mathematics-11-01796 (1)

athematics 2023, 11, x FOR PEER REVIEW
Mathematics 2023, 11, 1796 18 of 32
From the reported results, the ESMOA algorithm gives a slightly smaller ITAE value
Figure 6. Dynamic responses for CO, GSO, and the proposed ESMOA for Case 1.
(0.023431548) compared with that given by the GSO (0.026772795). The decreased amount
represents frequency 12.486% in area which 1. (b) declares Deviation a significant in frequency improvement in area percentage 2. (c) Deviation in this case. in transferred
On
the interconnected other side, as shown tie-line. in Figure 6, the corresponding outputs are reasonably coincident
regarding the change in frequency in area 1 (Figure 6a). However, slight improvement
is shown in the change in frequency in area 2 (Figure 6b). At the same time, significant
Figure 7 depicts the assessed four measures of the lowest, mean, m
mitigation is declared regarding the change in power transfer between the two areas
(Figure standard 6c). deviation produced by ITAE throughout several independent ope
vide Figure statistical 7 depicts comparability the assessed fourbetween measures CO, of theGSO, lowest, and mean, the maximum, suggested and ESMO
standard deviation produced by ITAE throughout several independent operations to
states the recommended ESMOA’s high efficacy and capability compared to
provide statistical comparability between CO, GSO, and the suggested ESMOA. This
figure The suggested states the recommended ESMOA ESMOA’s is used to high obtain efficacythe andsmallest capability compared measurements, to CO andas illus
GSO. the The smallest suggested minimum, ESMOA is used mean, to obtain maximum, the smallest measurements, and standard as illustrated. deviation It with
finds the smallest minimum, mean, maximum, and standard deviation with 0.023431548,
0.024266891, 0.02780845, and 0.001298062, respectively.
0.024266891, 0.02780845, and 0.001298062, respectively.
Figure 7. Statistical measures for CO, GSO, and the proposed ESMOA for Case 1.
Figure 7. Statistical measures for CO, GSO, and the proposed ESMOA for Case 1.
Table 4 contrasts the efficacy of the proposed ESMOA-based PD-PI controller with
variousTable previously 4 contrasts publishedthe controlling efficacy methods of the concerning proposed ITAE ESMOA-based and settling time. PD-PI c
As various shown, previously the proposed ESMOA-based published controlling PD-PI controller methods obtains the concerning minimum ITAE of and se
0.02343 where the conventional PI, PI-based-BFOA, PI-based-DE, PI-based-BFOA-PSO,
shown, the proposed ESMOA-based PD-PI controller obtains the minim
PI-based-GA, PI-based-FA, PI-based-PSO, PID-based-ARA, PI-based-FA, CO-based PD-PI
controller, 0.02343 and where GSO-based the conventional PD-PI controller find PI, 3.5795, PI-based-BFOA, 1.8379, 0.9911, 1.1865, PI-based-DE, 2.7475, 0.8695, PI-base
1.2142, PI-based-GA, 0.075401, 0.4714, PI-based-FA, 0.02392, andPI-based-PSO, 0.02677, respectively. PID-based-ARA, Regarding the lowest PI-based-FA,
ITAE
value, frequency settling time, and tie-line power variations, the proposed ESMOA-based
PI controller, and GSO-based PD-PI controller find 3.5795, 1.8379, 0.9911,
PD-PI controller beats the other previously published optimization strategies, as shown in
the 0.8695, table. 1.2142, 0.075401, 0.4714, 0.02392, and 0.02677, respectively. Regard
ITAE value, frequency settling time, and tie-line power variations, the prop
based PD-PI controller beats the other previously published optimization
shown in the table.
Table 4. Comparison of the proposed ESMOA outcomes with other reported res
ITAE and settling time.