stochastic weight trade-off particle swarm … valve point effects i. introduction optimal power...
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Journal of Automation and Control Engineering Vol. 2, No. 1, March 2014
31doi: 10.12720/joace.2.1.31-37©2014 Engineering and Technology Publishing
Stochastic Weight Trade-Off Particle Swarm
Optimization for Optimal Power Flow
Luong Dinh Le and Loc Dac Ho Faculty of Mechanical-Electrical-Electronic, Ho Chi Minh City University of Technology, HCMC, Vietnam
Email: [email protected], [email protected]
Jirawadee Polprasert and Weerakorn Ongsakul Energy Field of Study, School of Environment, Resources and Development, Asian Institute of Technology,
Pathumtnani 12120, Thailand
Email: [email protected], [email protected]
Dieu Ngoc Vo and Dung Anh Le Department of Power Systems, Ho Chi Minh City University of Technology, HCMC, Vietnam
Email: [email protected], [email protected]
Abstract—This paper proposes a stochastic weight trade-off
particle swarm optimization (SWT-PSO) method solving
optimal power flow (OPF) problem. The proposed SWT-
PSO is a new improvement of PSO method using a
stochastic weight trade-off for enhancing search its search
ability. The proposed method has been tested on the IEEE
30 bus and 57 bus systems and the obtained results are
compared to those from other methods such as conventional
PSO, genetic algorithm (GA), ant colony optimization
(ACO), evolutionary programming (EP), and differential
evolution (DE) methods. The numerical results have
indicated that the proposed SWT-PSO method is better
than the others in terms of total fuel costs, total loss and
computational times. Therefore, the proposed SWT-PSO
method can be a favorable method for solving OPF
problem.
Index Terms—optimal power flow, particle swarm
optimization, stochastic weight trade-off, quadratic fuel
function, valve point effects
I. INTRODUCTION
Optimal power flow (OPF) problem is the important
fundamental issues in power system operation. In essence,
it is the optimization problem and its main objective is to
reduce the total generation cost of units while satisfying
unit and system constraints. Although the OPF problem
developed long time ago but so far it has been extensively
studied due to its importance in power system operation.
There have been many methods developed to solve OPF
problem from classical methods such as Newton’s
method, gradient search, linear programming (LP),
nonlinear programming, quadratic programming (QP), etc
to methods based on artificial intelligence and
evolutionary based methods such as ant colony
Manuscript received July 1, 2014; revised November 21, 2014.
optimization (ACO), genetic algorithm (GA), improved
evolutionary programming (IEP), tabu search (TS),
simulated annealing (SA), etc. These methods have been
effectively for solving the problem.
In 1995, Eberhart and Kennedy suggested a particle
swarm optimization (PSO) method based on the analogy
of swarm of bird flocking and fish schooling [1]. Due to
its simple concept, easy implementation, and
computational efficiency when compared with
mathematical algorithm and other heuristic optimization
techniques, PSO has attracted many attentions and been
applied in various power system optimization problems
such as economic dispatch [2]-[5], reactive power and
voltage control [6]-[8], transient stability constrained
optimal power flow [9], and many others [10], [11]-[13].
In this paper, a stochastic weight trade-off particle
swarm optimization (SWT-PSO) algorithm is proposed
by improvement of conventional PSO method with new
parameter for better optimal solution and faster
computation. The proposed SWT-PSO method has been
tested on the IEEE 30-bus system with quadratic fuel cost
function and fuel cost function with valve point effects
fuel function and the IEEE 57-bus system. The obtained
results are compared to those from many other methods
in the literature such as genetic algorithm GA [20], ant
colony optimization (ACO) [21], improved evolutionary
programming (IEP) [22], evolutionary programming (EP)
[23], gravitational search algorithm (GSA) [24],
differential evolution (DE–OPF) [25], modified
differential evolution (MDE–OPF) [25], base-case [28],
and Matpower [28].
II. OPTIMAL POWER FLOW PROBLEM
The OPF problem can be described as an optimization
(minimization) problem with nonlinear objective function
Journal of Automation and Control Engineering Vol. 2, No. 1, March 2014
32©2014 Engineering and Technology Publishing
and nonlinear constraints. The general OPF problem can
be expressed as follows:
Minimize F (u, x) (1)
Subject to g (u, x) = 0 (2)
h (u, x) 0 (3)
where F (u, x) is the objective function, g (u, x) represents
the equality constraints, h (u, x) represents the inequality
constraints, and u is the vector of the control variables
such as generated active power, generation bus voltage
magnitudes, transformers taps, etc), and x is state
variables such as reactive power, load bus voltage
magnitude, bus voltage angle, etc).
The essence of the optimal power flow problem resides
in reducing the objective function and simultaneously
satisfying the load flow equations (equality constraints)
without violating the inequality constraints. The fuel cost
of generators in form of quadratic function is given by:
2
1
( ) ( )GN
i i Gi i Gi
i
F x a b P c P
(4)
where, NG is the number of generators including the slack
bus, PG is the generated active power at bus i, ai, bi and ci
are the unit costs curve for i
th generator.
The smooth quadratic fuel cost function without valve
point loadings of the generating units are given by (4),
where the valve-point effects are ignored. The generating
units with multi-valve steam turbines exhibit a greater
variation in the fuel-cost functions. Since the valve point
results in the ripples, a cost function contains higher order
nonlinearity. Therefore, the equation (4) should be
replaced by (5) for considering the valve-point effects.
The sinusoidal functions are thus added to the quadratic
cost functions as follows.
2
,min( ) sin( ( ))i i i i i i i i i i iF P a b P c P e f P P (5)
where ei and fi are the fuel cost coefficients of the ith
unit
with valve point effects. The shape of fuel cost function
with valve loading effects is given tin Fig. 1.
Figure 1. Example cost function with 6 valves [14]
While minimizing the cost function, it is necessary to
make sure that the generation still supplies the load
demands plus losses in transmission lines. Usually the
power flow equations are used as equality constraints
[14].
( , ) ( )0
( , ) ( )
i i
i i
i G Di
i i G D
P V P PP
Q Q V Q Q
(6)
where active and reactive power injection at bus i are
defined in the following equation
1
( , ) cos sinBN
i i j ij ij ij ij
j
P V VV G B
(7)
1
( , ) sin cosBN
i i j ij ij ij ij
j
Q V VV G B
(8)
The inequality constraints of the OPF reflect the limits
on physical devices in power systems as well as the limits
created to ensure system security. The most usual types
of inequality constraints are upper bus voltage limits at
generations and load buses, lower bus voltage limits at
load buses, reactive power limits at generation buses,
maximum active power limits corresponding to lower
limits at some generators, maximum line loading limits,
and limits on tap setting. The inequality constraints on the
problem are as follows:
Generation constraint: Generator voltages, real power
outputs, and reactive power outputs are restricted by their
upper and lower bounds:
,min ,maxGi Gi GiP P P
for i = 1, 2, . . . . . , NG (9)
,min ,maxGi Gi GiQ Q Q for i = 1, 2, . . . . . , NG (10)
,min ,maxGi Gi GiV V V for i = 1, 2, . . . . . , NG (11)
Shunt VAR constraint: Shunt VAR compensations are
restricted by their upper and lower bounds:
,min ,maxCi Ci CiQ Q Q
for i = 1, 2, . . . . . , NC (12)
where NC is the number of shunt compensators.
Tap changer constraint: Transformer tap settings are
restricted by their upper and lower bounds:
,min ,maxi i iT T T
for i = 1, 2, . . . . . , NT (13)
where NT is the number of transformer taps.
Security constraint: Voltage magnitudes at load buses
are restricted by their upper and lower bounds as follows
,min ,maxLi Li LiV V V for i = 1, 2, . . . . . , NL (14)
where NL is the number of load buses.
Journal of Automation and Control Engineering Vol. 2, No. 1, March 2014
33©2014 Engineering and Technology Publishing
III. IMPROVEMENT OF PSO
A. Overview of the PSO
The conventional PSO was originally introduced by
Kennedy and Eberhart as an optimization technique
inspired by swarm intelligence such as bird flocking, fish
schooling, and even human social behavior. Particles
representing candidate solutions change their positions
with time through search space. During the flight, each
particle adjusts its position according to its own
experience and the experience of neighboring particles as
a constructive cooperation by making use of the best
positions encountered by itself and its neighbors [1]. The
position mechanism of the particles in the search space is
updated by adding the velocity vector to its position
vector as given in equation (20) and as illustrated in Fig.
2 [15]. Let Xi = (xi1,…, xin) and Vi = (vi1,…, vin) be particle
position and its corresponding velocity in a n-dimensional
search space, respectively. The best position achieved by
a particle is recorded and denoted
by i1( ,..., )Pbest Pbest
i inPbest x x . The best particle among all
particles in the population is represented as
i1( ,..., )Gbest Gbest
i inGbest x x . The updated velocity and
position of a particle can be calculated by:
1 1k k ki i iX X V (15)
where Vik+1
is the velocity of individual i at iteration k+1
given by:
11 1 2 2( ) ( )k k k k k k
i i i i iV V c r Pbest X c r Gbest X (16)
Xik position of individual i at iteration k,
Xik+1
position of individual i at iteration k+1,
Vik velocity of individual i at iteration k,
c1 cognitive factors,
c2 social factors,
Pbestik the best position of individual i until
iteration k, Gbest
k the best position of the group until
iteration k,
r1, r2 random numbers between 0 and 1.
Figure 2. Concept of a searching point by PSO [15]
B. Stochastic Weight Trade-off
During the study PSO algorithm we found that the
expression affects the ability of the algorithm
convergence mainly falling into the velocity updating
(16). In this expression, the two components including
the cognitive factor c1 and the social factor c2 are
independent on each other. If both coefficients are too
large or small, there will be effects the convergence of the
algorithm. In the case both coefficients are too large, the
search space is too far beyond the seek region, making
difficulty for the algorithm convergence while both
factors are too small, the search space is too narrow,
leading to inexact optimal results.
To solve these problems, we propose several
improvements to the PSO algorithm. Among the
improved PSO methods, the stochastic weight trade-off
PSO (SWT-PSO) [29] is proposed to solve the optimal
power flow problem due to the goal of balancing between
particle experiences and social relationships as follows:
1) Improvement of r1 and r2 coefficients
r1 and r2 coefficients are also the additional factors (1 -
r1) and (1-r2) as in expression (17). The terms (1-r2) r1
and (1-r1) r2 will improve the algorithm efficiency to
converge faster to the optimal solution. When both r1 and
r2 are too large or too small, the term (1-r2) r1 will lead to
an imbalance for the algorithm. It is similar to the case
for the term (1-r1) r2. This method enables the algorithm
to create a balance between the two components of
personal experiences and learning from the community.
2) Improvement c1 and c2 coefficients
The coefficients c1 and c2 now are not constant as the
original PSO algorithm. We are recommend the time
varying coefficients c1 (k) and c2 (k) as in (18) and (19).
The two expressions will create the value of c1 and c2
factors large at the initialization and decrease them until
the maximum number of iterations reached. When
starting, the algorithm searches for large space to put to
the best possible area. At the algorithm termination, the c1
and c2 factors guide the algorithm converge to the optimal
result.
1
1 2 1 1
1 2 2
(1 ) ( ) ( )
+ (1 ) ( ) ( )
k k k k
i i i i
k k
i
V rV r c k r Pbest X
r c k r Gbest X
(17)
1max 1min
1 1min
max
( )( )
c cc k c k
k
(18)
2max 2min
2 2min
max
( )( )
c cc k c k
k
(19)
r1, r2 random numbers between 0 and 1,
c1min and c1max initial and final cognitive factors,
c2min and c2max initial and final social factors,
kmax maximum iteration number,
k current iteration number.
Journal of Automation and Control Engineering Vol. 2, No. 1, March 2014
34©2014 Engineering and Technology Publishing
C. SWT-PSO Procedure of OPF Problem
The implementation of SWT-PSO algorithm to solve
OPF problem can be described as follows:
Step 1:
Choose
the
population
size,
the
number
of
generations and
coefficients
c1min,
c1max,
c2min,
c2max.
Step 2:
Initialize
the
velocity
and
position
of
all
particles
by randomly
setting
their
values
within
the
pre-specified
boundaries. Set
the
value
of
particle
positions
to
Pbest
and the
particle
corresponding
to
the
best
case
to
Gbest.
Step 3:
Set
the
iteration
counter
k
=
1
and
particle
counter i =
1.
Step 4:
For
each
particle,
solve
AC
power
flow
using
Newton –
Raphson’s
method.
Step 5:
Evaluate
the
fitness
function
for
each
particle
according to
the
objective
function.
Step 6:
Compare
particle’s
fitness
evaluation
with
its
Pbesti. If
the
current
value
is
better
than
Pbesti,
set
Pbesti
to
the
current
value.
Identify
the
particle
with
the
neighborhood with
the
best
success
so
far,
and
assign
its
index to
Gbest.
Step 7:
Update
the
particle
velocity
by
using
the
global
best and
individual
best
of
each
particle
according
to
(17).
Step 8:
Update
particle
position
by
using
(15).
Step 9:
If
i <
total
number
of
particles,
i =
i +
1
and
return to
Step
4.
Step 10:
If
k <
Number
of
iterations,
set
i =
1
and
k =
k +
1, return
to
Step
4.
Step 11:
Stop
the
algorithm.
IV. NUMERICAL RESULTS
The proposed SWT-PSO method is tested on two
systems including the IEEE 30-bus system with quadratic
fuel function and valve point effects and the IEEE 57-bus
system with quadratic fuel function. The algorithm of the
SWT-PSO method is coded in Matlab platform and run
on a 2.5 GHz with 4 GB of RAM PC. The control
parameters of the SWT-PSO method for all test systems
are selected as follows: the cognitive and social
parameters are respectively set to c1(k) and c2(k) with
c1max = c2max
= 2.5, c1min = c2min
= 0.5, the velocity limit
coefficient is set to 0.15 (R = 0.15), the maximum
number of iterations ITmax is set to 200; the number of
particles Np is set to 15 for the IEEE 30-bus system and
25 for the IEEE 57-bus system. All penalty factors in the
fitness function are set to 106. For each system, the
proposed method is run 100 independent trials and the
obtained optimal results are compared to those from other
methods. The obtained results for the systems include
minimum total cost, power losses, and computational
time.
Case 1: The IEEE 30 bus system with quadratic fuel cost
function
TABLE I. OPTIMAL SOLUTION FOR THE IEEE 30-BUS SYSTEM WITH
QUADRATIC FUEL COST FUNCTION
Variable Min Max Optimal Solution
Pg1 (MW) 50 200 177.2529
Pg2 (MW) 20 80 48.3832
Pg5 (MW) 15 50 21.3497
Pg8 (MW) 10 35 21.4238
Pg11 (MW) 10 30 11.7042
Pg13 (MW) 12 40 12.0196
Vg1 (pu) 0.90 1.10 1.1000
Vg2 (pu) 0.90 1.10 1.0850
Vg5 (pu) 0.90 1.10 1.0520
Vg8 (pu) 0.90 1.10 1.0647
Vg11 (pu) 0.90 1.10 1.0857
Vg13 (pu) 0.90 1.10 1.0999
T11 (pu) 0.90 1.10 1.0317
T12 (pu) 0.90 1.10 0.9074
T15 (pu) 0.90 1.10 0.9865
T36 (pu) 0.90 1.10 0.9555
Qc10 (MVAr) 0 19 15.9942
Qc24 (MVAr) 0 4.3 4.2934
Ploss (MW) 8.7335
CPU Time (s) 12.184
Total Cost ($/h) 799.4637
To verify the feasibility of the proposed SWT-PSO
method, the standard IEEE 30-bus system [17] has been
used to test the OPF problem. The system line and bus
data are given in [18]. The system has six generators
located at buses 1, 2, 5, 8, 11, and 13 and four
transformers with off-nominal tap ratio in lines 6-9, 6-10,
4-12, and 28-27. The cost curve coefficients are given in
[19].
Table I shows the optimal dispatches of the generators.
Also note that all outputs of generator are within its
permissible limits. The obtained results of the SWT-PSO
are compared with those of other methods in Table II
including GA [20], ACO [21], IEP [22], and EP [23]. In
Table II, it is observed that SWT-PSO algorithm gives
better total cost than other methods in a fester manner.
These results have shown that the proposed method is
feasible and indeed capable of acquiring better solution.
Fig. 3 shows the convergence characteristic of SWT-PSO.
TABLE II. RESULT COMPARISON FOR THE IEEE 30 BUS SYSTEM WITH
QUADRATIC FUEL COST FUNCTION
Variable GA [20] ACO [21] IEP [22] EP [23] SWT-PSO
Pg1 (MW) 179.367 177.8635 176.2358 173.848 177.2529
Pg2 (MW) 44.24 43.8366 49.0093 49.998 48.3832
Pg5 (MW) 24.61 20.8930 21.5023 21.386 21.3497
Pg8 (MW) 19.90 23.1231 21.8115 22.630 21.4238
Pg11 (MW) 10.71 14.0255 12.3387 12.928 11.7042
Pg13 (MW) 14.09 13.1199 12.0129 12.000 12.0196
Ploss (MW) 9.5177 9.4616 9.5105 9.3700 8.7335
CPU Time (s) 315 20 99.013
(minutes) 51.4 12.184
Total Cost ($/h) 803.699 803.123 802.465 802.62 799.4637
Case 2: The IEEE 30-bus system with valve point effects
fuel function.
In this case, the generating units of buses 1 and 2 are considered to have the valve-point effects on their fuel cost characteristics. The fuel cost coefficients of these generators are taken from [28]. The fuel cost coefficients
Journal of Automation and Control Engineering Vol. 2, No. 1, March 2014
35©2014 Engineering and Technology Publishing
of the remaining generators have the same values as of
the test Case 1.
Figure 3. Convergence characteristic with quadratic fuel function for the IEEE 30-bus system
Table III shows the generation outputs of the best
solution. We have also observed that the solution
obtained by SWT-PSO always satisfies the equality and
inequality constraints. The result comparison in Table IV
has indicated that the SWT-PSO algorithm gives better
results than other methods with the percentage as follow
IEP [22] 3.3%, GSA [24] 0.82%, DE–OPF [25] 0.96%,
MDE–OPF [25] 0.93%. Therefore, the proposed SWT-
PSO is very effective for solving the OPF problem with
valve point loading effects.
Case 3: The IEEE 57 bus system
To evaluate the effectiveness and efficiency of the
proposed SWT-PSO approach in solving larger power
system, a standard IEEE 57-bus test system is considered.
The IEEE 57-bus system consists of 7 generation buses,
50 load buses, and 80 branches. The generators are
located at buses 1, 2, 3, 6, 8, 9, and 12 and 15
transformers are located at branches 19, 20, 31, 37, 41, 46,
54, 58, 59, 65, 66, 71, 73, 76, and 80. The system has also
3 switchable capacitor banks installed at buses 18, 25,
and 53. For dealing with this system, there 31 control
variables to be handled including real power output of 6
generators except the generator at the slack bus, voltage
at 7 generation buses, tap changer of 15 transformers, and
reactive power output of 3 switchable capacitor banks.
The total load demand of system is 1250.8 MW and
336.4 MVAR. The bus data, line data, cost coefficients,
and minimum and maximum limits of real power
generations are taken from [26], [27]. The maximum and
minimum values for voltages of all generator buses and
transformer tap settings are considered to be 1.1 and 0.9
in p.u. The maximum and minimum values for voltages
of all load buses are 1.06 and 0.94 in p.u [24].
Table V shows the optimal solution of the problem by
the conventional PSO and SWT-PSO methods. The
minimum cost obtained by this algorithm is compared
with BASE-CASE [28], MATPOWER [28], and
conventional PSO as presented in Table VI. The result
comparison has demonstrated that the proposed method
can give the lowest production cost within reasonable
time. The convergence characteristic of the best fuel cost
result obtained from the SWT-PSO approach is shown in
Fig. 4.
TABLE III. OPTIMAL SOLUTION FOR THE IEEE 30-BUS SYSTEM
WITH VALVE POINT EFFECTS
Variable Min Max Optimal Solution
Pg1 (MW) 50 200 191.7845
Pg2 (MW) 20 80 51.0558
Pg5 (MW) 15 50 15.3289
Pg8 (MW) 10 35 10.0758
Pg11 (MW) 10 30 14.2326
Pg13 (MW) 12 40 12.0021
Vg1 (pu) 0.90 1.10 1.0660
Vg2 (pu) 0.90 1.10 1.0534
Vg5 (pu) 0.90 1.10 1.0226
Vg8 (pu) 0.90 1.10 1.0057
Vg11 (pu) 0.90 1.10 1.0848
Vg13 (pu) 0.90 1.10 1.0347
T11 (pu) 0.90 1.10 0.9969
T12 (pu) 0.90 1.10 0.9603
T15 (pu) 0.90 1.10 0.9794
T36 (pu) 0.90 1.10 0.9480
Qc10 (MVAr) 0 19 12.4459
Qc24 (MVAr) 0 4.3 2.9992
Ploss (MW) 11.0797
CPU Time (s) 9.766
Total Cost ($/h) 922.1029
TABLE IV. RESULTS FOR THE VALVE POINT EFFECTS FOR IEEE 30-BUS SYSTEM WITH DIFFERENT METHOD
Variable IEP [22] GSA [24] DE–OPF
[25]
MDE–OPF
[25] SWT-PSO
Pg1 (MW) 149.7331 199.5994 196.989 197.426 191.7845
Pg2 (MW) 52.0571 51.9464 51.995 52.037 51.0558
Pg5 (MW) 23.2008 15.0000 15.000 15.000 15.3289
Pg8 (MW) 33.4150 10.0000 10.006 10.000 10.0758
Pg11 (MW) 16.5523 10.0000 10.015 10.001 14.2326
Pg13 (MW) 16.0875 12.0000 12.000 12.000 12.0021
Ploss (MW) 7.6458 15.1458 12.605 13.064 11.0797
CPU Time
(s)
93.583
(minutes) 9.8374 44.96 41.85 9.766 (s)
Total Cost
($/h) 953.573 929.7240 931.085 930.793 922.1029
TABLE V. OPTIMAL SOLUTIONS FOR THE IEEE 57-BUS SYSTEM
Variable PSO SWT-PSO Variable PSO SWT-PSO
Pg1 (MW) 139.1571 146.3227 T31 (pu) 1.00 1.0387
Pg2 (MW) 100.0000 78.5966 T37 (pu) 1.05 0.9734
Pg3 (MW) 75.8451 45.3091 T41 (pu) 0.99 0.9673
Pg6 (MW) 38.4932 71.2516 T46 (pu) 0.92 1.0544
Pg8 (MW) 455.5600 461.2314 T54 (pu) 0.99 1.0155
Pg9 (MW) 100.0000 86.5425 T58 (pu) 0.99 0.9561
Pg12 (MW) 360.2540 377.1499 T59 (pu) 0.95 0.9772
Vg1 (pu) 1.0399 1.0593 T65 (pu) 0.98 0.9674
Vg2 (pu) 1.0319 1.0477 T66 (pu) 1.02 0.9430
Vg3 (pu) 1.0378 1.0379 T71 (pu) 0.90 0.9776
Vg6 (pu) 1.0621 1.0596 T73 (pu) 1.00 0.9654
Vg8 (pu) 1.1000 1.0688 T76 (pu) 1.01 1.0246
Vg9 (pu) 1.0369 1.0402 T80 (pu) 0.97 1.0010
Vg12 (pu) 0.9892 1.0440 Qc18 (MVAr) 3.5 7.7338
T19 (pu) 1.02 0.9993 Qc25 (MVAr) 3.0 3.3608
T20 (pu) 1.04 0.9651 Qc53 (MVAr) 3.3 3.8498
Journal of Automation and Control Engineering Vol. 2, No. 1, March 2014
36©2014 Engineering and Technology Publishing
TABLE VI. RESULT COMPARISON FOR THE IEEE 57-BUS SYSTEM
Methods Total cost
($/h) Ploss (MW)
CPU time (s)
BASE-CASE [28] 51347.86 - -
MATPOWER [28] 41737.79 - -
PSO 42109.7231 1786.3245 8.814
SWT-PSO 41733.4425 15.6038 22.511
Figure 4.
Convergence characteristic
for
the
IEEE
57-bus
system
V.
CONCLUTION
In this
paper,
the
stochastic
weight
trade-off
particle
swarm optimization
method
has
been
presented
to
solve
the
OPF
problem.
The
improved
SWT-PSO
has
advantages
such
as
simple
algorithm
and
easy
to
use.
Moreover,
the
algorithm
can
be
implemented
in
the
whole problem
search
space
rather
than
individual
points,
leading faster
position
updating
function
of
particles.
The
proposed method
has
been
tested
on
the
IEEE
30
bus
and
57 bus
systems
and
the
obtained
results
are
compared
to
those
from
many
other
methods
in
the
literature.
The
numerical
results
show
the algorithm’s
flexibility
and
capability in
finding
the
optimal
solution.
Therefore,
the
proposed SWT-PSO
can
be
very
favorable
for
solving
OPF
problem,
especially
for
large
scale
systems
with
nonconvex objective
function.
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J. Kennedy
and
R.
C.
Eberhart,
“Particle
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International
Conference
on
Neural
Networks,
Perth,
Australia, vol.
IV,
1995,
pp.
1942-1948.
[2]
N. Sinha,
R.
Chakrabarti
and
P.
K.
Chattopadhyay,
“Evolutionary
programming techniques
for
economic
load
dispatch,”
IEEE
Trans.
Evolutionary Computation,
vol.
7,
no.
1,
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