Engineering, 2009, 1, 140-150
doi:10.4236/eng.2009.13017 Published Online November 2009 (http://www.scirp.org/journal/eng).
Multi-Area Unit Commitment Using Hybrid Particle
Swarm Optimization Technique with Import and Export
Constraints
S. R. P. CHITRA SELVI1, R. P. KUMUDINI DEVI2, C. CHRISTOBER ASIR RAJAN3
1Department of Electrical Engineering, Anna University, Chennai, India
2Department of Electrical Engineering, Anna University, Chennai, India
3Department of Electrical Engineering, Pondicherry Engg.College, Pondicherry, India
Received January 10, 2009; revised February 21, 2009; accepted February 23, 2009
Abstract
This paper presents a novel approach to solve the Multi-Area unit commitment problem using particle swarm
optimization technique. The objective of the multi-area unit commitment problem is to determine the optimal
or a near optimal commitment strategy for generating the units. And it is located in multiple areas that are
interconnected via tie lines and joint operation of generation resources can result in significant operational
cost savings. The dynamic programming method is applied to solve Multi-Area Unit Commitment problem
and particle swarm optimization technique is embedded for computing the generation assigned to each area
and the power allocated to all committed unit. Particle Swarm Optimization technique is developed to derive
its Pareto-optimal solutions. The tie-line transfer limits are considered as a set of constraints during the opti-
mization process to ensure the system security and reliability. Case study of four areas each containing 26
units connected via tie lines has been taken for analysis. Numerical results are shown comparing the cost so-
lutions and computation time obtained by using the Particle Swarm Optimization method is efficient than the
conventional Dynamic Programming and Evolutionary Programming Method.
Keywords: Multi-Area Unit Commitment, Evolutionary Programming, Dynamic Programming Method,
Particle Swarm Optimization Method
1. Introduction
In an interconnected system, the objective is to achieve
the most economical generation that could satisfy the
local demand without violating tie-line capacity con-
straints. Due to inter-area transmission constraints, multi-
area unit commitment problems (MAUC) are very com-
plicated when compared with single-area unit commit-
ment problems. Research explores that the application of
these existing single-area unit commitment to multi-area
unit commitment problem is required [1–4].
Furthermore, unit commitment is treated, as separately
from the economic dispatch, the linear fuel cost curve
may be an expensive operation schedule or a violation of
spinning reserve requirements. In multi-area systems,
local generations are not equal to local load demands.
Areas with lower fuel cost units may generate more
power than their demand and export the excessive energy
to the deficient areas; likewise, areas with higher fuel
cost units will generate less power than their demand and
import the additional energy from other areas with sur-
plus capacity. So, the unit commitment of an area should
comply with the local generation as well as the local load
demand. References [5–11] provide comprehensive
study on multi-area scheduling by relating unit commit-
ment and economic dispatch with tie-line constraints.
The following paragraph discusses some of the method,
which is adopted in the multi-area unit commitment
problem and their implications.
There are some drawbacks in implementing the simple
priority list method for unit commitment. Although the
technique was fast, the results are far from optimal, es-
pecially when there are massive on/off transitions. An-
other difficulty is in which did not deal with topological

S. R. P. CHITRA SELVI ET AL.141
k
connections in a multi-area system as it considered ex-
port/import limitations, which would cause infeasible
solutions in many applications. Another approach [6]
overcame the previous difficulties. It considered the
topological constraints and enhanced unit commitment
with economic dispatch .The λ iteration method takes
excessive time in finding the optimal solution in
large-scale power systems and the speed of the algorithm
required some improvement. In the iterative procedure
between unit commitment and economic dispatch, there
is a need to adjust the unit commitment according to the
required area generation. If we use Dynamic Program-
ming Sequential Combination (DP-SC) for unit com-
mitment in a power pool, the search for an optimal solu-
tion is very time consuming. If we adopt the priority list
method, there may be a solution gap between the resul-
tant schedule and the actual economic operation schedule.
If we repeat the process, we may reduce the operation
cost, but it will demand a longer execution time. The
DP-SC method is used for unit–commitment problem in
an interconnected area and particle swarm optimization
technique is embedded for assigning generation to each
area and modifying the economic dispatch schedule.
In this paper, we propose a more efficient approach to
the multi-area generation dispatch problem. The pro-
posed technique is used to improve the speed and reli-
ability of the optimal search process. Instead of using λ
iteration method in assigning power generation to each
area, we used particle swarm optimization to find the
optimal allocation of power generation in each area and
entire system. Using particle swarm optimization tech-
niques in each area and entire system, we can save time
in performing the economic dispatch and operating cost.
The meta-heuristic methods [12–19] are iterative tech-
niques that can search not only local optimal solutions
but also a global optimal solution depending on the
problem domain and time limit. In the meta-heuristic
methods, the techniques frequently applied to the UC
problem are genetic algorithm (GA), tabu search (TS),
evolutionary programming (EP), simulated annealing
(SA), particle swarm optimization (PSO), etc. They are
general-purpose search techniques based on the princi-
ples inspired from the genetic and evolution mechanisms
observed in natural systems and populations of living
beings. These methods have the advantage of searching
the solution space more thoroughly. The main difficulty
is their sensitivity to the choice of parameters.
In this paper, section one introduces that the mathe-
matical model of the multi-area unit commitment prob-
lem. In the problem formulation, DP method is used for
committing the unit in each area and λ iteration method
is used for importing and exporting power to other area
and minimizes the operating cost. Furthermore, tie-line
transfer capacities and area spinning reserve require-
ments are also incorporated in order to ensure system
security and reliability. The Reserve-sharing scheme is
used to enable the area without enough capacity to meet
its reserve demand. The objective of MAUC, constraints
and conditions of optimal solution are also discussed in
this section. Section 3 and 4 explains the EP and PSO
algorithm adopted for importing and exporting power to
other area. Section 5 gives the results of a case study
each one based on a four-area system. A four-area IEEE
test power system [6] is then used as an application ex-
ample to verify the effectiveness of the proposed method
through numerical simulations. A comparative study is
also made here to illustrate the different solutions ob-
tained based on conventional, EP and PSO methods.
Conclusions are presented in the last section.
2. Problem Formulation
The cost curve of each thermal unit is in quadratic form
2
() ()()
kkk kk
F
Pga Pgb Pgc
iii ii
 i 

:$/hr k=1 NA (1)
The incremental production cost is therefore
2kkk aPg b
ii i
  (2)
or
kk
P
gb
i
 
i
k
ai /2 (3)
The start up cost of thermal unit is an exponential
function of the time that the unit has been off
,
() (1
,
off
X
offi j
SXA Be
iji i
 )
(4)
2.1. Multi-Area Unit Commitment
The objective function for the multi-area unit commit-
ment is to minimize the entire power pool generation
cost as follows:
min[()(1) ()
,,,
,1 ,1
,11
1
N Ntk
Aoff
kk k
IFPgII SX
ij jijiji
ij ij
IP ji
k
 


(5)
and the following constraints are to be met for optimiza-
tion
1) System power balance constraints
; 1.......
kk
PgDW jt
jjj
kk
  (6)
where =
k
Pg j
k
,
k
Pgij
k

2) Spinning reserve constraints in each area
kkkk
PgD R EL
ijjj
i
k
j
;j=1…t (7)
3) Generation limits of each unit
,
kk
Pg PgPg
ij j
j
k
;i=1…Nk;j=1…t;k=1…NA (8)

S. R. P. CHITRA SELVI ET AL. 142
)0
j

)0
j

)
,
max
max
4) Minimum Up and Down time constraints
()*(
, ,1 ,1
off on
XTII
ii ij ij


(9)
()*(
, ,1 ,1
off off
XTII
ii ij ij


(10)
To decompose the problem in Equation (5), it is re-
written as
min[ ()]
,
1
tFPg
ij
Pj
 (11)
where
() (
,
1
Nkkk
FPgF Pg
ij ij
k

 (12)
subject to the constraints of Equation (6) and (8) and
following constraints.
5) Export/Import constraints
.
kkk
PgD E
ijj j
i
 (13)
,
kkk
PgD L
ijjj
ik
  (14)
0
kk
ELW
jjj
ik

 (15)
6) Area generation limits
,
kkk
k
Pg Pg R
iji j
ii
  ;=1… N
A
; (16) 1...jt
,
kk
P
gPg k
ij i
ii
  ;=1 N
A
;1...jt

(17)
Each for is represented in the
form of schedule tables, which is the solution of the
mixed variables optimisation problem
(
,
kk
FPg
ij
)1...,kN
A

min[()(1) ()
,,, , ,1
,
off kk k
IFPgII SX
ij iijijiij
ij
IPi
 (18)
Subject to constraints of Equation (7), (9-10) and ini-
tial on/off condition of each unit.
The multi-area unit commitment problem is solved by
Dynamic Programming Sequential Combination (DP-SC)
method to form the optimal generation scheduling ap-
proach. Among the available generating units in the in-
terconnected multi-area system and the proposed method
sequentially identifies, via a procedure that resembles
bidding, the most advantageous units to commit until the
multi-area system obligations are fulfilled and this
method has been explained [13].
2.2. Multi-Area Economic Dispatch
The objective of Multi-area Economic Dispatch (MAED)
is to determine the allocation of generation of each unit
in the system and power exchange between areas so as to
minimize the total production cost. The lamda–iteration
method is implemented in the MAED to include area
import and export constraints and tie-line constraints [15]
The objective is to select
s
ys

every hour to minimize
the operation cost.
kkk
PgD E L
jjj

k
j
j
(19)
where
, 1
N
kk
k
Pg Pg
ji i

Since the local demand k
j
Dis determined in accordance
with the economic dispatch within the pool , changes of
k
g
j
Pwill cause the spinning reserve constraint of Equa-
tion (7) to change accordingly and redefine Equa-
tion(18).
In this study, the iterative equal incremental cost
method (

method) was used to solve Equation (11)
and serve as a coordinator between unit commitments in
various areas. With the

iteration, the system would
operate at an optimal point if

for each unit is equal to
a system incremental cost
s
ys
 .Units may operate in one
of the following modes when commitment schedule and
unit generation limits are encountered:
1) Coordinate mode: The output of unit i is determined
by the system incremental cost
max,
min, sys i i
  (20)
2) Minimum mode: Unit i generation is at its mini-
mum level.
min,
s
ys
i
  (21)
3) Maximum mode: Unit i generation is at its maxi-
mum level.
max,
s
ys
i
  (22)
4) Shut down mode: Unit i is not in operation,
P
gi= 0.
Besides limitations on individual unit generations, in a
multi-area system, the tie-line constraints in Equation (9), (10)
and (14) are to be preserved. The operation of each area could
be generalized into one of three modes as follows:
Area coordinate mode
k
s
ys
  (23)
max max ,
kkkk k
DLP DE
jijj
i
   (24)
or
max max
,
kkk
LPgDE
ijj
i
  k (25)
a. Limited export mode
When the generating cost in one area is lower than the
cost in the remaining areas of the system, that area may
generate its upper limit according to Equation (13) or
(16), therefore,

S. R. P. CHITRA SELVI ET AL.143
k
s
ys
  (26)
k

is the optimal equal incremental cost which satisfies
the generation requirement in each area k.
b. Limited import mode
An area may reach its lower generation limit according
to Equation (14) or (17), because of the higher genera-
tion costs.
min
k
s
ys
  (27)
The proper generation schedule in multi-area will re-
sult by satisfying tie-line constraints and minimizing the
system generation cost.
2.3. Tie-Line Flow of Four Areas
An economically efficient area may generate more power
than the local demand, the excess power will be exported
to the other areas through the tie-lines. As shown in Fig.
1, assume area 1 has excess power, the line flows would
have directions from area 1 to other areas, and the
maximum power generation for area 1 would be the local
demand in area 1 plus the sum of all the tie-line capaci-
ties connected to area 1. If we fix the area 1 generation at
its maximum level, then the maximum power generation
in area 2 could be calculated in a similar way to area 1.
Since tie-line imports power at its maximum capacity,
this amount should be subtracted from the generation
limit of area 2. According to the system power balance
equation some areas must have a power generation defi-
ciency, and require generation imports. The minimum
generation level of these areas is the local demand, mi-
nus all the connected tie-line capacities. If any of these
tie lines is connected to an area with higher deficiencies,
then the flow directions should be reversed. The tie-line
flow details of four area and directional matrix were
presented in [9].
Directional matrix: It indicates power flow direction
from one area to another area.
,
Dlk [ 1 when line flows from to k l >k [ -1
when line flows from k to
l
l
0,
,.
DDD
lllkkl

,
initial are zero
.
Dlk
3. Evolutionary Programming Method
3.1. Introduction
EP is a mutation-based evolutionary algorithm applied to
discrete search spaces. D. Fogel (Fogel, 1988)] extended
the initial work of his father L. Fogel (Fogel, 1962)
[15–18] for applications involving real-parameter opti-
mization problems. Real-parameter EP is similar in prin
Figure 1. Flow chart for evolutionary algorithm.
ciple to evolution strategy (ES), in that normally distrib-
uted mutations are performed in both algorithms. Both
algorithms encode mutation strength (or variance of the
normal distribution) for each decision variable and a
self-adapting rule is used to update the mutation
strengths. Several variants of EP have been suggested
(Fogel, 1992).
3.2. Evolutionary Programming Algorithm
The original Evolutionary Programming involved evolv-
ing populations of extending algorithms to develop arti-
ficial intelligence [17]. In this technique a strong behav-
ioral link is sought between each parent and its offspring,
at the level of the species.Fig.1 shows e general scheme
of the EP algorithm.
3.3. Implementation of Evolutionary Algorithm
for Multi-Area Unit Commitment Problem
Step (1): Read in unit data, tie-line data, demand profile.
Step (2): Perform the dynamic programming to get the
initial commitment schedule for each area.

S. R. P. CHITRA SELVI ET AL. 144
Step (3): Initialization of parent population. The initial
parent population of size Np is randomly generated for
committed unit in each area:
1) To generate the initial parent population

(.......)];1,2, 3, 4 &1,2...
1;
kp kp
Ippkp N
p
p
gN g

(28)
2) To calculate the fuel cost for each population using
Equation (1)
2
[(()());1,2,3,4&1,2...
11
kp kp
K
F
CaPg bPgckp N
p
p

(29)
3) To calculate the start up cost for each population
using Equation (4)
4) To calculate the production cost Production
cost= kk
F
CSC
p
p
 (30)
5) To calculate the fitness function for each parent of
population
(
1
Nk kp )
K
K
FFCSC KPGD
i PPPi
 


k
j
(31)
The values of the penalty factor is chosen such that if
there are any constraints violations then the fitness func-
tion value corresponding to that parent will be ineffec-
tive.
Step (4): Mutation
1) To generate an offspring population Io of size from
Np from each parent Ip
[(........);1,2, 3, 4;01.....
1
ko ko
I
PgPg kN
p
N
O
in
i
(32)
generated as
2
(0,);1, 2......
KOKOK
PgPgNPg iN
ii i
 
Similarly all is generated for all areas subjected to Pgi
ko ko
Pg =Pg ; if Pg< Pg
i,mini,m
ii
ko KO
Pg=Pg ; if Pg > Pg
,max ,max
ii
i (33)
2
(0, )N

represents a normal random variable with zero
mean and standard deviation
(/ )(
max ,max ,min
FF
pi ij
Pg ij
i
   ) (34)
where

is scaling factor,
F
p
i
i
is the value of fitness
function corresponding to
I
and is the maxi-
mum fitness function value among parent population
max
F
2) To compute the fitness value corresponding to each
offspring using Equation (31)
Step (5): (competition and selection). The 2I individuals
compete with each other for selection using Equation (6).
A weight value is assigned to each individual as
follows:
i
W
1
I
Wt
it

W
se
(35)
{1,Wi
tf(/ )ufff tt i

{0,Wotherwi
t (36)
where t
f
is the fitness of the ith competitor randomly
selected from 2I individuals and u is a uniform random
number ranging over [0, 1].While computing the weight
for each individual, it is ensured that each individual is
selected only once from the combined population. Even
though relative fitness values are used during the process
of mutation, competition and selection, it leads to slow
convergence. This is because the ratio/( )
tt i
f
ff

is
always around 0.5 without uniform distribution between
0 and 1.Hence, the following strategy is followed in this
paper to assign weights:
{1,Wi
tf/() 0.5fff tt i
 (37)
{0,Wotherwis
te
This weight assignment is found to yield proper selec-
tion and good convergence. When all the 2I individuals
obtain their weights, they are ranked in descending order
and the first I individuals are selected as parents along
with their fitness values for next generation.
Steps (4) and Steps (5) are repeated until there is no
appreciable improvement in the minimum fitness value.
Step (6): Optimum generation schedule is obtained for
four areas using minimum fitness value. Check area gen-
eration with local demand
Step (7): Areas with lower fuel cost may export the
excessive generation to other areas with higher fuel cost
(deficiency areas) with tie line limit.
4. Particle Swarm Optimization
Particle swarm optimization (PSO) is inspired from the
collective behavior exhibited in swarms of social insects
[19]. It has turned out to be an effective optimizer in
dealing with a broad variety of engineering design prob-
lems. In PSO, a swarm is made up of many particles, and
each particle represents a potential solution (i.e., indi-
vidual). A particle has its own position and flight veloc-
ity, which are adjusted during the optimization process
based on the following rules:
1() ()() ()
12
P
PKPKP KP
VVCrandP PCrandP P
iii gii
bi

KP

   
(38)
1
K
PKPP
PPV
iii

 (39)
where 1
Vt

is the updated particle velocity in the next
iteration,V is the particle velocity in the current itera-
tion,
t

is the inertia dampener which indicates the im-

S. R. P. CHITRA SELVI ET AL.145
pact of the particle’s own experience on its next move-
ment, represents a uniformly distributed num-
ber within the interval [0, c1], which reflects how the
neighbours of the particle affects its flight,
1
Crand
K
P
bi
P is the
neighbourhood best position,
P
i
V is the current position
of the particle and represents a uniformly
distributed number within the interval [0, c2], which in-
dicates how the particle trusts the global best position,
2
C rand
K
P
g
i
P is the global best position, and is the up-
dated position of the particle. Under the guidance of
these two updating rules, the particles will be attracted to
move towards the best position found thus far. That is,
the optimal solutions can be sought out due to this driv-
ing force.
1P
i
V
The major steps involved in Particle Swarm Optimiza-
tion approach are discussed below:
1) Initialization
The initial particles are selected randomly and the ve-
locities of each particle are also selected randomly. The
size of the swarm will be (Np x n), where Np is the total
number of particles in the swarm and ‘n’ is the number
of stages.
2) Updating the Velocity
The velocity is updated by considering the current ve-
locity of the particles, the best fitness function value
among the particles in the swarm. The velocity of each
particle is modified by using Equation (28)
The value of the weighting factor

is modified by
following Equation (40) to enable quick convergence.
()iter/
mmax i

max
 
ax min
 ter (40)
The term

< 1 is known as the “inertia weight” and
it is a friction factor chosen between 0 and 1 in order to
determine to what extent the particle remains along its
original course unaffected by the pull of the other two
terms. It is very important to prevent oscillations around
the optimal value.
3) Updating the Position
The position of each particle is updated by adding the
updated velocity with current position of the individual
in the swarm
4.1. Algorithm of Particle Swarm Optimization
The step by step procedure to compute the global optimal
solution is followed.
Step (1): Initialize a population of particles with ran-
dom positions and velocities on d dimensions in the
problem space.
Step (2): For each particle, evaluate the desired opti-
mization fitness function in the variables.
Step (3): compare particles fitness evolution with par-
ticles . If current value is better then, then
set value equal to the current value, and the
location equal to the current location in the di-
mensional space.
Pbest Pbest
Pbest
stPbe
Step (4): Compare fitness evaluation with the popula-
tions overall previous. If current value is better
than
Pbest
g
best
[(
, then reset to the current particles array in-
dex and value.
Step (5): Change the velocity and position of the parti-
cle according to Equations (38) and (39) respectively.
Step (6): Loop to step 2 until a criterion is met, usually
a sufficiently good fitness or a maximum number of it-
erations.
4.2. Implementation of Particle Swarm
Optimization Algorithm for
Multi-Area Unit Commitment
The various steps of the PSO algorithm are given below
for solving multi area unit commitment problem:
Step (1): Read in unit data, tie-line data, load demand
profile.
Step (2): Perform the dynamic programming to get the
initial commitment schedule for each area.
Step (3): Initialization of particle .The initial particle
of size Np is generated randomly for committed unit in
each area :
1) Calculate the initial particle population
.....);1,2,3,4:1.....
12
kp kp
I
PPkp N
p
p
 (41)
2) Calculate the fuel cost for each particle using Equa-
tion (1)
2
)());1,2, 3,4;1,2...
11
kp kp
[( (
k
F
CaPbPckpN
p
p
 (42)
3) Calculate start up cost of each particle using Equa-
tion (4)
4) Calculate the production cost Production
Cost =k FC pk SC
p
 (43)
5) Calculate the fitness function for each particle of
population
k
FC ()
p1
Nkkp
k
FSCkP
pp i
i
  

k
D
j
(44)
6) To calculate the by using fitness function
values, If current value is better then previous,then
set value equal to the current value and compute
Pbest
Pbest
Pbest
g
best
1
if current value is.
Step (4): Updating the Velocity
The velocity is updated by considering the current velocity
of the particles, the best fitness function value among the
particles in the swarm using following Equation (45).
() ()() ()
12
P
PKPKP KP
CrandP PCrandP P
i gii bi
KP
V V
ii


   
(45)

S. R. P. CHITRA SELVI ET AL. 146
where

is weight factor, The weight

is computed
using Equation (40)
Step (5): Updating the particle position
The position of each particle is updated by adding the
updated velocity with current position of the individual
in the swarm.
1
K
PKPP
PPV
iii

 (46)
The steps described in sub Sections 3 to 5 are repeated
until a criterion is met, usually a sufficiently good fitness
the maximum generation count is reached. Step (6): Op-
timum generation schedule is obtained for four area us-
ing gbest particle. Check area generation with local de-
mand.
Step (7): Areas with lower fuel cost may export the
excessive generation to areas with higher fuel cost (defi-
ciency areas) with tie line limit
5. Test System and Simulation Results
The proposed MAUC algorithm has been implemented
in C++ environment and tested extensively. Test results
of a multi-area system are presented in this section. All
simulations are performed in a PC with Intel processor
(1.953 GHz) and 1012 MB of RAM.
As shown in Figure 2, a sample multi-area system
with four areas, IEEE reliability test system, 1996 data in
[9], are used to test the speed of solving the multi-area
UC and ED for a large-scale system with import/export
capability and tie line capacity constraints. In the sample
multi-area system, each area consists of 26 units. The
total number of units tested is 104, and their characteris-
tics are presented in [9]. There are some identical ther-
mal units also located in each area. The system contains
five tie lines four area interconnections as shown in Fig-
ure 4, and area one is the reference area. Figure 3 shows
the modified same load demand profile forecast used in
all four areas. The assumptions described in tie line ca-
pacity constraint are applied to the simulations.
The four areas have the same load demand profiles. As
the 1oad demand is same in these four areas, the eco-
nomical area will generate more power than expensive
areas. Figure 3 gives the changes in area 1 power genera-
tion, committed unit capacities, unit commitment pattern
of hour 7am and spinning reserve requirement of area 1
is 400MW, because the available unit capacities are not
more than the power generation plus the spinning reserve.
This phenomenon proves that the available capacity
should comply with the area power generation instead of
the local load demand.
The systems 1oad demand is 6800 MW, so area 1
generation increases steadily while that of area 2, 3 de-
creases. The incremental cost of area 2, 3 is higher than
Figure 2. Topological connections of four areas.
Figure 3. Load pattern for all four –area.
Figure 4.Tie-line flow pattern for 7am.
that of the other two areas since the tie flows to area 2, 3
are at their maximum capacities. This manifests that the
proposed method considers tie-line limits effectively.
Table I shows that parameter used in EP and PSO
method. Table 2, 3, 4 and Table V shows comparison
result of DP and EP, PSO. Figure 4 and 5 shows the
convergence characteristics for multi-area obtained using
proposed methodology.
Table 2 shows that the total production cost is obta-

S. R. P. CHITRA SELVI ET AL.147
Table 1. Parameter used in EP & PSO.
ined by using conventional method. Table 3 and 4 shows
that the total production cost is obtained for ten iterations
by using EP and PSO method. Figure 4 gives the plot of
EP average performance from 500 runs. Figure 5 gives
the plot of number of iteration versus the time taken to
complete those iterations and the maximum production
cost obtained under each iteration using PSO method.
As we indicated in the paper, the PSO algorithm has
also proved to be an efficient tool for solving the multi
–area unit commitment with economic dispatch problem.
There is no obvious limitation on the size of the problem
that must be addressed, for its data structure is such that
the search space is reduced to a minimum; no “relaxation
of constraints” is required; instead, populations of feasi-
ble solutions are produced at each generation and throu-
ghout the evolution process. The main advantages of the
proposed algorithm are speed.
The proposed PSO approach was compared to the re-
lated methods in the references indented to serve this
purpose, such as the DP with a zoom feature, and the EP
approaches. In addition, with the use of PSO method, the
status is improved by avoiding the entrapment in local
minima. By means of stochastically searching multiple
points at one time and considering trial solutions of suc
cessive generations, the PSO approach gives global
minima instead of entrapping in local optimum solutions.
The PSO method obviously displays a satisfactory per-
formance with respect to the quality of its evolved solu-
tions and to its computational requirements.
The final result of PSO would save 0.12% $2865.4 is
compared with the solution obtained by the conventional
method but it would require 33 seconds to complete the
computation .So, the EP method is reduced the operating
cost by 0.08 % than the conventional method but it re-
quires 36 seconds to complete this computation .From
these results, the PSO method had less total cost and con-
sumed also less CPU time compared to other method.
6. Conclusions
Application of PSO is a novel approach in solving the
MAUC problem. Results demonstrate that PSO is a very
competent method to solve the MAUC problem. PSO
Table 2. Operating cost of DP method.
generates better solutions than the other methods, mainly
because of its intrinsic nature of updates of positions
and velocities. The reason is due to the hourly basis solu-
tion. This is somehow similar to the “divide and con-
quer” strategy of solving a problem. Owning to this
Parameter EP PSO
Population size(p) 10 10
Mutation scaling factor(β) 0.03 -
Penalty factor(k1) 10000 1000
0
Maximum Generation 500 500
Learning factor(c1,c2) - 2
Hours
(24)
Area-1
(26 Unit)
Area-2
(26 Unit)
Area-3
(26 Unit)
Area-4
(26 Unit)
1
37115.330
08
24115.5214
8
28331.2265
6
22042.12
500
2
24747.960
94
23137.6396
5
22994.8997
4
19289.81
836
3
27995.107
42
23137.6396
5
23701.2568
4
19175.97
998
4
29576.867
19
18274.3261
7
26151.8378
9
18397.77
637
5
29347.660
16
18329.3261
7
25595.4296
9
18698.77
344
6
36118.037
11
18329.3261
7
23799.5097
7
19705.58
106
7
40483.162
11
28104.1445
3
21999.5986
3
24891.27
832
8
39248.855
47
32917.4687
5
19852.8554
7
21117.69
727
9
38728.734
38
34865.2382
8
18245.3730
5
21253.34
180
10
37215.339
84
32205.3750
0
22093.5957
0
24255.43
945
11
37193.468
75
32205.3750
0
20244.0820
3
23298.57
031
12
38310.472
66
32205.3750
0
20992.8925
8
21298.69
336
13
33225.353
52
34149.0293
0
18152.8222
7
26442.17
773
14
31623.279
30
37085.8281
3
17146.9394
5
25955.68
945
15
30595.626
95
33172.8613
3
17991.4726
6
23682.43
359
16
36312.250
00
32989.6523
4
22492.5781
3
25305.94
336
17
36925.175
78
32989.6523
4
23769.5800
8
25383.72
656
18
35682.320
31
39459.6250
0
27589.7597
7
19501.75
391
19
35682.320
31
39903.0585
9
23860.8418
0
22304.66
016
20
35682.320
31
32114.9414
1
21973.3906
3
15999.40
332
21
38042.478
52
29387.7168
0
19907.5390
6
20248.24
805
22
30190.896
48
15095.1718
8
21115.4316
4
21807.76
953
23
30923.708
98
18398.0820
3
19966.2128
9
22309.07
813
24
30202.210
94
15198.7812
5
19815.6132
8
18294.49
805
Total
cost
821168.93
75
677771.156
3
527784.731
2
520660.4
566

S. R. P. CHITRA SELVI ET AL. 148
Table 3. Opearting cost of EP method.
Table 4. Operating cost of PSO method
hourly solution, the complexity of the search is greatly
reduced. The total objective function is the sum of objec-
tives and constraints, which are fuel cost, start-up cost,
Table 4. Operating cost of PSO method.
spinning reserve, power demand, tie-line limit, and im-
port and export constraints. For a better solution, genera-
ted powers by N unit of generators and K areas, tie –line
limits are constantly checked so that feasible particles
can meet the power demand .This reduces the pressure of
Hours
(24)
Area-1
(26 Unit)
Area-2
(26 Unit)
Area-3
(26 Unit)
Area-4
(26 Unit)
1
37096.33008 24048.52148 28309.226
56
21998.125
00
2
24514.96094 23004.63965 22910.899
74
19251.818
36
3
27980.10742 23004.63965 23674.256
84
19145.979
98
4
29568.86719 18286.32617 26111.837
89
18374.776
37
5
29387.66016 18286.32617 25578.429
69
18671.773
44
6
35838.03711 18286.32617 23769.509
77
19673.581
06
7
40497.16211 28043.14453 21945.598
63
24858.278
32
8
39228.85547 32977.46875 19815.855
47
21081.697
27
9
38648.73438 34802.23828 18245.373
05
21201.341
80
10
37229.33984 32191.37500 22063.595
70
24199.439
45
11
37184.46875 32191.37500 20212.082
03
23272.570
31
12
38294.47266 32191.37500 20979.892
58
21262.693
36
13
33200.3535234120.0293 18127.822
27
26401.177
73
14
31630.27930 37051.82813 17124.939
45
25928.689
45
15
30578.62695 33162.86133 17978.472
66
23631.433
59
16
36281.25000 32960.65234 22459.578
13
25277.943
36
17
36949.17578 32960.65234 23748.580
08
25365.726
56
18
35766.32031 39439.62500 27569.759
77
19465.753
91
19
35766.32031 39811.05859 23839.841
8
22243.660
16
20
35766.32031 32081.94141 21943.390
63
15968.403
32
21
38122.47852 29353.71680 19897.539
06
20208.248
05
22
30177.89648 15065.17188 21073.431
64
21791.769
53
23
31583.70898 18379.08203 19966.212
89
22270.078
13
24
29449.21094 15159.78125 19816.613
28
18211.498
05
Total
cost
820740.9375 676860.1562 527162.73
96
519756.45
65
Hour
-s
(24)
Area-1
(26 Unit)
Area-2
(26 Unit)
Area-3
(26 Unit)
Area-4
(26 Unit)
1
37112.330
08
24093.521
48
28311.2265
6
22002.12500
2
24741.960
94
23127.639
65
22964.8997
4
19259.81836
3
27988.107
42
23127.639
65
23681.2568
4
19151.97998
4
29566.867
19
18254.326
17
26121.8378
9
18367.77637
5
29337.660
16
18309.326
17
25572.4296
9
18678.77344
6
36108.037
11
18309.326
17
23789.5097
7
19683.58106
7
40473.162
11
28084.144
53
21975.5986
3
24861.27832
8
39238.855
47
32897.468
75
19822.8554
7
21087.69727
9
38718.734
38
34841.238
28
18215.3730
5
21223.34180
10
37202.339
84
32185.375
00
22063.5957
0
24205.43945
11
37183.468
75
32185.375
00
20224.0820
3
23278.57031
12
38296.472
66
32185.375
00
20972.8925
8
21268.69336
13
33212.353
52
34129.029
30
18132.8222
7
26412.17773
14
31607.279
30
37063.828
13
17126.9394
5
25920.68945
15
30578.626
95
33152.861
33
17981.4726
6
23642.43359
16
36281.250
00
32969.652
34
22462.5781
3
25286.94336
17
36919.175
78
32969.652
34
23749.5800
8
25353.72656
18
35662.320
31
39439.625
00
27569.7597
7
19471.75391
19
35662.320
31
39893.058
59
23839.8418
0
22274.66016
20
35662.320
31
32094.941
41
21943.3906
3
15969.40332
21
38032.478
52
29365.716
8
19887.5390
6
20218.24805
22
30177.896
48
15065.171
88
21073.4316
4
21797.76953
23
30913.708
98
18387.082
03
19946.2128
9
22279.07813
24
30182.210
94
15168.781
25
19796.6132
8
18254.49805
Total
cost
820859.93
75
677300.15
62
527225.739
6
519950.4565

S. R. P. CHITRA SELVI ET AL.149
Figure 4. Convergence characteristics of EP method.
Figure 5. Convergence characteristics of PSO method.
Table 5. Comparison of DP, EP, PSO method.
the constraint violation of the total objective function.
Finally, the result obtained from the simulation is most
encouraging in comparison to the best-known solution so
far. In the future work, the power flow in each area can
be considered to further increase the system security.
Other issues such as transmission losses, transmission
costs, call and put options policies between and bilateral
contract areas can also be considered to reflect more re-
alistic situations in MAUC problems.
7. References
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S. R. P. CHITRA SELVI ET AL. 150
Appendix A
Nomenclature k
Pgi Upper limit of power generation of unit i in
area k
k
Dj Total load demand in area k at jth hour
,
k
Pgij Power generation of unit i in area k at j th
hour
k
Lj Total import power to area k at jth hour
k
Ej Total export power to area k at jth hour k
Rj Spinning reserve of area k at j th hour
,
k
I
ij
Commitment state (1 on, 0 for off) k
j
S Total commitment capacity for area k at j
th hour
Irlist List of committed units ascending pri-
ority order k
SD j Total system demand at j th hour Total
time span in hours
t
i Index for units
j Index for time
on
Ti Minimum up time of unit i
i

Lagrangian multiplier for unit
s
ys

Lagrangian multiplier for entire system off
Ti Minimum down time of unit i
N
A
Total number of areas
i

Time constant in start up cost function for
unit i
Nk Total number of units in area K
Oplist List of uncommitted units in descend-
ing order
Wj Net power exchange with outside systems
k
Pg j Power generation of area k at jth hour
k
Pgi Lower limit of power generation of
unit i in area k
/
,
on off
Xij Time duration for which unit i has been
on/off at jth hour