Load flow method for radial and weakly-meshed networks: Concept of duality

Abstract

Load flow method suitable for both radial and weakly-meshed network structures common to distribution systems is proposed in this paper. Here the concept duality of network has been exploited to obtain the solution. Unique data feeding technique and creation of single matrix [branch-voltage drop to node-current] based on dual nature of the electrical network are used in the procedural technique. A single informative forward path impedance [FPI] matrix is developed in the proposed method and it is found to be adequate for the analysis. The method first arranges constant input data and then proceeds further by collecting the information about forward path-impedances in matrix form. Then identify the dual of forward path-impedance matrix, which gives backward nodal-current injection information about the network. The solution is derived using forward path impedance and branch currents. This methodology is programmed for an IEEE 15-Node network using MATLAB Ver. 7.0. and it is demostrated that computational efficiency has been improved by the algorithm suggested. This paper also includes equations for two real-world issues like, when lines are modeled as Pi section and for network having tap changing transformer. The proposed method is flexible to extend for 3-phase networks also.

Full text

Load Flow Method for Radial and Weakly-
Meshed Networks: Concept of Duality
Dharmasa, C. Radhakrishna, and H.S. Jain, Senior Member, IEEE, P. Praveen Reddy, Student member,IEEE
Abstract__ Load flow method suitable for both radial
and weakly-meshed network structures common to
distribution systems is proposed in this paper. Here the
concept duality of network has been exploited to obtain
the solution. Unique data feeding technique and creation
of single matrix [branch-voltage drop to node-current]
based on dual nature of the electrical network are used in
the procedural technique. A single informative Forward
Path Impedance [FPI] matrix is developed in the
proposed method and it is found to be adequate for the
analysis. The method first arranges constant input data
and then proceeds further by collecting the information
about forward path–impedances in matrix form. Then
identify the dual of forward path-impedance matrix,
which gives backward nodal-current injection
information about the network. The solution is derived
using forward path impedance and branch currents. This
methodology is programmed for an IEEE 15-Node
network using MATLAB Ver. 7.0. and it is demostrated
that computational efficiency has been improved by the
algorithm suggested. This paper also includes equations
for two real-world issues like, when lines are modeled as
π
section and for network having tap changing
transformer. The proposed method is flexible to extend
for 3-phase networks also.
Index Terms__ Duality, information matrix, radial,
transpose, weakly-meshed, network.
Nomenclature
n = No. of nodes; n-1 = No. of branches
Pi = Path number; q = No. of iterations
[ZD] =Impedance matrix
[Si] =Power specified at ith node
[Ii] =Load current; [B] = Branch current
[V]ref =Reference voltage; [Vi]cal = Unknown voltages
1
I. INTRODUCTION
OAD FLOW is an important and fundamental tool
for the analysis of any power system in operation
as well as planning stages, particularly in modern
distribution system (MDS) and optimization of power
system. A power flow method must be robust and time
efficient to tackle the special features of distribution
system, such as: high R/X ratios of the line data; radial
or weakly meshed network structure; unbalanced
The financial support from SNIST, JNTU-AICTE, INDIA, Active
Career Advice etc., is gratefully acknowledged.
Dharmasa is with the SNIST-JNT U’ty, Ghatkesar, A.P, PIN-
501301, INDIA. (phone: +91-9490469903; fax:91-4027640394;
e-mail: rdharmasa@ yahoo.com,rdhar masa@hotmail.com).
C. Radhakrishna is with the Global Energy Consulting Engineers,
A.P, Hyderaba d, INDIA.(e-ail:radhakrishna.chebiyam@gmail.com).
H.S. Jain is with BHEL Corp. R&D, Hyderaba d, INDIA. (e-
mail:jain@bhelrnd.co.in).
P.Praveen Reddy is with the SNIST-JNT U’ty, A.P, PIN-501301,
INDIA. ( e-mail: praveenparava@yahoo.co.in).
distributed load; large number of nodes etc.
Unfortunately, the assumptions necessary for
simplification of the standard method [1] are often not
valid for the distribution systems. Various methods are
available to carry out power flow analysis in distribution
system. The first category consists of different versions of
Newton-Raphson method as in [2-3], other methods, like
ladder network theory and backward-forward methods
[4-7], which need improvement. The fast decoupled
power flow method submitted in [8] requires special
process of ordering the laterals input data as reported in
[9].The algorithms [10-11] are based on compensation
techniques, where the accuracy is controlled by the
tolerances of voltage mismatch between the nodes of a
break point. Also in these methods it is also observed
that, there is no direct mathematical relationship between
the system status and control variable, which makes
compensation based algorithms difficult. The algorithm
as in [12] for both radial and weakly meshed networks
requires two special matrices to obtain power flow
solution based on partial topological information. Also
this method requires KVL as well as KCL to obtain
solution for distribution networks.
It is well known that the information available in matrix
form can be easily ordered, accessed, modified and
transpose. The method both for radial and weakly meshed
networks is analysed based on the collection of graphical
information and application of concept of duality to the
distrbution network. The proposed algorithm develops a
single matrix i.e Forward Path Impedance [FPI] matrix
and then the same matrix is transposed mathematically to
obtain unknown load current injections at every node of a
given network. It is observed that load current injections
are duals of voltage drops in the network. This knowledge
is applied to find power flow solution even if the network
line is modelled as
π
sections or having tap changing
transformer in it.
II. PROPOSED METHOD FOR RADIAL NETWORK
Current injection model/technique [13-15] is
commonly used for distribution networks, where load
drawn at a node is referred as receiving end. Consider the
broad idea of constant input data. Fig. 1 specifies that the
current at ith node is Ii and it is expressed in matrix form
as the conjugate of ratio of power to voltage
[][]
()
)1(/.][ ∗
=ViSiIi
where, Si is the specified power injection at node i and
Vi is calculated voltage at ith node.
L
978-1-4244-3811-2/09/$25.00 ©2009 IEEE

In MATLAB the operator ‘./’ signifies element by
element division of variables in matrix form.
Usually voltage at reference node is given, where
as the unknown nodal voltages are calculated using
impedance and branch-path current. For that identify
topological information about the line data of a network
having dimension of (n-1)×1 and express as [ZD]. Then
develop key [FPI] matrix as explained.
A. Arrangement of line data
The branch impedances Zij and load currents Ii of a
typical distribution network are as shown in Fig.1.
Fig 1. A typical distribution network.
For the input information of a network as shown in
Fig. 1, the receiving end node numbers of a line are to
be tabulated in an ascending sequence, as in Table I.
TABLE I
TABULA TION OF INPUT DATA
Line Data Nodal
Power
Sending
Node
Receiving
Node
Branch
Impedance
1 2 Z12 S
2
2 3 Z23 S
3
3 i Z3,i S
i
r i+1 Z
r,i+1 S
i+1
k i+2 Z
k,i+2 S
i+2
.. .. .. ..
.. .. .. ..
j n Z
j,n S
n
Where number are assigned as i, i+1,i+2… receiving
end nodes and r, k, j are different to sending end nodes.
B. Formation of [ZD]
The impedance data [ZD] collected are from Table I
is of the order (n-1)×1.
[]
=
+
+
nj
2iK,
1ir,
i3,
23
12
Z
Z
Z
Z
Z
Z
ZD
,
..
..
Then, convert each element position of [ZD] from the
order (n-1) ×1 to diagonal (n-1) × (n-1) by filling zeros in
the off diagonal positions as shown below
[]
)2(=+
+
n,
2ik,
1ir,
i3,
23
12
Zj0000000
0..............
0..............
0....Z0000
0....0Z000
0....00Z00
0....000Z0
0....0000Z
ZD
C. Development of [FPI]
The paths Pi are marked for IEEE 15-Node network
data [16] as shown in Fig.2. The nodal voltages are to
be calculated from reference node to downstream nodes
as.
Fig.2. Represe ntation of gra phica l information of branch path
impedances for a radial distribution network.
Usually the voltage V1 is specified at reference node.
Consider the 2nd path having branch path -impedance
information Z12 in between node 1 and node 2.In the same
way proceeds further to collect path information about
line section i.e. 1-2-2-3-3-4 is having three branches Z12,
Z23 and Z34 between four nodes and tabulate this
information as shown in Table II.
Likewise, voltages at nodes 5, 6, ..…n information can
easily be collecteed, if all the branch currents are known.
Then, the generalized equation to compute node voltage

at the end of path Pi can be expressed using KVL as:
(Pi) node-path along drops
2,...niVoltageVVi =
−= 1
(3)xi
nodes athpP(i)
1x2,i
1ZBVV xii
=
==
−=
where, x= sending end nodes of a network.
The branch currents of (3) can be expressed in matrix
form in an ascending order by seeing their receiving
end node numbers only
[]
)4(
,
+
+
=
j,n
2k,i
1r,i
i3
23
12
B
..
..
B
B
B
B
B
B
Here also, i, i+1, i=2…receiving end nodes and r, k, j
are different numbers assigned to sending end nodes of
a network.
TABLE II
UNKNOWN VOLTAGE NODE AT THE END OF PATH, NUMBER OF NODES
AND BRANCHES IN PATH OF FIG.1
Branc
h Path
Pi
Total No. of Nodes in
the Path
Forward Branch-Path
Impedances ( Total No. of
Branches in the Path)
P2 1,2 (2) Z
12
(1)
P3 1,2,3 (3) Z12,Z23
(2)
P4 1,2,3,4 (4) Z12,Z23,Z34
(3)
P5 1,2,3,4,5 (5) Z
12,Z23,Z34,Z45
(4)
P6 1,2,3,4,5,6 (6) Z
12,Z23,Z34,Z45,Z56
(5)
P
7
1,2,3,4,5,6,7 (7) Z
12,Z23,Z34,Z45,Z56,Z67
(6)
P8 1,2,3,4,5,6,7,8 (8) Z
12,Z23,Z34,Z45,Z56,Z67, Z78
(7)
P9 1,2,3,4,5,6,7,8,9 (9) Z
12,Z23,Z34,Z45,Z56,Z67,Z78,Z89
(8)
P10 1,2,3,4,10 (5) Z
12,Z23,Z34,Z410
(4)
P11 1,2,3,4,10,11 (6) Z12,Z23,Z34,Z410,Z1011
(5)
P12 1,2,3,12 (4) Z
12,Z23,Z312
(3)
P13 1,2,3,12,13 (5) Z
12,Z23,Z312,Z1213
(4)
P14 1,2, 3,12,13,14 (6) Z
12,Z23,Z312,Z1213,Z1314
(5)
P15 1,2,3,12,13,14,15 (7) Z
12,Z23,Z312,Z1213,Z1415 (6)
Thus, it is seen that (3) can be formed as we traverse
the paths to know voltage at end of each path. The path
directed information about the nodes and branches are
to be filled in column 2 and 3 of Table II.
D. Procedure to build [FPI]
Obtain the first path branch impedance (Z12) from
Table II and mention as +1 at 1st row 1st column
position of [FPI]. Similarly place +1’s in row as per
branch information or voltage drop information along
path. The number of 1’s in the row of [FPI] is equal to
the total number of branches in that path sequence.
Then fill remaining position with zeros. As an example
[FPI] for IEEE 15-Node network data [16] can be
written as:
The suggested steps are as follows:
1. Create [FPI] null matrix of order n ×n.
2. Any path Pi, if Zi,i+1 is available in that path, then
place +1 in ith row and (i+1)th column position.
3. Nodes are numbered in an ascending order, with 1 as
incremental value.
4. Set +1 in the diagonal position of [FPI]n×n.
5. In [FPI] fill rowwise positions with 1’s as per the
nodal connectivity information available along the
path 1,..i,..(i+1)th, (i+2)th .…n. For that add ith row
node information to (i+1)th row position. In the same
way add rowwise for the subsequent positions.
6. The dimension of the [FPI] is reduced to (n-1) × (n-
1) by removing the first row and first column of
matrix, which is suitable to compute (n-1) unknown
nodal voltages.
[]
)5(=
11110000000011
01110000000011
00110000000011
00010000000011
00001000000011
00000100000011
00000011111111
00
000001111111
00000000111111
00000000011111
00000000001111
00000000000111
00000000000011
00000000000001
P
P
P
P
P
P
P
P
P
P
P
P
P
P
FPI
15
14
13
12
11
10
9
8
7
6
5
4
3
2
E. Development of backward node-current information
matrix [FPI]T
The [FPI]T is obtained by transpose of [FPI], which is
directly gives the backward node-current information
matrix.
To develop [ZD], [B], [V]ref and [Vi]cal for IEEE 15-
Node network are use expression (2-4).
Graphical information based voltage drops along the
various paths of network are expressed as:
[V] ref-[Vi]cal=[set of voltage drops]
Thus, voltage drop or voltage mismatch between the
reference and unknown nodes in each path of a network is
the product of [FPI], [ZD] and [B] as
[Voltage drops]=[FPI][ZD][B] (6)
[V] ref- [V]cal=[Voltage mismatch] (7)
[V]= [V] ref- [Vi] cal (8)
The simple transpose of [FPI] from (5) leads to [FPI]T.
This matrix contains straight away the backward
information about the load connected at each node of a
network. This can relate to the concept of duality used for

network. Two networks are said to be dual of each
other when the mesh equations of one are same as the
node equation of the other [17]. The two laws of
Kirchhoff (KVL and KCL) were almost same, word for
word, with voltage substituted for current, independent
loop for independent node pair. There is, hence no
need to apply KCL to obtain backward network
information, because the [FPI]T and [FPI] are dual in
nature. The branch current matrix ([B]) of a network is
equal to the product of backward load current
information matrix ([FPI]T) and load current matrix
([Ii]).
[B] = [FPI]T [Ii] (9a)
Case a. Radial network having lines section in
π
form: If the line sections are modeled in the
π
form
then equation (9a) can be expressed for the 4 node
radial part of the network as shown in fig 3.
Fig.3 A four node network sa mple represents conversion of radial
to weakly meshed network with the addition of link Bl.
)9b(
100
010
111
100
010
111
2
2
122
2
2
12
+=
nc
ic
c
n
i
n
i
I
I
I
I
I
I
B
B
B
)9(
100
010
111
100
010
111
2
2
2122
2
2
12
c
VY
VY
VY
I
I
I
B
B
B
nnc
iic
c
n
i
n
i+=
It is observed from equation (9c) that one more
constant [FPI] matrix to be added the line charging
currents as shown above.
Case b. Tap changing transformer in radial network:
Consider a line admittance with Yt in series with an
ideal transformer representing the off-nominal tap ratio
1: a as shown in fig.4.
In equation (1) load currents at node numbers 2 and n
are modified as follows:
At node 2 I20 current acts as constant impedance load
from reference [8].
()
20222 /. IVSI t+= ∗
()
∗−
+= a
a
YVVSI tt
1
/. 2222
At node ‘n’ in the same way I20 acts as constant
impedance load from reference [8]
()
0/. nnnnt IVSI += ∗
()
)d9(
2
1
/. ∗−
+= a
a
YVVSI tnnnnt
The [ZD] of equation (2) is modified as follows
=
n
i
Z
Z
Z
ZD
2
2
12
00
00
00
][
Because of transformer in between nodes 2 and n, the
value of branch impedance changes from Z2n to ×aYt
1.
For that equation (2) can be modified as
)9(
1
00
00
00
][ 2
12
e
aYt
Z
Z
ZD it
×
=
It is found that the equation (9a) is modified in the
column vector of the currents using new relations for
currents I2t and Int as expressed above
)9(
100
010
111 2
2
2
12
f
I
I
I
B
B
B
nt
i
t
n
i=
Fig. 4 A four node sample represents radial network with off nominal
tap changing transformer.

To obtain voltage mismatch matrix combine from (6) to
(9a-9d) as:
[V] = [FPI] [ZD] [FPI]T [Ii]
[Vq+1] = [LF] [Ii
q] (10)
where [LF] is Single load flow matrix
[Vq+1]= [Vq] + [Vq+1] (11)
The power flow solution for radial network can be
obtained by solving (1), (10), and (11) iteratively.
It is found that [FPI]T is the transpose of [FPI] for a
distribution network. The product of [FPI] and [ZD]
represents the relationship between forward branch
path- impedance matrix, where as [FPI]T and [Ii]
represent the relationship between backward node-
current information and load current respectively.
Hence, with the dual of backward load current
information matrix to forward path impedance
information, it is easy to obtain power flow solution. It
is inferred that more generally two points are sufficient
to obtain power flow solution for radial network by
having constant input data as follows:
• Network topology based collection of information
about the impedance along the branch-path and load
at each node.
• Dual of forward path impedance information to know
the backward load current information.
F. General procedure for radial network
The suggested general procedure for radial networks
is summarized in the following steps:
1. Assume voltages at all nodes as1 p.u.
2. Arrange the input data information of [ZD] and [Si]
of order (n-1) ×1 in an ascending order.
3. Convert branch impedance elements positions in
[ZD], place them in the order of (n-1) ×1 to (n-1)
×(n-1).
4. Build [FPI](n-1) × (n-1) and transpose the same to obtain
backward node-currents [FPI]T (n-1) × (n-1).
5. Use [Ii]=([Si]./[Vi])* to compute load currents, then
calculate first iterative voltage using [Vi]cal=[V]ref-
[FPI][ZD][FPI]T [Ii].
6. Update the nodal voltage iteratively.
III. EXTENDED PROCEDURE FOR WEAKLY MESHED
NETWORKS
In some distribution network loops are created by
closing normally open-tie-switches. In such cases,
formation of loops in the network does not affect the
nodal current injections. Power flow solution based on
transposed nature of the topological network can thus
be extended to deal with such weakly meshed
networks.
If a new branch Zl makes a loop in between any two
nodes of a radial structure, the network is modified as
shown in Fig. 3.
Fig.5. Representation of a loop in a radial distribution network.
This addition is represented by one row and one
column in [ZD] and in [FPI] discussed in Part II. The
modifications in proposed matrices are as follows:
A. Development of impedance matrix [ZD]
If a new branch Zl, is linked between ith and jth nodes,
simply substitute value of that new branch in the new
position of lth row and lth column of [ZD] as
[]
→
↓
=
l
weak
Z.0lrow0
.
.
0
l
column
0
ZD
ZD
0
0
.
B. [FPI] for weakly meshed networks
This matrix is modified in two steps:
1. The addition of branch Bl in Fig. 3 acts as a link or
co-tree in the tree structure of the network. Now the
loop starts flowing in the closed path i.e, 3-4-5-6-i-j-
13-12-3. The currents flowing information appear as
positive (+1) in [FPI] for the path-section 3-4, 4-5,
5-6, 6-i, due to the direction of loop current is same
as that of path-section, whereas in this path-section j-
13,13-12,12-3 it is negative (-1).
2. In [FPI] take the values of ith row at the lth row as
temporary position, then subtract the binary values of
ith row from jth row and substitute these informative
values in the rth row. Finally for one loop, fill a +1
value in the position of lth row and lth column as:

[]
−−−
=
101110000111100
001110000000011
000000000111111
011110000000011
000110000000011
000010000000011
0000011
00000111
000000100000111
000000011111111
000000001111111
000000000011111
000000000001111
000000000000111
000
000000000011
000000000000001
B
P
P
P
P
P
P
P
P
P
P
P
P
P
P
FPI
l
weak
j
i
15
13
12
11
10
9
8
6
5
4
3
2
If one more loop is added, using the same steps, we
can express [FPIweak] as
[]
=
101-1-1-1-0000111100
0101-1-1-0000111100
00
00
00
00
00
00
00
00
00
00
00
00
00
00
11110000000011
01110000000011
00110
000000011
00010000000011
00001100000111
00000100000111
00000011111111
00000001111111
00000000111111
00000000
011111
00000000001111
00000000000111
00000000000011
00000000000001
B
B
P
P
P
P
P
P
P
P
P
P
P
P
P
P
FPI
ll
l
n
j
13
12
11
10
9
8
i
6
5
4
3
2
weak
C. Matrix [B ]for weakly-meshed network
In matrix [B] of radial network place the new branch
current Bl in the row lth i.e after (n+1)th element
position to obtain weakly meshed branch current matrix
as:
[]
+
+
=
l
nj,
2ik,
1ir,
3i
23
12
weak
B
B
..
..
B
B
B
B
B
B
D. Calculation of mismatch voltage
Then voltage mismatch matrix is equal to
[Voltage mismatch] = [FPIweak] [ZDwe ak] [Bwea k] (12)
To call for [Bweak] again transpose the [FPIwea k],which
is [FPIweak]T. As in the earlier case, [Bweak ] is obtained as
a product of [FPIweak]T and [Ii-weak ] as
[Bwea k] for one loop:
[]
−
−
−
=
ll B
I
I
I
I
I
I
I
I
I
I
I
I
I
I
10000000000000
11000000000000
11100000000000
11110000000000
00001000000000
00001100000000
00000010
000000
00000011000000
00000011100000
00000011110000
00000011111000
00001111111100
11111111111110
11111111111
111
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
Bweak
15
14
13
12
11
10
9
8
7
6
5
4
3
2
15
13
12
11
10
9
8
6
5
4
3
2
14
7
100000000000000
0
1
1
1
0
0
0
0
1
1
1
1
0
0
[Bweak] for two loops:
[]
=
ll
l
15
14
13
12
11
10
9
8
7
6
5
4
3
2
ll
l
15
14
13
12
11
10
9
8
7
6
5
4
3
2
weak
B
B
I
I
I
I
I
I
I
I
I
I
I
I
I
I
10000000000000
11000000000000
11100000000000
11110000000000
00001000000000
00001100000000
0000
0010000000
00000011000000
00000011100000
00000011110000
00000011111000
00001111111100
11111111111110
1111111
1111111
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
1000000000000000
0100000000000000
1-0
1-1-
1-1-
1-1-
00
00
00
00
11
11
11
11
00
00
Case a. Analysis of
π
section of the line extended to
weakly meshed network. If the line section is modeled as
π
form then [Bweak]can be expressed for the 4 node radial
part of the network as shown in fig 3.
In this analysis a separate [FPIwea k] matrix is developed
for line charging currents along with the load currents.
−
+
−
=
cl
nc
ic
c
l
n
i
2n
2i
12
I
I
I
I
1000
1100
1110
0111
B
I
I
I
1000
1100
1010
0111
B
B
B
B
2
2
122
Where Icl=link line charging current
Case b. Tap changing transformer in weakly meshed
network: Use previously analyzed case. b equations for
radial network to update for weakly meshed networks.
If a tap changing transformer is added in a line having Yt
admittance, which creates a link. The branch currents [B]
changes to [Bweak].

The weakly meshed network with off nominal tap
changing transformer connected in between nodes i
and n of a line is expressed below using fig. 6
Fig. 6. Weakly meshed network with off nominal tap changing
transformer connected in between nodes i and n of a line.
tiiit Y
a
a
VII −
+= 1; tiint Y
a
a
VII −
+= 2
1
[Bweak]= −
=
l
nt
it
2n
2i
12
B
I
I
I
1000
1100
1010
0111
B
B
B
B2
The mismatch voltage matrix using (12) and [Bwe ak]
is constructed as:
[Vweak]=[FPIwe ak] [ZDwe ak] [FPIweak]T [Ii-weak] (13)
)14(=− ++
lB
I
I
.
.
I
I
I
I
I
SLF
0
V
V
.
.
V
V
V
V
V
0
V
V
.
.
V
V
V
V
V
15
14
1i
i
4
3
2
15
14
1i
i
4
3
2
1
1
1
1
1
1
1
Where, [SLF] is the constant simplified load flow matrix
[SLF] =[FPIwea k] [ZDweak ][FPIwe ak]T (15)
Here,[SLF] matrix is re-arranged in sub matrices form
[]
=
Sizes of the submatrices of [SLF] are
[SLF](n-1+l)×(n-1+l) ; [K](n-1)×(n-1); [LT]l×(n-1) ; [L](n-1) ×l
[M]l×l, Where l =the number of loops
As the addition of loops increases one by one, then row
and column dimensions in these matrices [ZDweak],
[FPIweak ] and [SLF] also increases proportionately.
E. Modification in the solution technique
It is necessary to re-arrange the network matrices, due
to addition of loops, though the nodes remain constant.
Afterwards apply Kron’s reduction in (15) to compute
voltages.
The general form of mismatch voltage before Kron’s
reduction technique is
[]
)16(=
ewn
radial-iradial
B
I
SLF
0
V
The mismatch voltage after Kron’s reduction technique
[Vweak]= [LF] (n-1) ×(n-1) [Ii-weak] (17)
It is observed that the proposed algorithm can also be
used to solve weakly meshed networks using single load
flow matrix LF.
IV. RESULTS AND DISCUSSIONS
The proposed power flow algorithm is implemented
using MATLAB Ver 7.0 and tested on a Windowsxp;
Pentium-IV; 2.8GHz; 248 MB of RAM system. Two
methods (proposed method and a direct approach
method) have been compared and the convergence
tolerance set at 0.001p.u. The methods are compared on
the basis of their performance both for radial and weakly
meshed network by modifying the data of [16]. The
results are presented in Table (III-VII) and The
performance comparison of methods can be obtained
plotting the graph between time in seconds verses in Fig.
7.
In the methods [4-7 and 12] use of three step approach
is made as follows:
Step 1) Calculation of nodal current using (1).
Step 2) Backward sweep to compute branch current
which is equivalent to [FPI]T×[Ii] of the
proposed method.
Step 3)Forward sweep to find the unknown voltage which
is equivalent to [FPI]×[ZD] of the proposed method.
In the proposed method, the forward ⇔backward
steps can be considered as a single step by utilizing the
concept of duality as network information. The proposed

method is also convenient to on-line applications.
TABLE III
ADVANTAGES OF THE PROPOSED METHOD
Sl.
No.
Radia l an d Wea kly- meshed Networ k
1 The sorted input (line data and bus data) data gives
sorted output (power flow solution) solution for easy
operations.
2 Algorithm uses single matrix Forward branch-path
impedance [FPI]
3 The lower triangular [FPI] matrix is the dual of upper
triangular [FPI]T matrix.
4 Both for radial and weakly meshed networks the
complexity of programming is reduced to 50% and
hence algorithm takes less net evaluation time to
converge the solution.
5 Unique data feeding technique and simple concepts
from graph theory leads to obtain more direct power
flow solution.
6 Once gr aphically pat h based KVL information are
calculated, then use of KCL is not necessary.
7 According to Graph theory of [17], easily we can
extend the radial network analysis to weakly meshed
network.
8 Equations are made available in matrix form for line
sections in
π
from.
9 Equations are also made available in matrix form for
the inclusion of tap changing transformer in the line.
TABLE IV
PERFORMANCE RESULTS FOR RADIAL NETWORK DATA
Methods/
Performance
Accuracy Iterations Memory
Proposed 0.001 3 3.07KB
Method [12] 0.001 3 3.70KB
TABLE V
PERFORMANCE RESULTS FOR WEAKLY NETWORK
Methods/
Performance
Accuracy Iterations Memory
Proposed 0.001 3 4.19KB
Method [12] 0.001 3 5.41KB
As the number of nodes in the network increases, the
dimension of both proposed method [FPI] and matrices
[BIBC], [BCBV] developed in [12] also increases.
The comparisons of computation time for various
dimensions of the matrix are mentioned in Table VI. To
find the computation time in proposed algorithm,
which is relative to the net execution time is divided
into:
1) Time taken to execute [FPI] program =ET.
2) Time to arrange [ZD] =TA.
3) Time to transpose [FPI] =TT.
4) Time to matrix multiplication=MT.
It is observed that in MATLAB software package
computation time taken by TA, TT, and MT are nearly
the same. Three matrix multiplication operations are
needed in proposed equation (10) against two matrices
required in equation (11) of method [12]. Although the
proposed method requires one extra multiplication
operation, the results in Table VI and Fig. 7 confirm
reduction in computational time. The saving in
computational time is expected to increase for real time
situations, where complexity is unlimited.
TABLE VI
COMPARISON OF RELATIVE NET EVALUATION TIME BETWEEN
PROPOSED AND METHOD [12]
Dimension
of [FPI]
Proposed Method Method
[12]
ET for
[FPI]
TA+TT
+MT
NET=
ET+TA+
TT+MT
NET=2*
ET
100×100 8.8906 0.0468 8.9374 17.7812
200×200 8.8906 0.1407 9.0313 17.7812
300×300 8.8906 0.3750 9.2656 17.7812
400×400 9.3594 0.7032 10.0626 18.7188
500×500 10.1406 0.8439 10.9845 20.2812
600×600 11.8438 1.5000 13.3438 23.6876
700×700 14.4063 2.5782 16.9845 28.8126
800×800 18.0154 3.7968 21.8122 36.0307
900×900 22.9063 5.0157 27.9220 45.8126
1000×1000 29.7656 6.6564 36.4220 59.5312
Fig.7. Relative net execution time verses number of nodes.
The conventional power flow algorithms are likely to
diverge, due to the complexity of the network. The
proposed algorithm robustness is validated by changing
range of resistance to reactance for a wide range of values
and the final results are presented in Table VII.
TABLE VII
CONVERSED POWER FLOW SOLUTION
Magnitude
of Voltage
Radial
Network
Weakly Meshed Network
No. of
Loop=1
No. of
Loops=2
V2 0.9750 0.9751 0.9753
V3 0.9743 0.9744 0.9746
V4 0.9624 0.9621 0.9643
V5 0.9601 0.9595 0.9640
V6 0.9577 0.9568 0.9639
V7 0.9560 0.9546 0.9643
V8 0.9553 0.9534 0.9655
V9 0.9552 0.9531 0.9662
V10 0.9574 0.9571 0.9593
V11 0.9456 0.9454 0.9476
V12 0.9723 0.9738 0.9662
V13 0.9710 0.9725 0.9648
V14 0.9694 0.9709 0.9633

V15 0.9692 0.9707 0.9630
V. CONCLUSIONS
An improved power flow solution approach suitable
for both radial and weakly-meshed electrical
distribution networks has been presented in this paper.
Topological structural information of network matrices
has been successfully exploited to achieve simpler
solutions. Basically, the forward-backward steps of
conventional procedures are merged into one step. The
method successfully applied the concept of duality for
the lines in
π
section form and also accommodates tap
changing transformer. The methodology proposed here
has been validated through a sample case study. The
solution approach proposed in this paper is found to be
faster, flexible and efficient. The algorithm can be
easily extended to 3-phase systems.
Acknowledgements
Authors express sincere thanks to Dr. P.S.R.Murthy,
Director, School of Electrical Engg. SNIST, India and
Prof.K.Pasupathi, former Dean (Engg), EIT, Eritrea,
Presently in EEE Dept. SNIST, Dr.S.V.Kulkarni,IIT,
Mumbai for their useful comments on the system
planning study.
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Dharmasa was born in 1971 in Karnataka, India.
He received B.E, M.E degrees in Electrical
Engg. from the U’ty Visvesvaraya College of
Engg., Bangalore, India, in 1996 and 2000
respectively. He was deputed during 2005-07 on
foreign assignment under Ministry of Education,
Eritrea.
Presently he is working as Professor in
Electrical Engg. Dept. at SNIST, India. He is
pursuing Ph.D in Electrical Engg. from JNT U’ty, A.P, India. His main
areas of interest: Electrical Distri. Syst. Planning, Automation, DSM
and Career planning.
C.Radhakrishna was born in 1944 in Andrapradesh, India. He
obtained the B.E, M.E degrees in Electrical Engg. from the National
Institute of Technology, Warangal, India in 1965 and 1967, respectively
and Ph.D in Electrical Engg. from IIT, Kanpur, India in 1981.
He is currently the Director (Technical) at Global Energy Consulting
Engineers, Hyderabad, Earlier, he served as Prof. & Head of Electrical
Engg. Dept. and Director of UGC-ASC at JNT U’ty, India a nd during
1996-98 as Dean of Studies and Director of Central Institute of Rural
Electrification Corp. under the Ministry of Power, Govt. of India. His
main areas of interest: Electrical Distr.Syst.planning, Auto.,
optimization and manage ment, load research, powe r quality, risk
management in power utilities.
H.S. Jain (M’94-SM’98) born’52, graduated in Electrical Engg.
(1973) from Jiwaji U’ty, Gwalior (MP) India. He obtained his Ph.D. on
his work on circuit breaker arcs from IIT, Bombay in 2001.
He works for Bharat Heavy Electrical Limited (BHEL) Corporate
Research & Development Division at Hyderabad (India) and specialises
in Vacuum/SF6 C.B and Gas Insulated Power Equipment. He is
currently placed as a General Manager responsible for developments in
H.V, Switchgear, Trans. & Prot. Systems and Insulation disciplines.
Dr.H.S.Jain is a senior member of IEEE.
P.Praveen Reddy is IV/IV Final year student of in Electrical and
Electronics Engg. student of SNIST, A.P, PIN-501301, INDIA. (e-mail:
praveenparava@yahoo.co.in). , IEEE student member.