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)
Optimization techniques | ZN-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.) | 4.9444 | 0.5532 | 0.2016 | 0.3285 | 0.3009 |
 Minimum undershoot (US) × 10−4 (p.u.) | − 14.6118 | − 38.9115 | − 2.2533 | − 5.1166 | − 5.2424 |
 Settling time (s) \(\left( {T_{\text{s}} } \right)\) | 10.02 | 4.22 | 4.99 | 1.96 | 1.4 |
Frequency deviation in area-2 \(\Delta F_{2}\) | |||||
 Maximum overshoot (OS) × 10−4(p.u.) | 4.6249 | 0.3729 | 0.0885 | 0.0008 | 0.0001 |
 Minimum undershoot (US) × 10−4 (p.u.) | − 9.259 | − 1.1566 | − 0.6219 | − 1.5482 | − 1.5194 |
 Settling time (s) \(\left( {T_{\text{s}} } \right)\) | 9.4 | 4.39 | 3.67 | 2.09 | 1.7 |
Tie-line power deviation \(\Delta P_{Tie}\) | |||||
 Maximum overshoot (OS) × 10−4 (p.u.) | 0.4528 | 0.0536 | 0.0259 | 0.0000 | 0.0000 |
 Minimum undershoot (US) × 10−4 (p.u.) | − 3.1312 | − 0.4437 | − 0.2593 | − 0.5077 | − 0.4908 |
 Settling time (s) \(\left( {T_{\text{s}} } \right)\) | 6.23 | 2.31 | 3.02 | 1.79 | 1.23 |
Terminal voltage deviation in area-1 \(\left( {\Delta V_{t1} } \right)\) | |||||
 Maximum overshoot (OS) × 10−4(p.u.) | 14.6051 | 2.6419 | 1.0969 | 2.3557 | 0.2391 |
 Minimum undershoot (US) × 10−4 (p.u.) | − 0.1870 | 0.0000 | − 0.4479 | − 0.3424 | − 0.0138 |
 Settling time (s) \(\left( {T_{\text{s}} } \right)\) | 8.17 | 7.7 | 7.4 | 3.66 | 0.167 |
Terminal voltage deviation in area-2 \(\left( {\Delta V_{t2} } \right)\) | |||||
 Maximum overshoot (OS) × 10−4 (p.u.) | 11.3510 | 2.0202 | 0.9069 | 1.2567 | 0.1330 |
 Minimum undershoot (US) × 10−4 (p.u.) | 0.0000 | − 0.0000 | − 0.1951 | − 0.0050 | − 0.0015 |
 Settling time (s) \(\left( {T_{\text{s}} } \right)\) | 10.02 | 8.2 | 7.1 | 3.2 | 1.013 |
ITAE | 0.1098 | 0.0139 | 0.0103 | 0.0032 | 0.0030 |