International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
17938163
560
Abstract—An Evolutionary Programming (EP) and
Efficient Particle Swarm Optimization (EPSO) techniques are
employed to solve Economic Dispatch (ED) problems including
transmission losses in power system is presented in this paper.
This paper is clearly justified with the results separately
obtained for the above two techniques and also provided with
the results by applying both the algorithms separately. With
practical consideration, ED will have nonsmooth cost
functions with equality and inequality constraints that make
the problem, a largescale highly constrained nonlinear
optimization problem. The proposed method expands the
original PSO to handle a different approach for satisfying
those constraints. In this paper, an Efficient Particle Swarm
Optimization (EPSO) technique is employed so that optimized
results are obtained, and by applying EP, faster convergence is
obtained . To demonstrate the effectiveness of the proposed
method it is being applied to test ED problems, one with
smooth and other with non smooth cost functions considering
valvepoint loading effects. Comparison with other
optimization and hybrid algorithm techniques showed the
superiority of the proposed EPEPSO approach and confirmed
its potential for solving nonlinear economic load dispatch
problems with losses.
Index Terms—Economic load dispatch , Evolutionary
Programming , Efficient Particle swarm optimization , Valve
point loading effect
I. I
NTRODUCTION
Economic Dispatch (ED) problem is one of the
fundamental issues in power system operation. In essence, it
is an optimization problem and its main objective is to
reduce the total generation cost of units, while satisfying
constraints.[1] Previous efforts on solving ED problems
have employed various mathematical programming
methods and optimization techniques excluding losses.
Recently, Eberhart and Kennedy suggested a Particle
Swarm Optimization (PSO) based on the analogy of swarm
Dr.K.Thauskkodi is with Director of Akshaya College of Engineering
and Technology , Coimbatore , Tamilnadu , 642 109 , India .email :
thanush_dr@rediffmail.com .
Manuscript received October 30 , 2009.
S.Muthu Vijaya Pandian is with Department of Electrical and
Electronics Engineering , V.L.B.Janakiammal College of Engineering and
Technology , Coimbatore , Tamil Nadu , 640 042 , India. (Phone : +91
98652 59633 ); Fax : +91 0422 2607152(email:
ajay_vijay@rediffmail.com).
of bird and school of fish. In PSO, each individual makes its
decision based on its own experience together with other
individual’s experiences. In artificial intelligence, an
evolutionary algorithm (EA) is a subset of evolutionary
computation, a generic populationbased metaheuristic
optimization algorithm. An EA uses some mechanisms
inspired by biological evolution: reproduction, mutation,
recombination, and selection. Candidate solutions to the
optimization problem play the role of individuals in a
population, and the fitness function determines the
environment within which the solutions "live" (see also cost
function). Evolution of the population then takes place after
the repeated application of the above operators. Artificial
evolution (AE) describes a process involving individual
evolutionary algorithms; EAs are individual components
that participate in an AE. The main advantages of the PSO
algorithm are summarized as: simple concept, easy
implementation, and computational efficiency when
compared with mathematical algorithm and other heuristic
optimization techniques[7]. The practical ED problems with
valvepoint loading effects are represented as a non smooth
optimization problem with equality and inequality
constraints. To solve this problem, many salient methods
have been proposed such as dynamic programming,
evolutionary programming, neural network approaches, and
genetic algorithm. In this paper, an alternative approach is
proposed to the non smooth ED problem using an Efficient
PSO (EPSO), which focuses on the treatment of the equality
and inequality constraints when modifying each
individual’s search. The equality constraint (i.e., the
supply/demand balance) is easily satisfied by specifying a
variable (i.e., a generator output) at random in each iteration
as a slag generator whose value is determined by the
difference between the total system demand (including
losses) and the total generation excluding the slag generator.
However, the inequality constraints in the next position of
an individual produced by the PSO algorithm can violate
the inequality constraints. In this case, the position of any
individual violating the constraints is set to maximum or
minimum depending on velocity evaluated.
Solving Economic Load Dispatch Problem
Considering Transmission Losses by a Hybrid
EPEPSO Algorithm for Solving both Smooth
and NonSmooth Cost Function
S. Muthu Vijaya Pandian and K. Thanushkodi
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
17938163
561
II. F
ORMULATION OF
E
CONOMIC
D
IAPATCH
P
ROBLEM
A. ED Problem with Smooth Cost Functions
Economic load dispatch (ELD) pertains to optimum
generation scheduling of available generators in an
interconnected power system to minimize the cost of
generation subject to relevant system constraints. Cost
equations are obtained from the heat rate characteristics
of the generating machine. Smooth costs functions are
linear, differentiable and convex functions. The most
simplified cost function of each generator can be
represented as a quadratic function as given in whose
solution can be obtained by the conventional
mathematical methods [2] :
C=∑F
j
P
j
(1)
F
j
P
j
=a
j
+b
j
P
j
+c
j
P
j
2
(2)
where C total generation cost
Fj cost function of generator j
aj bj cj cost coefficients of generator j
Fj cost function of generator j
aj bj cj cost coefficients of generator j
While minimizing the total generation cost, the total
generation should be equal to the total system demand
plus the transmission network loss.
The transmission loss is given by the equation,
PL= ∑ BojPj (3)
where B
oj
is the loss coefficient matrix.
The equality constraint for the ED problem can be
given by,
∑ Pj = D +∑PL (4)
where D is the total demand needed by the load or
consumer. The generation output of each unit should be
between its minimum and maximum limits. That is, the
following inequality constraint for each generator should
be satisfied
Pjmin<Pj<Pjmax (5)
where P
jmin
, P
jmax
are the minimum and maximum output
of individual generators.
B. ED Problem with Nonsmooth Cost Functions
In reality, the objective function of an ED problem has
non differentiable points according to valvepoint effects.
Therefore, the objective function should be composed of
a set of nonsmooth cost functions. In this paper, one case
of nonsmooth cost function is considered i.e. the valve
point loading problem where the objective function is
generally described as the superposition of sinusoidal
functions and quadratic functions [7].
C. Nonsmooth Cost Function with ValvePoint
Effects
The generator with multivalve steam turbines has very
different inputoutput curve compared with the smooth
cost function[6]. Typically, the valve point results in, as
each steam valve starts to open, the ripples like in to take
account for the valvepoint effects, sinusoidal functions
are added to the quadratic cost functions as follows:
FjPj=aj+bjPj+cjPj2+ejxsin(fjx(PjminPj)) (6)
III. E
VOLUTIONARY
P
ROGRAMMING
EP is a near global search stochastic optimization
method starting from multiple points, which placed
emphasis on the behavioural linkage between the parents
and their offsprings, rather than seeking to emulate
specific operators as observed in nature to find a solution.
However EP takes a long computation time to find a
solution and sometime EP suffers from the convergence
problem. On the other hand, EPSO is a gradient based
optimization method starting from a single point and
using gradient information to obtain a solution. The
solution obtained from EPSO is a local optimal solution.
In order to obtain a high quality solution , the first part,
EP is applied to obtain a near global solution. After the
specified termination criteria for EP is reached, EPSO is
applied in the second part by using the solution from EP
as an initial starting point and searches by using a
gradient information to obtain the final optimal solution
[2].
A. Evolutionary Programming Subproblem
1) Representation
For ‘m ,particles in the swarm and ‘n’ generators of the
systems, the array of control variable vectors (S) can be
shown as
P
11
P
12
…….. P
1m
P
21
P
22
…….. P
2m
P
n1
P
n2
………. P
nm
2) Initialization
To begin the population of chromosomes is uniform
randomly initialized within the operation range of the
generator [5].
3) Fitness Evaluation
The active power generations at all the buses except the
first bus in all intervals are control variable, which are
itself constrained. Equality constraint can be handled as
2
n
ij Dj ij
i
P P P
=
= −
∑
4) Creation of Offspring
A new population of solutions is produced from the
existing population by adding a guassian random number
with zero mean and predefined standard deviation as
follows:
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
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562
Pit’ = Pit + N(0,σit2) (7)
Where σ
it
can be calculated from the following equation:
σit = ? * fs/fmax(Pitmax Pitmin) (8)
Where ϐ is a scaling factor, which can be tuned during the
process of search for optimum.
5) Selection And Competition
The selection technique used here is the stochastic
tournament method. The 2P individuals compete with
each other for selection. A weight value ωs is assigned to
each individual as follows
1
p
j
s jω ω
=
=
∑
(9)
ωj = 1 if fs<fr
ωj =0 otherwise
In biological DNA systems, the basic units are the
adenine (A), thymine (T), guanine (G) and cytosine (C)
nucleotides that join the helical strands. In genetic
algorithms, the basic unit is called a symbol. The nature
of symbols depends on the particular genetic algorithm.
In gene expression programming, the symbols consist of
functions, variables and constants. Symbols for variables
and constants are called terminals, because they have no
arguments.
An ordered set of symbols form a gene, and an ordered
set of genes form a chromosome. In GEP programs, genes
typically have 4 to 20 symbols, and chromosomes are
typically built from 2 to 10 genes; chromosomes may
consist of only a single gene. The DNA strand for a
mammal typically contains about 5x10
9
nucleotides.
B. Gene Expression Programming
Gene Expression Programming is a procedure that
mimics biological evolution to create a computer program
to model some phenomenon. Gene expression
programming can be used to create many different types
of models including decision trees, neural networks and
polynomial constructs. The type of gene expression
programming implemented in DTREG is Symbolic
Regression so named because it creates a symbolic
mathematical or logical function [4].
Fig.1 Biological evolution of EP
DTREG provides a full implementation of the Gene
Expression Programming algorithm developed by
Cândida Ferreira. Here are some of the features of
DTREG’s implementation :
• Continuous and categorical target variables
• Automatic handling of categorical predictor
variables
• A large library of functions that you can select
for inclusion in the model
• Mathematical and logical (AND, OR, NOT, etc.)
function generation
• Choice of many fitness functions
• Both static linking functions and evolving
homeotic genes
• Fixed and random constants
• Nonlinear regression to optimize constants
• Parsimony pressure to optimize the size of
functions
• Automatic algebraic simplification of the
combined function
• Several forms of validation including cross
validation and holdout
C. Expression Trees and Karva
The key to GEP’s ability to quickly mutate valid
expressions is the way it encodes symbols in genes. This
notation is called the Karva Language. Expressions
encoded using Karva are called Kexpressions. Consider
the simple mathematical expression:
a*b+c (10)
This can be encoded as an expression tree of the form
Fig 2: Karvaexpressions encoding model 1
An expression tree is an excellent way to represent an
expression in a computer, because the tree can be
arbitrarily complex, and expression trees can be evaluated
quickly.
To convert an expression tree to the Karva notation,
start at the leftmost symbol in the top line of the tree and
scan symbols lefttoright and toptobottom. Each time a
symbol is encountered, add it to the Kexpression in left
toright order. When there are no more symbols on a line,
advance to the left end of the following line. Using this
method, the tree shown above is converted to the K
expression:
+*cab (11)
Note that + is the first symbol found on the first line, at
the end of that line scanning begins on the second line
and finds * followed by c. It then starts with the third line
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
17938163
563
and finds a and b.
As a second example, consider the expression
a*b+sqrt(c*d) (12)
The corresponding expression tree is
Fig 3: Karvaexpressions encoding model 2
Where ‘Q’ represents square root. This can be translated
to the Kexpression
+*Qab*cd (13)
The process of converting an expression tree to a K
expression can be carried out quickly by a computer. A
reverse process can quickly convert a Kexpression back
to an expression tree.
D. Implementation of PSO for ED Problems
The PSO algorithm searches in parallel using a group
of individuals, in a physical dimensional search space, the
position and velocity of individual i are represented as the
vectors Xi = (xib……….xin) and Vi = (vib……….vin) )
respectively, be the position of the individual i and its
neighbors’ best position so far. Using the information, the
updated velocity of individual i is modified under the
following equation in the PSO algorithm[3]:
V
i
k+1
=ωV
i
k
+c
1
rand1 x (P
besti
k
 x
i
k
)+c
2
rand
2
x (G
best
k
– x
i
k
)
Where
V
i
k+1
velocity of individual of iteration at k
ω weight parameter
C
1
, C
2
acceleration factors
rand
1
rand
2
random numbers between 0 and 1.
X
i
k
Position of individual i at iteration k
Pbest best position of group throughout iteration k
Each individual moves from the current position to the
next one by the using the following equation:
X
ik+1
=X
ik
+X
ik+1
The search mechanism of the PSO using the modified
velocity and position of the individual i based on (7)
and (8) is illustrated in fig (4)
Fig. 4 Search mechanism of PSO
IV. E
FFICIENT
PSO
FOR
ED
P
ROBLEMS
In this section, a new approach to implement the PSO
algorithm will be described while solving the ED
problems considering losses [3]. The main process of the
efficient PSO algorithm can be summarized as follows:
Step1) Initialization of a group at random while
satisfying constraints.
Step2) Velocity and position updates while satisfying
constraints
Step3) Update of Pbest and Gbest.
Step4) Calculate transmission losses for the obtained
Pbest and Gbest
Step5) Increment the demand with the transmission
losses
Step6) Go to Step 2 until satisfying stopping criteria.
In the subsequent sections, the detailed implementation
strategies of the EPSO are described.
A. Initialization of Individuals
In the initialization process, a set of individuals (i.e.,
generation outputs) is created at random. Therefore,
individual i position at iteration 0 can be represented as
the vector of n is the number of generators.[3] The
velocity of individual i is given by corresponds to the
generation update quantity covering all generators. The
following procedure is suggested for satisfying
constraints for each individual in the group:
Step1) Set j=1, i=1 element (i.e., generator) of an
individual i.
Step2) Select the jth element of the individual i.
Step3) Create the value of the element (i.e., generation
output) at random satisfying its inequality constraint.
Step4) If j=n1 then go to step 5; otherwise j=j+1 and go
to Step 2.
Step5) The value of the last element of an individual is
determined by subtracting ∑P from the total demand
Step6) If i=no of individuals go to step 7; otherwise put
i=i+1 and go to step 2.
Step7) Stop the initialization process.
After creating the initial position of each individual, the
velocity of each individual is also created at random. The
following strategy is used in creating the initial velocity:
(P
min
 €)P
ij
0
<v
ij
0
<P
max
€P
ij
0
(14)
Where e is a small positive real number
The velocity of element j of individual i is generated at
random within the boundary [8].
B. Velocity Update
To modify the position of each individual, it is
necessary to calculate the velocity of each individual in
the next stage, which is obtained from (7). In this velocity
updating process, the values of parameters such as w, c1
and c2 should be determined in advance. The weighting
function is defined as follows
w=w
max
(w
max
w
min
/iter
max
)*iter (15)
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
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564
Where
w
max
final weight
w
min
initial weight
iter
max
maximum number of iteration
iter current iteration number
C. Position Modification Considering Constraints
The position of each individual is modified by (8). The
resulting position of an individual is not always
guaranteed to satisfy the inequality constraints due to
over/under velocity [4]. If any element of an individual
violates its inequality constraint due to over/under speed
then the position of the individual is fixed to its maximum
or minimum operating point. Therefore, this can be
formulated as follows:
(16)
To resolve the equality constraint problem without
intervening the dynamic process inherent in the PSO
algorithm, we propose the following procedures:
Step1) Set j=1, i=1. Let the present iteration be k.
Step2) Select the jth element (i.e., generator) of an
individual i.
Step3) Modify the value of element j using (7), (8), and
(11).And satisfy inequality constraint.
Step4) If j=n1 then go to Step 5, otherwise j=j+ 1and go
to Step 2.
Step5) The value of the last element of an individual is
obtained by subtracting
∑P
ij
0
from the total system
demand.
Step6) If i=no. of individuals then go to step 7; otherwise
i=i+1 and go to Step2
Step7) Stop the modification procedure
D. Update of Pbest and Gbest
The Pbest of each individual at iteration k+1 is updated
as follows:
P
besti
k+1
=X
i
k+1
if TC
i
k+1
< TC
i
k
(17)
P
besti
k+1
=P
besti
k
if TC
i
k+1
> TC
i
k
(18)
Where
TC
i
– object function evaluated at the position of the
individual i.
Additionally, Gbest at iteration k+1 is set as the best
evaluated position among Pbest
k+1
V. S
IMULATED
R
ESULT
A
NALYSES
A. ED Problem with Non Smooth Cost Functions with
Valve point effect
TABLE
1:
I
NPUT DATA FOR
40
U
NIT
S
YSTEM
Generator P
jmin
P
jmax
a
i
b
i
c
i
e
i
f
i
1 36 114 0.00690 6.73 94.705 100 0.084
2 36 114 0.00690 6.73 94.705 100 0.084
3 60 120 0.02028 7.07 309.54 100 0.084
4 80 190 0.00942 8.18 369.54 150 0.063
5 47 97 0.01140 5.35 369.03 120 0.077
6 68 140 0.01142 8.05 148.89 100 0.084
7 110 300 0.01142 8.03 222.33 200 0.042
8 135 300 0.00357 6.99 287.71 200 0.042
9 135 300 0.00492 6.60 391.88 200 0.042
10 130 300 0.00573 12.9 455.76 200 0.042
11 94 375 0.00605 12.9 722.82 200 0.042
12 94 375 0.00515 12.8 635.20 200 0.042
13 125 500 0.00569 12.5 654.69 300 0.035
14 125 500 0.00421 8.84 913.40 300 0.035
15 125 500 0.00752 9.15 1760.4 300 0.035
16 125 500 0.00708 9.15 1728.3 300 0.035
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
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565
17 220 500 0.00708 7.97 1728.3 300 0.035
18 220 500 0.00313 7.95 647.83 300 0.035
19 242 550 0.00313 7.97 647.81 300 0.035
20 242 550 0.00313 7.97 647.85 300 0.035
21 254 550 0.00313 6.63 785.96 300 0.035
22 254 550 0.00218 6.63 785.96 300 0.035
23 254 550 0.00284 6.66 794.53 300 0.035
24 254 550 0.00284 6.66 794.53 300 0.035
25 254 550 0.00277 7.10 801.32 300 0.035
26 254 550 0.00277 7.10 801.32 300 0.077
27 10 150 0.52124 3.33 1055.1 120 0.077
28 10 150 0.52124 3.33 1055.1 120 0.077
29 10 150 0.52124 6.43 1055.1 120 0.077
30 47 97 0.01140 6.43 148.89 120 0.063
31 60 190 0.00160 6.43 222.92 150 0.063
32 60 190 0.00160 8.95 222.92 150 0.063
33 60 190 0.00160 8.62 222.92 150 0.042
34 90 200 0.00010 8.62 107.87 200 0.042
35 90 200 0.00010 5.88 116.58 200 0.042
36 90 200 0.00010 5.88 116.58 200 0.098
37 25 110 0.0161 5.88 307.45 80 0.098
38 25 110 0.0161 3.33 307.45 80 0.098
39 25 110 0.0161 3.33 307.45 80 0.098
40 242 550 0.00313 7.97 647.83 300 0.035
TABLE
2:
H
YBRID
R
ESULTS FOR
(NNEPSO)
40
U
NIT
S
YSTEMS
Unit Output (MW)
1 114
2 114
3
120
4
190
5
97
6
140
7
300
8
300
9
300
10
300
11
375
12
375
13
500
14
500
15
500
16
500
17
409.273
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
17938163
566
18
225
19
508
20
458
21
356
22
394
23
355
24
525
25
310
26
448
27
72
28
131
29
75
30
67
31
151
32
112
33
139
34
90
35
129
36
104
37
36
38
89
39
104
40
550
Demand = 10500 MW
Losses = 62.27 MW
Optimal cost = 130328.3256 $/hr
Elapsed Time = 8.3590 sec
The above table 2 given the output results for forty unit
systems hybrid Neural network and Efficient particle
swarm optimization method [8]
Fig 4: Convergence plot for 40 unit systems in
(NNEPSO) Method
B. Simulated result sfor forty Units hybrid EPEPSO
TABLE
3:
O
UTPUT
R
ESULTS FOR
F
ORTY
U
NITS
H
YBRID
(EPEPSO
)
M
ETHOD
Generator
Output (MW)
1
114
2
114
3
120
4
190
5
97
6
140
7
300
8
300
9
300
10
300
11
375
12
375
13
500
14
500
15
500
16
500
17
409.273
18
225
19
510
20
458
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
17938163
567
21
354
22
394
23
355
24
525
25
310
26
450
27
72
28
129
29
75
30
67
31
151
32
112
33
139
34
90
35
129
36
104
37
36
38
89
39
104
40
550
Demand = 10500 MW
Losses = 61.57 MW
Optimal cost = 130227.3256 $/hr
Elapsed Time = 7.7590 sec
1
2
3
4
5
6
7
8
9
10
1.26
1.27
1.28
1.29
1.3
1.31
1.32
x 10
5
ITERATIONS
TOTAL COST ($/hr)
ECONOMIC LOAD DISPATCH USING HYBRID EPEPSO
Fig 5: Convergence plot for 40 unit systems in
(EPEPSO) Algorithm
TABLE
4:
C
OMPARISON BETWEEN
H
YBRID
EPEPSO
AND
NEUROEPSO
M
ETHOD
Generator
Output Of Hybrid
NEUROEPSO (MW)
Output Of Hybrid EP
EPSO (MW)
1
114 114
2
114 114
3
120 120
4
190 190
5
97 97
6
140 140
7
300 300
8
300 300
9
300 300
10
300 300
11
375 375
12
375 375
13
500 500
14
500 500
15
500 500
16
500 500
17
409.273 409.273
18
225 225
19
508 510
20
458 458
21
356 354
22
394 394
23
355 355
24
525 525
25
310 310
26
448 450
27
72 72
28
131 129
29
75 75
30
67 67
31
151 151
32
112 112
33
139 139
International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010
17938163
568
34
90 90
35
129 129
36
104 104
37
36 36
38
89 89
39
104 104
40
550 550
TABLE
5
C
OMPARISION OF
T
OTAL
P
RODUCTION
C
OST AND
S
IMULATION
T
IME AMONG
NNEPSO
AND
EPEPSO
VI. C
ONCLUSION
In this paper, a new methodology for solving non
smooth ED problem including valve point loading using
EP combined with EPSO. The proposed algorithm
consists of two parts. The first part employs the property
of EP, which can provide a near global search region at
the beginning. When the specified termination criteria of
EP is reached, the local search EPSO is applied to tune
the control variables in order to obtain the final optimal
solution. It is clear from the Table 5 mean cost value and
simulation obtained by EPEPSO is comparatively less
compared to all other methods. Simulation results
demonstrate that the proposed method can give a cheaper
total production cost than those obtained from EPSO, EP,
NNEPSO and EPSQP. The resultant EPEPSO has been
suggested as a powerful optimization tool for nonconvex
ED problem.
R
EFERENCES
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SQP for Dynamic Economic Dispatch with Nonsmooth Fuel Cost
Function ” IEEE Transactions on Power Systems , vol 17 ,no.2 ,
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[3] J.B.Park , K.S.Lee , K.Y.Lee , “ A Patticle Swarm Optimization
for Economic Dispatch with Nonsmooth Cost Function ,” IEEE
Trans. Power Systems , vol 20 , Feb 2005.
[4] H.T.Yang , P.C.Yang , and C.L.Huang , “ Evolutionary
Programming Based Economic Dispatch for Units with Non
Smooth Fuel Cost Functions , ” IEEE Trans. Power Systems , vol
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[5] N.Sinha , R. Chakrabarti and P.K.chattopadhyay , “Evolutionary
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Trans. Evol. Comput,vol 7, pp.8394, Feb 2003.
[6] J.Nanda,A.Sachan,.L.Pradhan M.L.Kothari,,A.Koteswara Rao ,
“ Application of Artificial Neural Network to Economic Load
Dispatch , ” IEEE Trans. Power Systems , vol 22 , 1997.
[7] B.H.Chowdhury and S.Rahman , “ A Review of Recent Advances
in Economic Dispatch , ” IEEE Trans. Power Systems , vol
5 ,no.4 , pp.12481259 , Nov 1990.
[8] S.Muthu Vijaya Pandian and Dr.K.Thanushkodi , “ A Hybrid
(Neural Network – Efficient Particle Swarm Optimization (NN
EPSO) for Economic Dispatch Problems Considering
Transmission Losses with Nonsmooth Cost Functions ”
International Journal of Computer Science and Electrical
Engineering (IJCEE) no.3, vol1 June 2009 ,pg 207214.
S.Muthu Vijaya Pandian was born in Karaikudi , Tamil Nadu , India
on the 19th of June 1978. He received his M.B.A from Madurai
Kamaraj University , Madurai ,Tamil Nadu , India , in 2002 . He then
received his M.E in Power Systems from Government College of
Technology , Coimbatore , Tamil Nadu , India., in 2004 and his
currently pursuing his Ph.D in Anna university , Chennai , India . He is
currently Lecturer at the Department of Electrical and Electronics
Engineering ,V.L.B.Janakiammal College of Engineering and
Technology , Coimbatore , Tamil Nadu , India. He has published Three
International Journal , One International conference and his current
research interests include areas of Power Systems.
Dr.K.Thanushkodi was born in Theni District , Tamil Nadu , India in
1948. He received the B.E degree in Electrical and Electronics
Engineering and the M.sc.(Engg) degree from Madras University ,
Chennai , India in 1972 and 1974 , respectively , and the Ph.D degree in
Electrical and Electronics Engineering from Bharathiar University ,
Coimbatore , India , in 1994. He served as thirty three years of teaching
experience and guided 4 Ph.D . Now he guiding 12 Research scholars
in Anna University , Chennai and 30 Research scholar in Anna
University , Coimbatore , India. He published 75 papers in International ,
National Journals and Conferences. He is currently Director of Akshaya
College of Enggineering and Technology , Coimbatore and also
Syndicate member Anna University ,Chennai , India. His research
include computer modeling and simulation , computer networking and
power systems. .
Method Cost $/hr Time(s)
NN 146069.7350 28.07
EPSO 130330.3647 7.232
EP 143799.0000 9.242
NNEPSO 130328.3246 8.3529
EPEPSO 130227.3256 7.7590
EPSQP 1035748.0000 1251.0
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