Energy and Power Engineering, 2013, 5, 1005-1010
doi:10.4236/epe.2013.54B192 Published Online July 2013 (http://www.scirp.org/journal/epe)
Binary Gravitational Search based Algorithm for
Optimum Siting and Sizing of DG and Shunt Capacitors
in Radial Distribution Systems*
N. A. Khan1, S. Ghosh2, S. P. Ghoshal2
1Department of Electrical Engineering, Aliah University, Salt Lake, Kolkata, India
2Department of Electrical Engineering, National Institute of Technology, Durgapur, India
Received March, 2013
ABSTRACT
This paper presents a binary gravitational search algorithm (BGSA) is applied to solve the problem of optimal allot ment
of DG sets and Shunt capacitors in radial distribution systems. The problem is formulated as a nonlinear constrained
single-objective optimization problem where the total line loss (TLL) and the total voltage deviations (TVD) are to be
minimized separately by incorporating optimal placement of DG units and shunt capacitors with constraints which in-
clude limits on voltage, sizes of installed capacitors and DG. This BGSA is applied on the balanced IEEE 10 Bus dis-
tribution network and the results are compared with convention al binary particle swarm optimization.
Keywords: Normal Load Flow; Radial Distribution System; Distributed Generation; Shunt Capacitors; Binary Particle
Swarm Optimization; Binary Gravitational Search Algorithm; Total line Loss; Total Voltage Deviation
1. Introduction
Distribution systems are usually radial in nature for op-
erational simplicity. The Radial Distribution Systems
(RDS) are fed at only one point, which is the substation.
The substation receives power from the centralized gen-
erating stations through interconnected transmission
network. The end users of electricity receive electrical
power from the substation through RDS, which is a pas-
sive network. Hence, the power flow in the RDS is uni-
directional. The high (R/X) ratio of the distribution lines
results in large voltage drops, low voltage stability and
power losses. Under critical loading conditions in certain
industrial areas, RDS experiences sudden voltage col-
lapse due to low value of voltage stability index at most
of its nodes. The sizing and sitting of DG units and shunt
capacitors in distribution systems is a very complex
combinational optimization problem. This optimization
problem involves not only integer and binary decision
variables but also nonlinear, non-continuous,
non-differential objective functions and constraints. The
problems of this type are regarded as nondeterministic
polynomial-time hard (NP-hard) problems, which pose
computational complexities with some conventional
analytical optimization techniq ue s. In th e literatures, very
few papers [1-5] use the optimization of voltage profile
as objective functions the integration of DG and shunt
capacitor placement certain heuristic methods [1-5] have
been reported in the literature for obtaining promising
results. Recently, Rashedi et al. [6] proposed a new op-
timization algorithm called Gravitational Search Algo-
rithm (GSA), which has been demonstrated to be very
interesting to find solutions of unimodal and multimodal
functions. GSA is based on the law of attraction of
masses supported by the Newtonian gravity, which says
that” a particle in the universe attracts every other one
with a force that is directly pro portional to the pr oduct of
their masses and inversely proportional to the square of
the distance between them”. The original version of GSA
was designed for search spaces of real valued vectors.
However, many optimization problems are set in binary
discrete space, such as feature selection and dimensional-
ity reduction [7-11] data mining [12], unit commitment
[13], and cell formation [14], in which it is natural to
encode solutions as binary vector s. In addition, problems
defined in the real space, may be considered in the binary
space, too. The solution is to display real digits with
some bits in the binary mode. The binary search space is
considered as a hypercube in which an agent may move
to nearer and farther corners of the hypercube by flipping
various numbers of the bits. In the literatures, very few
papers use the optimization of voltage profile as objec-
tive functions. In this work, a binary version of GSA
(BGSA) [15] is utilized to decide the optimal lo cation s of
DGs and shunt capacitors to obtain an overall better vol-
tage profile for a radial distribution system. In the binary

N. A. KHAN ET AL.
1006
version of GSA (BGSA), the outcome of these forces is
converted into a pro bability valu e for each ele ment of the
binary vector, which guides whether that elements will
take on the value 0 or 1. The objective function is to mi-
nimize Total Line Loss (TLL) and maximize the lowest
voltage level of the system i.e nothing but minimize To-
tal Voltage Deviation (TVD) to reach a better voltage
profile. The locations of DGs and capacitors are formu-
lated by binary variables as decision variables in the con-
straints.
2. Power Flow Solution in Radial
Distribution System
The load flow solution is carried by the following set of
recursive equations (1) and (2) derived from the single
line diagram as shown in Figure 1.
22
11,1
2
.ii
iiLiii
i
PQ
PPPRV




 


(1)
22
11,1
2
.ii
iiLiii
i
PQ
QQQXV




 


(2)
where is the real power flow into the sending end of
branch connecting bus and bus ; 1
i
P
i1i1i
L
i
P

is
real component of load at bus ; ,1ii is the re-
sistance of line section between buses and i
1iR
i1

and
i is the bus voltage magnitude at bus i. i is the
reactive power flow into the sending end of branch
VQ 1i

connecting bus and bus ; 1
i1i
L
i is reactive
component of load at bus ; ,ii is the reactance
of line section between buses and i.
Q
1

1
i
i1
X
The problem of DG and shunt capacitors allotment
with their proper capacities is of great importance. The
installation of DG and shunt capacitors at non-optimal
places can result in an increase in system losses, voltage
deviations and costs. Therefore, a power system planning
engineer requires an efficient and fast optimization me-
thod capable of indicating the best solution for a given
Figure 1. Single line diagram of a Radial distribution sys-
tem.
distribution network. The selection of the best places for
installation and the preferable sizes DG and shunt ca-
pacitors banks in large distribution systems is a complex
discrete optimization proble m.
In order to incorporate the proposed method recursive
equations (1) and (2) are modifi ed as follows:
a) Real Power Flow with installation of DG
22
11,1 1
2
.ii
iiLiii i
i
PQ
PPPRAP
VP

 


  (3)
where 1i
A
P

is real power magnitude injected at bus
1i

; P

is real power multiplier, set to zero when
there is no real power source or set to 1 when there is DG
power source;
b) Reactive Power Flow with shunt capacitors place-
ment
22
11,1 1
2
.ii
iiLiii i
i
PQ
QQQXRPQ
V

 

 (4)
where 1i RP

is reactive capacitor power magnitude in-
jected at bus 1i

; Q

is reactive capacitor power
multiplier, set to zero when there is no capacitor power
source or set to 1 when there is a capacitor power source;
c) Computation of Bus Voltages
22
1,1,1
22
22
,1 ,12
2(.. )
()*
ii iiiiii
ii
ii ii
i
VV RPXQ
PQ
RX V


 

 (5)
3. Problem Formulation
The following sections describe the details of the pro-
posed problem formulation.
a) The Objective Functions
The main advantages of Shunt capacitors in the distri-
bution system are loss minimization in the feeders and
the improvement in the voltage profile, i.e. maintaining
the voltages at customer terminals with reactive power
compensation.
The following functions are computed using the pro-
posed algorithm: Total Line Loss (TLL), Total Voltage
Deviatio n (TVD).
b) Total Line Loss (TLL)
The installation capacitor banks should not result in an
increase in the system losses. The power loss of the line
section connecting buses and is computed as: i1i
22
,1 2
(,1)* ii
lossi i
i
PQ
Pii RV


 (6)

1,1 n
loss
i
TLLPi i




 (7)
c) Total Voltage Deviation (TVD)

N. A. KHAN ET AL. 1007
Voltage deviation can also be minimized with integra-
tion of Shunt capacitors. The total voltage deviation
(TVD) in the system, which is to be minimized, is ex-
pressed as:
1
11
n
i
i
TVD V


 (8)
where = 1, 2, 3……..n and i is the voltage of ith
bus in per unit for the system buses; the ideal magnitude
of each bus voltage is unity.
i V
d) Constraints
The following constraints are considered [16].
i) Total Power Conservation:
The algebraic summation of all incoming and outgoing
powers over the feeders, taking into consideration the
feeders’ losses and the powers supplied by Shunt capaci-
tors should be equal to the total demand at that bus.
ii) Distribution Feeder’s Thermal Capacity:
Power flows in feeders must be within their capacities.
iii) Distribution Substation’s Capacity:
The summation of total powers delivered to the net-
work by the substation’s transfo rmers must be with in the
substation’s capacity limit.
iv) Shunt capacitor Operation Limits:
The Shunt capacitor’s generated power must be within
the Shunt capacitor’s capacity.
v) Voltage Drop Limits:
The voltage levels at different buses must be within
predetermined v alues.
4. Proposed Binary Gravitational Search
Algorithm
The conventional GSA was originally designed to solve
problems in continuous valued space [6]. The search al-
gorithm is based on the metaphor of gravitational inter-
action between masses in the Newton theory. A j-th bit of
the i-th agent

ij
x
in a system is represented as a bit 0
or 1 where a combination of bits gives the i-th agent po-
sition.
The next agent’s velocity


ij
v is calculated based on
its current velocity and its acceleration as expressed in
(9). Then, a new agent’s position

ij
x
is updated using
a condition as shown in (11). However, the velocity is
limited in interv al [-6, 6] so as to achieve a good con ver-
gence rate.
  tatvrtv ijijij *1 (9)

d
i
v
d
ie
vSigmoid

1
1 (10)




otherwise
vsigmoidr
xd
i
d
i,1
if,0 
(11)
5. Simulation Results and Discussions
To demonstrate the performance of the proposed BGSA
in solving the optimal DG and shunt capacitor placement
problem, the IEEE 10 bus distribution system is used in
this study. In this paper, for this particular test system,
Total Line Loss (TLL) and Total Voltage Deviation
(TVD) were minimized and compared to the conven-
tional BPSO as to illustrate its p e rformance in solving the
same problem. All the optimization parameters are stan-
dardized where population size and maximum population
are set to 60 and 100, respectively. In the BPSO, two
positive coefficients are set to 2 () and inertia
weight, () monotonously decreases from 0.9 (max ) to
0.4 (min ). In the BGSA, the initial gravity constant, G0
is set to 100 and the best applying force, (
122cc
ww
st
w
be
K
) is mo-
notonously decreased from 100% (maxbest ) to 2.5%
(minbest ).The proposed BGSA algorithm has been im-
plemented on IEEE 10-bus radial distribution network.
K
K
a) Test System-I
IEEE 10 Bus [17] is a single line main feeder (Base
Voltage = 23 KV, Base MVA=100 MVA) without later-
als and sublaterals having total active and reactive pow-
ers of 12.368 MW and 8.372 MVAR, respectively.
Without any injection of DG active powers and Shunt
capacitors’ reactive powers, the normal load flow yields
Total Line Loss (TLL) and total voltage deviation (TVD)
as 78.3712 KW and 0.6989 p.u., respectively.
b) Total Line Loss (TLL) minimization
Figure 2(a) represents voltage profile of IEEE 10 bus
radial distribution system obtained by different optimiza-
tion techniques (BPSO, BGSA) and normal power flow
(NPF). TLL is improved more in the case of BGSA
(87.67%) as that of BPSO (81%) over NPF, it can be
seen from Table 1. It is observed, voltage profile is im-
proved as that of BPSO, lowest bus voltage increased
6.38% by BPSO whereas in Binary GSA, it is improved
by 8.69 % as shown in Table 3. Convergence character-
istic is depicted in Figure 2(b). Loss and corresponding
total voltage deviation is more reduced than that of
BPSO as observed from Figures 2(c) and (d) in TLL
minimization.
12345678910
0. 88
0. 9
0. 92
0. 94
0. 96
0. 98
1
1. 02
Bus nu mb e
r
Bus voltage m agnitude in p.u.
NPF
BPSO
BGSA
(a)

N. A. KHAN ET AL.
1008
020 40 6080100 120 140 160 180 200
5
10
15
20
25
30
35
Iterati on cyc l e s
T.L.L in kW
(b)
(c)
(d)
Figure 2. TLL Minimization Characteristics of IEEE 10
Bus Radial Distribution System; (a) Voltage profile ob-
tained by different algorithms, (b) Convergence character-
istic, (c) Comparison of TLL, (d) Comparison of TVD.
Table 1. Comparative study of Tll minimization of Ieee 10
bus test system.
Comparative Study Of TLL Minimization
Test System % Improvement in
BPSO over NPF % Improvement in BGSA over NPF
IEEE 10 Bus
system 81 87.67
From Table 2, it is observed that same capacities of DG
and shunt capacitor are used for minimizing the fitness
function (TLL) in th is wo rk. Two DG o f 100 0 KW (at 3rd
and 7th bus position) and two Shunt capacitors (at 2nd and
5th bus position), each of 400 KVAR are optimally
placed in BGSA.
c) Total Voltage Deviation (TVD) minimization
Figure 3(a) represents voltage profile of IEEE 10 bus
radial distribution system obtained by optimization tech-
niques (BPSO, BGSA) and normal power flow (NPF). It
can be seen that voltage profile is improved as that of
BPSO, lowest bus voltage increased 4.47% by BPSO
whereas in Binary GSA, it is improved by 8.67 % corre-
sponds to Table 4. Convergence characteristic of TVD
minimization is shown in Figure 3(b). Total voltage de-
viation and correspon ding line loss is decreased than that
of BPSO as observed from Figures 3(c) and (d). One DG
set of 1000 KW capacity (at 2nd bus position), and two.
TVD is improved more in the case of BGSA (8.1%) as
that of BPSO (4.47%) over NPF as presen ted in Table 6.
Table 2. Capacities of DG and shunt capacitors in Tll
minimization of Ieee 10 bus test system.
Comparative Stu dy by two algorithms
BPSO BGSA
Test SystemDG
(kW) Shunt Capaci- tors (kVAR) DG (kW)
Shunt Ca-
pacitors
(kVAR)
IEEE 10
Bus system2000 800 2000 800
Table 3. Lowest bus voltage improvement in Tll minimiza-
tion for Ieee 10 bus system
Comparative Study Of TLL Minimization
Test System% Improvement in
BPSO over NPF % Improvement in BGSA over NPF
IEEE 10 Bus
system 6.38 8.69
1 2 3 4 56 7 8 910
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Bus numb e r
Bus voltage m agnitude in p.u.
NPF
BPSO
BGSA
(a)

N. A. KHAN ET AL. 1009
050100 150 200250 300 350400
0. 2
0.25
0. 3
0.35
0. 4
0.45
0. 5
0.55
0. 6
0.65
0. 7
B us num ber
T.V .D in p.u.
(b)
(c)
(d)
Figure 3. TVD Minimization Characteristics of IEEE 10
Bus Radial Distribution System; (a)Voltage profiles ob-
ined by different algorithms, (b) Convergence characteristic,
(c) Comparison of total line loss, (d) Comparison of total
voltage deviation.
Table 4. Comparative study of Tvd Minimization of Ieee 10
bus test system.
Comparative Study Of TVD Minimization
Test System % Improvem ent in
BPSO over NPF % Improvement in BGSA over NPF
IEEE 10 Bus
system 4.47 13.47
Table 5. Capacities of DG and shunt capacitors In TVD
Minimization of Ieee 10 bus test system.
Comparative Stu dy by two algorithms
BPSO BGSA
Test System
DG
(kW)
Shunt
Capacitors
(kVAR)
DG
(kW)
Shunt Ca-
pacitors
(kVAR)
IEEE 10 Bus
system 1000800 1000 800
Table 6. Lowest bus voltage improvement in TVD minimi-
zation for Ieee 10 bus system.
Comparative Study Of TLL Minimization
Test System % Improvement in
BPSO over NPF % Improvement in BGSA over NPF
IEEE 10 Bus
system 4.47 8.10
Shunt Capacitors (at 4nd and 8th bus position), each of
400 KVAR capacity, are equivalent to total 800 KVAR
optimally placed in BGSA as indicated in Table 5. Same
capacity of DG and Shunt capacitors are used in BPSO
technique but for TVD minimization, required two DG
sets and two Shunt capacitors.
6. Conclusions
This paper presented a BGSA and a comparative per-
formance of BGSA and BPSO in solving the two sepa-
rate single-objective optimization problem for optimal
DG and Shunt capacitor placement in a radial distribu-
tion test systems. The optimization techniques have been
tested on IEEE 10 bus distribution test system for deter-
mining the best optimal DG and Shunt capacitor place-
ments for TLL and TVD minimization. The comparative
results showed that the proposed BGSA is the most ef-
fective and precise among the aforementioned optimiza-
tion techniques. In conclusion, the authors’ contribution
in this work is successful application of a binary GSA
algorithm for simultaneous solution of optimal number
and placements of DG powers and Shunt capacitors in a
balanced distribut ion syste m.

N. A. KHAN ET AL.
1010
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