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CHAPTER - 2
POWER FLOW ANALYSIS
_______________________________________________________________________________________________________
2.1 INTRODUCTION
Power flow analysis is a basic and necessary tool for any
electrical system under steady state condition to determine the exact
electrical performance. The load flow solution provide the real (kW)
and reactive power (kVAr) losses of the system and voltage magnitudes
and angles at different nodes of the system subject to the regulating
capability of generators, condensers and tap changing of transformers
under load as well as specified net interchange between individual
operating systems. This analysis is essential for the continuous
evaluation of the existing power system and effective planning of
alternatives for system expansion to meet increased load demand in
future. These analyses require the calculation of numerous load flows
for both normal and emergency operating conditions. The load flow
studies are helpful to confirm selected switchgear, transformer, and
cable sizing. These studies should also be used to confirm adequate
voltage profiles during different operating conditions, such as heavily
loaded and lightly loaded system conditions. Load flow studies can be
used to determine the optimum size and location of capacitors for
power factor correction. The results of load flow studies are also
starting points for other system studies.
The distribution power flow involves, first of all, finding all of the
node voltages. From these voltages, it is possible to directly compute
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currents, power flows, system losses and other steady state quantities.
Some applications, especially in the fields of optimization of power
system, distribution automation (i.e., VAR planning, network
optimization, state estimation, etc.) need repeated fast load flow
solutions. In these applications it is important that the load flow
problem is solved as efficiently as possible.
Das et al. [41] presented a simple method for distribution load
flow solution which solves a simple algebraic expression of voltage
magnitude. S. Ghosh and D. Das [50] proposed a method for solving
radial distribution networks which involves simple algebraic
expression. S. Mok et.al [53] proposed a new approach for power flow
analysis of balanced radial distribution systems. The convergence
characteristics and the effect of voltage dependency are analyzed. A
general load flow for meshed networks with more than one feeding
node is presented by Haque [54]. Ratio flow method based on forward
– backward for complex distribution system is presented in [65]. R.
Ranjan and D. Das proposed a simple algorithm based on circuit
theory using algebraic recursive expression to solve radial distribution
networks [66]. Load flow solution to distribution system is obtained by
using bus injection to branch current matrix and branch current to
bus voltage matrix and a simple multiplication [69]. A Load flow
technique to solve distribution networks based on sequential branch
numbering scheme by considering committed loads is presented [83].
A backward/ forward sweep load flow solution for three phase radial
distribution systems is proposed [86]. A load flow solution including
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voltage dependent load models based on forward – backward sweep
method is proposed [87]. Comparison of various distribution load flow
algorithms based on forward – backward sweep is presented in [105].
An iterative technique in which loads are simulated as impedances at
each iteration is proposed for radial distribution load flow [124].
In this chapter a backward – forward sweep method is proposed
for solving the load flow problem of a distribution system. The
proposed method is tested by taking 15, 33 and 69 node radial
distribution systems. The mathematical formulation and the
illustration of node identification of the proposed load flow method are
described in the following sections.
2.2 MATHEMATICAL FORMULATION
The load flow of a single source network can be solved
iteratively from two sets of recursive equations. The first set of
equations for calculation of the power flow through the branches
starting from the last branch and proceeding in the backward
direction towards the root node. The other set of equations are for
calculating the voltage magnitude and angle of each node starting
from the root node and proceeding in the forward direction towards
the last node. These recursive equations are derived as follows.
The fig. 2.1 shows the representation of 2 nodes in a
distribution line. Consider a branch ‘j’ is connected between the nodes
‘i’ and ‘i+1’.
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Fig.2.1. Representation of two nodes in a distribution line
The effective active ( iP ) and reactive ( iQ ) powers that of flowing
through branch ‘j’ from node ‘i' to node ‘i+1’ can be calculated
backwards from the last node and is given as,
'2 '2P +Qi+1 i+1'P = P +ri i+1 j 2Vi+1
(2.1)
'2 '2P + Qi+1 i+1'Q = Q + xi i+1 j 2Vi+1
(2.2)
where 'P = P + Pi+1 i+1 L i+1and 'Q = Q + Qi+1 i+1 Li+1
PLi+1and QLi+1
are loads that are connected at node ‘i+1’.
Pi+1 and Qi+1 are the effective real and reactive power flows from node
‘i+1’.
The voltage magnitude and angle at each node are calculated in
forward direction. Consider a voltage V δi i at node ‘i’ and V δi+1 i+1at
node ‘i+1’, then the current flowing through the branch ‘j’ having an
impedance , z = r + jxj j j connected between ‘i’ and ‘i+1’is given as,
V δ - V δi i i+1 i+1I =j r + jxj j(2.3)
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andP - j Qi iI =j V - δi i
(2.4)
On equating the equations (2.3) and (2.4), we have
V δ - V δP - jQ i i i+1 i+1i i =V - δ r j + j xi i j
(2.5)
2V - V V (δ - δ ) = (P - jQ )(r + jx )i i i+1 i+1 i i i j j (2.6)
By equating real and imaginary parts on both sides of equation (2.6),
we have
2V V c o s (δ - δ ) = V - (P r + Q x )i i+1 i+1 i i i j i j(2.7)
V V s i n (δ - δ ) = Q r - P xi i+1 i+1 i i j i j (2.8)
Squaring and adding equations (2.7) and (2.8), we get
2 2 2 2(V V ) = [V - (P r + Q x )] + [Q r - P x ]i i+1 i i j i j i j i j (2.9)
2 4 2 2 2 2 2(V V ) = V - 2V (P r + Q x )+ (r + x )(P + Q )i i+1 i i i j i j j j i i(2.10)
1/22 2(P + Q )2 2 2 i iV = V - 2(P r + Q x )+ (r + x )i+1 i i j i j j j 2Vi
(2.11)
and voltage angle, δi+1 can be derived on dividing equations (2.8) and
(2.7)
Q r - P xi j i jt a n (δ - δ ) =i+1 i 2V - (P r + Q x )i i j i j
(2.12)
Q r - P xi j i j-1δ = δ + ta ni+1 i 2V - (P r + Q x )i i j i j
(2.13)
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The magnitude and the phase angle equations can be used
recursively in a forward direction to find the voltage and angle
respectively of all nodes of radial distribution system.
The real and reactive power losses of branch ‘j’ can be
calculated as,
2 2P + Qi iPloss (j) = r j 2Vi
(2.14)
2 2P + Qi iQ loss (j) = x j 2Vi
(2.15)
The total real and reactive power loss of radial distribution system can
be calculated as,
2 2P + Qn b i iTP L = r j 2j=1 Vi
(2.16)
2 2P + Qnb i iTQL = x j 2j=1 Vi
(2.17)
Initially, a flat voltage profile is assumed at all nodes i.e., 1.0
pu. The branch powers are computed iteratively with the updated
voltages at each node. In the proposed load flow method, powers
summation is done in the backward walk and voltages are calculated
in the forward walk. The maximum difference of voltage magnitudes in
successive iterations is taken as convergence criteria and 0.0001 is
taken as tolerance value. For branch powers calculation the adjacent
nodes and branches of every node are identified with the help of node
identification algorithm which is explained in the following section.
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2 3 4 5
7
6
8
9
10
11
12
13
15
14
S/S1 2 3 4
5
6
7
9
8
10
11
12
13
14
Fig.2.2 Single line diagram of 15-node radial distribution system
2.3 ILLUSTRATION OF NODE IDENTIFICATION
Consider a 15 node radial distribution system as shown in
fig.2.2 for illustration of node identification. The formation of
various vectors (Adn, Adb, MF and MT) used in sparsity technique
for node identification is given below.
2.3.1 Algorithm for Node Identification
Following algorithm explains the methodology to identify the
nodes and branches connected to a particular node in detail,
which will help in finding the exact load feeding through that
particular node.
Step 1: Read system data
Step 2: Initialize vector MF with 1 & S=0
Step 3: Initialize the count for node i=1
Step 4: Initialize count for branch count j=1
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Step 5: if (i= = SE [j]) go to step 7 else go to step 6
Step 6: if (i= = RE[j] go to step 8 else go to step 9
Step 7: S=S+1
Adn [S] = RE [j]
Adb [S] = j
Step 8: S=S+1
Adn [S] =SE [j]
Adb [S] =j
Step 9: if (j<nb)
j=j+1 go to step 5 else go to step 10
Step 10: MT[i] =S
MF[i+1] =MT[i] +1
Step 11: if (i<=nd)
i=i+1 go to step 4 else go to step 12
Step 12: Stop
2.4 LOAD FLOW CALCULATION
The forward-backward distribution load flow method is
proposed in this thesis. Initially assume a flat voltage profile i.e.,
set the voltage equal to 1.0 pu at every node. The flow chart for
load flow is shown in fig.2.3. The backward forward load flow
algorithm is given in section 2.4.4. The adjacent branch and node
vectors for 15 node system are tabulated in table 2.1. The MF and
MT vectors are given in table 2.2.
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Table2.1. Adjacent branch & node vectors of fig.2.2
Table2.2. MF and MT vectors of fig.2.2
NodeNo.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
MF[i] 1 2 6 9 13 14 17 18 19 21 22 24 26 27 28
MT[i] 1 5 8 12 13 16 17 18 20 21 23 25 26 27 28
where,
MF[i] =Memory location from
MT[i] =Memory location to for a particular node ‘i’.
where, i=1 to nd
S. No. Node Adn Adb S. No. Node And Adb
1 1 2 1
14
6
2 7
15 7 8
16 8 9
2
2
1 1 17 7 6 8
3 3 2 18 8 6 9
4 6 7 199
2 5
5 9 5 20 10 6
6
3
2 2 21 10 9 6
7 4 3 2211
3 10
8 11 10 23 12 11
9
4
3 3 2412
11 11
10 5 4 25 13 12
11 14 13 26 13 12 12
12 15 14 27 14 4 13
13 5 4 4 28 15 4 14
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2.4.1 Backward Propagation
The updated effective power flows in each branch are
obtained in the backward propagation computation by considering
the node voltages of previous iteration. It means the voltage values
obtained in the forward path are held constant during the
backward propagation and updated power flows in each branch are
transmitted backward along the feeder using backward path. This
indicates that the backward propagation starts at the extreme end
node and proceeds towards source node. The active and reactive
power flows are calculated using equations (2.1) and (2.2).
2.4.2 Forward Propagation
The purpose of the forward propagation is to calculate the
voltages at each node starting from the feeder source node. The
feeder substation voltage is set at its actual value. During the
forward propagation the effective power in each branch is held
constant to the value obtained in backward walk. The node voltage
magnitudes are calculated using equation (2.11). The voltage angle
is calculated using equation (2.12).
2.4.3 Convergence Criterion
The voltages calculated in the previous and present iterations
are compared. In the successive iterations if the maximum mismatch
between the voltages is less than the specified tolerance i.e., 0.0001,
the solution is said to be converged. Otherwise new effective power
flows in each branch are calculated through backward walk with the
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present computed voltages and then the procedure is repeated until
the solution is converged. The algorithm for load flow solution of radial
distribution system is explained in the following section.
Fig.2.3. Flow chart for Radial Distribution Load Flow
2.4.4 Algorithm for Load Flow Calculation
Step 1: Read distribution system line and load data.
Assume initial node voltages are 1 pu and set ε = 0.0001.
Step 2: Start iteration count, IT =1.
Step 3: Initialize real power loss and reactive power loss vectors to
zero.
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Step 4: Calculate the effective real and reactive power flow in each
branch using equations (2.1) and (2.2).
Step 5: Calculate node voltages, real and reactive power loss of each
branch using equations (2.11), (2.14) and (2.15) respectively.
Step 6: Check for convergence i.e., ΔV <εmax in successive iterations.
If it is converged go to next step otherwise increment iteration number
and go to step 4.
Step 7: Calculate the real and reactive power losses for all branches
and also total real and reactive power loss.
Step 8: Print voltage at each node, the real and reactive power losses
of all branches and total loss.
Step 9: Stop.
2.5 ILLUSTRATIVE EXAMPLES
The effectiveness of the proposed method is illustrated through
three different examples consisting of 15, 33 and 69-node radial
distribution systems.
2.5.1 Example-1
The line and load data of 15-node, 11 kV radial distribution
system [41] shown in fig.2.2 is given in the table A.1. The voltage
magnitudes of the system are given in the table 2.3. The power losses
of the system are given in the table 2.4. The total real and reactive
power losses are 60.35 kW and 58.39 kVAr. The real and reactive
power losses are 4.92% and 4.64% of their respective loads. The
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minimum voltage is 0.9424 pu at node 13 and the voltage regulation
of the system is 5.76%.
Table2.3. Voltage magnitudes of 15-node system
Node No. Voltage Magnitude (pu) Angle (deg.)
1 1.0000 0
2 0.9713 0.0342
3 0.9547 -0.0599
4 0.9489 -0.0527
5 0.9479 -0.0405
6 0.9618 0.1481
7 0.9611 0.1566
8 0.9593 0.1792
9 0.9652 0.1081
10 0.9635 0.1289
11 0.9478 0.0147
12 0.9437 0.0658
13 0.9424 0.0821
14 0.9466 -0.0243
15 0.9465 -0.0222
Table2.4. Power loss of 15-node radial distribution system
Br.No.
Node Backward ForwardPloss(kW)
Qloss(kVAr)SE RE
P(kW)
Q(kVAr)
P(kW)
Q(kVAr)
1 1 2 1288.2 1308.5 1250.4 1271.67 37.72 36.90
2 2 3 735.48 748.61 724.19 737.57 11.34 14.01
3 3 4 397.24 404.92 395.2 402.53 2.05 2.40
4 4 5 44.16 45.03 44.1 44.99 0.06 0.04
5 2 9 355.27 362.3 350.5 357.41 4.77 4.89
6 9 10 140.39 143.09 140 142.83 0.39 0.27
34
7 2 6 70.11 71.49 70 71.41 0.11 0.08
8 6 7 114.63 116.76 114.16 116.44 0.47 0.32
9 6 8 44.16 45.03 44.1 44.99 0.06 0.04
10 3 11 256.95 261.23 254.77 259.69 2.18 1.54
11 11 12 114.78 116.86 114.17 116.45 0.6 0.41
12 12 13 44.17 45.04 44.1 44.99 0.07 0.05
13 4 14 70.2 71.55 70 71.41 0.2 0.14
14 4 15 140.44 143.12 140 142.83 0.44 0.29
Total loss 60.35 58.39
Fig.2.4 33-node radial distribution system
2.5.2 Example-2
The 33-node, 12.66 kV radial distribution system [20] is shown
in fig.2.4. The line and load data of this system are given in table A.2.
The voltage magnitude at different nodes of this system is given in
table 2.5. Table 2.6 gives the power loss results of the system. The
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total real and reactive power losses of the system are 201.54 kW and
132.11 kVAr respectively. These are 5.42% and 5.74% of their total
loads. The minimum voltage of the system is 0.9132 pu at node 18.
The maximum voltage regulation of system is 8.68%. Comparison of
load flow results between the proposed method and the existing
method [50] is given in table 2.7. The total real and reactive power
losses are reduced and the minimum voltage is improved in the
proposed method.
Table2.5. Voltage magnitudes of 33-node system
Node No. Voltage Magnitude (pu) Angle (deg.)
1 1.0000 0.0000
2 0.9970 0.0140
3 0.9830 0.0950
4 0.9755 0.1610
5 0.9681 0.2270
6 0.9499 0.1330
7 0.9462 -0.0970
8 0.9414 -0.0610
9 0.9351 -0.1340
10 0.9290 -0.1970
11 0.9283 -0.1900
12 0.9269 -0.1780
13 0.9208 -0.2690
14 0.9185 -0.3470
15 0.9171 -0.3850
16 0.9157 -0.4080
17 0.9137 -0.4850
18 0.9132 -0.4950
19 0.9965 0.0030
36
20 0.9929 -0.0640
21 0.9922 -0.0840
22 0.9916 -0.1040
23 0.9794 0.0640
24 0.9727 -0.0250
25 0.9694 -0.0680
26 0.9478 0.1720
27 0.9452 0.2280
28 0.9337 0.3110
29 0.9254 0.3890
30 0.9220 0.4950
31 0.9178 0.4100
32 0.9169 0.3870
33 0.9166 0.3790
Table2.6. Power loss of 33-node radial distribution system
Br.No.
Node Backward ForwardPloss(kW)
Qloss(kVAr)SE RE
P(kW)
Q(kVAr)
P(kW)
Q(kVAr)
1 1 2 3917.6 2435.2 3905.4 2428.9 12.24 6.33
2 2 3 3444.2 2207.8 3392.4 2181.4 51.79 26.38
3 3 4 2362.8 1684.2 2342.9 1674.0 19.90 10.14
4 4 5 2222.9 1594.0 2204.2 1584.5 18.69 9.52
5 5 6 2144.23 1554.5 2105.98 1521.48 37.12 33.02
6 6 7 1095.24 527.87 1093.32 521.54 1.92 6.33
7 7 8 893.33 421.54 888.49 419.95 4.84 1.59
8 8 9 688.49 319.95 684.31 316.94 4.18 3.00
9 9 10 624.31 296.95 620.75 294.42 3.56 2.52
10 10 11 560.75 274.42 560.2 274.24 0.55 0.18
11 11 12 515.2 244.24 514.32 243.95 0.88 0.29
12 12 13 454.32 208.95 451.66 206.85 2.69 2.09
13 13 14 391.66 171.85 390.93 170.89 0.73 0.96
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14 14 15 270.9 90.9 270.58 90.58 0.36 0.32
15 15 16 210.58 80.58 210.3 80.37 0.28 0.21
16 16 17 150.3 60.37 150.05 60.04 0.25 0.34
17 17 18 90.05 40.04 90 40 0.05 0.04
18 2 19 361.14 161.08 360.98 160.93 0.16 0.15
19 19 20 270.98 120.93 270.14 120.18 0.83 0.75
20 20 21 180.14 80.18 180.04 80.06 0.10 0.12
21 21 22 90.04 40.06 90 40 0.04 0.06
22 3 23 939.61 457.24 936.43 455.07 3.18 2.17
23 23 24 845.43 404.07 841.28 401.01 4.14 3.06
24 24 25 421.28 201.01 420 200 1.28 1.01
25 6 26 950.75 973.61 948.15 972.28 2.60 1.33
26 26 27 888.15 947.28 884.82 945.59 3.33 1.69
27 27 28 824.82 920.59 813.52 910.62 11.30 6.86
28 28 29 753.52 890.62 745.69 883.8 7.84 5.82
29 29 30 625.69 813.8 621.8 811.82 3.88 1.98
30 30 31 421.8 211.84 420.21 210.26 1.59 1.58
31 31 32 270.22 140.26 270 140.02 0.21 0.24
32 32 33 60.01 40.02 60 40 0.01 0.02
Total loss 201.54 132.11
Table2.7. Comparison of load flow results of 33-node system
Description
Total Loss MinimumVoltage and
it’s nodenumber
Real power(kW)
Reactivepower(kVAr)
Existingmethod [50]
202.67 135.140.9131 at
node 18
Proposedmethod
201.54 132.110.9132 at
node 18
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2.5.3 Example-3
The fig 2.5 shows a 69-node, 12.66 kV radial distribution
system [17]. The line and load data of this system is given in table A.3.
Table 2.8 gives comparison of the voltage magnitudes (pu) of the
system obtained by the proposed method with the existing method
[50]. The minimum voltage is 0.9094 pu at node 65 and maximum
voltage regulation is 9.06%. The line losses of the system are given in
table 2.9. The total real and reactive power losses of this system are
224.45 kW and 107.14 kVAr respectively. The real and reactive power
losses are 5.91% and 3.78% of their total loads. The load flow results
of the proposed method are compared with the existing method [50] in
table 2.10. The total real and reactive power losses are reduced and
minimum voltage is improved by the proposed method.
Fig.2.5 69-node radial distribution system
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Table2.8. Voltage magnitudes of 69-node system
NodeNo.
Voltage Magnitude (pu) Angle (deg.)
Proposedmethod
Existing method[50]
Proposedmethod
1 1.0000 1.0000 0.0000
2 1.0000 0.9999 -0.0010
3 0.9999 0.9999 -0.0020
4 0.9998 0.9998 -0.0060
5 0.9990 0.9990 -0.0180
6 0.9901 0.9901 0.0500
7 0.9808 0.9808 0.1220
8 0.9786 0.9786 0.0900
9 0.9775 0.9775 0.0820
10 0.9726 0.9725 0.1660
11 0.9715 0.9714 0.1850
12 0.9684 0.9682 0.2380
13 0.9653 0.9653 0.2850
14 0.9624 0.9624 0.3310
15 0.9596 0.9595 0.3770
16 0.9590 0.9590 0.3860
17 0.9581 0.9581 0.4000
18 0.9580 0.9581 0.4000
19 0.9578 0.9576 0.4090
20 0.9575 0.9573 0.4140
21 0.9569 0.9568 0.4230
22 0.9569 0.9568 0.4230
23 0.9569 0.9568 0.4240
24 0.9566 0.9566 0.4270
25 0.9565 0.9564 0.4300
26 0.9565 0.9564 0.4320
27 0.9563 0.9563 0.4320
40
28 0.9999 0.9999 -0.0030
29 0.9998 0.9999 -0.0050
30 0.9997 0.9997 -0.0030
31 0.9997 0.9997 -0.0030
32 0.9996 0.9996 -0.0009
33 0.9994 0.9994 0.0040
34 0.9992 0.9990 0.0090
35 0.9990 0.9990 0.0100
36 0.9999 0.9999 -0.0030
37 0.9997 0.9998 -0.0090
38 0.9996 0.9996 -0.0120
39 0.9996 0.9995 -0.0120
40 0.9995 0.9995 -0.0130
41 0.9988 0.9988 -0.0240
42 0.9987 0.9986 -0.0280
43 0.9985 0.9985 -0.0290
44 0.9985 0.9985 -0.0290
45 0.9984 0.9984 -0.0310
46 0.9984 0.9984 -0.0310
47 0.9998 0.9998 -0.0080
48 0.9986 0.9985 -0.0530
49 0.9948 0.9947 -0.1920
50 0.9942 0.9942 -0.2110
51 0.9788 0.9785 0.0900
52 0.9788 0.9785 0.0910
53 0.9747 0.9747 0.1030
54 0.9716 0.9714 0.1290
55 0.9669 0.9669 0.1650
56 0.9627 0.9626 0.2000
57 0.9402 0.9401 0.5970
58 0.9291 0.9290 0.8000
41
59 0.9248 0.9248 0.8810
60 0.9197 0.9197 0.9860
61 0.9126 0.9123 1.0550
62 0.9124 0.9121 1.0580
63 0.9118 0.9117 1.0610
64 0.9112 0.9098 1.0790
65 0.9094 0.9092 1.0850
66 0.9715 0.9713 0.1860
67 0.9715 0.9713 0.1860
68 0.9679 0.9679 0.2440
69 0.9679 0.9679 0.2440
Table2.9. Power loss of 69-node radial distribution system
Br.No.
Node Backward ForwardPloss(kW)
Qloss(kVAr)SE RE
P(kW)
Q(kVAr)
P(kW)
Q(kVAr)
1 1 2 4016.3 2785.3 4016.2 2785.1 0.08 0.18
2 2 3 4016.2 2785.1 4016.1 2784.9 0.08 0.18
3 3 4 3748.9 2591.8 3748.7 2591.3 0.20 0.47
4 4 5 2897.9 1980.1 2896 1977.9 1.94 2.27
5 5 6 2896 1977.9 2867.8 1963.5 28.31 14.41
6 6 7 2865.2 1961.3 2836 1946.5 28.41 14.98
7 7 8 2795.6 1916.5 2788.7 1913 6.91 7.10
8 8 9 2669.6 1828 2666.2 1826.2 3.39 2.92
9 9 10 779.5 523.23 774.78 521.67 4.75 1.58
10 10 11 746.78 511.67 745.77 511.33 1.02 0.34
11 11 12 564.77 381.33 562.59 380.61 2.19 0.73
12 12 13 361.56 236.61 360.29 236.18 1.29 0.43
13 13 14 352.28 231.18 351.04 230.77 1.25 0.41
14 14 15 343.04 225.27 341.84 224.8 1.21 0.39
15 15 16 341.84 224.87 341.62 224.8 0.23 0.07
42
16 16 17 296.62 194.8 296.3 194.7. 0.32 0.11
17 17 18 236.3 159.7 236.3 194.7 0.00 0.00
18 18 19 176.3 124.7 176.2 159.7 0.11 0.04
19 19 20 176.2 124.66 176.13 124.66 0.07 0.02
20 20 21 175.13 124.04 175.02 124.64 0.11 0.04
21 21 22 61.02 43.01 61.02 124.01 0.00 0.00
22 22 23 56.02 40.01 56.02 43.01 0.01 0.00
23 23 24 56.02 40.01 56.01 40.01 0.01 0.00
24 24 25 28.01 20 28 20 0.01 0.00
25 25 26 28 20 28 20 0.00 0.00
26 26 27 14 10 14 10 0.00 0.00
27 3 28 81.53 64.01 81.53 64.01 0.00 0.00
28 28 29 55.53 46.01 55.52 46.01 0.00 0.01
29 29 30 29.52 28.01 29.52 28.01 0.01 0.00
30 30 31 29.52 28.01 29.52 28.01 0.00 0.00
31 31 32 29.52 28.01 29.52 28.01 0.01 0.00
32 32 33 29.52 28.01 29.51 28.01 0.01 0.00
33 33 34 15.51 18 15.5 18 0.01 0.00
34 34 35 6 4 6 4 0.00 0.00
35 3 36 185.76 129.16 185.76 129.16 0.00 0.00
36 36 37 159.76 110.61 159.74 110.57 0.02 0.04
37 37 38 133.74 92.02 133.72 92 0.02 0.02
38 38 39 133.72 92 133.72 91.99 0.01 0.01
39 39 40 109.72 74.99 109.72 74.99 0.00 0.00
40 40 41 85.72 57.99 85.67 57.93 0.05 0.06
41 41 42 84.47 56.93 84.45 56.91 0.02 0.02
42 42 43 84.45 56.91 84.45 56.91 0.00 0.00
43 43 44 78.45 52.61 78.45 52.61 0.00 0.00
44 44 45 78.45 52.61 78.44 52.6 0.01 0.01
45 45 46 39.22 26.3 39.22 26.3 0.00 0.00
46 4 47 850.76 611.16 850.73 611.11 0.02 0.06
47 47 48 850.73 611.11 850.15 609.68 0.58 1.43
43
48 48 49 771.15 553.28 769.52 549.28 1.63 4.00
49 49 50 384.82 274.78 384.7 274.5 0.12 0.28
50 8 51 44.1 31 44.1 31 0.00 0.00
51 51 52 3.6 2.7 3.6 2.7 0.00 0.00
52 9 53 1856.8 1281.0 1851 1278.1 5.79 2.95
53 53 54 1846.6 1274.6 1839.9 1271.2 6.73 3.43
54 54 55 1813.5 1252.2 1804.4 1247.5 9.15 4.66
55 55 56 1780 1230.3 1771.2 1225.8 8.81 4.49
56 56 57 1771.2 1225.8 1721.5 1209.2 49.78 16.71
57 57 58 1721.5 1209.2 1697.0 1200.9 24.54 8.23
58 58 59 1697.0 1200.9 1687.5 1197.8 9.52 3.15
59 59 60 1587.5 1125.8 1576.87 1122.6 10.69 3.25
60 60 61 1576.9 1122.6 1562.9 1115.41 14.05 7.16
61 61 62 318.93 227.47 318.82 227.41 0.11 0.06
62 62 63 286.82 204.41 286.68 204.34 0.14 0.07
63 63 64 286.68 204.34 286.02 204.01 0.66 0.34
64 64 65 59.04 42.02 59 42 0.04 0.02
65 11 66 36 26 36 26 0.00 0.00
66 66 67 18 13 18 13 0.00 0.00
67 12 68 56.02 40.01 56 40 0.02 0.01
68 68 69 28 20 28 20 0.00 0.00
Total loss 224.45 107.14
Table2.10. Comparison of load flow results of 69 node system
Description
Total LossMinimum
voltage and it’snode number
Realpower(kW)
Reactivepower(kVAr)
Existing method [50] 224.96 114.15 0.9092 at node 65
Proposed method 224.45 107.14 0.9094 at node 65
44
2.6 CONCLUSIONS
The iterative techniques commonly used in transmission
networks are not suitable for distribution power flow analysis because
of poor convergence characteristics. In this work the distribution
power flow is carried out by the backward and forward propagation
iterative equations. The effective branch powers are calculated in
backward propagation. In forward propagation voltage magnitudes at
each node are calculated. The illustration of node identification is
used for calculating the effective powers of a branch in backward
propagation. It was found that the proposed load flow method is
suitable for fast convergence characteristics.