Tables at a glance

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Figures at a glance

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Table 1

KP	KI	KD
1	0.25	0.28

Table 1: Coefficients of the reference PID controller [8]

Table 2

	Transfer function	Parameter limits	Used parameter values
Amplifire	Ka/(1+sτa)	10≤Ka≤40, 0.02≤τa≤0.1	Ka =10, τa=0.1
DC Exciter	Ke/(1+sτe)	1≤Ke≤10, 0.4≤τe≤1.0	Ke =1, τe=0.4
Generator	Kg/(1+sτg)	0.7≤Kg≤1.0, 1.0≤τg≤2.0	Kg =1, τg=1
Sensor	Ks/(1+sτs)	Ks=1, 0.001≤τs≤0.06	Ks =1, τs=0.05

Table 2: Parameters of the AVR system model

Table 3

Yi limitation

2-3

τi limitation

0.4-0.7

Table 3: Limitation of optimized parameters of CDM

Table 4

Ƴ ₁	Ƴ ₂	Ƴ ₃	Ƴ ₄	Ƴ ₅	τ(s)
2.0475	2.0349	2.4618	2.1936	2.8168	0.6334

Table 4: Optimized key parameters of CDM

Table 5

Control strategy	Ts(sec) , Vt	Tp(sec) , Vt	ITSE (Ve)	Oscillatory modes	MDR (ζ)
PID	1.2604	1.5500	0.0552	-0.1080 ±4.4191i	0.0244
Optimal CDM	1.0024	1.2300	0.0105	-2.13e+00 ± 6.01e+00i	0.3338

Table 5: Numerical comparison of the outcomes of various control techniques

Table 6

Parameter variation	Nominal	K_A	T_A	K_E	T_E	K_G	T_G	K_S	T_S
% change	0	+ 5 0 % -50%	+ 5 0 % -50%	+ 5 0 % -50%	+50% -50%	+ 5 0 % -50%	+ 5 0 % -50%	+ 5 0 % -50%	+ 5 0 % -50%
MDRPID	0.0244	-0.0643 0.2026	-0.0169 0.0929	-0.0643 0.2026	0.0347 0.0316	-0.0643 0.2026	0.0884 -0.0541	-0.0643 0.2026	-0.0070 0.0636
MDROptimal CDM	0.3338	0.1946 0.4251	0.2150 0.5390	0.1946 0.4251	0.3030 0.2412	0.1946 0.4251	0.4136 0.1634	0.1946 0.4251	0.2429 0.4535

Table 6: Sensitivity analysis with changes in the range of dynamic parameters

FIGURE 1

FIGURE 2

Figure 2: The practical representation of the AVR system

FIGURE 3

Figure 3: The block diagram of AVR system

FIGURE 4

Figure 4: The block diagram of the structure of CDM controller

FIGURE 5

Figure 5: A whirlpool that is formed by (a) a centripetal force that is caused by rotating motion, and (b) the action of centrifugal force, as well as the updating of the locations of particles

FIGURE 6

Figure 6: the block diagram of the AVR control system in conjunction with the proposed CDM-TFWO controller

FIGURE 7

Figure 7: The flowchart of designing the optimal CDM using the TFWO method for the AVR system

FIGURE 8

Figure 8: Convergence curve of TFWO for CDM controller

FIGURE 9

Figure 9: Convergence curve of TFWO for CDM controller

FIGURE 10

Figure 10: Coefficient diagram of designed proposed controller

FIGURE 11

Figure 11: Generator terminal voltage with two strategies: Reference PID control and the proposed optimal CDM