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IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 6, NO. 1, WINTER-SPRING 2007

1682-0053/07$10 2007 JD

31

AbstractThis paper presents a new, efficient and robust

three-phase load flow for unbalanced distribution systems.

The proposed method uses equivalent current injections,

based on Newton-Raphson method. The load flow issue is

considered as an optimization problem and is decoupled into

two subproblems without using any assumptions. The

Jacobian matrix is set to be constant, thus, the method is fast,

robust and efficient. The proposed method has been tested on

the IEEE 13 Bus Radial Distribution Test Feeder with

satisfactory results.

Index TermsThree-Phase load flow, unbalanced

distribution networks, fast decoupled load flow.

NOMENCLATURE

The three phases, neutral and ground are referred to with

the superscript ,n,c,b,a and g,respectively.

x The subscript x denotes the nodes of three phase

system

y The subscript y denotes the nodes of three phase

system

r The superscript rdenotes the real parts

i The superscript i denotes the imaginary partsag

xV Voltage of phase a , at node x with respect to

groundaxyI Current flowing through phase a , between nodes

x and y aa

xyZ Self-impedances between nodes x and y in

phase a ab

xyZ Mutual-impedance between phases a and b between nodes x and y

abc

xV Voltage of phases a , b , and c at node x with

respect to groundabc

xyI Current flowing through phases a , b , and c

between nodes x and y

ECI Equivalent current injectiona

eqsp,k

I Specified ECI of phase a, at k-th bus][n,a

eqsp,kI Specified ECI of phase a, at k-th bus and at

n -th iterationspa

kS )( Complex power load at phase a at k-th busspa

kP )( Real power load at phase a at k-th busspa

kQ )( Reactive power load at phase a at k-th busa

kP0 Real multicipants of the total load at phase a , at

k-th busakQ 0 Reactive multicipants of the total load at phase a,

at k-th bus

1A Proportion of constant power component of theactive load

Manuscript received March 15, 2006; revised August 9, 2006.

The authors are with the Electrical Engineering Department, TarbiatModares University, Tehran, Iran (e-mail: hatamisharif@modares.ac.ir,

parsa@modares.ac.ir).Publisher Item Identifier S 1682-0053(07)0478

1R Proportion of constant power component of the

reactive load

2A Proportion of constant current component of the

active load

2R Proportion of constant current component of the

reactive load

3A Proportion of constant impedance component of

the active load

3R Proportion of constant impedance component of

the reactive loadabc

cal,kI The calculated three-phase current at k -th bus.

abckI The mismatch three-phase current at k -th bus.abce Real parts of voltage of phases a , b , and c abcf Imaginary parts of voltage of phases a , b ,

and c r,abcI Real parts of ECI mismatches of phases a , b ,

and c i,abcI Imaginary parts of ECI mismatches of phases a ,

b , and c abcabc f,e Vector of corrections of bus voltages of

phases a , b , and c , ,/ /abc r abc abc i abc abcI e I f G = = Submatrix of the

Jacobian matrix, ,

/ /

abc i abc abc r abc abc

I e I f B = = Submatrix of theJacobian matrixabcabcabc

bus B,G,Y Three phase admittance matrix, real

and imaginary parts of three phase admittance

matrix, respectively

I. INTRODUCTION

HE ANALYSIS of power distribution systems is an

important area of research activities due to the vital

role of distribution systems as the final link between the

bulk power system and consumers. Load flow is an

important tool for the analysis of distribution systems. This

tool must be able to model the special features ofdistribution systems such as unbalanced loads,

untransposed lines, radial and weakly meshed topology,

grounded or ungrounded systems, high resistance to

reactance(R/X) ratios, and single, two or three phase lines.

Due to the high R/X ratios and unbalanced operation in

distribution systems, the Newton-Raphson and ordinary

Fast Decoupled Load Flow method may provide inaccurate

results and may not be converged. Therefore, conventional

load flow methods cannot be directly applied to distribution

systems. In many cases, the radial distribution systems

include untransposed lines which are unbalanced because

of single phase, two phase and three phase loads. Thus,

load flow analysis of balanced radial distribution systems[1], [2] will be inefficient to solve the unbalanced cases and

the distribution systems need to be analyzed on a three-

phase basis instead of single phase basis.

Three-Phase Fast Decoupled Load Flow for

Unbalanced Distribution SystemsA. R. Hatami and M. Parsa Moghaddam

T

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IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 6, NO. 1, WINTER-SPRING 200732

There have been a lot of interests in the area of three-

phase distribution load flows. A fast decoupled power flow

method has been proposed in [3]. 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 node 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. In [4], a method for solving

unbalanced radial distribution systems based on the

Newton-Raphson method has been proposed. Thukaram et

al. [5] have proposed a method for solving three-phase

radial distribution networks. This method uses the forward

and backward propagation to calculate branch currents and

node voltages. A three-phase fast decoupled power flow

method has been proposed in [6]. This method uses

traditional Newton-Raphson algorithm in a rectangular

coordinate system. In [7], a method for the solution of

unbalanced three-phase power systems using the Newton-

Raphson has been proposed in which, three-phase currentinjection equations are written in rectangular coordinate

system. However, these methods are very cumbersome and

need large computational time.

A fast decoupled G-matrix method for power flow,

based on equivalent current injections, has been proposed

in [8] .This method uses a constant Jacobian matrix which

needs to be inverted only once. However, the Jacobian

matrix is formed by omitting the reactance of the

distribution lines with the assumption that R>>X; and fails

if X>R. In [9], a method has been suggested for three-

phase power flow analysis in distribution networks by

combining the implicit Z-bus method [10] and the Gauss-Seidel method. This method uses fractional factorization of

Y-bus matrix. Thus, large computational time is necessary

for this method. Ranjan et al. [11] have proposed a method

for load flow solution of unbalanced radial distribution

network. This method uses basic principles of circuit

theory and solves a simple algebraic recursive expression

of voltage magnitude. However, the power loss of the lines

must be calculated in each iteration and therefore needs

large computational time.

In this paper, the method proposed in [2] for balanced

distribution system is developed for threephase

unbalanced distribution systems. Furthermore, in the

proposed method any type of loads can be modeled easilyin the problem formulation. The proposed method is based

on equivalent current injections (ECI) and uses Newton-

Raphson technique in rectangular coordinates. Here, the

load flow is considered as an optimization problem and is

decoupled into two subproblems without using any

assumptions such as voltage magnitude and R/X ratios.

II.MODELING OF DISTRIBUTION SYSTEM COMPONENTS

A.Line Model

Most of the distribution systems are of unbalanced

operational feature and include untransposed lines.Therefore, it is necessary to introduce an accurate method

for calculating the line impedance. Line modeling of three-

phase unbalanced distribution system has been discussed in

Fig. 1. Three-Phase line section model.

[11]. A general form of three-phase feeder line section is

shown in Fig. 1. The line constants could be obtained by

the method developed by Carson and Lewis [12]. The

voltage equations, can be written as

ag aa ab ac aag V z z z I V y xy xy xy xyx

bg bg ba bb bc bV V z z z I x y xy xy xy xycg c

cg ca cb ccV IVx xyz z zy xy xy xy

=

(1)

Or in a compact form as

[ ] [ ] [ ] [ ]abc abc abc abcV V Z I x y xy xy = (2)

The, three-phase current is then obtained by

[ ] [ ] ([ ] [ ])abc abc abc abcI Y V Vxy xy x y= (3)

B.Load Models

In this paper, the loads are considered to be three-phase

(balanced or unbalanced) or single-phase. Three-phase

loads can be connected in wye or delta forms whilesingle-phase loads can be connected as line-to-ground or

line-to-line fashions. All loads can be modeled as a

combination of constant power, constant impedance or

constant current loads.

C.Power Equipments

Other equipments could be modeled in phase quantities

with or without mutual coupling. Details of three-phase

modeling of distribution system components are given in

[12].

III. THE ECI-BASEDNRALGORITHM

The ECI-based technique uses current instead of power

for modeling formulation. For load buses the specified

constant power, constant impedance or constant current can

be converted into specified equivalent current injection

(Isp-eq). For example, for a three-phase star connected load

model, which is shown in Fig. 2, the k-th bus specified ECI

at the n-th iteration is determined by

,[ ] ,[ ] ,[ ] ,[ ]

, - , - , - , -[ ] [ ]Tabc n a n b n c n

k sp eq k sp eq k sp eq k sp eqI I I I= (4)

,[ ] * *

, - ,[ ] ,[ ]

( ) ( )( ) ( )

a sp a a sp

a n k k k

k sp eq a n a n

k k

S P QI

V V

+= = (5)

Where

2,[ ] ,[ ]

0 1 2 3( ) ( )a sp a a n a n

k k k k P P A A V A V= + + (6)

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HATAMI AND PARSA MOGHADDAM: THREE-PHASE FAST DECOUPLED LOAD FLOW FOR UNBALANCED 33

Fig. 2. A three-phase wye connected load.

2,[ ] ,[ ]

0 1 2 3( ) ( )a sp a a n a n

k k k k Q Q R R V R V = + + (7)

1 2 3

1 2 3

1

1

A A A

R R R

+ + =

+ + = (8)

The calculated current at k-th bus,,

ab c

k c a l I , is given by

, 0[ ] [ ] [ ]abc abc abc

k cal k km

m l

I I I

= + (9)

where,00

[ ] [ ] [ ]k

abc abc abc

k kI Y V= = current drawn by the shunt

admittance,0

abc

kY , connected between k-th bus and neutral

and [ ] [ ] ([ ] [ ])km

abc abc abc abc

km k mI Y V V= , m= set of incident

lines at k-th bus.

Now let the mismatch current be defined as,

, , -[ ] [ ] [ ]abc abc abc

k k cal k sp eqI I I = (10)

The above mismatch current function can be rewritten as

, ,

,

, , ,

abc r abc r

abc r abcabc abc

abc i abcabc i abc i

abc abc

I II ee f

I fI I

e f

=

(11)

where

, ,[ ] [ ] [ ]abc abc r abc iI I j I= + (12)

and

[ ] [ ] [ ]abc abc abcV e j f = + (13)

Now if the three-phase admittance be defined as

[ ] [ ] [ ]abc abc abcbus

Y G j B= + (14)

Then, the mismatch current can be determined by the

following equation

,

,

-abc r abc abc abc

abc i abc abc abc

I G B e

I B G f

=

(15)

IV. PROPOSED METHOD

The load flow problem proposed here is considered as an

optimization problem in which, the main objective is to

minimize the sum of squares of specified values of ECI

from their calculated values. The problem is formulated as

- -

- -

( )

Tr r

abc abc abc abc

sp eq cal sp eq cal

i iabc abc abc abc

sp eq cal sp eq cal

I I I IMin

I I I I

=

(16)

Data in ut

Voltage initialization

Form ][Band][Gabcabc

Use equation (20) to compute

[H] and then invert it

Use equations (9), (10), (15) to

calculateiabc,rabc, IandI

Use equations (22) and (23) to

calculateiabc,

mod

rabc,

mod IandI

Solve equations (19) and (20) for

]f[and]e[ abcabc

Update the voltages

]f[]f[][f

]e[]e[][e

abcabcabc

abcabcabc

+=

+=

abcabc f,e

Sto

Yes

No

Fig. 3. Flowchart of the proposed method.

The optimal voltage corrections which minimizes the

function can be written as

1

0

0

,

,

abce abcT abc abcT abc

G G B B

abc T abc abc T abcabc G G B B

f

abc rI

abcT abc T G B

abcT abc TB G abc iI

+

+

=

(17)

or

1 ,

mod

,

mod

0

0

abc rabc

abc iabc

H Ie

H If

=

(18)

This equation can be decoupled into two equations

representing active and reactive loops as follows

-1 ,

mod[ ] [ ] [ ]abc abc r e H I = (19)

-1 ,

mod[ ] [ ] [ ]abc abc if H I = (20)

where[ ] [ ] [ ] [ ] [ ]abc T abc abc T abcH G G B B= + (21)

The real and imaginary components of the mismatch

current are represented by the following equations

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IRANIAN JOURNAL OF ELECTRICAL AND COMPUTER ENGINEERING, VOL. 6, NO. 1, WINTER-SPRING 200734

TABLE I

RESULTS OF THE PROPOSED METHOD AND COMPARISON WITH THE ACTUAL VALUES OF IEEE13BUS DISTRIBUTION FEEDER

, , ,

mod[ ] [ ] [ ] [ ] [ ]abc r abc T abc r abc T abc iI G I B I = + (22)

and, , ,

mod[ ] [ ] [ ] [ ] [ ]abc i abc T abc i abc T abc r I G I B I = (23)

Equations (19) and (20) are solved iteratively until the

convergence is obtained. In the above equations, H is a

constant matrix and is inverted only once. Since the load

flow problem is decoupled into two sub problems without

using any assumptions such as voltage magnitude and R/X

ratios therefore, the proposed method is a robust, efficient

and computationally fast. The flowchart of the proposed

algorithm is shown in Fig. 3.

V.CASE STUDY

In this paper, the standard IEEE 13 Bus Radial

Distribution Test Feeder is used to evaluate the

performance of the proposed load flow algorithm. The

standard IEEE 13 Bus Radial Distribution Test Feeder,

which is shown in Fig. 4, was developed by the IEEE

Distribution System Analysis Subcommittee to have a

common set of valid data to verify solutions of developed

programs [13]. In this system, there is one substation

voltage regulator consisting of three single-phase units

connected in wye fashion. Voltage regulators are assumedto be step-type and can be connected in the substation

and/or to a specified line segment. In this study the effect

of voltage regulator is not considered. The per unit voltage

magnitudes and phase angles of the buses are shown in

Table I. Dashes (-) indicate that a phase is not present at

that bus. The percentage error is calculated using (24),

where the calculated values are the results of the proposed

load flow algorithm and the actual values are from [13]. As

it can be seen in Table I, maximum error percentage is

about 1%. Therefore, the proposed load flow algorithm

gives accurate results and is a valid and efficient three-

phase load flow for unbalanced distribution systems.

100valueactual

valuecalcualted-valueactualerrorpercentage = (24)

Node Actual Voltage Values Calculated Voltage Values

Bus Mag.(pu)Phase

Angle(Degree)Mag.(pu)

PhaseAngle(Degree)

Percentage

Error

a 1.0000 0.00 1.0000 0.00 0

b 1.0000 -120 1.0000 -120 0650

c 1.0000 120 1.0000 120 0

a 1.0625 0.00 1.0625 0.00 0

b 1.0500 -120.00 1.0500 -120.00 0RG60c 1.0687 120.00 1.0687 120.00 0

a 1.0210 -2.49 1.0290 -2.50 0.78

b 1.0420 -121.72 1.0480 -121.9 0.56632

c 1.0174 117.83 1.0190 118.5 0.15

a 1.0180 -2.56 1.020 -3.0 0.2

b 1.0401 -121.77 1.045 -122.0 0.47633

c 1.0148 117.82 1.015 118.10 0.02

a 0.9940 -3.23 0.9960 -3.90 0.2

b 1.0218 -122.22 1.0290 -122.5 0.70634

c 0.9960 117.34 0.9980 118 0.2

a ----

b 1.0329 -121.90 1.039 -121.00 0.6645

c 1.0155 117.86 1.014 118.00 0.15

a ----

b 1.0311 -121.98 1.035 -122.00 0.38646c 1.0134 117.90 1.013 117.5 0.03

a 0.9900 -5.30 0.9950 -5.8 0.50

b 1.0529 -122.34 1.045 -122.00 0.75671

c 0.9778 116.02 0.9765 116.50 0.13

a 0.9900 -5.30 0.985 -5.00 0.50

b 1.0529 -122.34 1.043 -122.00 0.94680

c 0.9778 116.02 0.9740 115 0.39

a 0.9881 -5.32 0.9822 -5.2 0.60

b -----684

c 0.9758 115.92 0.9720 114.5 0.39

a ----

b ----611

c 0.9738 115.78 0.9700 114.9 0.39

a 0.9825 -5.25 0.9800 -5.3 0.25

b ----652c -----

a 0.9900 -5.31 0.9950 -5.8 0.50

b 1.0529 -122.34 1.045 -122.00 0.75692

c 0.9777 116.02 0.9765 116.50 0.13

a 0.9835 -5.56 0.9800 -5.2 0.035

b 1.0553 -122.34 1.046 -122.2 0.88675

c 0.9758 116.03 0.9700 115 0.60

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HATAMI AND PARSA MOGHADDAM: THREE-PHASE FAST DECOUPLED LOAD FLOW FOR UNBALANCED 35

Fig. 4. IEEE 13 Bus radial distribution test feeder.

VI. CONCLUSION

A new, three-phase load flow for unbalanced distribution

systems has been presented in this paper. The load flowproblem is considered as an optimization problem and is

decoupled into two subproblems without using any

assumptions. The method is capable of modeling different

types of loads. Also, the algorithm is simple and uses a

constant Jacobian matrix. Therefore, the proposed method

is a fast, robust and efficient method. The load flow

algorithm was tested on the standard IEEE 13 Bus Radial

Distribution Test Feeder. The results confirm the validity

of the method.

REFERENCES

[1]

D. Das, P. Kothari, and A. Kalam, "Simple and efficient method forload flow solution of radial distribution systems," Electrical Powerand Energy Systems, vol. 17, no. 5, pp. 335-346, Oct. 1995.

[2] P. Aravindhababu, "A new fast decoupled power flow method for

distribution systems," Electric Power Components and Systems,vol. 31, no. 9, pp. 869-878, Sept. 2003.

[3] R. D. Zimmerman and H. D. Chiang, "Fast decoupled power flow for

unbalanced radial distribution systems," IEEE-PES Winter Meeting,paper no. 95, New York, 1995.

[4] S. K. Goswami and S. K. Basu, "Direct solution of distributionsystems,"IEE Proc., pt. C, vol. 188, no. 1, pp. 78-88, 1999.

[5] D. Thukaram, H. M. Wijekoon Banda, and J. Jerome, "A Robustthree phase power flow algorithm for radial distribution systems,"

Electric Power System Research, vol. 50, no. 3, pp. 227-236,Jun. 1999.

[6] W. M. Lin, Y. S. Su, H. C. Chin, and J. H. Teng, "Three-Phase

unbalanced distribution power flow solutions with minimum datapreparation," IEEE Trans. on Power Systems, vol. 14, no. 3,

pp. 1178-1183, Aug. 1999.

[7]

P. A. N. Garcia, J. L. R. Pereira, S. Carnerio, V. M. da Costa, andN. Martins, "Three-Phase power flow calculations using the current

injection method," IEEE Trans. on Power Systems ,vol. 15, no. 2,pp. 508-514, May 2000.

[8] W. M. Lin and J. H. Teng, "Three-Phase distribution networks fastdecoupled power flow solutions," Electric Power and Energy

Systems, vol. 22, no. 5, pp. 375-380, Jun. 2000.

[9] J. H. Teng, "A Modified gauss-seidel algorithm of threephasepower flow analysis in distribution network," Electrical Power andEnergy Systems, vol. 24, no. 2, pp. 97-102, Feb. 2002.

[10] T. H. Chen, M. S. Chen, K. J. Hwang, P. Kotas, and E. A. Chebli,"Distribution system power flow analysis-a rigid approach," IEEETrans. on Power Delivery, vol. 6, no. 3, pp. 1146-1153, Jul. 1991.

[11] R. Ranjan, B. Venkatesh, A. Chaturvedi, and D. Das, "Power flow

solution of three-phase unbalanced radial distribution network,"Electric Power Components and Systems, vol. 32, no. 4, pp. 421-433, Apr. 2004.

[12] W. H. Kersting, Distribution System Modeling and Analysis, CRCPress, 2002.

[13] IEEE Distribution System Analysis Subcommittee, "Radialdistribution test feeders," IEEE Trans. on Power Systems, vol. 6,no. 3, pp. 975-985, Aug. 1991.

A. R. Hatami received the B.Sc. degree in electrical engineering fromAmirkabir University of Technology, Tehran, Iran, in 1995, and M.Sc.from Tarbiat Modares University, Tehran, Iran, in 1998.

From 2000 to 2002 he was with Chabahar Maritime University,Chabahar, Iran, and since 2002, he has been with Bu Ali Sina University,

Hamedan, Iran, as a lecturer. He is currently a Ph.D. student in TarbiatModares University. His research interests include power distributionsystems, reliability and electricity market.

M. Parsa Moghaddamrecived his B.Sc. degree in Electrical Engineeringfrom Sharif University of Technology, Tehran, Iran, in 1980, M.Sc. degreefrom Toyohashi University of Technology, Japan, in 1985; Ph.D. degree

from Tohoku University, Japan, in 1988.Currently, he is an Assosiate Professor in the Department of Electrical

Engineering, Tarbiat Modares University, Tehran, Iran. His researchinterests include power system planning and control, optimization, andrestructuring.

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