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. 2026 Jan 27;16:3741. doi: 10.1038/s41598-025-34481-7

Table 21.

Optimal controllers’ settings for the LFC loop in the first and second area.

Controller	Optim. Algorithm
Controller	Optim. Algorithm	K_P	K_I	K_D1	N ₁	K_D2	N ₂	K_E	K_CE	K_P	K_I	K_D1	N ₁	K_D2	N ₂	K_E	K_CE
PID	PSO	0.1	1.4	1.3	0.5	-	-	-	-	0.2	1.3	1.3	0.2	-	-	-	-
PID	GTO	0.1	0.3	1.7	0.3	-	-	-	-	0.1	0.2	0.5	0.3	-	-	-	-
PID	MPA	0.1	1.2	1.2	0.5	-	-	-	-	0.1	1.4	1.1	0.2	-	-	-	-
PIDD²	PSO	81.5	2.0	2.0	2.0	9.7	67.7	-	-	80.5	2.0	1.7	1.5	7.9	83.1	-	-
PIDD²	GTO	40.0	2.0	1.8	1.3	9.2	61.1	-	-	34.3	2.0	2.0	1.1	9.1	68.8	-	-
PIDD²	MPA	99.8	2.0	2.0	1.5	10.0	51.2	-	-	100.0	2.0	2.0	2.0	10.0	49.0	-	-
FPIDD²+PIDD²	PSO	100.0	2.0	2.0	2.0	10.0	58.7	2.0	0.1	83.9	2.0	2.0	2.0	6.9	96.5	1.9	0.1
FPIDD²+PIDD²	GTO	100.0	2.0	0.1	2.0	10.0	100.0	2.0	0.0	100.0	2.0	2.0	2.0	10.0	0.1	2.0	0.1
FPIDD²+PIDD²	MPA	99.8	2.0	2.0	2.0	9.6	60.6	2.0	0.0	48.6	2.0	1.9	0.1	3.6	100.0	2.0	0.0