Energy and Power En gi neering, 2011, 3, 135-143
doi:10.4236/epe.2011.32017 Published Online May 2011 (http://www.SciRP.org/journal/epe)
Bacterial Foraging Algorithm Based Parameter
Estimation of Thr ee W inding Tr ansformer
Srikrishna Subramanian, Seeni Padma
Department of Electrical Engineering, FEAT, Annamalai University,
Annamalainagar, India
Received February 7, 2011; revised March 23, 2011; accepted April 2, 2011
Abstract
Transformers are one of the main components of any power system. An accurate estimation of system be-
haviour, including load flow studies, protection, and safe control of the system calls for an accurate equiva-
lent circuit parameters of all system components such as generators, transformers, etc. This paper presents a
methodology to estimate the equivalent circuit parameters of the Three Winding Transformer (TWT) using
Bacterial Foraging Algorithm (BFA). The estimation procedure is based on load test data at one particular
operating point namely supply voltage, load currents, input power. The performance characteristics, such as
efficiency and voltage regulation are considered along with the name plate data in order to minimize the er-
ror between the estimated and measured data. The estimation procedure is demonstrated with a sample three
winding transformer and the results are compared against the directly measured performance of TWT and
genetic algorithm optimization results. The simulation results show the ability of the proposed technique to
capture the true values of the machine parameters and the superiority of the results obtained using the bacte-
rial foraging algorithm.
Keywords: Parameter Estimation, Three Winding Transformer, Bacterial Foraging Algorithm
1. Introduction
Three winding transformers (TWT) are widely used in
power system and power electronic applications. Deter-
mination of its equivalent circuit parameters is useful in
performance computations, power system load flow
studies etc. Measurement or computation of equivalent
circuit parameters of TWT is difficult and unreliable due
to the complexity of the geometry of the windings.
Various methodologies were applied for parameter esti-
mation of transformer equivalent circuit. The equivalent
circuit parameters of three winding transformer are de-
termined based on Genetic Algorithm (GA) [1] has been
discussed. The ferroresonance of the transformer has
been predicted or confirmed and its severity can be
evaluated by using transformer equivalent circuit models
[2]. A topology-based and duality derived three-phase
three winding core type transformer model has been de-
veloped and it treats the leakage inductances and the
coupling effects of the core in a straightforward and in-
tegrated way [3]. The method based on Genetic Algo-
rithm (GA) has been developed for the identification of
synchronous machine parameters from short circuit tests
[4]. GA has also been applied to determine the electric
parameters of an induction machine using Park model [5].
The Park model electric parameters of an induction ma-
chine [6] are used in control techniques for variable
speed drives, have been estimated by GA.
The average winding temperature rise under its field
operation conditions and rise in winding temperature has
been determined from the estimated values of winding
resistance [7]. The parameters of a saturation model of
transformer are also estimated by using the data from
transformer inrush tests and steady state operation [8].
An alternative approach to conventional open and short-
circuit test for determining the parameters of N-windings
transformer operating at power frequency on an on-line
mode. The method is based on linear Least Error Square
(LES) algorithm and uses the digitized samples of the
input current and voltage as well as the output current
and voltage of the transformer windings [9]. The GA
based method has also been suggested to identify the
parameters of an induction motor [10].
The conventional model for multiwinding transform-

136 S. SUBRAMANIAN ET AL.
ers is difficult to relate to its physical construction, and
the measurement of the model parameters is also difficult
and unreliable, [11] hence a physically based electrical
model of a high voltage multiwinding transformers has
been developed. In this model, each component corre-
sponds to a physical quantity of the transformer and the
leakage inductance for nonuniformly spaced windings,
which store significant energy in the flux in the radial
field, has also been easily calculated.
Differential evolution algorithm [12] has been applied
for parameter identification of an induction motor. Pa-
rameter identification of an induction machine using GA
[13] has been discussed for variable speed applications.
The general mathematical model of the motor based
upon Kron’s voltage equations has been considered to
estimate the parameters and the start-up performance of
the motor has been used as the measurement for identifi-
cation process. A multi-stage transformer model [14] for
high frequency transient operation is established, and the
equivalent circuit parameters are estimated by using their
mathematical formulation. The modified version of GA
namely, enhanced GA [15] which operates on real-val-
ued parameter sets and provides an improvement in the
solution quality, has been applied to determine the
equivalent circuit parameters of induction motors.
Sensitivity of estimated parameters in transformer
thermal modeling has been discussed [16]. Least Squares
Method [17] has been applied to estimate the transformer
equivalent circuit parameters, also determined the opti-
mal approximation polynomial functions for each pa-
rameter. Artificial Neural Networks (ANN) based method
has been suggested for estimation of electrical losses in
the three-phase distribution transformer [18]. Electronic
transformer model has been developed and also estimate
the equivalent circuit parameters are presented in [19].
Recursive least squares routine [20] has been applied to
estimate the on line dynamic parameters for transformer.
The modern heuristic search technique, called Bacte-
rial Foraging Algorithm (BFA) has been developed
based on modelling of bacteria E. coli behavior present
in human intestine and it has been proven that is efficient
[21-26] for various engineering optimization problems.
In this article, BFA has been applied to estimate the
equivalent circuit parameters of three phase transformer.
The effectiveness of the proposed BFA approach has
been tested with the suitable transformer.
2. Problem Description and Formulation
The parameter estimation is one of the important prob-
lems to solve in the system studies. The conventional
method of parameter determination using short-circuit
test data provides an approximate equivalent circuit.
Equivalent circuit model of a TWT is shown in Figure 1.
This method requires a minimum of two tests namely
short circuit test and direct current resistance test are
conducted at supply conditions different from normal
operation. In addition to that empiricism exists in the
allocation of leakage reactance between primary, secon-
dary and tertiary windings. The performance evaluation
of transformer using the equivalent circuit parameters is
inaccurate because the change in the winding resistance
due to temperature rise which is caused by loading effect
is ignored in the conventional method. Apart from that
the stray-load loss component is not accounted. For the
above reason, the exact equivalent circuit model of
transformer has been formulated by including the load
impedance as shown in Figure 2.
The following equations are used to estimate the
equivalent circuit parameter. Let the impedance of the
primary secondary, tertiary windings referred to primary
side are
22
11
zrx
1

 (1)
 
22
222 2l
zrr x

 
 (2)

2
333 3l
zrr x
2

 
 (3)
Then the admittance of the magnetizing winding is
_
+
Z2
Z3
Z1
Zc
_
+
V1
V2
V3
_
+
Figure 1. Equivalent circuit model of TWT.
23
II



r
1
x
1
x
23
I
1
R
c
X
c
_
+
V
1
2'X
2
r

2l
r

3
r

3l
r

3'X
Figure 2. Exact equivalent circuit model of a TWT.

S. SUBRAMANIAN ET AL.
137
11
c
cc
YRX  (4)
1
c
c
ZY  (5)
The equivalent impedance of secondary and tertiary
winding referred to primary side is
23
23 23
23
Z
Z
ZZ
Z
Z




(6)
The estimated value of the equivalent impedance as
referred to primary side is
23
11
23
c
est
c
Z
Z
ZZ
Z
Z



  (7)
The estimated value of primary voltage be
11est est
VZ1 I
3
(8)
And the estimated value of input power be
1 1232est corecucucu
PPPPPPP (9)
The objective of the parameter problem is to find a set
of equivalent circuit parameters that minimizes the error.
The equivalent circuit parameters are to be estimated by
minimizing the following objective function,

22
12
f
Xf f (10)
where
11 23223 3
,,,,,,, ,
cc
X
rx xr xrxRX


11
1
1
*1 00
mes est
mes
VV
fV

 (11)
11
2
1
*1 00
mes est
mes
PP
fP

 (12)
3. Bacterial Foraging Optimization
The selection behaviour of bacteria tends to eliminate
poor foraging strategies and improve successful foraging
strategies. After many generations a foraging animal
takes actions to maximize the energy obtained per unit
time spent foraging. This activity of foraging led the re-
searchers to use it as optimization process. The E coli
bacterium has a control system that enables it to search
for food and try to avoid noxious substances. The bacte-
ria distributed motion can model as the following four
stages:
3.1. Swarming and Tumbling via Flagella (Ns)
The flagellum is a left-handed helix configured so that as
the base of the flagellum (i.e. where it is connected to the
cell) rotate counter clockwise, from the free end of the
flagellum looking towards the cell, it produces a force
against the bacterium pushing the cell. This mode of mo-
tion is called swimming. A bacterium swims either for
maximum number of steps Ns or less depending on the
nutrition concentration and environment condition. Dur-
ing clockwise rotation each flagellum pulls on the cell
shown in Figure 3. So that the net effect is that each fla-
gellum operates relatively independently of the others
and so the bacterium “tumbles”.
3.2. Chemotaxis (Nc)
A chemotaxis step is a set of consequence swim steps
following by a tumble. A maximum of swim steps with a
chemotactic step is predefined by Ns. The actual number
of swim steps is determined by the environment. If the
environment shows good nutrients concentration in the
direction of the swim, the bacteria swim more steps.
When the swim steps is stopped a tumble action takes
place.
3.3. Reproduction (Nre)
After Nc chemotactic steps, a reproduction step is taken.
Let Nre be the number of reproduction steps to be taken.
It is assumed that half of the population members have
sufficient nutrients so that they will reproduce with no
mutations. For reproduction, the population is sorted in
order of ascending accumulated cost accumulated cost
represents that it did not get as many nutrients during its
lifetime of foraging and hence, is not as “healthy” and
(a) (b)
Figure 3. Swarming and tumbling behaviour.

138 S. SUBRAMANIAN ET AL.
thus unlikely to reproduce).Least healthy group of bacte-
ria dies out and the other healthiest splits into two.
3.4. Elimination and Dispersal (Ned)
Elimination event may occur for example when local
significant increases in heat kill a population of bacteria
that are currently in a region with a high concentration of
nutrients. A sudden flow of water can dispose bacteria
from one place to another. The effect of elimination and
dispersal event is possibly destroying chemotactic pro-
gress, but they also have the effect of assisting in Chemo-
taxis, since dispersal may place bacteria near good food
sources. The flowchart for BFA is depicted in Fig u re 4.
4. BF Algorithm to Estimation of
Transformer Equivalent Parameter
The proposed method is employed to search the optimal
equivalent circuit parameters for the three winding
transformer. Each bacterium contains nine members: r1,
x1, 23
x
, 2, 2
r
x
, 3, 3
r
x
, Rc and Xc. If there is s num-
ber of bacteria in a population, then the dimension of
population is s × 9. The process of estimate the equiva-
lent circuit parameters of the transformer can be ex-
plained as follows: First input the bacterial foraging pa-
rameters and conventional measured data, and also spec-
ify lower and upper limits of the equivalent circuit pa-
rameters. Generate the positions of the equivalent
circuit parameter randomly and evaluate the objective
value of each bacterium. After evaluating the objective
function, modify the position of the equivalent circuit
parameters for all the bacteria using the tumbling/
swimming process and perform reproduction and elimi-
nation operation. The output of equivalent circuit pa-
rameters are obtained when the maximum steps is
reached. Finally, compute the operating performances of
the transformer such as efficiency and regulation. In
proposed method, the process of “chemotaxis” enables
bacteria to obtain a satisfactory ability of local search. It
is worth notice that the individuals in bacterial foraging
algorithm could converge rapidly without information
sharing between each other, which is di fferent from other
m
i
), and Xi. Also initialize all the counter
va
l + 1
1
ion value and
e corresponding
function is cal-
of input samples.

i = 1, 2, s, take the tumbling/swimming
de
each ele-
m, 2, p, a random number.
Move: Let
ethods.
The algorithm for proposed method as follows
Step 1: In tialize parameters P, s, Nre, Ned, Ped, C(i)
(i = 1, 2, , s
lues to zero.
Step 2: Elimination-dispersal loop: l =
Step 3: Reproduction loop: k = k +
Step 4: Chemotaxis loop: j = j + 1
1) For i = 1, 2, , s, calculate cost funct
efficiency- for each bacterium i as follows.
 Nis signal samples are passed through the model.
 The output is then compared with th
desired signal to calculate the error.
 The same of the squared error averaged over Nis is
finally stored in J(i, j, k, l). The cost
culated for number
End of for loop.
2) For ,
cision
Tumble: Generate a random vector ∆(i) with
ent ∆m(i) m = 1,


 
,, ,,
ii
T
i
jlkljklCi
ii


 

(13)
direction of tumble for bacterium
i i
pute J(i, j + 1, k, l) and then
Let
Fixed step size in the
s considered.
Com






,1,, ,1,,
1, ,,1, ,
i
cc
JswijklJijkl
J
jklPjkl 

 
(14)
(have not climbed down too long)
st (if doing better), let Jlast=
Jsw (i, j + 1, k, l) and Let
Swim:
a) Let m = 0; (counter for swim length)
b) While m < Ns
 Let m = m + 1
 If Jsw (i, j + 1, k, l) < Jla


 
,, ,,
ii
T
i
jlkljklCi
ii


 

(15)
use this θ(j + 1, k, l) to compute the new J (i, j +
et m = Ns. This is the end of the while state-
i+1) if i ≠ s (i.e. go to b) to
pr
e
Ch of the bacteria is not over.
r the
ascending cost Jhealth
(hi
e made are placed at the same location as
th
o the
And i
1, k, l)
 Else, l
ment.
3) Go to next bacterium (
ocess the next bacterium.
Step 5: If j < Nc, go to step 4. In this case, continu
emotaxis since the life
Step 6: Reproduction:
1) Fogiven k and l, and for each i = 1, 2, , s,
Let i
health
J= min Jsw(i, j, k, l) be the health of the bac-
terium i (a measure of how many nutrients it got over its
life time and how successful it was at avoiding noxious
substance). Sort bacteria in order of
gher cost means lower health).
2) The Sr = s/2 bacteria with highest Jhealth values die
and other Sr bacteria with the best value split (and the
copies that ar
eir parent)
Step 7: If k < Nre go to 3. In this case, the number of
specified reproduction steps has not been reached, s

S. SUBRAMANIAN ET AL.
139
ne ed
then go to step 2; otherwise,
rint the results and stop.
. Results and Discussions
TW
ain the optimal value of equiva-
le
est
m
ance curve of efficiency and regulation obtained from
xt generation of the chemotactic loop is start.
Step 8: Elimination-dispersal: For i = 1, 2, , s, with
probability Ped, eliminates and disperses each bacterium
(this keeps the number of bacteria in the population con-
stant). To do this, if a bacterium is eliminated, simply
disperse another one to a random location on the optimi-
zation domain. If l < Ned,
p
5
To validate the feasibility, the proposed method have
been employed for parameter estimation of single phase,
5 KVA, 220 V, 50 Hz, three winding transformer. Load
test data and computed performance characterises of
T half to full load condition are presented in Table 1.
The equivalent circuit parameters are also determined
by performing the suitable tests and the predetermined
efficiency and voltage regulation are compared with the
load test values which are presented in Table 2. Com-
parsion of these two performances shows that there is a
deviation in the values of efficiency and regulation. By
estimating the accurate value of equivalent circuit pa-
rameters, the deviation can be minimized. Hence the
problem has been formulated as an error minimization
problem and the modern heuristic search technique, BFA
has been applied to obt
nt circuit parameters.
The parameters of the BFA used for the simulation
studies are summarized in Table 3. The best results are
obtained from 20 trail runs and are reported in Table 4.
For the sake of comparison, rated load condition is con-
sidered and the efficiency and voltage regulation at rated
load are predetermined by using the estimated equivalent
circuit parameters. The simulation results obtained by the
proposed method are compared with actual load test
measurements and Genetic Algorithm (GA) based results,
the comparison is given in Table 4. From the compari-
son, it is revealed that an improvement in the predeter-
mined efficiency and voltage regulation. The simulation
has been performed to various load conditions. For each
load conditions, the efficiency and voltage regulation are
computed. Comparative studies with actual load t
easurement, GA and BFA are presented in Table 5.
In addition, the percentage of error over the load test
measurement is computed and is also presented in Table
5. The comparison clearly shows the reduction in error
between the actual and estimated data. BFA based esti-
mation of equivalent circuit parameters values are close
to directly measured values. This facts lead to a conclu-
sion that the proposed methodology provide to global
optimum solutions. Figures 5 and 6 shows the perform-
N
o
Yes
No
Ye s
No
Yes
Yes
No
Yes
Start
Initialize all variables. Set all loop-counters
and bacterium index i equal to 0.
Increase elimination–dispersion loop
counter l = l_+1
l < N
ed
?
Increase reproduction loop counte
r
k = k + 1
k < N
re
?
m < N
s
?
Increase bacterium index i = i + 1
Increase chemotactic loop counter j = j + 1
i < s
Compute the objective function value for the i
th
bacterium J(i, j, k, l), adding the cell to cell attractant
effect to nutrient concentration and set J
last
= J(i, j, k, l)
Tumble (let the i
th
bacterium take the step of height
C(i) along a randomly generated tumble vector (i))
Compute the objective function value J(i, j + 1, k, l)
taking in to account the cell to cell attractant effect
Set swim count m = 0
m = m + 1
Set
J
last
=
J
(i,
j
+1,
k
,l) and swim (let the i
th
bacterium take a set of height C(i) along the
direction of the same tumble vector  (i))
Perform
Elimination-
dispersal
Print the
results and
stop
j < N
c
?
Perform
Reproduction
No
Setm=
N
s
Yes
No
J
(i,j+1,
k
,l) <
J
last
?
Figure 4. Flow chart for baraging algorithm. cterial fo

S. SUBRAMANIAN ET AL.
140
Table 1. Load test data of a single phase TWT at 220 V, 50 Hz supply.
Load (%) I1 (A) P1 (W) 2
V(V) 2
I
(A) P2 (W) 3 V

(V) 3
I

(V) P3 (W) Efficiency (%) Voltage regulation (%)
50 11.55 2510 210.8 5.60 1175 210.8 5.90 1240 96.22 4.18
60 13.50 2940 210.0 6.50 1360 210.0 6.95 1455 95.75 4.55
70 15.30 3350 209.0 7.45 1550 209.0 7.85 1640 95.22 5.00
80 18.00 3900 206.4 8.95 1840 206.4 9.05 1860 94.87 6.18
90 20.20 4400 205.6 10.20 2095 205.6 10.00 2050 94.21 6.55
100 22.10 4800 204.4 11.05 2255 204.4 11.00 2240 93.65 7.09
Table 2. Directly measured parameters and performance.
Parameters () r1 () x1 () 23
x
() 2
r() 2
x

()3 r

()3
x

()Rc ()Xc ()Efficiency (%) Voltage regulation (%)
Measureda 0.3073 0.3914 - 0.5700 0.27960.64000.05171058.0263.393.05 6.70
Measuredb - - - -- - - - - - 93.65 7.09
aParameters measured from OC and SC tests...; bFull load performance directly measured from load test.
Table 3. Parameter used for BFA method.
Parameter Value
Number of bacterium (s) 20
Number of chemotatic steps (Nc) 10
Swimming length (Ns) 4
Number of reproduction steps (Nre) 4
Number of elimination and dispersal events (Ned) 5
Depth of attractant (dattract) 0.1
Width of attractant ( 
attract) 0.2
Height of repellent (hrepellant) 0.1
Width of repellent ( 
repellant) 10
Probability of elimination-dispersal events (Ped) 0.02
Table 4. Comparison of estimated parameters of TWT us-
ing GA, BFA with measured data.
Estimated
Parameters Measured
GA BFA
r1 () - 0.2733 0.2810
x1 () - 0.4381 0.4223
23
x

() - 0.3987 0.3572
2
r

() - 0.5376 0.5468
2
x

() - 0.4981 0.5122
3
r

() - 0.6038 0.5943
3
x

() - 0.4491 0.4238
Rc () - 1121.1 1121.1
Xc () - 265.3 264.2
Efficiency (%) 93.65 93.57 93.68
Voltage regulation (%) 7.09 7.10 7.02
Figure 5. Performance curve of efficiency. Figure 6. Performance curve of regulation.

S. SUBRAMANIAN ET AL.
141
Table 5. Comparison of performance of TWT using GA, BFA against with directly measured data.
Efficiency (%) Regulation (%)
% Load Measured GA % Error BFA % ErrorMeasuredGA % Error BFA % Error
50 96.22 95.52 –0.72 96.29 0.07 4.18 4.15 –0.58 4.14 –0.96
60 95.75 95.26 –0.51 95.81 0.06 4.55 4.53 –0.37 4.510 –0.89
70 95.22 94.89 –0.34 95.35 0.13 5 4.99 –0.17 4.97 –0.6
80 94.87 94.51 –0.37 94.97 0.04 6.18 6.14 –0.55 6.12 –0.97
90 94.21 94.01 –0.21 94.30 0.09 6.55 6.54 –014 6.49 –0.91
100 93.65 93.57 –0.08 93.68 0.03 7.09 7.10 0.14 7.02 –0.98
actual load test values, GA and BFA method. It is obvi-
ous that the performance characteristics of the sample
transformer using BFA based parameter estimation
method shows the better performance than other optimi-
zation method.
6. Conclusions
In this article, the BFA has been suggested to estimate
the equivalent parameters of TWT. The equivalent cir-
cuit parameters obtained by OC and SC tests. The calcu-
lated performance characteristics such as efficiency and
voltage regulation by using these parameters differ with
the values of load test which indicates that the calculated
parameters are inaccurate. The problem has been formu-
lated as an error minimization problem and the modern
heuristic search technique namely; BFA has been applied
to estimate the accurate equivalent circuit parameters.
The feasibility of the proposed technique has been tested
with single phase three winding transformer, and the
results are compared with actual load test values and GA
based results. From this comparative study, it clearly
indicates that the proposed method provides an accurate
estimate of equivalent circuit parameters hence an im-
provement in the performance characteristics. The pro-
posed method having the merits such as less mathemati-
cal burden, accurate estimate, high quality solution, fast
convergence and less computational time. The proposed
method can be applied to any capacity of transformer.
The BFA technique promises to be quite efficient in
solving highly nonlinear optimization problem so its ap-
plication in some other fields may also be tried.
7. Acknowledgements
The authors are grateful to the authorities of Annamalai
University for providing all facilities to carry out this
work.
8. References
[1] S. H. Thilagar and G. S. Rao, “Parameter Estimation of
Three-Winding Transformer Using Genetic Algorithm,”
Engineering Applications of Artificial Intelligence, Vol.
15, No. 5, 2002, pp. 429-437.
doi:10.1016/S0952-1976(02)00087-8
[2] B. A. Mork, “Five-Legged Wound-Core Transformer
Model: Derivation, Parameters, Implementation, and
Evaluation,” IEEE Transactions on Power Delivery, Vol.
14, No. 4, 1999, pp. 1519-1525. doi:10.1109/61.796249
[3] X. S. Chen and S. S. Venkata, “A Three-Phase Three-
Winding Core-Type Transformer Model for Low-Fre-
quency Transient Studies,” IEEE Transactions on Power
Delivery, Vol. 12, No. 2, 1997, pp. 775-781.
doi:10.1109/61.584369
[4] H. K. M. Youssef and K. M. Naggar, “Genetic-Based
Algorithm for Identification Synchronous Machine Pa-
rameter Using Short Circuit Tests,” International Journal
of Energy Research, 2000, Vol. 24, No. 10, pp. 877-885.
doi:10.1002/1099-114X(200008)24:10<877::AID-ER630
>3.0.CO;2-8
[5] P. Pillay, V. Levin, P. Otaduy and J. Kueck, “In-Situ
Induction Motor Efficiency Determination Using the Ge-
netic Algorithm,” IEEE Transactions on Energy Conver-
sion, Vol. 13, No. 4, 1998, pp. 326-333.
doi:10.1109/60.736318
[6] B. abdelhadi, A. Benoudijit and N. Nait-Said, “Applica-
tion of Genetic Algorithm with a Novel Adaptive Scheme
for the Identification of Induction Machine Parameters,”
IEEE Transactions on Energy Conversion, Vol. 20, No. 2,
2005, pp. 284-291.
doi:10.1109/TEC.2004.841508
[7] S. H. Thilagar and G. S. Rao, “In-Situ Determination of
Transformer Winding Temperature Rise Using Genetic
Algorithm Based Parameter Estimation,” Proceedings of
the International Conference Electrical and Computer
Engineering, Dhaka, 26-28 December 2002, pp. 91-94.
[8] B. L. Peacock, M. Steurer, J. Langston, T. Baldwin and S.
Henry, “Estimation of Parameters for Digital Modelling
of Saturation Effects for a Transformer Model for a
Real-Time Simulator,” Proceedings of the 41st South-
eastern Symposium System Theory, University of Ten-
nessee Space Institute Tullahoma, USA, 15-17 March
2009, pp. 306-310. doi: 10.1109/SSST.2009.4806811
[9] S. A. Soliman, R. A. Alammari and M. A. Mostafa,
“On-Line Estimation of Transformer Model Parameters,”
Proceedings of the Large Engineering System, Halifax,

142 S. SUBRAMANIAN ET AL.
28-30 July 2004, pp. 170-178.
doi: 10.1109/LESCPE.2004.1356295
[10] F. Alonge, F. D’lppolito, G. Ferrante and F. M. Raimondi,
“Parameter Identification of Induction Motor Model Us-
ing Genetic Algorithms,” IEE Proceedings on Control
Theory Applications, Vol. 145, No. 6, 1998, pp. 587-597.
doi:10.1049/ip-cta:19982408
[11] J. Wang, A. F. Witulski, J. L. Vollin, T. K. Phelps and G.
I. Cardwell, “Derivation, Calculation and Measurement
of Parameters for a Multi-Winding Transformer Electri-
cal Model,” Proceedings of Applied Power Electronics,
Dallas, 14-18 March 1999, pp. 220-226.
doi: 10.1109/APEC.1999.749513
[12] R. K. Ursem and Vadstrup, “Parameter Identification of
Induction Motors Using Differential Evolution,” Pro-
ceedings of Evolutionary Computation, Canberra, 8-12
December 2003, pp. 790-796.
doi: 10.1109/CEC.2003.1299748
[13] K. S. Huang, W. Kent, Q. H. Wu and D. R. Turner, “Pa-
rameter Identification of an Induction Machine Using
Genetic Algorithms,” Proceedings of the Computer Aided
Control System Design, USA, 22-27 August, 1999, pp.
510-515. doi: 10.1109/CACSD.1999.808700
[14] A. Keyhani, A. Tsai and S. Sebo, “Modeling and Pa-
rameter Estimation of Power Transformers for the Study
of High Frequency System Transients,” Proceedings of
Circuits and Systems, Champaign, 14-16 August 1989, pp.
258-264. doi: 10.1109/MWSCAS.1989.101842
[15] P. Y. Chung, M. Dolent and R. D. Lorenz, “Parameter
Identification for Induction Machines by Continuous Ge-
netic Algorithms,” Proceedings ANNIE, St. Louis, 5-8
November 2000, pp. 1-13.
[16] O. Amoda, D. Tylavsky, G. McCulla and W. Knuth,
“Sensitivity of Estimated Parameters in Transformer
Thermal Modeling,” Proceedings NAPS, Starkville, 4-6
October 2009, pp. 1-6. doi: 10.1109/NAPS.2009.5484079
[17] D. Meister and M. A. G. de Oliveira, “The Use of the
Least Squares Method to Estimate the Model Parameters
of a Transformer,” Proceedings of Electrical Power
Quality and Utilization, Lodz, 15-17 September 2009, pp.
1-6. doi: 10.1109/EPQU.2009.5318853
[18] A. Suppitaksakul and V. Saelee, “Application of Artifi-
cial Neural Networks for Electrical Losses Estimation in
Three-Phase Transformer,” Proceedings of Electrical
Engineering/Electronics, Computer, Telecommunications
and Information Technology, Pattaya, 6-9 May 2009, pp.
248-251. doi: 10.1109/ECTICON.2009.5137002
[19] J. J. Wu, H. Sugimoto and C. K. Wang, “Parameter Esti-
mate Modeling of Electronic Transformer,” Proceedings
of Power Electronics and Motion Control, Shanghai,
14-16 August 2006, pp. 1-8.
doi: 10.1109/IPEMC.2006.4778192
[20] M. R. Feyzi and M. Sabahi, “Online Dynamic Parameter
Estimation of Transformer Equivalent Circuit,” Proceed-
ings of Power Electronics and Motion Control, Shanghai,
14-16 August 2006, pp. 1-5.
doi: 10.1109/IPEMC.2006.4778186
[21] B. K. Panigrahi and V. R. Pandi, “Bacterial Foraging
Optimisation: Nelder-Mead Hybrid Algorithm for Eco-
nomic Load Dispatch,” IET Generation, Transmission
and Distribution, Vol. 2, No. 4, 2008, pp. 556-565.
doi:10.1049/iet-gtd:20070422
[22] M. Tripathy and S. Mishra, “Bacteria Foraging Based
Solution to Optimize Both Real Power Loss and Voltage
Stability Limit,” IEEE Transaction on Power Systems,
Vol. 22, No. 1, 2007, pp. 240-248.
doi:10.1109/TPWRS.2006.887968
[23] W. J. Tang, Q. H. Wu and J. R. Saunders, “Bacterial
Foraging Algorithm for Dynamic Environments,” Pro-
ceedings of the IEEE Congress Evolutionary Computa-
tion, Canada, 16-21 July 2006, pp. 1324-1330.
doi: 10.1109/CEC.2006.1688462
[24] D. P. Acharya, G. Panda, S. Mishra and Y. V. S. Lakshmi,
“Bacteria Foraging Based Independent Component
Analysis,” Proceeding of the Computational Intelligence
and Multimedia Applications, Sivakasi, 13-15 December,
2007, pp. 527-531. doi: 10.1109/ICCIMA.2007.126
[25] G. Noriega, J. Restrepo, V. Guzmán, M. Gimenez and J.
Aller, “On-Line Parameter Estimation of Electric Sys-
tems Using the Bacterial Foraging Algorithm,” Proceed-
ings of the Power Electronics and Applications, Barce-
lona, 8-10 September 2009, pp. 1-7.
[26] P. G. Kou, J. Z. Zhou, C. S. Li, Y. Y. He and H. He,
“Identification of Hydraulic Turbine Governor System
Parameters Based on Bacterial Foraging Optimization
Algorithm,” Proceedings of the Natural Computation,
ICNC, Shandong, 10-12 August 2010, pp. 3339-3343.
doi: 10.1109/ICNC.2010.5583639

S. SUBRAMANIAN ET AL.
143
Nomenclature
C(i) Step size
i Bacterium number
j Counter for chemotactic step
J (i, j, k, l) Cost at the location of ith bacterium
Jcc Swarm attractant cost
i
health
J Health of bacterium i
k Counter for reproduction step
l Counter for elimination-dispersal step
m Counter for swimming locomotion
Nc Maximum number of chemotactic steps
Ned Number of elimination-dispersal events
Nre Maximum number of reproduction steps
Ns Maximum number of swims
P Dimension of the optimization problem
Ped Probability of occurrence of elimination-
dispersal events
s Population of the E. coli bacteria
i

(j, k, l) Location of the ith bacterium at jth
chemotactic step, kth reproduction step,
and l the elimination-dispersal step

attract Width of attractant

repellant Width of repellent
hrepellent Height of repellent
dattract Depth of attract
r1 Resistance of the primary winding (Ω)
x1 Leakage reactance of the primary wind-
ing (Ω)
23
x
 Mutual leakage reactance between sec-
ondary and tertiary windings referred to
primary side (Ω).
2
r Resistance of the secondary winding re-
ferred to primary side (Ω)
2
x
 Leakage reactance of the secondary
winding referred to primary side (Ω)
3
r

Resistance of the tertiary winding re-
ferred to primary side (Ω)
3
x

Leakage reactance of the tertiary winding
referred to primary side (Ω)
Rc Core loss equivalent resistance (Ω)
Xc Magnetizing reactance (Ω)
2l
r

and 3l r

Load resistances of secondary and terti-
ary side referred to primary side (Ω)
z1, z2, z3 Measured Primary, secondary and terti-
ary side impedances respectively(Ω)
2
z

and 3 z

Secondary and tertiary side impedance
referred to primary side (Ω)
z1est Estimated Primary side impedance (Ω)
V1mes,2mes V

,3mes V

Measured value of primary, secon-
dary and tertiary voltage referred to
primary side (V)
I1,2
I

,3
I

Measured value of primary, secondary
and tertiary current referred to primary
side (A)
V1est Estimated value of primary voltage (V)
E1 Voltage across the magnetizing winding
(V)
Im Current through the magnetizing winding
(A)
P1mes Measured value of power in primary side
(W)
P1est Estimated value of power in primary side
(W)
P1, P2, P3 Measured values of power primary, sec-
ondary and tertiary side (W)
P1cu, P2cu, P3cu Copper loss components of primary,
secondary and tertiary side (W)
Pcore Core loss component (W)