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Table 6 Comparative performance of ITAE value (for \(t_{\text{sim}} = 10\) s), settling times (for 2% tolerance band) and peak overshoots for multi-area multi-unit power system (test system-2)

From: Combined load frequency and terminal voltage control of power systems using moth flame optimization algorithm

Optimization techniquesZN-tuned PID [39]SA PID (ISE) [39]MFO PID (ISE)MFO PID (ITAE)MFO FOPID (ITAE)
Frequency deviation in area-1 \(\Delta F_{1}\)
 Maximum overshoot (OS) × 10−4 (p.u.)4.94440.55320.20160.32850.3009
 Minimum undershoot (US) × 10−4 (p.u.)− 14.6118− 38.9115− 2.2533− 5.1166− 5.2424
 Settling time (s) \(\left( {T_{\text{s}} } \right)\)10.024.224.991.961.4
Frequency deviation in area-2 \(\Delta F_{2}\)
 Maximum overshoot (OS) × 10−4(p.u.)4.62490.37290.08850.00080.0001
 Minimum undershoot (US) × 10−4 (p.u.)− 9.259− 1.1566− 0.6219− 1.5482− 1.5194
 Settling time (s) \(\left( {T_{\text{s}} } \right)\)9.44.393.672.091.7
Tie-line power deviation \(\Delta P_{Tie}\)
 Maximum overshoot (OS) × 10−4 (p.u.)0.45280.05360.02590.00000.0000
 Minimum undershoot (US) × 10−4 (p.u.)− 3.1312− 0.4437− 0.2593− 0.5077− 0.4908
 Settling time (s) \(\left( {T_{\text{s}} } \right)\)6.232.313.021.791.23
Terminal voltage deviation in area-1 \(\left( {\Delta V_{t1} } \right)\)
 Maximum overshoot (OS) × 10−4(p.u.)14.60512.64191.09692.35570.2391
 Minimum undershoot (US) × 10−4 (p.u.)− 0.18700.0000− 0.4479− 0.3424− 0.0138
 Settling time (s) \(\left( {T_{\text{s}} } \right)\)8.177.77.43.660.167
Terminal voltage deviation in area-2 \(\left( {\Delta V_{t2} } \right)\)
 Maximum overshoot (OS) × 10−4 (p.u.)11.35102.02020.90691.25670.1330
 Minimum undershoot (US) × 10−4 (p.u.)0.0000− 0.0000− 0.1951− 0.0050− 0.0015
 Settling time (s) \(\left( {T_{\text{s}} } \right)\)10.028.27.13.21.013
ITAE0.10980.01390.01030.00320.0030