IEEE Systems, Man and Cybernetics Magazine - July 2021 - 34

It is also noted from Table S1 that the variations in the
settling time of system responses are also significantly
small. The comparative responses of the dynamic performances
of the area 1 frequency with both the nominal
and perturbed system parameters are demonstrated in
Fig ure 2(a). The comparative responses of the area 2 frequency
and tie-line power deviations are available in
" Additional Information " [Figure S1(a) and (b), respectively].
A
ps ps
,, ,,
t
sg
!25 % variation in KT TT and R is considered
to demonstrate the system response under the
parametric uncertainty.
It is noted that the nominal and perturbed parameter
responses are stable and very close to each other. Therefore,
the optimal controller gains obtained by the proposed
method with the nominal parameters are robust
and do not need to change with the given variations in
the system parameters. Nevertheless, the maximum
parameter variations in the system are tested for parameter
R, keeping other parameters unchanged (see the
0 × 10-3
Nominal Parameters
25% of Nominal
-1
-2
-3
-4
04 812
Time (s)
(a)
0 × 10-3
Nominal Parameters
+25% of Nominal
Kps, Tps, Tt, Tsg, and R
-25% of Nominal
Kps, Tps, Tt, Tsg, and R
-0.5
Kps, Tps, Tt, Tsg, and R
-25% of Nominal
Kps, Tps, Tt, Tsg, and R
0 × 10-3
-1
-2
-3
-4
01 2
16
20
" Parameter Values for PV Integrated Thermal Power Systems "
section in " Additional Information " ). Considering
the optimal controller gains from case A, it is noted that
the parameter R should be lower than 1.40366262. Any
further increment of R will result in system instability for
that particular controller gain and system parameters.
Case C: Sensitivity Analysis of the Controller
Against Variable Load Disturbances
This section demonstrates the sensitivity of the proposed
controller against variable load fluctuations. A load variation,
as available in [19], is considered in area 2. Further
details of the load variations are available in " Additional
Information " (Figure S2).
In this case, both the nominal and perturbed system
parameters are considered with the variable load disturbances.
Here,
ps ps
,, ,,
t
considered. The controller gains for this scenario are the
same as the gains obtained from the nominal parameters.
The performance of the system is tabulated in Table S3.
It is observed that the variations in the maximum peak,
settling time, and error index are remarkably low in comparison
to the respective nominal performances. For example,
considering the variable load disturbance with nominal
system parameters, Mp
of area 1 is -0.0008455 Hz.
However, with a +25% variation in the considered parameters,
Mp
ference in Mp between these two cases is 0.0000896 Hz,
which is exceptionally low and within tolerance limits. Similar
behaviors are also observed for the other responses.
The responses of the system with all such variations
are shown in Figure 2(b) as well as Figure S1(c) and (d). It
is evident from the responses that the optimal controller
gains obtained by the proposed method with the nominal
parameters are maintaining system stability in the presence
of both load and system parameter variations. Thus,
the performance of the proposed controller is highly effective
and powerful when there is a chance of change in load
disturbances and/or system parameters.
-1
010203040
Time (s)
(b)
50
60
70
Extension to Four-Area Power Systems
The proposed method is extended to an interconnected
four-area power system taken from the existing literature
[29]. The authors of [29] proposed a direct synthesis
method to design PID controller gains in the frequency
domain for the considered system. A simplified diagram
of the interconnected four-area power system is shown in
Figure 3. The four-area power system with the proposed
feedback controller is available in " Additional Information "
(Figure S3).
The considered system has three reheater thermal
Figure 2. The output responses of a two-area PV
integrated thermal power system considering the
nominal and perturbed system parameters: the area
1 frequency deviations (a) for a fixed load disturbance
and (b) considering variable load disturbances.
34 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE July 2021
power plants in areas 1, 2, and 3 as well as one nonreheater
thermal power plant in area 4. These areas are interconnected
by four tie lines. The components of the
reheater plants are the governor, turbine, reheater, and
generator load models. The components of the
of area 1 is -0.0009351 Hz. Thus, the absolute dif!25
% variations in KT TTsg and R are
∆f1 (Hz)
∆f1 (Hz)

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