CHAPTER - 2

LOAD FLOW METHOD FOR RADIAL DISTRIBUTION SYSTEMS

2.1 INTRODUCTION

The modern power distribution network is constantly being faced with an ever-

growing load demand. Distribution networks experience distinct change from a low to high

load level every day. In order to evaluate the performance of a power distribution system and

to examine the effectiveness of proposed modifications to a system in the planning stage, it is

essential that a load flow analysis of the system is to be repeatedly carried-out. It basically

gives the steady state operating condition of a distribution system corresponding to a

specified load on the system. Certain applications, particularly in the distribution automation

(i.e., VAR planning, state estimation, etc.) and optimization of power system require repeated

load flow solutions.

Load flow studies are the back bone of power system analysis and design. It gives the

steady state solution of a power system network for normal operating condition which helps

in continuous monitoring of the current state of the system. It is also employed for planning,

optimization and stability studies. Hence an efficient and fast load flow method is required.

Generally distribution networks have high R/X ratio. Due to this reason, popularly

used Newton – Raphson [2] and Fast Decoupled load flow algorithms [3] may provide

inaccurate results and may not be converge. Hence, conventional load flow methods cannot

be directly applied to obtain the load flow solution of radial distribution systems.

Kersting and Mendive [6] and Kersting [11] have developed load flow techniques

based on ladder theory. Baran and Wu [18] and Chiang [21] have developed a load flow

solution of distribution system based on the Newton – Raphson method. They have computed

system Jacobian matrix using chain rule in their method. Chiang [21] has also proposed three

different algorithms for solving radial distribution network based on the method of Baran

and Wu. He has proposed decoupled, fast decoupled and very fast decoupled distribution load

flow algorithms. The first two methods are similar to the method proposed by Baran and Wu.

The very fast decoupled distribution load flow method is very attractive, since it does not

require any Jacobian matrix construction and factorization but more computations are

involved because it solves three fundamental equations. representing real power, reactive

power and voltage magnitudes.

Many researchers [38, 52, 74] have proposed different methods to analyze the radial

distribution system based on forward and backward sweep current injection methods.

Venkatesh and Ranjan [72] have proposed a method using data structures to find the load

flow solution of radial distribution system.

In this chapter, a simple method of load flow technique for radial distribution system

is proposed. The proposed method involves the solution of simple algebraic equation in

receiving end voltages. The mathematical formulation of the proposed method is explained in

Section 2.2. In this Section, derivation of voltage, angle, real and reactive power losses from

phasor diagram of single line diagram of distribution system is discussed. The bus

identification using data structure of distribution system is described in Section 2.3. The steps

of load flow algorithm are presented in Section 2.4. Also the effectiveness of the proposed

method is tested with different examples of distribution system and the results are compared

with the existing methods. In Section 2.5, conclusions of the proposed method are presented.

2.2. MATHEMATICAL FORMULATION

Assumptions:

Radial distribution networks are balanced and can be represented by their

equivalent single line diagram.

Half line charging susceptances of distribution lines are negligible and these

distribution lines are represented as short lines.

Shunt capacitor banks are treated as loads.

Consider a branch connected between buses 1 and 2 as shown in Fig. 2.1.

1 2

V1∟0 I V2∟-δ2

R1+jX1

P2+jQ2

Fig. 2.1 Electrical equivalent of a typical branch 1

In Fig. 2.1, V1∟0 and V2∟-δ2 are the voltage magnitudes and phase angles of two

buses 1 and 2. Let the current flowing through branch 1 is I. The substation voltage is

assumed to be 1+j0 p.u. Let power factor angle of load P2+jQ2 be θ2. Let R1 and X1 be the

resistance and reactance of the branch 1 respectively. The phasor diagram of Fig. 2.1 is

shown in Fig. 2.2.

Fig. 2.2 Phasor diagram of a branch 1 connected between buses 1 and 2

From Fig. 2.2, the Eqns. are

2X

2

R2

2

1 VVVV

… (2.1)

2

2121

2

21212

2

1 sinIRcosIXsinIXcosIRVV … (2.2)

To eliminate ‘I’ in Eqn. (2.2)

2

22

V

PcosI

2

22

V

QsinI

where

P2 = Total active power load of all buses beyond bus 2 including local load and

active power losses beyond bus 2

Q2 = Total reactive power load of all buses beyond bus 2 including local load and

reactive power losses beyond bus 2

The Eqn. (2.2) becomes

2

2

1212

2

2

12122

2

1V

RQXP

V

XQRPVV

2

2

1212

2

22

2

1212

22

2

1

V

RQXP

V

1X2Q1R2P

XQRP2VV

1212

2

12122

21212

242

22

2

1 RQXPXQRPV2XQRPVVV

0VVXRQPXQRPV2V 2

2

1

221

21

2

22212122

2

2

4

0XRQP2

VXQRPV2V

21

21

2

222

1

2

12122

2

2

4

… (2.3)

0cVbV 2

2

2

4 … (2.4)

where

XQ2RP2Vb 12121

2

… (2.5)

XRQPc 21

21

2

222 … (2.6)

where

| | = substation voltage (taken as 1.0 p.u.)

R1 = resistance of branch 1

X1 = reactance of branch 1

The four possible solutions for the voltage, V2 from Eqn. (2.4) are

i) c4bb2

1 22/1

2/1

ii) ­ c4bb2

1 22/1

2/1

iii) - c4bb2

1 22/1

2/1

iv) c4bb2

1 22/1

2/1

It is found for realistic systems, when P2, Q2, R1, X1 and V are expressed in p.u., ‘b’ is

always positive because the term (2 + 2 is extremely small as compared to. V1

2

In addition the term ‘4c’ is negligible compared to b2. Therefore, {b

2 – 4c}

1/2 is nearly equal

to ‘b’ and hence the first two solutions of V2 are nearly equal to zero and third solution is

negative and hence not feasible. The fourth solution of V2 is positive and hence it is only

the possible feasible solution. Therefore, the possible feasible solution of Eqn. (2.4) is

c4bb2

1V

22/1

2/1

2 … (2.7)

In general, the solution for V 1i is

c4b2/1

b2

1 2/1V i

2ii1i … (2.8)

where

XQ2RP2Vb k1ik1ii

2

i

XRQPc2k

2k

2

1i2

1ii

where

i =1, 2……nbus.

k =1, 2, 3…..nbus-1

nbus = total number of buses.

The real and reactive power loss of branch ‘k’ is given by

V

QPRP k

1i

2

2

1i2

1ik

loss

… (2.9)

V

QPXQ k

1i

2

2

1i2

1ik

loss

… (2.10)

The Total Active and Reactive Power Losses (TPL, TQL) are given by

1nbus

1k

loss kPTPL … (2.11)

kQTQL1nbus

1k

loss

… (2.12)

The phase angle (δ2) of voltage V2 can be calculated as follows

From Fig. 2.2,

VV

Vtan

R2

X2

21212

2121

2sinXIcosRIV

sinRIcosXItan … (2.13)

On simplification we will get

VXQRP

RQXPtan

2

2

1212

121212 … (2.14)

In general

VXQRP

RQXPtan

1i

2

k1ik1i

k1ik1i11i … (2.15)

Usually, the substation voltage V1 is known and is taken as 1.0 ∟00p.u. Initially,

Ploss[k] and Qloss[k] are set to zero for all k. Then the initial estimate of Pi+1 and Qi+1 will be

the sum of the loads of all the buses beyond bus ‘i' plus the local load of bus ‘i' plus the losses

beyond bus ‘i' Compute Vi+1,Ploss[k], Qloss[k], δi+1using Eqns. (2.8), (2.9), (2.10) and (2.15).

This will complete one iteration of the solution. Update the loads P(i+1) and Q(i+1) (by

including losses) and repeat the same procedure until the voltage mismatch reach a tolerance

level of 0.0001 p.u. in successive iterations.

2.3 BUS IDENTIFICATION USING DATA STRUCTURES

In this section, the data structure is used to identify the inter connection of buses and

branches of radial distribution system, which avoids the conventional method of unique

lateral bus and branch numbering process. An algorithm is developed and the methodology of

identifying the buses and branches connected to a particular bus in detail, which will help in

finding the exact load feeding through that particular bus, is presented in this section.

The proposed method initially forms the Bus Incidence Matrix (BIM) of the radial

distribution system and then can be processed to create a Data structure. This method can

handle the system data with any random bus and line numbering scheme except the slack bus

being numbered as 1. Also this algorithm provides dynamically declared and alterable Data

structure. The illustration of the algorithm is given in Section 2.3.4.

2.3.1 Formation of Bus Incidence Matrix

Consider the radial distribution system with multiple laterals, branches and buses. The

bus numbers have been marked in a random order with substation as bus number 1.The

elements of BIM are considered as follows:

BIM is of size m x n, where m is number of lines and n is the number of buses.

If ith

and jth

buses are connected through a branch, k then

BIMij = 1

Otherwise BIMij = 0

2.3.2 Formation of Data structure

After forming the BIM, then the Data structure can be created as follows.

First select a bus, identify the paths emanating from it and each path is represented by

a contour. Form the data structure which gives the information about bus numbers, real and

reactive powers at those buses and the respective branch numbers corresponding to each

3

I

IX

V

X

VII

VI

I

VII

II

X

IV

S/S 1 2

4

5

6

7

9

8

11

12

10 14

13

contour. It also gives the number of contours emanating from the end bus and their respective

index numbers of the contours.

2.3.3 Algorithm steps for bus identification

Step 1 : Read system branch data.

Step 2 : Form the Bus incidence matrix (BIM)

Step 3 : Create the Data structure using BIM

Step 4 :Stop

2.3.4 Illustration

Consider the single line diagram of 15 bus radial distribution system shown in Fig 2.3.

Consider bus 1 as the reference bus. The Bus Incidence Matrix for this system is as shown in

Table 2.1.

10 9

14 2 3 4 5 1

7 11 15 6

12 8 13

Fig. 2.3 Single line diagram of 15 bus radial distribution system

1, 2, 3 ,….. represent bus numbers

, ,….represent branch numbers

, , …. represent contour numbers

Table 2.1 The Bus Incidence Matrix of the 15 bus radial distribution system

Bus No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Branch No.

1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

3 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

4 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0

5 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0

7 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

9 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0

10 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0

11 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0

12 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0

13 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0

14 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1

Consider a contour with several buses and several connections emanating from its

terminal bus, and then a generic data structure would look like the illustration below in Fig.

2.4.

As an example, the data structure for the system is

.

b.n

o.ob.n

o.o

b.n

o.o

c.no

.

Br.

no.Br.

no.

Br.

no.

D.n

o.

P

i P

i

P

i

Q

i Q

i

Q

i

D.n

o.

D.n

o.

1 2

I I

I

Fig. 2.4 The general data structure of the system

In Fig. 2.4, b.no. stands for bus number, br.no. stands for branch number, Pj, Qj

stands for real and reactive power at that buses, c.no. stands for number of contours present

adjacent to that bus i, D.no. stands for data structure corresponding to the emanating contours

(i.e., adjacent contours to the bus, i).

For 15 bus system 11 contours are identified and are shown in Fig. 2.3 with dotted lines. The

data structure of contour 1 is as shown below.

bus : [1 2] (bus no.s are 1 and 2)

c.no. : 3 (no. of contours emanating from the end bus of this contour

or adjacent contours of the bus, 2)

Index number of that contours: [II V VIII] (numbers of adjacent contours of bus, 2)

Branch : 1 (branch connected between buses 1 and 2)

Real power : [0 44.1] (Real power load at buses 1and 2)

Reactive power : [0 44.99] (Reactive power load at buses 1and 2)

Resistance of branch 1: 0.8523 p.u. (resistance of branch 1 connected between 1and 2)

Reactance of branch 1: 0.8336 p.u. (reactance of branch 1 connected between 1and 2)

The data structure of the entire system is shown in Table 2.2

Table 2.2 Data Structure of all contours of 15 bus Radial Distribution System

Contour I

b.no. 1 2

br. no. 1

real power P1 P2

reac. Power Q1 Q2

Contour II

b.no. 2 3

br. no. 2

real power P2 P3

reac. Power Q2 Q3

3

II, V, VIII

2

III, IX

Contour III

b.no. 3 4

br. no. 3

real power P3 P4

reac. Power Q3 Q4

Contour IV

b.no. 4 5

br. no. 4

real power P4 P5

reac. Power Q4 Q5

Contour V

b.no. 2 6

br. no. 7

real power P2 P6

reac. Power Q2 Q6

Contour VI

b.no. 6 7

br. no. 8

real power P6 P7

reac. Power Q6 Q7

Contour VII

b.no. 6 8

br. no. 9

real power P6 P8

reac. Power Q6 Q8

Contour VIII

b.no. 2 9 10

br. no. 5 6

real power P2 P9 P10

reac. Power Q2 Q9 Q10

Contour IX

b.no. 3 11 12 13

br. no. 10 11 12

real power P3 P11 P12 P13

reac. Power Q3 Q11 Q12 Q13

Contour X

b.no. 4 14

br. no. 13

real power P4 P14

reac. Power Q4 Q14

Contour XI

b.no. 4 15

br. no. 14

real power P4 P15

reac. Power Q4 Q15

2.4 LOAD FLOW CALCULATION

Once all the buses and branches are identified, it is very easy to calculate the effective

load at each bus including losses and then solve for the bus voltage magnitudes and angles

3

IV, X, XI

0

2

VI, VII

0

0

0

0

0

0

using the Eqns. (2.8) and (2.15). Then compute the real and reactive power losses in each

branch using Eqns. (2.9) and (2.10). The convergence criterion of the proposed algorithm is

that if change in the magnitude of the two bus voltages in successive iterations is less than

0.0001 p.u., the solution is said to be converged and the total real and reactive power losses

are computed using Eqns. (2.11) and (2.12).

2.4.1 Algorithm for Load flow calculation:

Step 1 : Read line and load data of radial distribution system. Initialize TPL, TQL

to zero. Assume bus voltages 1 p.u., set convergence Criterion ΔV ≤ ε

Step 2 : Start iteration count=1.

Step 3 : Build BIM and Data Structure of the system.

Step 4 : Calculate effective load at each bus starting from the last bus.

Step 5 : Initialize real power loss and reactive power loss to zero.

Step 6 : Find effective losses at each bus.

Step 7 : Calculate load at each bus including losses.

Step 8 : Calculate bus voltages (magnitude and angles), real and reactive power

loss of each branch using Eqns. (2.8), (2.15), (2.9) and (2.10) respectively.

Step 9 : Calculate the value of change in bus voltages i.e. ΔV ≤ ε in successive

iterations. If ΔV≤ε go to step 11 otherwise go to step 10.

Step 10: Increment iteration number and go to step 6.

Step 11: Calculate TPL and TQL using Eqns. (2.8) and (2.9).

Step 12: Print voltages at each bus, TPL, TQL and number of iterations.

Step 13: Stop.

2.4.2 Flow Chart for the proposed method

Read Distribution System data

Start

Assume bus voltages Vi=1+j0, for i=1,2,….nbus

Set iteration count (IC) =1, convergence criterion (ε) =0.0001,

Initialize total real and total reactive power losses to zero

Build Bus Incidence matrix (BIM)

Build Data Structure of the system using BIM

Calculate real and reactive power loss of

each branch using Eqns. (2.9) & (2.10)

Compute bus voltages (magnitude and angle) using Eqns.

(2.8) & (2.15)

Calculate effective load at each bus including losses

IC=IC+1

Fig. 2.5 Flow chart of load flow method

2.5 ILLUSTRATIVE EXAMPLES

To illustrate the effectiveness of the proposed method three different examples

consisting of 15, 33 and 69 bus radial distribution systems are considered.

2.5.1 Example – 1

The single line diagram of 15 bus, 11 kV radial distribution system is shown in Fig.

2.3. The line and load data of this system are given in Appendix – A (Table A1). The voltage

profile of the system is given in Table 2.3. The line flows of 15 bus system are given in Table

2.4. The total real and reactive power losses are 61.7933 kW and 57.2967 kVAr. The

minimum voltage of this system is 0.9445 p.u. occurred at bus 13 and voltage regulation is

5.55%. The number of iterations taken for this system is 3. The CPU time taken for the

solution is 0.14 sec. The solution obtained by proposed method is compared with the existing

method [104] and voltages are approximately matching both in magnitude and phase angles.

Table 2.3 Voltage profile of 15 bus radial distribution system

Bus No.

Proposed method Existing method [104]

Voltage

Magnitude (p.u.)

Angle

(deg.)

Voltage

Magnitude (p.u.) Angle (deg.)

1 1.0000 0.0000 1.0000 0.0000

2 0.9713 0.0320 0.9712 0.0320

3 0.9567 0.0493 0.9547 0.0493

4 0.9509 0.0565 0.9489 0.0565

5 0.9499 0.0687 0.9478 0.0687

6 0.9582 0.1894 0.9582 0.1894

7 0.9560 0.2166 0.9560 0.2166

8 0.9570 0.2050 0.9569 0.2050

9 0.9680 0.0720 0.9679 0.0720

10 0.9669 0.0850 0.9669 0.0850

11 0.9500 0.1315 0.9477 0.1317

12 0.9458 0.1824 0.9437 0.1824

13 0.9445 0.1987 0.9424 0.1987

14 0.9486 0.0848 0.9466 0.0849

15 0.9484 0.0869 0.9464 0.0869

Table 2.4 Line flows of 15 bus radial distribution system

Branch

No.

From

(i)

To

(j)

Pij

(kW)

Qij

(kVAr)

Pji

(kW)

Qji

(kVAr)

Ploss

(kW)

Qloss

(kVAr)

1 1 2 1250.49 1271.34 1212.8 1234.47 37.69 36.87

2 2 3 724.19 737.35 712.9 726.31 11.29 11.04

3 3 4 394.80 402.45 392.36 400.06 2.44 2.39

4 4 5 44.10 44.98 44.04 44.94 0.06 0.04

5 2 9 114.16 116.42 113.69 116.1 0.47 0.32

6 9 10 44.10 44.98 44.04 44.94 0.06 0.04

7 2 6 350.51 357.34 344.56 353.45 5.95 3.89

8 6 7 140.00 142.80 139.61 142.53 0.39 0.27

9 6 8 70.00 71.40 69.89 71.32 0.11 0.08

10 3 11 254.78 259.64 252.6 258.17 2.18 1.47

11 11 12 114.17 116.43 113.57 116.02 0.60 0.41

12 12 13 44.10 44.98 44.03 44.93 0.07 0.05

13 4 14 70.00 71.40 69.8 71.26 0.20 0.14

14 4 15 140.00 142.80 139.56 142.5 0.44 0.30

2.5.2 Example – 2

The 33 - bus, 12.66 kV radial distribution system [19] is shown in Fig. 2.6. The line

and load data of this system is given in Appendix A (Table A2). The voltage profile of the

system is given in Table 2.5. The line flows of 33 bus system are given in Table 2.6. The total

real and reactive power losses are 202.5022 kW and 135.1286 kVAr. The minimum voltage

of this system is 0.9131p.u. at bus 18 and voltage regulation is 8.69%. The number of

iterations taken for this system is 3. The CPU time has taken to obtain the solution is 0.12

sec. The computational efficiency of the proposed method is given in Table 2.7 and is

compared with other known methods [11, 18]. The results obtained by the proposed method

and the existing methods [71, 104] are matching both in voltage magnitude and phase angle.

Fig. 2.6 Single line diagram of 33 bus radial distribution system

Table 2.5 Voltage profile of 33 bus radial distribution system

Bus

No.

Proposed method Existing method

[71] Existing method [104]

Voltage

Magnitude (p.u.)

Angle

(deg.)

Voltage

Magnitude (p.u.)

Voltage

Magnitude (p .u.)

Angle

(deg.)

1 1.0000 0.0000 1.00000 1.0000 0.0000

2 0.9970 0.0132 0.99703 0.9970 0.0141

3 0.9829 0.0929 0.98289 0.9828 0.0939

4 0.9755 0.1585 0.97538 0.9753 0.1595

5 0.9681 0.2252 0.96796 0.9680 0.2261

6 0.9497 0.1307 0.94948 0.9495 0.1317

7 0.9462 -0.0996 0.94595 0.9461 -0.0986

8 0.9413 -0.0635 0.93230 0.9412 -0.0625

9 0.9351 -0.1366 0.92597 0.9349 -0.1356

10 0.9293 -0.1992 0.92009 0.9291 -0.1981

11 0.9284 -0.1919 0.91922 0.9283 -0.1909

12 0.9269 -0.1804 0.91771 0.9266 -0.1794

13 0.9207 -0.2707 0.91153 0.9207 -0.2707

14 0.9185 -0.3494 0.90924 0.9182 -0.3494

15 0.9171 -0.3871 0.90782 0.9171 -0.3871

16 0.9157 -0.4103 0.90643 0.9157 -0.4103

17 0.9137 -0.4876 0.90439 0.9137 -0.4876

18 0.9131 -0.4972 0.90377 0.9130 -0.4971

19 0.9965 0.0023 0.99650 0.9965 0.0033

20 0.9929 -0.0646 0.99292 0.9929 -0.0637

21 0.9922 -0.0840 0.99221 0.9922 -0.0830

22 0.9916 -0.1043 0.99158 0.9915 -0.1034

23 0.9794 0.0603 0.97931 0.9792 0.0613

24 0.9727 -0.0284 0.97264 0.9726 -0.0274

25 0.9694 -0.0721 0.96931 0.9692 -0.0711

26 0.9477 0.1702 0.94755 0.9476 0.1712

27 0.9452 0.2263 0.94499 0.9452 0.2273

28 0.9337 0.3093 0.93354 0.9337 0.3103

29 0.9255 0.3870 0.92532 0.9255 0.3882

30 0.9220 0.4925 0.92177 0.9218 0.4934

31 0.9178 0.4080 0.91760 0.9178 0.4090

32 0.9169 0.3850 0.91669 0.9169 0.3859

33 0.9166 0.3773 0.91640 0.92 0.38

Table 2.6 Line flows of 33 bus radial distribution system

Branch

No.

From

(i)

To

(j)

Pij

(kW)

Qij

(kVAr)

Pji

(kW)

Qji

(kVAr)

Ploss

(kW)

Qloss

(kVAr)

1 1 2 3905.25 2418.80 3893.04 2412.5

8 12.21 6.22

2 2 3 3392.47 2171.41 3340.82 2145.5

0 51.65 26.31

3 3 4 2342.98 1674.06 2323.08 1663.9

3 19.90 10.13

4 4 5 2204.29 1584.53 2185.59 1575.0

1 18.70 9.52

5 5 6 2106.04 1521.52 2067.80 1488.5

1 38.24 33.01

6 6 7 1093.35 521.56 1091.44 515.23 1.91 6.33

7 7 8 888.51 419.96 883.67 418.36 4.84 1.60

8 8 9 684.33 316.96 680.15 313.96 4.18 3.00

9 9 10 620.77 294.43 670.21 291.91 3.56 2.52

10 10 11 560.22 274.25 559.67 274.07 0.55 0.18

11 11 12 514.34 243.96 513.46 243.67 0.88 0.29

12 12 13 451.67 206.56 449.00 204.76 2.67 2.10

13 13 14 390.94 170.90 390.21 169.94 0.73 0.96

14 14 15 270.59 90.58 270.53 90.26 0.36 0.32

15 15 16 210.30 80.38 210.02 80.17 0.28 0.21

16 16 17 150.05 60.04 149.80 59.7 0.25 0.34

17 17 18 90.00 40.00 89.95 39.96 0.05 0.04

18 2 19 360.98 160.93 360.82 160.78 0.16 0.15

19 19 20 270.14 120.18 269.31 119.43 0.83 0.75

20 20 21 180.04 80.06 179.94 79.94 0.10 0.12

21 21 22 90.00 40.00 89.96 39.94 0.04 `0.06

22 3 23 936.43 445.07 933.28 442.91 3.15` 2.16

23 23 24 841.29 401.01 836.15 396.95 5.14 4.06

24 24 25 420.00 200.00 418.71 198.99 1.29 1.01

25 6 26 948.18 972.31 945.58 970.99 2.60 1.32

26 26 27 864.85 945.61 881.52 943.92 3.33 1.69

27 27 28 813.55 910.65 802.95 900.69 11.30 9.96

28 28 29 745.71 883.83 737.88 877.01 7.83 6.82

29 29 30 621.82 811.84 617.92 809.86 3.90 1.98

30 30 31 420.23 210.27 418.64 208.87 1.59 1.57

31 31 32 270.01 140.02 269.80 139.77 0.21 0.25

32 32 33 60.00 40.00 59.99 39.98 0.01 0.02

Table 2.7 Comparison of computational efficiency of proposed method with other

methods

Method System Execution time, (sec.) No. of iterations

Proposed method 0.12 3

Kersting Method [11] 0.14 4

Baran and Wu [18] 0.13 3

2.5.3 Example – 3

The 69 bus, 11 kV radial distribution system [76] is shown in Fig. 2.7. The line and

load data of this system is given in Appendix A (Table A3). The voltage profile of the system

is given in Table 2.8. The line flows of 69 bus system are given in Table 2.9. The total real

and reactive power losses are 224.9457 kW and 102.1397 kVAr. The minimum voltage of

this system is 0.9092 p.u. at bus 65 and voltage regulation is 9.082%. The number of

iterations taken for this system is 3. The CPU time has taken to obtain the solution is 0.13

sec. The performance in terms of computational efficiency of the proposed method is given in

Table 2.10 and is compared with other known methods [11, 18]. The solution obtained by

proposed method is compared with the existing methods [71, 104] and results are found to be

approximately matching both in voltage magnitude and phase angles.

Fig. 2.7 Single line diagram of 69 bus radial distribution system

Table 2.8 Voltage profile of 69 bus radial distribution system

Bus

No.

Proposed method Existing method

[71] Existing method [104]

Voltage

Magnitude

(p.u.)

Angle

(deg.)

Voltage

Magnitude

(p.u.)

Voltage

Magnitude

(p.u.)

Angle

(deg.)

1 1.0000 0.0000 1.00000 1.0000 0.0000

2 1.0000 -0.0012 0.99997 0.9999 -0.0012

3 0.9999 -0.0024 0.99993 0.9999 -0.0025

4 0.9998 -0.0059 0.99984 0.9998 -0.0059

5 0.9990 -0.0185 0.99902 0.9990 -0.0185

6 0.9901 0.0493 0.99009 0.9901 0.0492

7 0.9808 0.1210 0.98079 0.9808 0.1210

8 0.9786 0.1382 0.97858 0.9786 0.1382

9 0.9774 0.1471 0.97745 0.9774 0.1470

10 0.9724 0.2317 0.97245 0.9725 0.2316

11 0.9713 0.2505 0.97135 0.9713 0.2504

12 0.9682 0.3033 0.96819 0.9682 0.3032

13 0.9653 0.3496 0.96526 0.9653 0.3495

14 0.9624 0.3958 0.96237 0.9624 0.3957

15 0.9595 0.4418 0.95950 0.9595 0.4417

16 0.9590 0.4504 0.95897 0.9589 0.4503

17 0.9581 0.4645 0.95809 0.9581 0.4645

18 0.9581 0.4647 0.95808 0.9581 0.4646

19 0.9576 0.4732 0.95761 0.9576 0.4731

20 0.9573 0.4788 0.95732 0.9573 0.4787

21 0.9568 0.4877 0.95683 0.9568 0.4876

22 0.9568 0.4878 0.95683 0.9568 0.4877

23 0.9568 0.4891 0.95676 0.9567 0.4890

24 0.9566 0.4920 0.95660 0.9566 0.4919

25 0.9564 0.4952 0.95643 0.9564 0.4951

26 0.9564 0.4965 0.95636 0.9564 0.4964

27 0.9563 0.4968 0.95634 0.9563 0.4967

28 0.9999 -0.0027 0.99993 0.9999 -0.0027

29 0.9999 -0.0047 0.99973 0.9998 -0.0053

30 0.9998 -0.0021 0.99985 0.9997 -0.0032

31 0.9997 -0.0016 0.99971 0.9997 -0.0028

32 0.9997 0.0006 0.99961 0.9996 0.0009

33 0.9995 0.0061 0.99935 0.9993 0.0035

34 0.9992 0.0140 0.99901 0.9990 0.0940

35 0.9992 0.0150 0.99895 0.9989 0.0104

36 0.9999 -0.0030 0.99992 0.9999 -0.0030

37 0.9997 -0.0094 0.99975 0.9997 -0.0094

38 0.9996 -0.0118 0.99959 0.9995 -0.0118

39 0.9995 -0.0125 0.99954 0.9995 -0.0125

40 0.9995 -0.0125 0.99884 0.9995 -0.0125

41 0.9988 -0.0235 0.99884 0.9988 -0.0235

42 0.9986 -0.0282 0.99855 0.9985 -0.0282

43 0.9985 -0.0288 0.99850 0.9985 -0.0288

44 0.9985 -0.0289 0.99850 0.9985 -0.0289

45 0.9984 -0.0307 0.99841 0.9984 -0.0307

46 0.9984 -0.0307 0.99840 0.9984 -0.0307

47 0.9998 -0.0077 0.99979 0.9997 -0.0077

48 0.9985 -0.0525 0.99854 0.9985 -0.0525

49 0.9947 -0.1916 0.99470 0.9947 -0.1916

50 0.9942 -0.2114 0.99415 0.9941 -0.2114

51 0.9785 0.1385 0.97854 0.9785 0.1385

52 0.9785 0.1387 0.97853 0.9785 0.1386

53 0.9747 0.1690 0.97466 0.9746 0.1689

54 0.9714 0.1946 0.97142 0.9714 0.1945

55 0.9669 0.2302 0.96694 0.9669 0.2300

56 0.9626 0.2651 0.96257 0.9625 0.2650

57 0.9401 0.6617 0.94010 0.9401 0.6615

58 0.9290 0.8643 0.92904 0.9290 0.8641

59 0.9248 0.9452 0.92476 0.9248 0.9451

60 0.9197 1.0497 0.91974 0.9197 1.0495

61 0.9123 1.1188 0.91234 0.9123 1.1186

62 0.9121 1.1215 0.91205 0.9120 1.1213

63 0.9117 1.1252 0.91166 0.9117 1.1250

64 0.9098 1.1430 0.90976 0.9098 1.1428

65 0.9092 1.1484 0.90919 0.9092 1.1482

66 0.9713 0.2517 0.97129 0.9713 0.2516

67 0.9713 0.2517 0.97129 0.9713 0.2516

68 0.9679 0.3093 0.96786 0.9678 0.3092

69 0.9679 0.3093 0.96786 0.9678 0.3092

Table 2.9 Line flows of 69 bus radial distribution system

Branch

No.

From

(i)

To

(j)

Pij

(kW)

Qij

(kVAr)

Pji

(kW)

Qji

(kVAr)

Ploss

(kW)

Qloss

(kVAr)

1 1 2 4016.32 2785.2 4016.25 2785.02 0.07 0.18

2 2 3 4016.32 2785.2 4016.25 2785.02 0.07 0.18

3 3 4 3748.77 2591.38 3748.58 2590.91 0.19 0.47

4 4 5 2896.08 1977.96 2894.15 1975.7 1.93 2.26

5 5 6 2867.94 1963.63 2839.8 1949.3 28.14 14.33

6 6 7 2836.1 1946.53 2806.86 1931.64 29.24 14.89

7 7 8 2788.83 1913.03 2781.96 1909.53 6.87 3.5

8 8 9 2666.37 1826.32 2663.01 1824.61 3.36 1.71

9 9 10 774.79 521.67 770.08 520.11 4.71 1.56

10 10 11 745.78 511.34 744.77 511.01 1.01 0.33

11 11 12 562.59 380.62 560.41 379.9 2.18 0.72

12 12 13 360.29 236.19 359.01 235.77 1.28 0.42

13 13 14 351.05 230.78 349.81 230.37 1.24 0.41

14 14 15 341.85 224.88 340.65 224.48 1.2 0.4

15 15 16 341.63 224.81 341.41 224.74 0.22 0.07

16 16 17 296.31 194.7 295.99 194.59 0.32 0.11

17 17 18 236.3 159.7 236.3 159.7 0 0

18 18 19 176.2 124.67 176.1 124.64 0.1 0.03

19 19 20 176.13 124.64 176.06 124.62 0.07 0.02

20 20 21 175.03 124.01 174.92 123.97 0.11 0.04

21 21 22 61.03 43.01 61.05 43.01 0 0

22 22 23 56.02 40.01 56.01 40.01 0.01 0

23 23 24 56.01 40 56 40 0.01 0

24 24 25 28 20 27.99 20 0.01 0

25 25 26 28 20 28 20 0 0

26 26 27 14 10 14 10 0 0

27 3 28 81.53 64.01 81.53 64.01 0 0

28 28 29 55.52 46.01 55.52 66 0 0.01

29 29 30 29.52 28.01 29.52 28.01 0 0

30 30 31 29.52 28.01 29.52 28.01 0 0

31 31 32 29.52 28.01 29.52 28.01 0 0

32 32 33 29.51 28 29.5 28 0.01 0

33 33 34 15.5 18 15.49 18 0.01 0

34 34 35 6 4 6 4 0 0

35 3 36 185.76 129.16 185.76 129.16 0 0

36 36 37 159.74 110.57 159.72 110.53 0.02 0.04

37 37 38 133.72 92 133.7 91.98 0.02 0.02

38 38 39 133.72 91.99 133.71 91.98 0.01 0.01

39 39 40 109.72 74.99 109.72 74.99 0 0

40 40 41 85.67 57.93 85.62 57.87 0.05 0.06

41 41 42 84.45 56.91 84.43 56.89 0.02 0.02

42 42 43 84.45 56.91 84.45 56.91 0 0

43 43 44 78.45 52.61 78.45 52.61 0 0

44 44 45 78.44 52.6 78.43 52.59 0.01 0.01

45 45 46 39.22 26.3 39.22 26.3 0 0

46 4 47 850.73 611.11 850.71 611.05 0.02 0.06

47 47 48 850.15 609.68 849.57 608.25 0.58 1.43

48 48 49 769.52 549.28 767.89 545.28 1.63 4

49 49 50 384.7 274.5 384.58 274.22 0.12 0.28

50 8 51 44.1 31 44.1 31 0 0

51 51 52 3.6 2.7 3.6 2.7 0 0

52 9 53 1851.08 1278.14 1845.3 1275.2 5.78 2.94

53 53 54 1840.02 1271.22 1833.31 1268.8 6.71 2.42

54 54 55 1804.49 1247.57 1795.36 1242.92 9.13 4.65

55 55 56 1771.3 1225.9 1762.51 1221.42 8.79 4.48

56 56 57 1721.63 1209.22 1671.96 1192.55 49.67 16.67

57 57 58 1697.14 1201.01 1672.66 1192.79 24.48 8.22

58 58 59 1687.64 1197.86 1678.14 1194.72 9.5 3.14

59 59 60 1576.97 1122.63 1566.3 1119.39 10.67 3.24

60 60 61 1562.95 1115.48 1548.93 1108.34 14.02 7.14

61 61 62 318.84 227.43 318.32 227.37 0.52 0.06

62 62 63 286.7 204.36 286.57 204.29 0.13 0.07

63 63 64 286.04 204.02 285.38 203.68 0.66 0.34

64 64 65 59 42 58.96 41.98 0.04 0.02

65 11 66 36 26 36 26 0 0

66 66 67 18 13 18 13 0 0

67 12 68 56 40 55.98 39.99 0.02 0.01

68 68 69 28 20 28 20 0 0

Table 2.10 Comparison of computational efficiency of proposed method with other

methods

Method System Execution time (sec.) No. of iterations

Proposed method 0.13 3

Kersting Method [11] 0.37 4

Baran and Wu [18] 0.29 3

2.6 CONCLUSIONS

A simple method for the load flow solution of radial distribution network has been

proposed. This method involves the concept of data structure to specify the configuration of

the system instead of commonly used a unique branch and bus numbering scheme to calculate

the cumulative loads of the system. This method is very simple and gives direct solution of the

radial distribution network based on the solution of a simple algebraic equation obtained from

the phasor diagram of the system. The effectiveness of the system is tested with15, 33 and 69

bus radial distribution systems. It is found that the solution obtained by this method matches

with the solution obtained by existing methods.