mathematics-11-01796 (1)

Ki 0.88462493 1.597201322 2.639381164
KP2 Area 2
2.881063135 0.731447135 3
Kd 3 2.903096059 2.095134679
n 392.6175818 40.09581819 283.8518794
ITAE Value 0.023431548 0.026772795 0.023922459
ITAE improvement percent compared to the proposed ESMOA - 12.48 2.0521
Mathematics 2023, 11, 1796 16 of 32
• Case Figure 4: 6 Step also shows load disturbance the dynamic only responses in areafor 1 frequency considering variations non-linearities each in region the
and power tie-line system. transfer power. As seen in Figure 6, the suggested ESMOA outperforms the
CO and GSO in reducing the objective function. Even though the GSO algorithm delivers
4.2.1. the quickest Application reaction for Case when 1 approaching the stable region, multiple oscillations are present,
Area particularly 2 remains the unaltered frequency in this deviation case, but in area a 0.12.
p.u step load augmentation hike in
area 1From is assessed. the reported The suggested results, the ESMOA (Figure algorithm 3), GSO, gives and a slightly CO techniques smaller are ITAE applied; value
Figure (0.023431548) 5 showscompared the corresponding with that converging given by the characteristics, GSO (0.026772795). and Table The decreased 3 summarizes amount the
simulated represents data 12.486% for the which investigated declares a ITAE significant minimizing improvement approaches. percentage The summarized in this case. data On
accompany the other side, theas ITAE shown target in Figure scores6, and the the corresponding related parameter outputs settings are reasonably of K p1 , Kcoincident
i , K p2 , K d ,
and regarding n each the area. change Asin shown, frequency the proposed in area 1 ESMOA (Figure 6a). achieves However, the lowest slight value improvement regardingis
the shown ITAEin objective the change of 0.023431, in frequency where in area GSO2 and (Figure CO obtain 6b). At ITAE the same objective time, values significant of 0.023922 mitigation
0.02677, is declared respectively. regarding the change in power transfer between the two areas (Figure and 6c).
Figure 5. Converging characteristics of the suggested ESMOA, GSO, and CO techniques for Case 1.
1.
Table 3. Results for Case 1.
Controller
parameters
Algorithm
Proposed
ESMOA
GSO
K P1 3 3 3
K i 1.198019018 1.882598713 1.125704786
K P2 Area 1
3 3 3
K d 1.575698137 1.163053801 1.712103293
CO
n 500 198.3903318 500
0.167226621 3 2.477149042
K P1
K P2 Area 2 2.881063135 0.731447135 3
K i 0.88462493 1.597201322 2.639381164
K d 3 2.903096059 2.095134679
n 392.6175818 40.09581819 283.8518794
ITAE Value 0.023431548 0.026772795 0.023922459
ITAE improvement percent compared to the
proposed ESMOA
- 12.48 2.0521