CHAPTER - 3

LOAD FLOW METHOD FOR UNBALANCED SYSTEMS

3.1 INTRODUCTION

Load flow analysis is an important task for power system planning and operational

studies. Certain applications, particularly in distribution automation and optimization of a

power system, require repeated load flow solution and in these applications, it is very

important to solve the load flow problem as efficiently as possible. As the power distribution

networks become more and more complex, there is a higher demand for efficient and reliable

system operation. Consequently, the most important system analysis tool, load flow studies,

must have the capability to handle various system configurations with adequate accuracy and

speed. In many cases, it is observed that the radial distribution systems are unbalanced

because of single-phase, two-phase and three-phase loads. Thus, load flow solution for

unbalanced case, special treatment is required.

Conventional load flow methods cannot be directly applied to distribution systems as

discussed in previous chapter. Moreover, the techniques for three phase power flow analysis

for unbalanced systems cannot be developed by simply extending the single phase balanced

methods to three phase systems. A three phase load flow method has to address issues like

modeling of different types of component connections, determining starting point for three

phase power flow solution as there are phase shifts and transformation ratios for each phase

and at different buses. For untransposed lines and cables the balanced models are no longer

useful. The symmetrical component transformation can decouple the three phases. For three

phase networks an impedance matrix is to be obtained.

Chen et al. [24 - 26] have proposed a method to calculate power losses in unbalanced

radial distribution systems. Also proposed a method, how to model the transformer for power

flow analysis. Zimmerman and Chiang [39] have proposed Fast decoupled power flow

method for unbalanced radial distribution systems. This method orders the laterals instead of

buses into layers, thus reducing the problem size to the number of laterals. Using of lateral

variables instead of bus variables makes this method more efficient for a given system

topology, but it may add some difficulties if the network topology is changed regularly,

which is common in distribution systems because of switching operations. Thukaram et al.

[50] have proposed a method for solving three-phase radial distribution networks. This

method uses the forward and backward propagation to calculate branch currents and bus

voltages.

Garcia et al. [53] have proposed a method based on the Three-phase Current Injection

Method (TCIM), in which the current injection equations are written in rectangular

coordinates and is a full Newton method. Also it presents quadratic convergence properties

and convergence is obtained for all except some extremely ill conditioned cases. Lin and

Teng [54] have proposed a fast decoupled method which uses G-matrix for power flow based

on equivalent current injections. Teng [55] has proposed a method based on the Network

Topology which uses two matrices, viz. Bus Injection to Branch Current (BIBC) and Branch-

Current to Bus-Voltage (BCBV) matrices, to find out the solution. Kersting [56] has

proposed modeling of transformer and other components of distribution systems. Jen-Hao

Teng [78] has proposed direct method of load flow solution of unbalanced radial distribution

networks.

Mamdouh Abdel-Akher et al. [93] have proposed improved load flow method for

unbalanced RDS using sequence components. Peng Xiao et al. [96] have proposed a method

to model the different connections of transformer in as unbalanced radial distribution

networks. Subrahmanyam [100] has proposed simple three phase load flow method by

solving simple algebraic recursive expression of voltage magnitude. Many researchers [114,

127] have proposed different methods to solve load flow solution of unbalanced radial

distribution networks.

An efficient method for load flow analysis plays a critical role in automation

algorithms of RDS whose scope encompasses fault isolation, network reconfiguration and

service restoration. The ability of automation algorithms to handle these complex tasks that

require frequent topological changes in the RDS demands a dynamic topology processor

based on a well-defined data structure. In this chapter, a simple method of load flow

technique for unbalanced radial distribution system is proposed using data structure. The

proposed method involves the solution of simple algebraic equation of receiving end

voltages. However, most of the methods reported require a unique lateral branch and node

numbering method that needs to be pre-processed before the actual load flow can be carried

out using the recursive voltage equation. This unique numbering method is very essential in

obtaining the final solution.

The mathematical formulation of the proposed method is explained in the Section 3.2.

In this section, the modeling of different components of distribution system such as line, load

and transformer etc. are described. The steps of load flow algorithm are explained in Section

3.3. The effectiveness of the proposed method is tested with different examples of

distribution systems in Section 3.4 and the results are compared with the existing methods. In

Section 3.5, conclusions based on the solutions obtained by the proposed method are

presented.

3.2 MODELING OF UNBALANCED RADIAL DISTRIBUTION SYSTEM

Unbalanced radial distribution system can be modeled as a network of buses

connected by distribution lines (with or without voltage regulators), switches or transformers.

Each bus may also have a corresponding load, shunt capacitor and/or co-generator connected

to it. This model can be represented by a radial interconnection of copies of the basic building

block shown in Fig. 3.1. Since a given branch may be single-phase, two-phase, or three-

phase, each of the labeled quantities is respectively a complex scalar, 2 1 or 3 1 complex

vector.

Fig. 3.1 Basic building block of Distribution System

3.2.1 Distribution Systems Line Model

For the analysis of power transmission line, two fundamental assumptions are made,

namely:

Three-phase currents are balanced.

Transposition of the conductors to achieve balanced line parameters.

However, distribution systems do not lend themselves to either of the two

assumptions. Because of the dominance of single-phase loads, the assumption of balanced

three-phase currents is not applicable. Distribution lines are seldom transposed, nor can it be

assumed that the conductor configuration is an equilateral triangle. When these two

assumptions are invalid, it is necessary to introduce a more accurate method of calculating

the line impedance.

A general representation of a distribution system with N conductors can be formulated

by resorting to the Carson’s equations [56], leading to an N×N primitive impedance matrix.

For most application, the primitive impedance matrices containing the self and mutual

impedances of each branch need to be reduced to the same dimension. A convenient

representation can be formulated as a 3×3 matrix in the phase frame, consisting of the self

and mutual equivalent impedances for the three phases. The standard method used to form

this matrix is the Kron’s reduction [56], based on the Kirchhoff’s laws. For instance, a four-

wire grounded wye overhead distribution line shown in Fig. 3.2 results in a 4×4 impedance

matrix. The corresponding equations are

a

ijI aa

ijZ

b

ijI bb

ijZ ab

ijZ

a

iV

c

ijI

b

iV cc

ijZ

bc

ijZ ac

ijZ

an

ijZ b

jV

a

jV

c

iV nn

ijZ

cn

ijZ

bn

ijZ

c

jV

n

ijI

Fig. 3.2 Model of the three-phase four-wire distribution line

n

ij

c

ij

b

ij

a

ij

nn

ij

nc

ij

nb

ij

na

ij

cn

ij

cc

ij

cb

ij

ca

ij

bn

ij

bc

ij

bb

ij

ba

ij

an

ij

ac

ij

ab

ij

aa

ij

n

j

c

j

b

j

a

j

n

i

c

i

b

i

a

i

I

I

I

I

ZZZZ

ZZZZ

ZZZZ

ZZZZ

V

V

V

V

V

V

V

V

… (3.1)

It can also viewed in matrix form as

j-bus

i-bus

A B C N

A B C N

n

ij

abc

ij

nn

ij

Tn

ij

n

ij

abc

ij

n

j

abc

j

n

i

abc

i

I

I

ZZ

ZZ

V

V

V

V … (3.2)

If neutral is grounded, the voltages n

iV and n

jV are considered to be equal. Then from the first

row of Eqn. 3.2, the value of n

ijI can be obtained as

abc

ij

Tn

ij

1nn

ij

n

ij IZZI

… (3.3)

Substituting Eqn. (3.3) into Eqn. (3.2), the Kron’s reduction [56] of voltage equation reduces

to

abc

ij

abc

ij

abc

j

abc

i IZeVV … (3.4)

where,

cc

ij

cb

ij

ca

ij

bc

ij

bb

ij

ba

ij

ac

ij

ab

ij

aa

ijTn

ij

1nn

ij

n

ij

abc

ij

abc

ij

ZeZeZe

ZeZeZe

ZeZeZe

ZZZZZe

… (3.5)

3.2.2 Changing of phase impedances into sequence impedances

Phase components can be changed into sequence components as follows:

AZAZ abc

ij

1012

ij

… (3.6)

abc

i

1012

i VAV … (3.7)

abc

ij

1012

ij IAI … (3.8)

where A is symmetrical component transformation matrix given by

2

2

aa1

aa1

111

A and a= 01201

Now,

22

ij

21

ij

20

ij

12

ij

11

ij

10

ij

02

ij

01

ij

00

ijTn

ij

1nn

ij

n

ij

012

ij

012

ij

ZeZeZe

ZeZeZe

ZeZeZe

ZZZZZe

… (3.9)

The sequence voltage at the jth

bus can be obtained as:

012

ij

012

ij

012

i

012

j IZeVV

i.e.,

2

ij

1

ij

0

ij

22

ij

21

ij

20

ij

12

ij

11

ij

10

ij

02

ij

01

ij

00

ij

2

i

1

i

0

i

2

j

1

j

0

j

I

I

I

ZeZeZe

ZeZeZe

ZeZeZe

V

V

V

V

V

V

… (3.10)

3.2.3 Load Models

The different load models that have been considered are given below.

3.2.3.1 Lumped Load Model

In distribution systems loads can exist in one, two and three phase loads with wye or

delta connections. Also loads can be categorized into four types depending on the load

characteristics; constant power, constant impedance, constant current and complex loads. In

this chapter, the loads considered are of constant power type and can be mathematically

represented as follows

Wye connected loads

*

nph

i

nph

iph

iV

SIL

… (3.11)

where

ph

iIL = load current per phase at ith

bus

nph

iS = power per phase at ith

bus

nph

iV = phase to neutral voltage at ith

bus

Delta connected loads

*

phph

i

nph

iph

iV

SIL

… (3.12)

where

ph

iIL = load current per phase at ith

bus

nph

iS = power per phase at ith

bus

phph

iV = phase voltage at ith

bus

3.2.3.2 Distributed Load Model

Sometimes the primary feeder supplies loads through distribution transformers tapped

at various locations along line section. If every load point is modeled as a bus, then there will

be a large number of buses in the system. Hence these loads are represented as lumped loads:

At one-fourth length of line from sending end a dummy bus is created at which two

thirds of the load is assumed to be connected.

The remaining one-third load is assumed to be connected at the receiving end bus.

3.2.4 Transformer Model

The impact of the transformers in a distribution system is significant. Transformers

affect system loss, zero sequence current, method of grounding and protection strategy.

Three-phase transformer is represented by two blocks as shown in Fig. 3.3, one block

represents the per unit leakage admittance matrixabc

TY , and the other block models the core

loss as a function of voltage on the secondary side.

Fig. 3.3 General Three-phase Transformer Model

3.2.4.1 Core Loss

Core

Loss

Primary Secondary

The core loss of a transformer is approximated by shunt core loss functions on each

phase of the secondary terminal of the transformer. These core loss approximation functions

are based on the results of EPRI load modeling research which state that real and reactive

power losses in the transformer core can be expressed as functions of the terminal voltage of

the transformer. Transformer core loss functions represented in per unit at the system power

base [25] are:

2VC2

eBVABaseSystem

RatingKVA.u.pP

… (3.13)

2VF2

eEVDBaseSystem

RatingKVA.u.pQ

… (3.14)

where,

V is voltage magnitude in per unit.

It is to be noted that the coefficients A, B, C, D, E, and F are transformer dependent

constants.

3.2.4.2 Leakage Admittance Matrix: abc

TY

The admittance matrix part of the proposed three-phase transformer models follow the

methodology derived by [25, 56], for simplification, a single three-phase transformer is

approximated by three identical single-phase transformers connected appropriately. This

assumption is not essential; however, it simplifies the ensuing derivation and explanation.

Based upon this assumption, the characteristic sub matrices used in forming the three phase

transformer admittance matrices can be developed. The Table 3.1 shows the sub-matrices of

TY for the nine most common step-down transformer connection types. The Table 3.2 shows

the matrices for step-up transformer connection types.

Table 3.1 Sub-matrices of TY for common step-down transformer connections [96]

Transformer connection

Bus p Bus s

Self Admittance abc

ss

abc

pp YY

Mutual Admittance abc

sp

abc

ps YY

Table 3.2 Sub-matrices of TY for common step-up transformer connections [96]

where,

t

t

t

1

Y00

0Y0

00Y

Y ,

ttt

ttt

ttt

2

Y2YY

YY2Y

YYY2

3

1Y ,

tt

tt

tt

3

Y0Y

YY0

0YY

3

1Y

tY - Per-unit leakage admittance.

If the transformer has an off-nominal tap ratio α: β between the primary and

secondary windings, where α and β are tappings on the primary and secondary sides

respectively, then the sub-matrices are modified as follows:

Wye-G Wye-G 11 YY

11 YY

Wye-G Wye 22 YY

22 YY

Wye-G Delta 21 YY t

33 YY

Wye Wye-G 22 YY

22 YY

Wye Wye 22 YY

22 YY

Wye Delta 22 YY t

33 YY

Delta Wye-G 12 YY

3

t

3 YY

Delta Wye 22 YY t

33 YY

Delta Delta 22 YY

22 YY

Transformer connection

Bus p Bus s

Self Admittance abc

ss

abc

pp YY

Mutual Admittance abc

sp

abc

ps YY

Wye-G Wye-G 11 YY

11 YY

Wye-G Wye 22 YY

22 YY

Wye-G Delta 21 YY t

33 YY

Wye Wye-G 22 YY

22 YY

Wye Wye 22 YY

22 YY

Wye Delta 22 YY t

33 YY

Delta Wye-G 12 YY

3

t

3 YY

Delta Wye 22 YY

3

t

3 YY

Delta Delta 22 YY

22 YY

a) Divide the self admittance matrix of the primary by α2.

b) Divide the self admittance matrix of the secondary by β2.

c) Divide the mutual admittance matrices by αβ

With the nodal admittance matrix abc

TY , the transformer voltage-current relationship

can be expressed as:

abc

s

pabc

T

abc

s

P

V

VY

I

I

… (3.15)

where,

sssp

psppabc

T YY

YYY

The matrix abc

TY is divided into four 3×3 sub-matrices: Ypp, Yps, Ysp and Yss. Vp and

Vs are the line-to-neutral bus voltages and Ip and Is are injection currents at the primary and

secondary sides of the transformer respectively.

The sequence component of primary and secondary currents can be calculated as

abc

p

1012

p IAI

… (3.16)

abc

s

1012

s IAI … (3.17)

Then for three phase unbalanced system, the total current through branch k, abc

kI is

summation of current at bus j due to self load and the cumulative current of all the branches

connected to bus j.

The equation for 012

jV can be written as

012

k

012

k

012

i

012

j IZVV

… (3.18)

where

012

kI = sequence current vector in kth

branch = abc

k

1IA

012

iV and012

jV are the voltage vectors of ith

and jth

buses respectively.

AZAZ abc

k

1012

k

… (3.19)

where

cc

k

cb

k

ca

k

bc

k

bb

k

ba

k

ac

k

ab

k

aa

k

abc

k

ZZZ

ZZZ

ZZZ

Z

Then the phase voltages can be calculated as:

012abc AVV … (3.20)

These calculations are to be carried out till the bus voltages are converged within the

specified tolerance limits. Then the real and reactive power loss in branch k connected

between i and j buses can be expressed as:

)kSRe(kP abc

ij

abc

Loss … (3.21)

)kS(ImkQ abc

ij

abc

Loss … (3.22)

where

*abc

k

abc

j

abc

i

abc

ij IVVkS

kSabc

ij is a complex power loss in the branch, k connected between buses i and j

abc

iV andabc

jV are voltage at buses i and j respectively and

abc

kI is the current through branch k connected between buses i and j.

The total active power loss (TPL) and total reactive power loss (TQL) are calculated

by using the Eqns. (2.11) and (2.12) given in Chapter 2.

3.3 ALGORITHM FOR LOAD FLOW CALCULATION

Step 1 : Read line and load data of an unbalanced radial distribution system.

Step 2 : Initialize the bus voltages as 1p.u and bus voltage angles for phase – a,

phase - b and phase - c as000 120,120,0 respectively. Initialize TPL, TQL to zero.

Step 3 : Set iteration count =1 and tolerance = 0.0001.

Step 4 : Build BIM and Data Structure of the system using Section 2.3.

Step 5 : Convert phase component of voltages and impedances into sequence components

using Eqns. (3.6) – (3.7).

Step 6 : Calculate sequence component load currents at all buses.

Step 7 : Calculate Sequence voltages (V012

) at all buses using Eqn. (3.18). The phase voltages

(Vabc

) can be calculated using Eqn. (3.20).

Step 8 : Check for the convergence, if the difference between the voltage magnitudes in two

consecutive iterations is less than , then go to Step 9else set count = count+1 and

go to Step 6.

Step 9 : Calculate losses in each branch using Eqn. (3.21) and (3.22) and compute

TPL and TQL.

Step 10 : STOP.

3.4 FLOW CHART FOR THE PROPOSED METHOD

Read Distribution System.

Start

Assume bus voltages for phase system are , , .

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

reactive power losses to zero

Build Bus Incidence matrix (BIM)

Fig. 3.4 Flow chart of load flow method for unbalanced systems

3.5 ILLUSTRATIVE EXAMPLES

To illustrate the effectiveness of the proposed method is tested with two different

examples consisting of 25 and IEEE 37 bus unbalanced radial distribution systems.

3.5.1 Example – 1

The proposed algorithm is tested on 25-bus unbalanced radial distribution system

whose single line diagram is shown in Fig. 3.5. For the load flow the base voltage and base

MVA are chosen as 4.16 kV and 30 MVA respectively. The line and load data are given in

Appendix B (Table B.1). The voltage profile of the system obtained using load flow solution

is given in Table 3.3. The summary of load flow result of 25 bus system is given in Table 3.4.

Fig. 3.5 Single line diagram of 25 bus URDS

Table 3.3 Load flow result of 25 bus unbalanced radial distribution system

Bus

no.

Proposed method Existing method[127]

|Va|

(p.u.)

Angle

(Va)

(Deg)

|Vb|

(p.u.)

Angle

(Vb)

(Deg)

|Vc|

(p.u.)

Angle

(Vc)

(Deg)

|Va|

(p.u.)

|Vb|

(p.u.)

|Vc|

(p.u.)

1 1.0000 0.0000 1.0000 -2.0944 1.0000 2.0944 1.0000 1.0000 1.0000

2 0.9702 -0.0099 0.9711 -2.1016 0.9755 2.0824 0.9702 0.9711 0.9755

3 0.9632 -0.0122 0.9644 -2.1034 0.9698 2.0796 0.9632 0.9644 0.9698

4 0.9598 -0.0134 0.9613 -2.1043 0.9674 2.0783 0.9598 0.9613 0.9674

5 0.9587 -0.0133 0.9603 -2.1043 0.9664 2.0783 0.9587 0.9603 0.9664

6 0.955 -0.0097 0.9559 -2.1006 0.9615 2.082 0.955 0.9559 0.9615

7 0.9419 -0.0097 0.9428 -2.0997 0.9492 2.0816 0.9419 0.9428 0.9492

8 0.9529 -0.0097 0.9538 -2.1005 0.9596 2.082 0.9529 0.9538 0.9596

9 0.9359 -0.0097 0.9367 -2.0993 0.9438 2.0815 0.9359 0.9367 0.9438

10 0.9315 -0.0097 0.9319 -2.099 0.9395 2.0813 0.9315 0.9319 0.9395

11 0.9294 -0.0097 0.9296 -2.0989 0.9376 2.0813 0.9294 0.9296 0.9376

12 0.9284 -0.0097 0.9284 -2.0988 0.9366 2.0814 0.9284 0.9284 0.9366

13 0.9287 -0.0097 0.9287 -2.0989 0.9368 2.0814 0.9287 0.9287 0.9368

14 0.9359 -0.0096 0.937 -2.0992 0.9434 2.0814 0.9359 0.937 0.9434

15 0.9338 -0.0096 0.9349 -2.099 0.9414 2.0814 0.9338 0.9349 0.9414

16 0.9408 -0.0097 0.9418 -2.0996 0.9483 2.0816 0.9408 0.9418 0.9483

17 0.9347 -0.0096 0.936 -2.0991 0.942 2.0815 0.9347 0.936 0.942

18 0.9573 -0.0122 0.9586 -2.103 0.9643 2.0795 0.9573 0.9586 0.9643

19 0.9548 -0.0122 0.9563 -2.1029 0.962 2.0795 0.9524 0.9544 0.96

20 0.9535 -0.0122 0.9547 -2.1028 0.9603 2.0795 0.9548 0.9563 0.962

21 0.9538 -0.0121 0.9549 -2.1029 0.9605 2.0797 0.9537 0.9549 0.9605

22 0.9518 -0.0121 0.9525 -2.1028 0.9585 2.0799 0.9518 0.9525 0.9585

23 0.9565 -0.0133 0.9584 -2.1043 0.9648 2.0783 0.9565 0.9584 0.9648

24 0.9544 -0.0133 0.9565 -2.1043 0.9631 2.0782 0.9544 0.9565 0.9631

25 0.952 -0.0132 0.9547 -2.1044 0.9612 2.0783 0.952 0.9547 0.9612

From the results the minimum voltages in phases a, b and c are 0.9284 p.u., 0.9284

p.u. and 0.9366 p.u. respectively. Total active power losses are observed as 52.7 kW, 55.41

kW and 41.83 kW in phases a, b and c respectively. Total Reactive power losses are as

58.2048 kVAr, 53.2694 kVAr and 55.6711 kVAr in phases a, b and c respectively. The

voltage regulation is 7.16%, 7.16% and 6.34% in phases a, b and c respectively. The solution

obtained by the proposed method is compared with solution obtained by the existing method

[127] and results are confirmed exactly. The total demand in each phase also can be observed

in Table 3.4.

Table 3.4 Summary of load flow result of 25 bus system

Description Proposed method Existing method [127]

Phase phase A phase B phase C phase A phase B phase C

Total Real power

loss in kW 52.7 55.41 41.83 52.82 55.44 41.86

Total Reactive

power loss in kVAr 58.2048 53.2694 55.6711 58.32 53.29 55.69

Total Real power

demand in kW 1126.000 1138.7102 1125.1284 1126.12 1138.74 1125.16

Total Reactive

power demand in

kVAr 850.2048 854.2695 855.6711 850.32 854.29 855.69

Minimum Voltage

in p.u. 0.9284 0.9284 0.9366 0.9284 0.9284 0.9366

3.5.2 Example – 2

The proposed algorithm is tested on IEEE 37-bus unbalanced radial distribution

system whose single line diagram is shown in Fig. 3.6. For the load flow the base voltage and

base MVA are chosen as 4.8 kV and 100 MVA respectively. The line and load data are given

in Appendix B (Table B.2 to B.6). The voltage profile of the system obtained using load flow

solution is given in Table 3.5. The summary of load flow result of 25 bus system is given in

Table 3.6.

Fig. 3.6 Single line diagram of IEEE 37 bus URDS

Table 3.5 Load flow result of 37 bus unbalanced radial distribution system

Bus

no.

Proposed method Existing method [132]

|Va| (p.u) |Vb|

(p.u)

|Vc| (p.u) |Va| (p.u) |Vb|

(p.u)

|Vc| (p.u)

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 0.9863 0.9855 0.9817 0.9863 0.9855 0.9817

3 0.9781 0.9772 0.9719 0.9781 0.9772 0.9719

4 0.9709 0.9715 0.9645 0.9709 0.9715 0.9645

5 0.9652 0.9667 0.9588 0.9652 0.9667 0.9588

6 0.9634 0.9651 0.9571 0.9634 0.9651 0.9571

7 0.9607 0.9631 0.9547 0.9607 0.9631 0.9547

8 0.9582 0.9621 0.9527 0.9582 0.9621 0.9527

9 0.9547 0.9606 0.9494 0.9547 0.9606 0.9494

10 0.9512 0.9596 0.9472 0.9512 0.9596 0.9472

11 0.9501 0.9592 0.9461 0.9501 0.9592 0.9461

12 0.9498 0.9590 0.9451 0.9498 0.9590 0.9451

13 0.9497 0.9589 0.9448 0.9497 0.9589 0.9448

14 0.9763 0.9749 0.9697 0.9763 0.9749 0.9697

15 0.9740 0.9718 0.9672 0.9740 0.9718 0.9672

16 0.9727 0.9683 0.9647 0.9727 0.9683 0.9647

17 0.9726 0.9679 0.9646 0.9726 0.9679 0.9646

18 0.9725 0.9675 0.9645 0.9725 0.9675 0.9645

19 0.9761 0.9746 0.9701 0.9761 0.9746 0.9701

20 0.9757 0.9738 0.9699 0.9757 0.9738 0.9699

21 0.9697 0.9709 0.9635 0.9697 0.9709 0.9635

22 0.9690 0.9705 0.9631 0.9690 0.9705 0.9631

23 0.9686 0.9704 0.9630 0.9686 0.9704 0.9630

24 0.9634 0.9651 0.9571 0.9634 0.9651 0.9571

25 0.9632 0.9642 0.9569 0.9632 0.9642 0.9569

26 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

27 0.9542 0.9595 0.9478 0.9542 0.9595 0.9478

28 0.9541 0.9593 0.9473 0.9541 0.9593 0.9473

29 0.9497 0.9588 0.9445 0.9497 0.9588 0.9445

30 0.9737 0.9717 0.9671 0.9737 0.9717 0.9671

31 0.9723 0.9714 0.9667 0.9723 0.9714 0.9667

32 0.9709 0.9629 0.9631 0.9709 0.9629 0.9631

33 0.9707 0.9624 0.9629 0.9707 0.9624 0.9629

34 0.9705 0.9619 0.9629 0.9705 0.9619 0.9629

35 0.9686 0.9701 0.9627 0.9686 0.9701 0.9627

36 0.9536 0.9578 0.9475 0.9536 0.9578 0.9475

37 0.9751 0.9737 0.9691 0.9751 0.9737 0.9691

From the results the minimum voltages in phases a, b and c are 0.9497 p.u., 0.9588

p.u. and 0.9445p.u. respectively. Total active power losses are observed as 31.5612 kW,

23.6734 kW and 30.4408 kW in phases a, b and c respectively. Total reactive power losses

are as 24.0121 kVAr, 22.3235 kVAr and 29.1918 kVAr in phases a, b and c respectively. The

voltage regulation is 5.03%, 4.12% and 5.55% in phases a, b and c respectively. The solution

obtained by the proposed method is compared with solution obtained by the existing method

[132] and results are confirmed exactly. The total demand in each phase also can be observed

in Table 3.6.

Table 3.6 Summary of load flow result of 37 bus unbalanced radial distribution

system

Description Proposed method Existing method [132]

Phase phase A phase B phase C phase A phase B phase C

Total Real power

loss in kW 31.5612 23.6734 30.4408 31.56 23.67 30.44

Total Reactive

power loss in

kVAr 24.0121 22.3235 29.1918 24.01 22.32 29.19

Total Real power

demand in kW 885.56 789.67 1163.44 885.56 789.67 1163.44

Total Reactive

power demand in

kVAr 442.01 397.32 521.81 442.01 397.32 521.81

Minimum Voltage

in p.u. 0.9497 0.9588 0.9445 0.9497 0.9588 0.9445

3.6 CONCLUSIONS

In this chapter, the implementation of Data structures for bus identification and load

flow solution for Unbalanced Radial distribution systems has been presented. The modeling

of major components such as line, transformer and load related to unbalanced radial

distribution system is presented. The proposed method is demonstrated with the two different

three phase unbalanced radial distribution systems and the results obtained are compared with

results of existing method.