IEEE Systems, Man and Cybernetics Magazine - January 2023 - 53

The global best solution is defined as
()
q
g
=
kN
nnm
!
!
"
"
1
1
,,
,,
f
f
,
Or min 6On qk rnand
,
b()@ !! .
(42)
After the completion of nm number of iterations, ()
Drq
will be the optimal value of the decision variable. A pseudocode
of the proposed method is given in algorithm S1 (available
at https://doi.org/10.1109/MSMC.2022.3208393).
Results
First, the results of the LFC system are presented by considering
the four variations in the objective function.
Next, the effects of variations in the input disturbances
considering the nominal and the perturbed system parameters
are demonstrated, followed by a comparative study
of the proposed method with a few existing works. The
proposed method is simulated 10 times, and the best
results among those are presented. In the GSA algorithm,
N ,20=
b 20 ,=
G ,. ,
0 100 0 0001v== and n 10m
= [15] are
considered. The system parameter values are given in
Appendix B (avai lable at https://doi.org/10.1109/
MSMC.2022.3208393) [11], [25]. The obtained range to
maintain the stability of the considered system is
2 ..
ub
d 2 8108179 Here, e in dlb is considered as 11 ,0 10
-
1
lb
#
and dd .8=+ The initialization and the update of the
positions of the agents from the given range ensure system
stability. Therefore, checking the stability of the system
for each agent in each iteration (unlike the methods in
[2], [11], and [23]) is now obliterated. The performances of
the LFC system are shown with a load disturbance of
0.01 p.u. [11], [25] at time t = 0 s. The zoomed-in output
responses of the system are featured in the inset of the
corresponding figures (Figures 2-4).
The optimal controller gains for objective function O1
., ., and k3 =- 449 336
., ., and k3 =- 459 828
., ., and k3 =- .;42 784 and
., ., and k3 =- 42 78
are kk8 163
12 14 273==they
are kk8 263
O3, they are kk0 366
12 14 343==12
8 803==for
O4, they are kk0 384
12 8 815==.;
for O2,
.; for
..
The performances are demonstrated through the maximum
peak value of output response (),Mp
the settling time
(ts), the steady state error (ess), and the error index integral
square error (ISE).
The performances considering these gain values for all
of the objective functions are tabulated in Table 1. It is
observed that the values of ISE for all of the objective
functions are very low. Furthermore, an improvement in
the Mp
-8 .498 10 4
-
#
#
in comparison
Objective
O1
O2
O3
O4
parameters have improved the overall system performances
in comparison with objectives like O1 or O3. The
proposition is corroborated by the responses of the system
as shown in Figure 2. Therefore, it is observed that
the proposed method helps to maintain system stability
and improves the steady state and the transient performances
of the output responses.
Performances of the LFC System Against
Parametric Uncertainties With Variable
Load Disturbances
The performances of the system considering two timevarying
load disturbances along with the nominal and
perturbed system parameters are presented. The performances
are demonstrated through the maximum overshoot
of the output response
^h the maximum
M ,os
p
undershoot ^h ts, and ISE. First, a variable step load
M ,us
p
disturbance as shown in Figure S1(a) (available at https://
doi.org/10.1109/MSMC.2022.3208393) is considered for the
analysis [2], [26]. Next, a sinusoidal load disturbance is
considered according to .( .) .
3 =+
Pt
Table 1. The Performances of the LFC
System for the Proposed Method.
Mp (Hz) (-ve)
(#10-4)
8.498
8.412
31.817
31.659
ts (s)
2.843
1.967
9.836
9.818
ess (-ve)
(#10-5)
4.956
4.85
44.67
44.73
ISE
(#10-8)
9.49
9.28
558.78
530.64
D 00346005
sinsin
0.003
0.002
0.001
−0.001
−0.002
−0.003
−0.004
value is observed in the system output responses
considering objectives O2 and O4 in comparison with
objectives O1 and O3, respectively. For instance, the Mp
value corresponding to O1 is
with the Mp value of -8 .412 10 4
-
corresponding to O2.
Similar improvements in the ts value are observed with O2
and O4 in comparison with O1 and O3, respectively. It is
noted that the objective functions (O2 or O4) equipped
with the steady state and the transient performance
0.001
−0.001
0246 810
12
Reference
Objective O1
Objective O2
Objective O3
Objective O4
Figure 2. The output responses of the LFC system by the
proposed method with objectives O1, O2, O3, and O4.
January 2023 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 53
∆F (Hz)
https://www.doi.org/10.1109/MSMC.2022.3208393 https://www.doi.org/10.1109/MSMC.2022.3208393 https://www.doi.org/10.1109/MSMC.2022.3208393 https://doi.org/10.1109/MSMC.2022.3208393 https://doi.org/10.1109/MSMC.2022.3208393
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