1 Introduction
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Each area should be capable of supporting its own load at steady state;
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Throughout a sudden load disruption and any interruptions, the system must be kept under control;
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To improve system stability, frequency and tie-line power variations in terms of undershoot, overshoot, and settling time must be lessened.
2 System modeling
2.1 Modeling of the photovoltaic unit
2.2 Modeling of the reheat thermal unit
3 Controller-based performance function
3.1 Controller structure
3.1.1 Outer control loop
3.1.2 Inner control loop
3.2 Design of the performance function
4 Chaos game optimization
4.1 Inspiration
4.2 Mathematical model
5 Simulation results and discussion
5.1 Scenario 1: application of CGO to LFC studies
Algorithm | Maximum | Minimum | Average | Standard deviation |
---|---|---|---|---|
PSO | 0.428 | 0.398 | 0.417 | 0.01517 |
MFO | 0.416 | 0.3931 | 0.404 | 0.01301 |
CGO | 0.393 | 0.374 | 0.382 | 0.00954 |
5.2 Scenario 2: 10% change in the demands of areas 1 and 2
Controller | Area 1 | Area 2 | ||||||
---|---|---|---|---|---|---|---|---|
Kp1 | Ki1 | Kd1 | N1 | Kp2 | Ki2 | Kd2 | N2 | |
Linear phase | ||||||||
CGO:PI | − 0.3088 | − 0.2570 | – | – | − 2.8646 | − 0.3938 | – | – |
CGO:PIDn | − 1.5032 | − 0.2309 | − 3.8656 | 155.4926 | − 4 | − 4 | − 0.6173 | 72.0845 |
CGO:PIDn-PI | ||||||||
Outer control | − 1.7517 | − 0.1332 | 2.6022 | 0.1941 | − 3.9969 | − 4 | − 0.4613 | 127.0962 |
Inner control | 1.3048 | 2.9875 | – | – | 4 | 4 | – | – |
Nonlinear phase | ||||||||
CGO:PI | − 0.0099 | − 0.0047 | – | – | − 0.7325 | − 0.8139 | – | – |
CGO:PIDn | − 0.0101 | − 0.0058 | 0.0835 | − 196.4229 | − 0.5760 | − 0.5821 | − 0.3201 | 195.8083 |
CGO:PIDn-PI | ||||||||
Outer control | 1e−8 | 0 | 0.1500 | 110.6970 | − 0.6231 | 2.1e−9 | − 0.0682 | 126.5505 |
Inner control | 4 | 0.4927 | – | – | 0.1902 | 1.9616 |
Controller | Objective functions | Settling time \(\mathrm{Ts}\) \((\mathrm{Sec})\) at 0.002 band | |||||
---|---|---|---|---|---|---|---|
\(\mathrm{ITAE}\) | \(\mathrm{IAE}\) | \(\mathrm{ISE}\hspace{0.17em}\)× 10−2 | \(\mathrm{ITSE}\hspace{0.17em}\)× 10−2 | ∆F1 | ∆F2 | ∆\({P}_{tie}\) | |
GA:PI (Abd-Elazim and Ali 2018) | 10.9780 | 2.3394 | 30.0598 | 86.1671 | 25.59 | 22.12 | 16.31 |
FA:PI (Abd-Elazim and Ali 2018) | 6.8292 | 1.7434 | 22.1473 | 46.2696 | 21.26 | 23.25 | 16.72 |
MFO:PI (Sharma et al. 2018) | 2.6985 | 1.0261 | 9.8102 | 16.9840 | 8.19 | 12.27 | 6.91 |
CGO:PI (proposed) | 2.2130 | 0.8178 | 7.0157 | 10.4382 | 10.24 | 13.51 | 7.10 |
MWOA:PIDn (Abd-Elazim and Ali 2018) | 1.6160 | 0.5589 | 5.8976 | 4.4147 | 10.27 | 10.08 | 11.05 |
MFO:PIDn (Sharma et al. 2018) | 0.3933 | 0.2146 | 1.2699 | 0.5700 | 6.31 | 5.60 | 6.15 |
CGO:PIDn (proposed) | 0.3740 | 0.2080 | 1.3880 | 0.6122 | 6.00 | 5.28 | 6.27 |
CGO:PIDn-PI (proposed) | 0.05597 | 0.04444 | 0.1188 | 0.01966 | 0.67 | 2.24 | 0.82 |
Controller | \(\Delta F\)1 × 10−2 | \(\Delta F\)2 × 10−2 | \(\Delta Ptie\) × 10−2 | ||||||
---|---|---|---|---|---|---|---|---|---|
Us Hz | Os Hz | RMS | Us Hz | Os Hz | RMS | Us Hz | Os Hz | RMS | |
GA:PI (Abd-Elazim and Ali 2018) | 30.17 | 17.59 | 7.372 | 29.46 | 14.99 | 6.782 | 5.461 | 4.533 | 1.283 |
FA:PI (Abd-Elazim and Ali 2018) | 30.63 | 15.65 | 6.280 | 27.57 | 13.76 | 5.771 | 5.054 | 3.640 | 1.042 |
MFO:PI (Sharma et al. 2018) | 14.13 | 0.022 | 3.828 | 21.84 | 0.305 | 4.218 | 1.858 | 2.275 | 0.512 |
CGO:PI (proposed) | 12.97 | 0.105 | 3.129 | 20.07 | 1.612 | 3.664 | 2.094 | 1.669 | 0.430 |
MWOA:PIDn (Abd-Elazim and Ali 2018) | 25.70 | 1.11 | 3.277 | 20.55 | 1.053 | 2.981 | 0.708 | 1.104 | 0.242 |
MFO:PIDn (Sharma et al. 2018) | 11.77 | 0.464 | 1.496 | 11.78 | 0.427 | 1.411 | 0.297 | 0.050 | 0.126 |
CGO:PIDn (proposed) | 12.46 | 0.380 | 1.554 | 12.36 | 0.356 | 1.485 | 0.304 | 0.066 | 0.134 |
CGO:PIDn-PI (proposed) | 11.16 | 0.921 | 0.466 | 5.57 | 1.89 | 0.440 | 0.249 | 0.302 | 0.049 |
5.3 Scenario 3: performance and uncertainty of area 2 for a 10% SLP
5.4 Scenario 4: RLPs are applied in areas 1 and 2
Controller | Objective functions | |||
---|---|---|---|---|
\(\mathrm{ITAE}\) | \(\mathrm{IAE}\) | \(\mathrm{ISE}\) | \(\mathrm{ITSE}\) | |
CGO:PI (proposed) | 872.2577 | 12.6551 | 1.7409 | 125.7148 |
CGO:PIDn (proposed) | 226.5000 | 3.2824 | 0.3268 | 22.89 |
CGO:PIDn-PI (proposed) | 44.9000 | 0.6607 | 0.02741 | 1.9450 |
5.5 Scenario 5: performance analysis considering nonlinearities
Controller | Objective functions | Settling time Ts \((\mathrm{Sec})\) at 0.002 band | |||||
---|---|---|---|---|---|---|---|
ITAE | IAE | ISE × 10−2 | ITSE × 10−2 | ∆F1 | ∆F2 | ∆\({P}_{tie}\) | |
CGO:PI (Proposed) | 40.310 | 1.748 | 2.641 | 41.780 | 66.2 | 65.10 | 57.6 |
CGO:PIDn (Proposed) | 36.300 | 1.735 | 2.933 | 44.730 | 41.1 | 65.04 | 57.4 |
CGO:PIDn-PI (Proposed) | 5.459 | 0.477 | 0.5473 | 3.740 | 6.51 | 32.02 | 35.95 |