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Optimal Power Flow Incorporating FACTS Devices
using Gravitational Search Algorithm
Yusuf SONMEZ
Department of Electrical Technology,
Gazi Vocational Collage, Gazi University,
Ankara, TURKEY
ysonmez@gazi.edu.tr
Ugur GUVENC
Department of Electrical and Electronic Engineering,
Technology Faculty, Duzce University
Duzce, TURKEY
ugurguvenc@duzce.edu.tr
Serhat DUMAN
Department of Electrical Education,
Technical Education Faculty, Duzce University
Duzce, TURKEY
serhatduman@duzce.edu.tr
Nuran YORUKEREN
Department of Electrical Engineering,
Engineering Faculty, Kocaeli University
Kocaeli, TURKEY
nurcan@kocaeli.edu.tr
Abstract—This paper aims to solve the optimal power problem
(OPF) incorporating flexible AC transmission systems (FACTS)
devices using Gravitational Search Algorithm (GSA) that
minimizes the fuel cost function in the problem. In the
optimization problem, Thyristor controlled series compensation
(TCSC) and thyristor controlled phase shifter (TCPS) FACTS
devices are considered to find their optimum location in
transmission lines. In order to evaluate the effectiveness of
proposed algorithm, it has been tested on modified IEEE 30 bus
system and compared with particle swarm optimization (PSO)
and hybrid tabu search and simulated annealing (TS/SA)
approach which are used in solving the same problem and
reported before in the literature. Results show that GSA
produces better results than others and has fast computing time
for solving OPF problem with FACTS.
Keywords-optimal power flow; flexible AC transmission
systems; gravitational search algorithm; optimization
I. INTRODUCTION
Optimal Power Flow (OPF) is generally designated to a
non-linear optimization problem. It is aimed in this
optimization problem that control variables for a power
network are set to their optimum values minimizing chosen
objective functions such as fuel cost or network loses while
meeting some equality and inequality constraints. Therefore,
OPF problem solution has great attention last three decades in
power markets and many researchers have studied on this topic
using different methods like conventional mathematical models
or evolutionary algorithms [1-3].
Flexible AC transmission system (FACTS) is a power
electronics based system composed of static devices such as
SVC (static VAR compensation), TCSC (thyristor controlled
series compensation) and TCPS (thyristor controlled phase
shifter) etc. used for the AC transmission of electrical energy.
These FACTS devices can increase transmission line
capability, improve the system stability and security of
transmission system [4]. Also, in the classical OPF problem,
since there is an increasing in load demands, it causes an
overloading of transmission lines and losing optimality due to
forecasted errors [5]. Therefore, in recent years, researchers
have considered using FACTS incorporated with OPF in order
to improve the static performance of power systems. There are
several papers in dealing with OPF incorporating FACTS.
However, classical OPF problem is already a non-differential,
non-linear and non-convex problem. Moreover, inclusion of
FACTS devices into the OPF, it makes the problem more
complex. In order to overcome this negativity, researchers have
intensified their efforts on artificial intelligence and
evolutionary methods for determining the parameters of
FACTS. There are some studies in the literature using these
methods such as GA [7,8], PSO [8, 9] non-linear interior point
[10], fuzzy logic [5] and some hybrid methods created by
combining a few of them [11, 12, 4]. But there is a main
disadvantage that these methods doesn’t exactly converge the
local optimum and have huge computational time.
In this paper, a new method is proposed to solve OPF with
FACTS problem using Gravitational Search Algorithm. In the
OPF problem, TCSC and TCPS devices are considered to
minimize the fuel cost function while meeting the constraints,
namely; equality constraints, in equality constraints, generation
constraints, transformer constraints, security constraints and
FACTS devices constraints. In order to show the effectiveness
of proposed algorithm, it is tested on IEEE 30 bus system with
two different cases and compared the PSO reported in [9] for
test case1 and TS/SA (tabu search/simulated annealing)
reported in [4] for the other test case. Results show that GSA
produce better solution than others in solving such a problem.
978-1-4673-1448-0/12/$31.00 ©2012 IEEE
II. OPF INCORPORATING FACTS DEVICES
A. Modeling of TCSC and TCPS
In this paper, static TCSC and TCPS FACTS devices are
used to minimize the cost function based on [4,9]. In the TCSC
model, a series capacitor bank can be seen as a controllable
reactor by thyristor. Equivalent circuit of TCSC is illustrated in
Fig. 1.
-jX
C
R
ij
+jX
ij
ij
V
i
V
j
Figure 1. Equivalent circuit of TCSC
Cij XXX −=
mod (1)
() ()()
ijijijijjiijiij bgVVgVP
δδ
sincos
2+−= (2)
() ()()
ijijijijjiijiij bgVVbVQ
δδ
cossin
2−−−= (3)
() ()()
ijijijijjiijjji bgVVgVP
δδ
sincos
2−−= (4)
() ()()
ijijijijjiijjji bgVVbVQ
δδ
cossin
2++−= (5)
where Vi and Vj are voltage magnitude of i-th and j-th
buses, δij is the voltage angle difference between i-th and j-th
bus and, gij and bij are described as follows.
()
2
mod
2XR
R
g
ij
ij
ij +
= (6)
()
2
mod
2
mod
XR
X
b
ij
ij +
= (7)
In the TCPS model, there is a phase shifting transformer
having a control parameter which is voltage shift angle φ.
Equivalent circuit of TCPS is illustrated in Fig. 2 [4].
R
ij
+jX
ij
ij
V
i
V
j
ϕ
∠1:1
Figure 2. Equivalent circuit of TCPS
The power flow equations [4] are described as follows.
() ()()
ϕδϕδ
+++−= ijijijij
jiiji
ij bg
K
VV
K
gV
Psincos
2
2
(8)
() ()()
ϕδϕδ
+−+−
−
=ijijijij
jiiji
ij bg
K
VV
K
bV
Qcossin
2
2
(9)
() ()()
ϕδϕδ
+−+−= ijijijij
ji
ijjji bg
K
VV
gVP sincos
2 (10)
() ()()
ϕδϕδ
++++−= ijijijij
ji
ijjji bg
K
VV
bVQ cossin
2(11)
where )cos(
ϕ
=K. The TCPS effect on a power system
is generally represented by the injected power model shown in
Fig 3 [4].
Figure 3. Injected model of TCPS
() ()()
ijijijijjiijiis bgmVVgVmP
δδ
cossin
22 −−−= (12)
() ()()
ijijijijjiijiis bgmVVbVmQ
δδ
sincos
2
2++= (13)
(
)
(
)
(
)
ijijijijjijs bgmVVP
δ
δ
cossin +−= (14)
(
)
(
)
(
)
ijijijijjijs bgmVVQ
δ
δ
sincos −−= (15)
where )tan(
ϕ
=m
B. OPF Formulation
In this paper, The OPF with FACTS devices problem aims
to minimize the fuel cost function. The mathematical model of
the problem [4] is described as follows.
()
∑
=
++=
NG
i
GiiGiii PcPbaF
1
2
min i=1, 2, …,NG (16)
where ai, bi and ci are fuel cost parameters and PGi are
active power generations at i-th bus and NG is the number of
total generator.
subject to:
Equality constraints:
() ()
() ()
()
∑
∑∑
=
==
=−−
+−
bus
TCPSbus
N
j
ijCijCijji
N
i
ii
N
i
DiGi
XXYVV
PtcPP
1
11
0cos
δθ
ϕ
bus
Ni ∈∀ (17)
() ()
() ()
()
∑
∑∑
=
==
=−+
+−
bus
bus TCPS
N
i
ijCijCijji
N
i
N
i
iiDiGi
XXYVV
QtcQQ
1
11
0sin
δθ
ϕ
bus
Ni ∈∀ (18)
where QGi is reactive power generations at i-th bus, PDi and
QDi are active and reactive power demand at i-th bus, Ptci and
Qtci are injected active and reactive power, φi is phase shift
angle of i-th TCPS, Yij(XC) and θij(XC) are magnitude and angle
of ij-th elements in Ybus matrix, Nbus is the set of bus indices,
NTCPS is number of TCPSs.
Inequality constraints:
max,min, GiGiGi PPP ≤≤ i=1, 2, …, NG (19)
max,min, GiGiGi QQQ ≤≤ i=1, 2, …, NG (20)
max,min, iii VVV ≤≤ i=1, 2, …, Nbus (21)
max,ii SlSl ≤ i=1, 2, …, Ntl (22)
max,
0ii CC XX ≤≤ i=1, 2, …, NTCSC (23)
max,
0ii
ϕ
ϕ
≤≤ i=1, 2, …, NTCPS (24)
where Ntl is number of transmission lines, NTCSC is number of
TCSCs.
III. IMPLEMENTAION OF GRAVITATIONAL SEARCH
ALGORITHM
GSA is a brand new heuristic optimization algorithm based
on law of gravity and law of motion [14,15]. GSA is different
from other optimization methods like PSO and GA and a useful
method for solving non-linear functions. It is reported in some
studies [14-26] that GSA improves the solution quality for
solving non-linear optimization problems when compared other
methods. In GSA, a set of agents are described as objects and
their masses are introduced by using law of gravity and law of
motion in order to find the local optima in solution. Here, it is
expressed that step by step how the GSA works based on [14].
GSA searches the best fitness value for a given function.
The searching process consists of six steps. In the first step,
assume that there is a system with N agents; the position
(masses) of the i-th agents is described as follows.
),...,,..,( 1n
i
d
iii xxxX = i=1, 2, …, N (25)
where xi
d is the position of the i-th masses in the d-th
dimension and n is the total number of agents. In the second
step, the fitness evolution for all agents at each cycle is
performed. It is done via calculating best and worst fitness
values at each cycle. For a minimization problem it is described
as follows.
() ()
tfittbest i
min= Ni ∈∀ (26)
() ()
tfittworst i
max= Ni ∈∀ (27)
where fiti(t) is the fitness value of the i-th agent at t time. In
the third step, the gravitational constant G(t) at time t is
calculated using a function G of the initial value G0 and time t
described as follows.
() ( )
tGGtG ,
0
= (28)
()
T
t
eGtG
α
−
=0 (29)
In the fourth step, mass of each agents Mi(t) are updated as
follows.
NiMMMM iiipiai ,...,2,1, ==== (30)
() () ()
() ()
tworsttbest
tworsttfit
tm i
i−
−
= (31)
() ()
()
∑
=
=N
k
k
i
i
tm
tm
tM
1
(32)
In the fifth step, in order to calculate the acceleration of an
agent, the force acting on i-th mass Fi
d(t) is computed first
based on the law of gravity as follows.
() () () ()
() () () ()
()
txtx
tXtX
tMtM
tGrandtF d
i
d
j
ji
ij
N
ijkbestj
j
d
i−
+
=∑
≠=
ε
2
,,
.(33)
where kbest is the set of first k agents with the best fitness
value and biggest mass, randj is a random number in the
interval of [0,1], Mi(t) and Mj(t) are the gravitational masses of
the i-th agent and the j-th agent,
2
)(),( tXtX ji is the
Euclidian distance between i-th and j-th agents, ε is a small
constant. Then acceleration of an agent ai
d(t) is calculated as
follows.
() ()
()
tM
tF
ta
i
d
i
d
i= (34)
In the last step of the searching process, the next velocity
and the position of the agents are updated as follows.
( ) () ()
tatvrandtv d
i
d
ii
d
i+×=+1 (35)
() () ()
11 ++=+ tvtxtx d
i
d
i
d
i (36)
The searching process of GSA expressed above repeats
until the stopping criterion is reached. Thus, the optimum
solution is obtained for a specified function when the searching
process stops. The flow diagram of the GSA is illustrated in
Fig. 4 [14].
IV. SIMULATIO N RESULTS
The proposed meta-heuristic approach has been applied to
solve the OPF problem incorporating flexible AC transmission
system. In order to verify the effectiveness of the proposed
GSA approach which is tested on standard IEEE 30-bus test
system shown in Fig. 2 for different operating scenarios and the
test system data is given in [27]. In this simulation study, G0 is
set to 100 and α is set to 10, and T is the total number of
iterations. Maximum iteration numbers are 200 for operating
scenarios.
Case 1 is the OPF with TCSC and TCPS at line 3-4, which
is simulated by the GSA, the results obtained from the
proposed approach are compared to GA, SA, TS and TS/SA,
respectively. The results of this comparison and the average
and maximum results of the GSA technique are given in Table
I and Table II. In case study, the reactance limits of the TCSC
and phase shifting angle of the TCPS are considered to be 0-
0.02 in p.u and 0-0.1 radian, respectively [4]. Fig. 5 shows the
convergence of the best total cost result obtained from the GSA
approach.
Figure 4. Flow diagram of the GSA
Figure 5. Convergence of GSA for case 1
TABLE I. THE RESULTS OF GSA APPROACH FOR CASE 1
Method Total cost ($/h)
Min. Average Max.
GSA 803.31234015 803.31234017 803.31234022
The minimum fuel cost obtained from the proposed
approach is 803.31234015 $/h. GSA is approximately less by
0.09885%, compared to previously reported results 804.1072
$/h.
In Case 2, the OPF is considered with two TCSCs at
branches 4 and 24 together with two TCPSs at branches 4 and
8. Minimum and maximum limits for the control variables,
TCSCs and TCPSs are taken from [9]. The results obtained
from the GSA are compared to the other method in the
literature.
TABLE II. BEST CONTROL VARIABLE S SETTINGS FOR CASE 1
Control variables settings GA [4] SA [4] TS [4]
P1 (MW) 192.5105 192.5105 192.5105
P2 (MW) 48.3951 48.3951 48.3951
P5 (MW) 19.5506 19.5506 19.5506
P8 (MW) 11.6204 11.6204 11.6204
P11 (MW) 10.0000 10.0000 10.0000
P13 (MW) 12.0000 12.0000 12.0000
TCSC line 3-4 0.0200 0.0200 0.0200
TCPS line 3-4 0.0141 0.0141 0.0141
Total PG (MW) 294.0766 294.0766 294.0766
Ploss (MW) 10.6766 10.6766 10.6766
Cost($/h) 804.1072 804.1072 804.1072
CPU time (min) 10:24 6:40 6:24
Control variables settings TS/SA [4] GSA
P1 (MW) 192.5105 177.18623041
P2 (MW) 48.3951 48.89254087
P5 (MW) 19.5506 21.49421605
P8 (MW) 11.6204 21.56424278
P11 (MW) 10.0000 12.07831745
P13 (MW) 12.0000 12.00000000
TCSC line 3-4 0.0200 0.02000000
TCPS line 3-4 0.0141 0.01228762
Total PG (MW) 294.0766 293.2155476
Ploss (MW) 10.6766 9.81554756
Cost($/h) 804.1072
803.31234015
CPU time (min) 4:43 1:05
TABLE III. BEST CONTROL VARIABLE S SETTINGS FOR CASE 2
Control variables settings
(p.u.) PSO [9] GSA
P1 1.7472 1.74782623
P2 0.4851 0.46409423
P5 0.2380 0.21501375
P8 0.2106 0.22838850
P11 0.1210 0.13032508
P13 0.1200 0.13132134
V1 1.0811 1.08513531
V2 1.0618 1.04658378
V5 1.0314 1.03541025
V8 1.0391 1.07661053
V11 1.0617 0.95395895
V13 1.0715 0.96257641
T11 1.0040 1.08481924
T12 0.9918 1.09820187
T15 0.9955 1.04395236
T36 0.9686 1.06035021
TCSC4 0.1687 0.13302452
TCSC24 0.2903 0.23014370
TCPS4 -0.0272 0.02345993
TCPS8 -0.0336 0.03369910
Cost($/h) 800.931
799.056077
Voltage deviations 1.070 0.87277107
The results of this comparison and the average and
maximum results of the GSA technique are given in Table III
and Table IV. The convergence characteristic curve of the best
total fuel cost result obtained from the GSA is shown in Fig.6.
From the results in Table III, it can be seen that the best fuel
cost result obtained from the proposed method is 799.056077
$/h, which is less in comparison to reported result the literature.
Figure 6. Convergence of GSA for case 2
TABLE IV. THE RESULTS OF GSA APPROACH FOR CASE 2
Method Total cost ($/h)
Min. Average Max.
GSA 799.056077 799.615973 800.129561
V. CONCLUSION
In this paper a new method is proposed to solve the optimal
power flow problem with FACTS devices using Gravitational
Search Algorithm. In the optimization problem fuel cost
function is minimized by fixing location of TCSC and TCPS
FACTS devices. Experimental studies show that GSA has fast
computing time and finds better results than compared others in
solving such a problem. In our future work, GSA will be
implemented on more case studies with different FACTS
devices and compared other popular optimization algorithms.
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