ δ− = − (2.8)

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

( ) ( )2 2 22

i ii j ii i j i j i i j i jV R X Q R XQV V P P= + − + − (2.9)

( ) ( ) ( )( )2 2 24 2 2 22i j i i j i j ii i ii i j i j

V R XQ QVV V P R X P= − + + + + (2.10)

24

( ) ( ) ( )1

222

2 2 2

22

i i

iij i j i ji i ji ji

QPXQRV V P R X

V= +

− + + +

(2.11)

and voltage angle , jδ can be derived on diving equations (2.8) and (2.7)

( )( )2

tani i j i i j

j i

i ii i j i j

Q R P X

V R XQPδ δ

−− =

− +

(2.12)

( )

1

2tan

ii i j i jj i

i ii i j i j

Q R XP

V R XQPδ δ −

−= +

− +

(2.13)

The magnitude and the phase angle equations can be used recursively in a forward direction to

find the voltage and angle respectively of all buses of radial distribution system.

2.2.4.3 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 present computed

voltages and then the procedure is repeated until the solution is converged

2.2.4.4 Active and Reactive power losses calculation

The active and reactive power losses of branch ‘ ij ’can be calculated as,

22

2( , )

iL o s s

ii j

i

QPi jP R

V

+= ⋅ (2.14)

22

2( , )

i ii jL o s s

i

QPi jQ X

V

+= ⋅ (2.15)

The total active and reactive power loss of radial distribution system can be calculated as,

( ) ( )2 22 21

, 2 21 1

n nbi ii i

T Loss i j i ji j i j

i i

Q QP PP R R

V V

−

= =

+ += ⋅ = ⋅∑ ∑ (2.16)

( ) ( )2 22 2

1

, 2 21 1

n nbi ii i

i j i jT Lossi j i j

i i

Q QP PQ X X

V V

−

= =

+ += ⋅ = ⋅∑ ∑ (2.17)

25

2.2.4.5 Voltage deviation calculation

The voltage deviation of the system is defined as

,

1

n

i nom ii

V VVD=

= −∑ (2.18)

where , , 1 .i nom i rated p uV V= =

(2.19)

2.2.4.6 Voltage stability index calculation

From Fig.2.1, current that of flowing through branch ‘ ij ’from bus ‘ i ’ to bus ‘ j ’ can

be calculated as

i ji j

i j i j

V VI

jR X

−=

+ (2.20)

*' '

i jjj jj V IQP − = (2.21)

Eq. (2.22) gives the voltage stability index at all buses in RDS, was proposed by Chakravorty

et al [67]. Using eq. (2.20) and (2.21) :

2 24 4 4j j ijj ij ijj ji iij

V VQ QX R RSI P P X = − − − + (2.22)

0jSI > , indicates stable operation of RDS.

Objective function for improving voltage stability index is given by (2.23):

1

j

VSISI

=

, 2, 3, ......j n= (2.23)

The maximum value of jSI gives minimum value of objective function (VSI ). So, minimum

values of objective function indicate improvement of voltage stability index. .

26

2.3 Algorithm for Load Flow Calculation

The flow chart for load flow is shown in fig.2.2. The backward forward sweep load

flow algorithm is given below

Start

Read line & load data

Set flat voltages [1 p.u] for all buses (nodes)

Computer effective real & reactive power flows of

all branches using backward sweep from equations

[2.1] & [2.2]

Update bus voltages & phase angles using forward sweep

from equations [2.11] &[2.13]

Is load flow

converged

Compute branch power losses, total system losses, total system voltage

deviation, total system voltage stability index & print the results.

Stop

No

Yes

Fig.2.2: Flow chart for Radial Distribution Load Flow

27

Step 1 : Read distribution system line and load data. Assume initial bus voltages are 1

p.u and set ε = 0.0001.

Step 2 : Start iteration count, IT =1.

Step 3 : Initialize active power loss and reactive power loss vectors to zero.

Step 4 : Calculate the effective active and reactive power flow in each branch using

equations (2.1) and (2.2).

Step 5 : Calculate bus voltages, active and reactive power loss of each branch using

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

Step 6 : Check for convergence i.e., max

V∆ < ε in successive iterations.

If it is converged go to next step otherwise increment iteration number and go to step 4.

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

reactive power loss, voltage deviation of each bus, total voltage deviation,

voltage stability index of each bus and voltage stability index of system.

Step 8 : Print voltage at each node, the active and reactive power losses of all branches,

total active and reactive loss, total voltage deviation and voltage stability index

of system.

Step 9 : Stop.

2.4 Simulation Results and Analysis

The proposed method computes the load flow solution for the given RDS. The

effectiveness of the proposed method is tested on 33 and 69 bus RDS.

2.4.1 Test system -1 (33 bus RDS)

The test system -1 is a 12.66 kV, 33 bus RDS consisting of 33 buses configured with

one substation, one main feeder, 3 laterals and 32 branches. The total active and reactive loads

on this system are 3715 kW and 2300 kVAr, respectively. It is demonstrated in Fig. A.1 [68].

The line and load data of this system is given in appendix Table A.1 & A.2 .

28

Table 2.1 gives results of voltage magnitude and angles at different buses of this system. The

voltage magnitude and angles at different buses of this system are given in Fig.2.3 & Fig.2.4

respectively. The minimum voltage is 0.9042 p.u at bus 18 and maximum voltage regulation

is 9.58%.

Table 2.1: Voltage magnitudes and angles of 33 bus RDS

Bus No. Voltage Magnitude (p.u) Voltage Angle

1 1.0000 0

2 0.9970 0.0002

3 0.9829 0.0017

4 0.9754 0.0028

5 0.9680 0.0040

6 0.9496 0.0024

7 0.9462 -0.0017

8 0.9326 -0.0046

9 0.9262 -0.0060

10 0.9204 -0.0073

11 0.9195 -0.0072

12 0.9180 -0.0070

13 0.9119 -0.0088

14 0.9096 -0.0102

15 0.9082 -0.0109

16 0.9068 -0.0114

17 0.9048 -0.0128

18 0.9042 -0.0130

19 0.9965 0.0000

20 0.9929 -0.0011

21 0.9922 -0.0015

22 0.9916 -0.0018

23 0.9793 0.0011

24 0.9727 -0.0004

25 0.9693 -0.0012

26 0.9477 0.0031

27 0.9452 0.0042

28 0.9338 0.0058

29 0.9256 0.0073

29

30 0.9220 0.0092

31 0.9179 0.0078

32 0.9169 0.0074

33 0.9167 0.0073

Fig.2.3: Bus voltage magnitude (p. u.) of each bus in 33 bus RDS

Fig.2.4: Bus voltage angle (deg.) of each bus in 33 bus RDS

0 5 10 15 20 25 30 350.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Bus No.

Voltage m

agnitude (p.u.)

0 5 10 15 20 25 30 35-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Bus No.

Voltage angle (deg.)

30

Table 2.2 gives results of active power and reactive power losses of each branch of the

system. The real power and reactive power losses of each branch of the system are given in

Fig. 2.5 & Fig. 2.6 respectively. The total active and reactive power losses of the system are

210.07 kW and 142.53 kVAr respectively. The active and reactive power losses are 5.65%

and 6.19% of their total loads.

Table 2.2: Power loss of 33-bus RDS

Br. No. Bus Active power loss

PLoss (kW)

Reactive power loss

Q Loss (kVAr) SE RE

1 1 2 12.2467 6.3359

2 2 3 51.8257 26.3964

3 3 4 19.9247 10.1474

4 4 5 18.7226 9.5357

5 5 6 38.2964 33.0593

6 6 7 1.9441 6.4265

7 7 8 11.8608 8.5598

8 8 9 4.2612 3.0614

9 9 10 3.6291 2.5724

10 10 11 0.5642 0.1865

11 11 12 0.8980 0.2969

12 12 13 2.7177 2.1382

13 13 14 0.7433 0.9784

14 14 15 0.3641 0.3241

15 15 16 0.2870 0.2096

16 16 17 0.2566 0.3426

17 17 18 0.0542 0.0425

18 18 19 0.1610 0.1536

19 19 20 0.8322 0.7499

20 20 21 0.1008 0.1177

21 21 22 0.0436 0.0577

22 22 23 3.1816 2.1739

23 23 24 5.1438 4.0618

24 24 25 1.2875 1.0074

25 25 26 2.5950 1.3218

26 26 27 3.3223 1.6916

31

27 27 28 11.2810 9.9462

28 28 29 7.8210 6.8135

29 29 30 3.8896 1.9812

30 30 31 1.5934 1.5748

31 31 32 0.2132 0.2485

32 32 33 0.0132 0.0205

Total loss 210.0756 142.5337

Fig.2.5: Active power loss (kW) of each branch in 33 bus RDS

Fig.2.6: Reactive power loss ( kVAr) of each branch in 33 bus RDS

0 5 10 15 20 25 30 350

10

20

30

40

50

60

Branch No.

Active power loss

( kW

)

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

Branch No.

Reactive power loss

( kW

)

32

Comparison of load flow results between the proposed method and the existing method [69] is

given in Table 2.3. The total active and reactive power losses are reduced and the minimum

voltage is improved in the proposed method.

Table 2.3: Comparison of load flow results of 33 bus RDS

Description

Total active

power loss ,

P Loss (kW)

Total reactive

power loss,

Q Loss ( kVAr )

Minimum

Voltage and it’s

bus number

CPU

time(s)

Existing method [69] 210.80 143.11 0.9038 at bus 18 2.944659

Proposed method 210.07 142.53 0.9042 at bus 18 2.822436

Table 2.4 gives results of voltage deviation and voltage stability index of the system. The

voltage deviations of each bus of the system is given in Fig. 2.7. The voltage stability index of

each bus of the RDS is given in Fig. 2.8.

Table 2.4 : Voltage deviation & Voltage stability index of 33 bus RDS

System VD in p.u. VSI

33 bus 0.1338 1.4960

Fig. 2.7: Bus voltage deviation ( VD ) in p.u of each bus in 33 bus RDS

0 5 10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Bus No.

Volt

age d

evia

tion ( p

.u. )

33

Fig.2.8: Voltage stability index (VSI ) of each bus in 33 bus RDS

2.4.2 Test system -2 (69 bus RDS)

The test system -2 is a 12.66 kV, 69 bus large scale RDS consisting of 69 buses

configured with one substation, one main feeder, 7 laterals and 68 branches. The total active

and reactive loads on this system are 3802.19 kW and 2694.6 kVAr, respectively. It is

demonstrated in Fig. A.2 [70]. The line and load data of this system is given in appendix

Table A.3 & A.4 .

Table 2.5 gives results of voltage magnitude and angles at different buses of this system. The

voltage magnitude and angles at different buses of this system are given in Fig.2.9 & Fig.2.10

respectively. The minimum voltage is 0.9101 pu at bus 65 and maximum voltage regulation is

8.99%.

Table 2.5: Voltage magnitudes and angles of 69 bus RDS

Bus No. Voltage Magnitude Voltage Angle

1 1.0000 0

2 1.0000 -0.0000

3 0.9999 -0.0000

4 0.9998 -0.0001

5 0.9990 -0.0003

6 0.9901 0.0012

7 0.9809 0.0027

8 0.9787 0.0031

9 0.9776 0.0033

0 5 10 15 20 25 30 351

1.1

1.2

1.3

1.4

1.5

1.6

Bus No.

Voltage sta

bility index (p.u)

34

10 0.9726 0.0048

11 0.9715 0.0052

12 0.9683 0.0062

13 0.9654 0.0070

14 0.9626 0.0079

15 0.9597 0.0087

16 0.9592 0.0089

17 0.9583 0.0092

18 0.9583 0.0092

19 0.9578 0.0093

20 0.9575 0.0094

21 0.9570 0.0096

22 0.9570 0.0096

23 0.9570 0.0096

24 0.9568 0.0097

25 0.9566 0.0097

26 0.9566 0.0098

27 0.9566 0.0098

28 0.9999 -0.0000

29 0.9999 -0.0001

30 0.9998 -0.0000

31 0.9997 -0.0000

32 0.9997 0.0000

33 0.9995 0.0001

34 0.9992 0.0002

35 0.9992 0.0003

36 0.9999 -0.0001

37 0.9997 -0.0002

38 0.9996 -0.0002

39 0.9995 -0.0002

40 0.9995 -0.0002

41 0.9988 -0.0004

42 0.9986 -0.0005

43 0.9985 -0.0005

44 0.9985 -0.0005

45 0.9984 -0.0005

46 0.9984 -0.0005

47 0.9998 -0.0001

48 0.9985 -0.0009

35

49 0.9947 -0.0034

50 0.9942 -0.0037

51 0.9786 0.0031

52 0.9786 0.0031

53 0.9748 0.0038

54 0.9716 0.0044

55 0.9671 0.0052

56 0.9628 0.0060

57 0.9407 0.0140

58 0.9298 0.0180

59 0.9256 0.0196

60 0.9206 0.0217

61 0.9133 0.0233

62 0.9130 0.0233

63 0.9126 0.0234

64 0.9107 0.0238

65 0.9102 0.0240

66 0.9714 0.0052

67 0.9714 0.0052

68 0.9680 0.0063

69 0.9680 0.0063

Fig.2.9: Bus voltage magnitude (p. u.) of each bus in 69 bus RDS

0 10 20 30 40 50 60 70

0.92

0.94

0.96

0.98

1

1.02

1.04

Bus No.

Voltage m

agnitude (p.u.)

36

Fig.2.10: Bus voltage angle (deg.) of each bus in 69 bus RDS

Table 2.6 gives results of active power and reactive power losses of each branch of the

system. The active power and reactive power losses of each branch of the system are given in

Fig. 2.11 & Fig. 2.12 respectively. The total active and reactive power losses of the system are

224. 54 kW and 101.96 kVAr respectively. The active and reactive power losses are 5.91%

and 3.78% of their total loads.

Table 2.6: Power loss of 69 bus RDS

Br.

No.

Bus P Loss (kW) Q Loss (kVAr)

SE RE

1 1 2 0.0746 0.1792

2 2 3 0.0746 0.1792

3 3 4 0.1947 0.4673

4 4 5 1.9333 2.2645

5 5 6 28.1910 14.3574

6 6 7 29.2967 14.9213

7 7 8 6.8821 3.5082

8 8 9 3.3687 1.7151

9 9 10 4.7716 1.5771

10 10 11 1.0135 0.3351

11 11 12 2.1886 0.7233

12 12 13 1.2837 0.4237

13 13 14 1.2449 0.4114

0 10 20 30 40 50 60 70-0.04

-0.02

0

0.02

0.04

0.06

Bus No.

Voltage angle (deg.)

37

14 14 15 1.2040 0.3978

15 15 16 0.2237 0.0740

16 16 17 0.3202 0.1059

17 17 18 0.0026 0.0009

18 18 19 0.1041 0.0344

19 19 20 0.0669 0.0219

20 20 21 0.1074 0.0355

21 21 22 0.0005 0.0002

22 22 23 0.0051 0.0017

23 23 24 0.0112 0.0037

24 24 25 0.0060 0.0020

25 25 26 0.0025 0.0008

26 26 27 0.0003 0.0001

27 3 28 0.0003 0.0007

28 28 29 0.0021 0.0051

29 29 30 0.0041 0.0014

30 30 31 0.0007 0.0002

31 31 32 0.0036 0.0012

32 32 33 0.0087 0.0029

33 33 34 0.0060 0.0020

34 34 35 0.0005 0.0002

35 3 36 0.0014 0.0034

36 36 37 0.0151 0.0369

37 37 38 0.0173 0.0202

38 38 39 0.0050 0.0058

39 39 40 0.0002 0.0002

40 40 41 0.0487 0.0569

41 41 42 0.0201 0.0235

42 42 43 0.0027 0.0031

43 43 44 0.0005 0.0006

44 44 45 0.0061 0.0077

45 45 46 0.0000 0.0000

46 4 47 0.0233 0.0575

47 47 48 0.5828 1.4265

48 48 49 1.6334 3.9968

49 49 50 0.1159 0.2835

50 8 51 0.0018 0.0009

51 51 52 0.0000 0.0000

52 9 53 5.7695 2.9378

38

53 53 54 6.6977 3.4116

54 54 55 9.1058 4.6362

55 55 56 8.7717 4.4685

56 56 57 49.5807 16.6423

57 57 58 24.4380 8.2011

58 58 59 9.4858 3.1370

59 59 60 10.6484 3.2323

60 60 61 13.9966 7.1293

61 61 62 0.1118 0.0569

62 62 63 0.1346 0.0685

63 63 64 0.6597 0.3360

64 64 65 0.0411 0.0209

65 11 66 0.0026 0.0008

66 66 67 0.0000 0.0000

67 12 68 0.0233 0.0077

68 68 69 0.0000 0.0000

Total loss 224. 5407 101. 9661

Fig.2.11: Active power loss (kW) of each branch in 69 bus RDS

0 10 20 30 40 50 60 700

10

20

30

40

50

60

Branch No.

Real pow

er loss

(kW

)

39

Fig. 2.12 : Reactive power loss ( kVAr) of each branch in 69 bus RDS

The load flow results of the proposed method are compared with the existing method

[69] in Table 2.7. The total active and reactive power losses are reduced and the minimum

voltage is improved by the proposed method.

Table 2.7: Comparison of load flow results of 69 bus RDS

Description

Total active

power loss ,

P Loss (kW)

Total reactive

power loss ,

Q Loss (kVAr)

Minimum

Voltage and it’s

bus number

CPU

time (s)

Existing method [69] 224.79 102.28 0.9092 at bus 65 4.471

Proposed method 224.54 101.96 0.9102 at bus 65 3.936

Table 2.8 gives results of voltage deviation and voltage stability index of the system. The

voltage deviations of each bus of the system is given in Fig. 2.13. The voltage stability index

of each bus of the RDS is given in Fig. 2.14.

Table 2.8: Voltage deviation & Voltage stability index of 69 bus RDS

System VD in p.u. VSI

69 bus 0.0993 1.4571

0 10 20 30 40 50 60 700

5

10

15

20

Branch No.

Reactive p

ow

er loss

( k

W )

40

Fig.2.13: Bus voltage deviation ( VD ) of each bus in 69 bus RDS

Fig.2.14: Voltage stability index (VSI ) of each bus in 69 bus RDS

2.5 Conclusions

In this chapter radial distribution load flow method is described. It is tested over two

balanced radial distribution systems. The proposed backward and forward sweep technique

gives advantages over the other load flow techniques. It does not employ complicated

calculations, i.e. the derivatives of the power flow equations. It is flexible and easily

accommodates changes that may occur in any RDS. These changes could be modifications or

0 10 20 30 40 50 60 700

0.02

0.04

0.06

0.08

0.1

Bus No.

Voltage devia

tion (p.u.)

0 10 20 30 40 50 60 701

1.1

1.2

1.3

1.4

1.5

Bus No.

Voltage sta

bility index (p.u)

41

additions of transformers, capacitors, distributed generators, other systems or both to the

distribution system.

The proposed backward and forward sweep technique is easy to program and has the

fastest CPU computation time when compared to other radial and conventional power flow

methods. Such advantages make the backward and forward sweep technique a suitable choice

for planning and real-time computations. The iterative techniques commonly used in

transmission networks are not suitable for distribution power flow analysis because of poor

convergence characteristics. It is found that the propose load flow method is suitable for fast

convergence characteristics.