hematics 2023, 11, x FOR PEER REVIEWMathematics 2023, 11, 1796 24 of 3224 o(b)(c)Figure Figure 12. Dynamic 12. Dynamic responses for CO, CO, GSO, GSO, and and the proposed the proposed ESMOA ESMOA for Casefor 3. (a) Case Deviation 3. (a) inDeviationfrequency frequency in area in area 1. 1. (b) (b) Deviation in in frequency in area in area 2. (c) 2. Deviation (c) Deviation in transferred in transferred power through power throuinterconnected tie-line.For this case, Figure 13 depicts the assessed four measures of the lowest, mean, mimum, and standard deviation of the produced ITAE throughout several independenterations to provide statistical comparability between CO, GSO, and the suggesESMOA. The suggested ESMOA is used to obtain the smallest measurements, as illtrated. It finds the smallest minimum, mean, maximum, and standard deviation w0.044279, 0.04573, 0.04841, and 0.001272, respectively.
Mathematics 2023, 11, x FOR PEER REVIEWMathematics 2023, 11, 1796 25 of 32Figure 13. 13. Statistical measures measures for CO, for GSO, CO, andGSO, the proposed and the ESMOA proposed for Case ESMOA 3. for Case 3.Table 8 contrasts the efficacy of the proposed ESMOA-based PD-PI controller withvarious previously published controlling methods concerning ITAE. As shown, the proposedESMOA-based PD-PI controller obtains the minimum ITAE of 0.04428, where theconventional PID-based-PSO, PID-based-ARA, PID-based-JAYA, PI-based-DE, PID-based-SAMPE-JAYA, CO-based PD-PI controller, and GSO-based PD-PI controller find 0.2354,0.146308, 0.2272, 0.2021, 0.1726, 0.075148, and 0.082937, respectively.Table 8 contrasts the efficacy of the proposed ESMOA-based PD-PI controvarious previously published controlling methods concerning ITAE. As shown,posed ESMOA-based PD-PI controller obtains the minimum ITAE of 0.04428, wconventional PID-based-PSO, PID-based-ARA, PID-based-JAYA, PI-based-Dbased-SAMPE-JAYA, CO-based PD-PI controller, and GSO-based PD-PI contro0.2354, 0.146308, 0.2272, 0.2021, 0.1726, 0.075148, and 0.082937, respectively.Table 8. Comparison of the proposed ESMOA outcomes with other reported results in terms of ITAEfor Case 3.designed based on the proposed ESMOA, outperforms the same controller baseGSO and CO. Additionally, the presented cascaded PD-PI controller, designed bthe proposed ESMOA, outperforms the PI and PID controllers, which were dTable 8. Comparison of the proposed ESMOA outcomes with other reported results inController Optimization Technique Reference ITAE Objective ValueITAE for Case 3.PID PSO [40] 0.2354Controller PID Optimization ARA Technique [40] Reference 0.146308ITAE ObjectivePID JAYA PSO [40] [40] 0.2272 0.2354PID DE ARA [40] [40] 0.2021 0.146308PID SAMPE-JAYA JAYA [40] [40] 0.1726 0.2272PD-PI PID CO DE Applied [40] 0.075148 0.2021PD-PI PID GSOSAMPE-JAYA Applied [40] 0.082937 0.1726PD-PI Proposed ESMOACO AppliedApplied0.0442790.075148PD-PI GSO Applied 0.082937Then, the proposed ESMOA results obtained using the cascaded PD-PI controller arePD-PI Proposed ESMOA Applied 0.044279compared with the other reported results that use different types of controllers, namely, PIand PID. The parameters of the compared controllers were optimized in previous articlesbased on Then, otherthe recent proposed algorithms ESMOA which simulated results obtained several operating using the scenarios cascaded of the PD-PI same contrpower compared systemwith model. the Therefore, other reported all the controllers results that areuse adequately different designed types with of controllers, thebest parameters. This point demonstrates that the presented cascaded PD-PI controller,PI and PID. The parameters of the compared controllers were optimized in previdesigned based on the proposed ESMOA, outperforms the same controller based on theGSO cles and based CO. on Additionally, other recent the presented algorithms cascaded which PD-PI simulated controller, several designed operating based onscenarithe same proposed power ESMOA, system outperforms model. Therefore, the PI and PID all the controllers, controllers whichare wereadequately designed based designedon best other parameters. previous algorithms. This point demonstrates that the presented cascaded PD-PI cobased on other previous algorithms.
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