IEEE Systems, Man and Cybernetics Magazine - July 2021 - 35

nonreheater thermal plant are the governor, turbine, and
generator load models.
In Figure S3, each of the areas has a feedback controller,
which minimizes the frequency variations in the system
and power variations in the tie lines. The controller in
area 1 is
kk
Kk [ ,,,],kkk,1
1 234
area
78
,],
=
area 4 is
in area 3 is
area
=
t
Ot fP dt The parameter
i a RRnt
=+==
:
min #
k
n
k
k
The objective function for this system is taken as
()
[ TT .
11 tie,k
Here, the value of nt is four.
D
l =- 10 ,
values considered for this system are given in the
" Parameter Values for Interconnected Four-Area Power Systems "
section in " Additional Information. " Here, km
k =
u 10
m
For the GSA, N is considered as 30; ()G 0c
, 6f= ,, are considered in this problem.
and b are set at 30
and md1
and 10, respectively; and the maximum number of iterations is
500 [1], [40]. The performance indices are represented by the
maximum peak value (),Mp
settling time (),ts
d1
and ISE.
To demonstrate the performance of the proposed controller,
load disturbances in two areas are considered.
.
,
.
=
T =P 001 p.u. MW is considered in area 1 at time t 5s=
and T =d3P 001 p.u. MW is considered at area 3 at time
t 100 s [29].
At first, the system is simulated using the proposed
method considering the nominal system parameters. The
optimal controller gains obtained by the proposed method
are
K 23 89156 [.=
, 45.5814, 14.0617, 38.3191, 7.0475, -3.112,
20.6083, 8.4636, 22.087, 2.72064, -1.4539, -1.2643, 9.8369,
-2.9005, 22.691].
Then, these optimal gains are considered along with
the perturbed system parameters to obtain the sensitive
performances of the controller. Here, ±25% variations in
,,, ,,,,
KKKK TTT Tand ps
ps
ps
ps ps ps ps
check the sturdiness of the proposed controller. The performance
obtained from the system for all such variations
is tabulated in Table S4.
It is observed from Table S4 that the frequency fluctuations
and tie-line power deviations are very small
cons ider i ng the nomi nal sys t em paramet er s .
Furthermore, simultaneous variations in the power
Area 2
RT
Area 3
RT
ps123 4123 4 are considered to
050
100
Time (s)
(a)
2 × 10-5
1
-3
-2
-1
050
Area 1
RT
Area 4
NRT
Figure 3. A simplified diagram of a four-area
interconnected power system [29]. The system with
the proposed controller is available in " Additional
Information. " NRT: nonreheater thermal plant; RT:
reheater thermal plant.
100
Time (s)
(b)
Figure 4. The output responses of a four-area power
system considering the nominal and perturbed
system parameters: the (a) frequency variations in
area 1 and (b) tie-line power deviations between
areas 1 and 2.
July 2021 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 35
150
200
Nominal Parameters
+25% of Nominal Kps and Tps
-25% of Nominal Kps and Tps
150
200
-5
Nominal Parameters
+25% of Nominal Kps and Tps
-25% of Nominal Kps and Tps
=
Kk [ ,,].kk
,4
13 14 15
Kk [ ,,,],kkk,3 9101112
area
in area 2 is Kk k [, ,
and, finally, in
area,2 56
=
system gains and time constants have a negligible effect
on system performances. For example, the maximum
peak value of the area 4 frequency deviation, with nominal
system parameters, is -0.00006585 Hz. However, a
+25% variation in the considered system parameters
yields a maximum peak of -0.00006586 Hz. Therefore,
the difference in the peak values between these two
conditions is almost negligible.
Similar performances are also observed for the frequency
and power deviations in other areas and tie lines.
The responses of area 1 frequency deviations and the
power deviations in the tie line between areas 1 and 2
with all such variations are shown in Figure 4. The frequency
responses of the remaining areas and the remaining
tie-line power deviations are available in " Additional
Information " (Figure S4). It is noted from Figures 4 and S4
that the responses of the system with the nominal and
perturbed parameters are stable. Thus, the proposed
method is competent enough to operate with an
5 × 10-4
∆Ptie12 (p.u. MW)
∆f1 (Hz)
IEEE Systems, Man and Cybernetics Magazine - July 2021

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