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

Table 13.

Optimal controllers’ settings for the AVR 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	0.5	0.6	0.1	1.9	-	-	-	-	0.7	0.7	0.2	1.4	-	-	-	-
PID	GTO	0.7	0.9	0.2	0.5	-	-	-	-	0.6	0.9	0.7	0.7	-	-	-	-
PID	MPA	0.2	0.9	0.4	0.7	-	-	-	-	0.9	1.9	1.2	1.8	-	-	-	-
PIDD²	PSO	2.0	2.0	2.0	2.0	2.0	100.0	-	-	2.0	2.0	2.0	2.0	2.0	100.0	-	-
PIDD²	GTO	1.7	2.0	1.5	0.2	0.4	7.6	-	-	2.0	2.0	1.4	2.0	0.6	100.0	-	-
PIDD²	MPA	0.8	2.0	2.0	0.3	2.0	0.1	-	-	2.0	2.0	2.0	0.1	2.0	0.9	-	-
FPIDD²+PIDD²	PSO	1.0	1.3	2.0	1.3	2.0	0.0	-	-	1.5	2.0	1.0	1.2	2.0	100.0	-	-
FPIDD²+PIDD²	GTO	2.0	2.0	0.8	2.0	1.1	0.0	-	-	1.9	2.0	1.1	1.4	0.7	100.0	-	-
FPIDD²+PIDD²	MPA	2.0	2.0	2.0	2.0	0.2	77.4	-	-	0.8	2.0	1.4	2.0	0.6	14.9	-	-