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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: [email protected],
[email protected]).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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