Post on 10-Mar-2018
CHAPTER - 2
LOAD FLOW METHOD FOR RADIAL DISTRIBUTION SYSTEMS
2.1 INTRODUCTION
The modern power distribution network is constantly being faced with an ever-
growing load demand. Distribution networks experience distinct change from a low to high
load level every day. In order to evaluate the performance of a power distribution system and
to examine the effectiveness of proposed modifications to a system in the planning stage, it is
essential that a load flow analysis of the system is to be repeatedly carried-out. It basically
gives the steady state operating condition of a distribution system corresponding to a
specified load on the system. Certain applications, particularly in the distribution automation
(i.e., VAR planning, state estimation, etc.) and optimization of power system require repeated
load flow solutions.
Load flow studies are the back bone of power system analysis and design. It gives the
steady state solution of a power system network for normal operating condition which helps
in continuous monitoring of the current state of the system. It is also employed for planning,
optimization and stability studies. Hence an efficient and fast load flow method is required.
Generally distribution networks have high R/X ratio. Due to this reason, popularly
used Newton – Raphson [2] and Fast Decoupled load flow algorithms [3] may provide
inaccurate results and may not be converge. Hence, conventional load flow methods cannot
be directly applied to obtain the load flow solution of radial distribution systems.
Kersting and Mendive [6] and Kersting [11] have developed load flow techniques
based on ladder theory. Baran and Wu [18] and Chiang [21] have developed a load flow
solution of distribution system based on the Newton – Raphson method. They have computed
system Jacobian matrix using chain rule in their method. Chiang [21] has also proposed three
different algorithms for solving radial distribution network based on the method of Baran
and Wu. He has proposed decoupled, fast decoupled and very fast decoupled distribution load
flow algorithms. The first two methods are similar to the method proposed by Baran and Wu.
The very fast decoupled distribution load flow method is very attractive, since it does not
require any Jacobian matrix construction and factorization but more computations are
involved because it solves three fundamental equations. representing real power, reactive
power and voltage magnitudes.
Many researchers [38, 52, 74] have proposed different methods to analyze the radial
distribution system based on forward and backward sweep current injection methods.
Venkatesh and Ranjan [72] have proposed a method using data structures to find the load
flow solution of radial distribution system.
In this chapter, a simple method of load flow technique for radial distribution system
is proposed. The proposed method involves the solution of simple algebraic equation in
receiving end voltages. The mathematical formulation of the proposed method is explained in
Section 2.2. In this Section, derivation of voltage, angle, real and reactive power losses from
phasor diagram of single line diagram of distribution system is discussed. The bus
identification using data structure of distribution system is described in Section 2.3. The steps
of load flow algorithm are presented in Section 2.4. Also the effectiveness of the proposed
method is tested with different examples of distribution system and the results are compared
with the existing methods. In Section 2.5, conclusions of the proposed method are presented.
2.2. MATHEMATICAL FORMULATION
Assumptions:
Radial distribution networks are balanced and can be represented by their
equivalent single line diagram.
Half line charging susceptances of distribution lines are negligible and these
distribution lines are represented as short lines.
Shunt capacitor banks are treated as loads.
Consider a branch connected between buses 1 and 2 as shown in Fig. 2.1.
1 2
V1∟0 I V2∟-δ2
R1+jX1
P2+jQ2
Fig. 2.1 Electrical equivalent of a typical branch 1
In Fig. 2.1, V1∟0 and V2∟-δ2 are the voltage magnitudes and phase angles of two
buses 1 and 2. Let the current flowing through branch 1 is I. The substation voltage is
assumed to be 1+j0 p.u. Let power factor angle of load P2+jQ2 be θ2. Let R1 and X1 be the
resistance and reactance of the branch 1 respectively. The phasor diagram of Fig. 2.1 is
shown in Fig. 2.2.
Fig. 2.2 Phasor diagram of a branch 1 connected between buses 1 and 2
From Fig. 2.2, the Eqns. are
2X
2
R2
2
1 VVVV
… (2.1)
2
2121
2
21212
2
1 sinIRcosIXsinIXcosIRVV … (2.2)
To eliminate ‘I’ in Eqn. (2.2)
2
22
V
PcosI
2
22
V
QsinI
where
P2 = Total active power load of all buses beyond bus 2 including local load and
active power losses beyond bus 2
Q2 = Total reactive power load of all buses beyond bus 2 including local load and
reactive power losses beyond bus 2
The Eqn. (2.2) becomes
2
2
1212
2
2
12122
2
1V
RQXP
V
XQRPVV
2
2
1212
2
22
2
1212
22
2
1
V
RQXP
V
1X2Q1R2P
XQRP2VV
1212
2
12122
21212
242
22
2
1 RQXPXQRPV2XQRPVVV
0VVXRQPXQRPV2V 2
2
1
221
21
2
22212122
2
2
4
0XRQP2
VXQRPV2V
21
21
2
222
1
2
12122
2
2
4
… (2.3)
0cVbV 2
2
2
4 … (2.4)
where
XQ2RP2Vb 12121
2
… (2.5)
XRQPc 21
21
2
222 … (2.6)
where
| | = substation voltage (taken as 1.0 p.u.)
R1 = resistance of branch 1
X1 = reactance of branch 1
The four possible solutions for the voltage, V2 from Eqn. (2.4) are
i) c4bb2
1 22/1
2/1
ii) c4bb2
1 22/1
2/1
iii) - c4bb2
1 22/1
2/1
iv) c4bb2
1 22/1
2/1
It is found for realistic systems, when P2, Q2, R1, X1 and V are expressed in p.u., ‘b’ is
always positive because the term (2 + 2 is extremely small as compared to. V1
2
In addition the term ‘4c’ is negligible compared to b2. Therefore, {b
2 – 4c}
1/2 is nearly equal
to ‘b’ and hence the first two solutions of V2 are nearly equal to zero and third solution is
negative and hence not feasible. The fourth solution of V2 is positive and hence it is only
the possible feasible solution. Therefore, the possible feasible solution of Eqn. (2.4) is
c4bb2
1V
22/1
2/1
2 … (2.7)
In general, the solution for V 1i is
c4b2/1
b2
1 2/1V i
2ii1i … (2.8)
where
XQ2RP2Vb k1ik1ii
2
i
XRQPc2k
2k
2
1i2
1ii
where
i =1, 2……nbus.
k =1, 2, 3…..nbus-1
nbus = total number of buses.
The real and reactive power loss of branch ‘k’ is given by
V
QPRP k
1i
2
2
1i2
1ik
loss
… (2.9)
V
QPXQ k
1i
2
2
1i2
1ik
loss
… (2.10)
The Total Active and Reactive Power Losses (TPL, TQL) are given by
1nbus
1k
loss kPTPL … (2.11)
kQTQL1nbus
1k
loss
… (2.12)
The phase angle (δ2) of voltage V2 can be calculated as follows
From Fig. 2.2,
VV
Vtan
R2
X2
21212
2121
2sinXIcosRIV
sinRIcosXItan … (2.13)
On simplification we will get
VXQRP
RQXPtan
2
2
1212
121212 … (2.14)
In general
VXQRP
RQXPtan
1i
2
k1ik1i
k1ik1i11i … (2.15)
Usually, the substation voltage V1 is known and is taken as 1.0 ∟00p.u. Initially,
Ploss[k] and Qloss[k] are set to zero for all k. Then the initial estimate of Pi+1 and Qi+1 will be
the sum of the loads of all the buses beyond bus ‘i' plus the local load of bus ‘i' plus the losses
beyond bus ‘i' Compute Vi+1,Ploss[k], Qloss[k], δi+1using Eqns. (2.8), (2.9), (2.10) and (2.15).
This will complete one iteration of the solution. Update the loads P(i+1) and Q(i+1) (by
including losses) and repeat the same procedure until the voltage mismatch reach a tolerance
level of 0.0001 p.u. in successive iterations.
2.3 BUS IDENTIFICATION USING DATA STRUCTURES
In this section, the data structure is used to identify the inter connection of buses and
branches of radial distribution system, which avoids the conventional method of unique
lateral bus and branch numbering process. An algorithm is developed and the methodology of
identifying the buses and branches connected to a particular bus in detail, which will help in
finding the exact load feeding through that particular bus, is presented in this section.
The proposed method initially forms the Bus Incidence Matrix (BIM) of the radial
distribution system and then can be processed to create a Data structure. This method can
handle the system data with any random bus and line numbering scheme except the slack bus
being numbered as 1. Also this algorithm provides dynamically declared and alterable Data
structure. The illustration of the algorithm is given in Section 2.3.4.
2.3.1 Formation of Bus Incidence Matrix
Consider the radial distribution system with multiple laterals, branches and buses. The
bus numbers have been marked in a random order with substation as bus number 1.The
elements of BIM are considered as follows:
BIM is of size m x n, where m is number of lines and n is the number of buses.
If ith
and jth
buses are connected through a branch, k then
BIMij = 1
Otherwise BIMij = 0
2.3.2 Formation of Data structure
After forming the BIM, then the Data structure can be created as follows.
First select a bus, identify the paths emanating from it and each path is represented by
a contour. Form the data structure which gives the information about bus numbers, real and
reactive powers at those buses and the respective branch numbers corresponding to each
3
I
IX
V
X
VII
VI
I
VII
II
X
IV
S/S 1 2
4
5
6
7
9
8
11
12
10 14
13
contour. It also gives the number of contours emanating from the end bus and their respective
index numbers of the contours.
2.3.3 Algorithm steps for bus identification
Step 1 : Read system branch data.
Step 2 : Form the Bus incidence matrix (BIM)
Step 3 : Create the Data structure using BIM
Step 4 :Stop
2.3.4 Illustration
Consider the single line diagram of 15 bus radial distribution system shown in Fig 2.3.
Consider bus 1 as the reference bus. The Bus Incidence Matrix for this system is as shown in
Table 2.1.
10 9
14 2 3 4 5 1
7 11 15 6
12 8 13
Fig. 2.3 Single line diagram of 15 bus radial distribution system
1, 2, 3 ,….. represent bus numbers
, ,….represent branch numbers
, , …. represent contour numbers
Table 2.1 The Bus Incidence Matrix of the 15 bus radial distribution system
Bus No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Branch No.
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
3 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0
5 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0
7 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
9 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0
10 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
11 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0
12 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0
13 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
14 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
Consider a contour with several buses and several connections emanating from its
terminal bus, and then a generic data structure would look like the illustration below in Fig.
2.4.
As an example, the data structure for the system is
.
b.n
o.ob.n
o.o
b.n
o.o
c.no
.
Br.
no.Br.
no.
Br.
no.
D.n
o.
P
i P
i
P
i
Q
i Q
i
Q
i
D.n
o.
D.n
o.
1 2
I I
I
Fig. 2.4 The general data structure of the system
In Fig. 2.4, b.no. stands for bus number, br.no. stands for branch number, Pj, Qj
stands for real and reactive power at that buses, c.no. stands for number of contours present
adjacent to that bus i, D.no. stands for data structure corresponding to the emanating contours
(i.e., adjacent contours to the bus, i).
For 15 bus system 11 contours are identified and are shown in Fig. 2.3 with dotted lines. The
data structure of contour 1 is as shown below.
bus : [1 2] (bus no.s are 1 and 2)
c.no. : 3 (no. of contours emanating from the end bus of this contour
or adjacent contours of the bus, 2)
Index number of that contours: [II V VIII] (numbers of adjacent contours of bus, 2)
Branch : 1 (branch connected between buses 1 and 2)
Real power : [0 44.1] (Real power load at buses 1and 2)
Reactive power : [0 44.99] (Reactive power load at buses 1and 2)
Resistance of branch 1: 0.8523 p.u. (resistance of branch 1 connected between 1and 2)
Reactance of branch 1: 0.8336 p.u. (reactance of branch 1 connected between 1and 2)
The data structure of the entire system is shown in Table 2.2
Table 2.2 Data Structure of all contours of 15 bus Radial Distribution System
Contour I
b.no. 1 2
br. no. 1
real power P1 P2
reac. Power Q1 Q2
Contour II
b.no. 2 3
br. no. 2
real power P2 P3
reac. Power Q2 Q3
3
II, V, VIII
2
III, IX
Contour III
b.no. 3 4
br. no. 3
real power P3 P4
reac. Power Q3 Q4
Contour IV
b.no. 4 5
br. no. 4
real power P4 P5
reac. Power Q4 Q5
Contour V
b.no. 2 6
br. no. 7
real power P2 P6
reac. Power Q2 Q6
Contour VI
b.no. 6 7
br. no. 8
real power P6 P7
reac. Power Q6 Q7
Contour VII
b.no. 6 8
br. no. 9
real power P6 P8
reac. Power Q6 Q8
Contour VIII
b.no. 2 9 10
br. no. 5 6
real power P2 P9 P10
reac. Power Q2 Q9 Q10
Contour IX
b.no. 3 11 12 13
br. no. 10 11 12
real power P3 P11 P12 P13
reac. Power Q3 Q11 Q12 Q13
Contour X
b.no. 4 14
br. no. 13
real power P4 P14
reac. Power Q4 Q14
Contour XI
b.no. 4 15
br. no. 14
real power P4 P15
reac. Power Q4 Q15
2.4 LOAD FLOW CALCULATION
Once all the buses and branches are identified, it is very easy to calculate the effective
load at each bus including losses and then solve for the bus voltage magnitudes and angles
3
IV, X, XI
0
2
VI, VII
0
0
0
0
0
0
using the Eqns. (2.8) and (2.15). Then compute the real and reactive power losses in each
branch using Eqns. (2.9) and (2.10). The convergence criterion of the proposed algorithm is
that if change in the magnitude of the two bus voltages in successive iterations is less than
0.0001 p.u., the solution is said to be converged and the total real and reactive power losses
are computed using Eqns. (2.11) and (2.12).
2.4.1 Algorithm for Load flow calculation:
Step 1 : Read line and load data of radial distribution system. Initialize TPL, TQL
to zero. Assume bus voltages 1 p.u., set convergence Criterion ΔV ≤ ε
Step 2 : Start iteration count=1.
Step 3 : Build BIM and Data Structure of the system.
Step 4 : Calculate effective load at each bus starting from the last bus.
Step 5 : Initialize real power loss and reactive power loss to zero.
Step 6 : Find effective losses at each bus.
Step 7 : Calculate load at each bus including losses.
Step 8 : Calculate bus voltages (magnitude and angles), real and reactive power
loss of each branch using Eqns. (2.8), (2.15), (2.9) and (2.10) respectively.
Step 9 : Calculate the value of change in bus voltages i.e. ΔV ≤ ε in successive
iterations. If ΔV≤ε go to step 11 otherwise go to step 10.
Step 10: Increment iteration number and go to step 6.
Step 11: Calculate TPL and TQL using Eqns. (2.8) and (2.9).
Step 12: Print voltages at each bus, TPL, TQL and number of iterations.
Step 13: Stop.
2.4.2 Flow Chart for the proposed method
Read Distribution System data
Start
Assume bus voltages Vi=1+j0, for i=1,2,….nbus
Set iteration count (IC) =1, convergence criterion (ε) =0.0001,
Initialize total real and total reactive power losses to zero
Build Bus Incidence matrix (BIM)
Build Data Structure of the system using BIM
Calculate real and reactive power loss of
each branch using Eqns. (2.9) & (2.10)
Compute bus voltages (magnitude and angle) using Eqns.
(2.8) & (2.15)
Calculate effective load at each bus including losses
IC=IC+1
Fig. 2.5 Flow chart of load flow method
2.5 ILLUSTRATIVE EXAMPLES
To illustrate the effectiveness of the proposed method three different examples
consisting of 15, 33 and 69 bus radial distribution systems are considered.
2.5.1 Example – 1
The single line diagram of 15 bus, 11 kV radial distribution system is shown in Fig.
2.3. The line and load data of this system are given in Appendix – A (Table A1). The voltage
profile of the system is given in Table 2.3. The line flows of 15 bus system are given in Table
2.4. The total real and reactive power losses are 61.7933 kW and 57.2967 kVAr. The
minimum voltage of this system is 0.9445 p.u. occurred at bus 13 and voltage regulation is
5.55%. The number of iterations taken for this system is 3. The CPU time taken for the
solution is 0.14 sec. The solution obtained by proposed method is compared with the existing
method [104] and voltages are approximately matching both in magnitude and phase angles.
Table 2.3 Voltage profile of 15 bus radial distribution system
Bus No.
Proposed method Existing method [104]
Voltage
Magnitude (p.u.)
Angle
(deg.)
Voltage
Magnitude (p.u.) Angle (deg.)
1 1.0000 0.0000 1.0000 0.0000
2 0.9713 0.0320 0.9712 0.0320
3 0.9567 0.0493 0.9547 0.0493
4 0.9509 0.0565 0.9489 0.0565
5 0.9499 0.0687 0.9478 0.0687
6 0.9582 0.1894 0.9582 0.1894
7 0.9560 0.2166 0.9560 0.2166
8 0.9570 0.2050 0.9569 0.2050
9 0.9680 0.0720 0.9679 0.0720
10 0.9669 0.0850 0.9669 0.0850
11 0.9500 0.1315 0.9477 0.1317
12 0.9458 0.1824 0.9437 0.1824
13 0.9445 0.1987 0.9424 0.1987
14 0.9486 0.0848 0.9466 0.0849
15 0.9484 0.0869 0.9464 0.0869
Table 2.4 Line flows of 15 bus radial distribution system
Branch
No.
From
(i)
To
(j)
Pij
(kW)
Qij
(kVAr)
Pji
(kW)
Qji
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
1 1 2 1250.49 1271.34 1212.8 1234.47 37.69 36.87
2 2 3 724.19 737.35 712.9 726.31 11.29 11.04
3 3 4 394.80 402.45 392.36 400.06 2.44 2.39
4 4 5 44.10 44.98 44.04 44.94 0.06 0.04
5 2 9 114.16 116.42 113.69 116.1 0.47 0.32
6 9 10 44.10 44.98 44.04 44.94 0.06 0.04
7 2 6 350.51 357.34 344.56 353.45 5.95 3.89
8 6 7 140.00 142.80 139.61 142.53 0.39 0.27
9 6 8 70.00 71.40 69.89 71.32 0.11 0.08
10 3 11 254.78 259.64 252.6 258.17 2.18 1.47
11 11 12 114.17 116.43 113.57 116.02 0.60 0.41
12 12 13 44.10 44.98 44.03 44.93 0.07 0.05
13 4 14 70.00 71.40 69.8 71.26 0.20 0.14
14 4 15 140.00 142.80 139.56 142.5 0.44 0.30
2.5.2 Example – 2
The 33 - bus, 12.66 kV radial distribution system [19] is shown in Fig. 2.6. The line
and load data of this system is given in Appendix A (Table A2). The voltage profile of the
system is given in Table 2.5. The line flows of 33 bus system are given in Table 2.6. The total
real and reactive power losses are 202.5022 kW and 135.1286 kVAr. The minimum voltage
of this system is 0.9131p.u. at bus 18 and voltage regulation is 8.69%. The number of
iterations taken for this system is 3. The CPU time has taken to obtain the solution is 0.12
sec. The computational efficiency of the proposed method is given in Table 2.7 and is
compared with other known methods [11, 18]. The results obtained by the proposed method
and the existing methods [71, 104] are matching both in voltage magnitude and phase angle.
Fig. 2.6 Single line diagram of 33 bus radial distribution system
Table 2.5 Voltage profile of 33 bus radial distribution system
Bus
No.
Proposed method Existing method
[71] Existing method [104]
Voltage
Magnitude (p.u.)
Angle
(deg.)
Voltage
Magnitude (p.u.)
Voltage
Magnitude (p .u.)
Angle
(deg.)
1 1.0000 0.0000 1.00000 1.0000 0.0000
2 0.9970 0.0132 0.99703 0.9970 0.0141
3 0.9829 0.0929 0.98289 0.9828 0.0939
4 0.9755 0.1585 0.97538 0.9753 0.1595
5 0.9681 0.2252 0.96796 0.9680 0.2261
6 0.9497 0.1307 0.94948 0.9495 0.1317
7 0.9462 -0.0996 0.94595 0.9461 -0.0986
8 0.9413 -0.0635 0.93230 0.9412 -0.0625
9 0.9351 -0.1366 0.92597 0.9349 -0.1356
10 0.9293 -0.1992 0.92009 0.9291 -0.1981
11 0.9284 -0.1919 0.91922 0.9283 -0.1909
12 0.9269 -0.1804 0.91771 0.9266 -0.1794
13 0.9207 -0.2707 0.91153 0.9207 -0.2707
14 0.9185 -0.3494 0.90924 0.9182 -0.3494
15 0.9171 -0.3871 0.90782 0.9171 -0.3871
16 0.9157 -0.4103 0.90643 0.9157 -0.4103
17 0.9137 -0.4876 0.90439 0.9137 -0.4876
18 0.9131 -0.4972 0.90377 0.9130 -0.4971
19 0.9965 0.0023 0.99650 0.9965 0.0033
20 0.9929 -0.0646 0.99292 0.9929 -0.0637
21 0.9922 -0.0840 0.99221 0.9922 -0.0830
22 0.9916 -0.1043 0.99158 0.9915 -0.1034
23 0.9794 0.0603 0.97931 0.9792 0.0613
24 0.9727 -0.0284 0.97264 0.9726 -0.0274
25 0.9694 -0.0721 0.96931 0.9692 -0.0711
26 0.9477 0.1702 0.94755 0.9476 0.1712
27 0.9452 0.2263 0.94499 0.9452 0.2273
28 0.9337 0.3093 0.93354 0.9337 0.3103
29 0.9255 0.3870 0.92532 0.9255 0.3882
30 0.9220 0.4925 0.92177 0.9218 0.4934
31 0.9178 0.4080 0.91760 0.9178 0.4090
32 0.9169 0.3850 0.91669 0.9169 0.3859
33 0.9166 0.3773 0.91640 0.92 0.38
Table 2.6 Line flows of 33 bus radial distribution system
Branch
No.
From
(i)
To
(j)
Pij
(kW)
Qij
(kVAr)
Pji
(kW)
Qji
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
1 1 2 3905.25 2418.80 3893.04 2412.5
8 12.21 6.22
2 2 3 3392.47 2171.41 3340.82 2145.5
0 51.65 26.31
3 3 4 2342.98 1674.06 2323.08 1663.9
3 19.90 10.13
4 4 5 2204.29 1584.53 2185.59 1575.0
1 18.70 9.52
5 5 6 2106.04 1521.52 2067.80 1488.5
1 38.24 33.01
6 6 7 1093.35 521.56 1091.44 515.23 1.91 6.33
7 7 8 888.51 419.96 883.67 418.36 4.84 1.60
8 8 9 684.33 316.96 680.15 313.96 4.18 3.00
9 9 10 620.77 294.43 670.21 291.91 3.56 2.52
10 10 11 560.22 274.25 559.67 274.07 0.55 0.18
11 11 12 514.34 243.96 513.46 243.67 0.88 0.29
12 12 13 451.67 206.56 449.00 204.76 2.67 2.10
13 13 14 390.94 170.90 390.21 169.94 0.73 0.96
14 14 15 270.59 90.58 270.53 90.26 0.36 0.32
15 15 16 210.30 80.38 210.02 80.17 0.28 0.21
16 16 17 150.05 60.04 149.80 59.7 0.25 0.34
17 17 18 90.00 40.00 89.95 39.96 0.05 0.04
18 2 19 360.98 160.93 360.82 160.78 0.16 0.15
19 19 20 270.14 120.18 269.31 119.43 0.83 0.75
20 20 21 180.04 80.06 179.94 79.94 0.10 0.12
21 21 22 90.00 40.00 89.96 39.94 0.04 `0.06
22 3 23 936.43 445.07 933.28 442.91 3.15` 2.16
23 23 24 841.29 401.01 836.15 396.95 5.14 4.06
24 24 25 420.00 200.00 418.71 198.99 1.29 1.01
25 6 26 948.18 972.31 945.58 970.99 2.60 1.32
26 26 27 864.85 945.61 881.52 943.92 3.33 1.69
27 27 28 813.55 910.65 802.95 900.69 11.30 9.96
28 28 29 745.71 883.83 737.88 877.01 7.83 6.82
29 29 30 621.82 811.84 617.92 809.86 3.90 1.98
30 30 31 420.23 210.27 418.64 208.87 1.59 1.57
31 31 32 270.01 140.02 269.80 139.77 0.21 0.25
32 32 33 60.00 40.00 59.99 39.98 0.01 0.02
Table 2.7 Comparison of computational efficiency of proposed method with other
methods
Method System Execution time, (sec.) No. of iterations
Proposed method 0.12 3
Kersting Method [11] 0.14 4
Baran and Wu [18] 0.13 3
2.5.3 Example – 3
The 69 bus, 11 kV radial distribution system [76] is shown in Fig. 2.7. The line and
load data of this system is given in Appendix A (Table A3). The voltage profile of the system
is given in Table 2.8. The line flows of 69 bus system are given in Table 2.9. The total real
and reactive power losses are 224.9457 kW and 102.1397 kVAr. The minimum voltage of
this system is 0.9092 p.u. at bus 65 and voltage regulation is 9.082%. The number of
iterations taken for this system is 3. The CPU time has taken to obtain the solution is 0.13
sec. The performance in terms of computational efficiency of the proposed method is given in
Table 2.10 and is compared with other known methods [11, 18]. The solution obtained by
proposed method is compared with the existing methods [71, 104] and results are found to be
approximately matching both in voltage magnitude and phase angles.
Fig. 2.7 Single line diagram of 69 bus radial distribution system
Table 2.8 Voltage profile of 69 bus radial distribution system
Bus
No.
Proposed method Existing method
[71] Existing method [104]
Voltage
Magnitude
(p.u.)
Angle
(deg.)
Voltage
Magnitude
(p.u.)
Voltage
Magnitude
(p.u.)
Angle
(deg.)
1 1.0000 0.0000 1.00000 1.0000 0.0000
2 1.0000 -0.0012 0.99997 0.9999 -0.0012
3 0.9999 -0.0024 0.99993 0.9999 -0.0025
4 0.9998 -0.0059 0.99984 0.9998 -0.0059
5 0.9990 -0.0185 0.99902 0.9990 -0.0185
6 0.9901 0.0493 0.99009 0.9901 0.0492
7 0.9808 0.1210 0.98079 0.9808 0.1210
8 0.9786 0.1382 0.97858 0.9786 0.1382
9 0.9774 0.1471 0.97745 0.9774 0.1470
10 0.9724 0.2317 0.97245 0.9725 0.2316
11 0.9713 0.2505 0.97135 0.9713 0.2504
12 0.9682 0.3033 0.96819 0.9682 0.3032
13 0.9653 0.3496 0.96526 0.9653 0.3495
14 0.9624 0.3958 0.96237 0.9624 0.3957
15 0.9595 0.4418 0.95950 0.9595 0.4417
16 0.9590 0.4504 0.95897 0.9589 0.4503
17 0.9581 0.4645 0.95809 0.9581 0.4645
18 0.9581 0.4647 0.95808 0.9581 0.4646
19 0.9576 0.4732 0.95761 0.9576 0.4731
20 0.9573 0.4788 0.95732 0.9573 0.4787
21 0.9568 0.4877 0.95683 0.9568 0.4876
22 0.9568 0.4878 0.95683 0.9568 0.4877
23 0.9568 0.4891 0.95676 0.9567 0.4890
24 0.9566 0.4920 0.95660 0.9566 0.4919
25 0.9564 0.4952 0.95643 0.9564 0.4951
26 0.9564 0.4965 0.95636 0.9564 0.4964
27 0.9563 0.4968 0.95634 0.9563 0.4967
28 0.9999 -0.0027 0.99993 0.9999 -0.0027
29 0.9999 -0.0047 0.99973 0.9998 -0.0053
30 0.9998 -0.0021 0.99985 0.9997 -0.0032
31 0.9997 -0.0016 0.99971 0.9997 -0.0028
32 0.9997 0.0006 0.99961 0.9996 0.0009
33 0.9995 0.0061 0.99935 0.9993 0.0035
34 0.9992 0.0140 0.99901 0.9990 0.0940
35 0.9992 0.0150 0.99895 0.9989 0.0104
36 0.9999 -0.0030 0.99992 0.9999 -0.0030
37 0.9997 -0.0094 0.99975 0.9997 -0.0094
38 0.9996 -0.0118 0.99959 0.9995 -0.0118
39 0.9995 -0.0125 0.99954 0.9995 -0.0125
40 0.9995 -0.0125 0.99884 0.9995 -0.0125
41 0.9988 -0.0235 0.99884 0.9988 -0.0235
42 0.9986 -0.0282 0.99855 0.9985 -0.0282
43 0.9985 -0.0288 0.99850 0.9985 -0.0288
44 0.9985 -0.0289 0.99850 0.9985 -0.0289
45 0.9984 -0.0307 0.99841 0.9984 -0.0307
46 0.9984 -0.0307 0.99840 0.9984 -0.0307
47 0.9998 -0.0077 0.99979 0.9997 -0.0077
48 0.9985 -0.0525 0.99854 0.9985 -0.0525
49 0.9947 -0.1916 0.99470 0.9947 -0.1916
50 0.9942 -0.2114 0.99415 0.9941 -0.2114
51 0.9785 0.1385 0.97854 0.9785 0.1385
52 0.9785 0.1387 0.97853 0.9785 0.1386
53 0.9747 0.1690 0.97466 0.9746 0.1689
54 0.9714 0.1946 0.97142 0.9714 0.1945
55 0.9669 0.2302 0.96694 0.9669 0.2300
56 0.9626 0.2651 0.96257 0.9625 0.2650
57 0.9401 0.6617 0.94010 0.9401 0.6615
58 0.9290 0.8643 0.92904 0.9290 0.8641
59 0.9248 0.9452 0.92476 0.9248 0.9451
60 0.9197 1.0497 0.91974 0.9197 1.0495
61 0.9123 1.1188 0.91234 0.9123 1.1186
62 0.9121 1.1215 0.91205 0.9120 1.1213
63 0.9117 1.1252 0.91166 0.9117 1.1250
64 0.9098 1.1430 0.90976 0.9098 1.1428
65 0.9092 1.1484 0.90919 0.9092 1.1482
66 0.9713 0.2517 0.97129 0.9713 0.2516
67 0.9713 0.2517 0.97129 0.9713 0.2516
68 0.9679 0.3093 0.96786 0.9678 0.3092
69 0.9679 0.3093 0.96786 0.9678 0.3092
Table 2.9 Line flows of 69 bus radial distribution system
Branch
No.
From
(i)
To
(j)
Pij
(kW)
Qij
(kVAr)
Pji
(kW)
Qji
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
1 1 2 4016.32 2785.2 4016.25 2785.02 0.07 0.18
2 2 3 4016.32 2785.2 4016.25 2785.02 0.07 0.18
3 3 4 3748.77 2591.38 3748.58 2590.91 0.19 0.47
4 4 5 2896.08 1977.96 2894.15 1975.7 1.93 2.26
5 5 6 2867.94 1963.63 2839.8 1949.3 28.14 14.33
6 6 7 2836.1 1946.53 2806.86 1931.64 29.24 14.89
7 7 8 2788.83 1913.03 2781.96 1909.53 6.87 3.5
8 8 9 2666.37 1826.32 2663.01 1824.61 3.36 1.71
9 9 10 774.79 521.67 770.08 520.11 4.71 1.56
10 10 11 745.78 511.34 744.77 511.01 1.01 0.33
11 11 12 562.59 380.62 560.41 379.9 2.18 0.72
12 12 13 360.29 236.19 359.01 235.77 1.28 0.42
13 13 14 351.05 230.78 349.81 230.37 1.24 0.41
14 14 15 341.85 224.88 340.65 224.48 1.2 0.4
15 15 16 341.63 224.81 341.41 224.74 0.22 0.07
16 16 17 296.31 194.7 295.99 194.59 0.32 0.11
17 17 18 236.3 159.7 236.3 159.7 0 0
18 18 19 176.2 124.67 176.1 124.64 0.1 0.03
19 19 20 176.13 124.64 176.06 124.62 0.07 0.02
20 20 21 175.03 124.01 174.92 123.97 0.11 0.04
21 21 22 61.03 43.01 61.05 43.01 0 0
22 22 23 56.02 40.01 56.01 40.01 0.01 0
23 23 24 56.01 40 56 40 0.01 0
24 24 25 28 20 27.99 20 0.01 0
25 25 26 28 20 28 20 0 0
26 26 27 14 10 14 10 0 0
27 3 28 81.53 64.01 81.53 64.01 0 0
28 28 29 55.52 46.01 55.52 66 0 0.01
29 29 30 29.52 28.01 29.52 28.01 0 0
30 30 31 29.52 28.01 29.52 28.01 0 0
31 31 32 29.52 28.01 29.52 28.01 0 0
32 32 33 29.51 28 29.5 28 0.01 0
33 33 34 15.5 18 15.49 18 0.01 0
34 34 35 6 4 6 4 0 0
35 3 36 185.76 129.16 185.76 129.16 0 0
36 36 37 159.74 110.57 159.72 110.53 0.02 0.04
37 37 38 133.72 92 133.7 91.98 0.02 0.02
38 38 39 133.72 91.99 133.71 91.98 0.01 0.01
39 39 40 109.72 74.99 109.72 74.99 0 0
40 40 41 85.67 57.93 85.62 57.87 0.05 0.06
41 41 42 84.45 56.91 84.43 56.89 0.02 0.02
42 42 43 84.45 56.91 84.45 56.91 0 0
43 43 44 78.45 52.61 78.45 52.61 0 0
44 44 45 78.44 52.6 78.43 52.59 0.01 0.01
45 45 46 39.22 26.3 39.22 26.3 0 0
46 4 47 850.73 611.11 850.71 611.05 0.02 0.06
47 47 48 850.15 609.68 849.57 608.25 0.58 1.43
48 48 49 769.52 549.28 767.89 545.28 1.63 4
49 49 50 384.7 274.5 384.58 274.22 0.12 0.28
50 8 51 44.1 31 44.1 31 0 0
51 51 52 3.6 2.7 3.6 2.7 0 0
52 9 53 1851.08 1278.14 1845.3 1275.2 5.78 2.94
53 53 54 1840.02 1271.22 1833.31 1268.8 6.71 2.42
54 54 55 1804.49 1247.57 1795.36 1242.92 9.13 4.65
55 55 56 1771.3 1225.9 1762.51 1221.42 8.79 4.48
56 56 57 1721.63 1209.22 1671.96 1192.55 49.67 16.67
57 57 58 1697.14 1201.01 1672.66 1192.79 24.48 8.22
58 58 59 1687.64 1197.86 1678.14 1194.72 9.5 3.14
59 59 60 1576.97 1122.63 1566.3 1119.39 10.67 3.24
60 60 61 1562.95 1115.48 1548.93 1108.34 14.02 7.14
61 61 62 318.84 227.43 318.32 227.37 0.52 0.06
62 62 63 286.7 204.36 286.57 204.29 0.13 0.07
63 63 64 286.04 204.02 285.38 203.68 0.66 0.34
64 64 65 59 42 58.96 41.98 0.04 0.02
65 11 66 36 26 36 26 0 0
66 66 67 18 13 18 13 0 0
67 12 68 56 40 55.98 39.99 0.02 0.01
68 68 69 28 20 28 20 0 0
Table 2.10 Comparison of computational efficiency of proposed method with other
methods
Method System Execution time (sec.) No. of iterations
Proposed method 0.13 3
Kersting Method [11] 0.37 4
Baran and Wu [18] 0.29 3
2.6 CONCLUSIONS
A simple method for the load flow solution of radial distribution network has been
proposed. This method involves the concept of data structure to specify the configuration of
the system instead of commonly used a unique branch and bus numbering scheme to calculate
the cumulative loads of the system. This method is very simple and gives direct solution of the
radial distribution network based on the solution of a simple algebraic equation obtained from
the phasor diagram of the system. The effectiveness of the system is tested with15, 33 and 69
bus radial distribution systems. It is found that the solution obtained by this method matches
with the solution obtained by existing methods.