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Table 1 Statistical analysis of different agents for GOA, PSO and GA optimized PDF plus (1+ PI) controllers

From: Grasshopper optimization algorithm optimized multistage controller for automatic generation control of a power system with FACTS devices

Functions Test Functions Parameters GOA PSO GA
F1 Shifted Sphere Function
\( {f}_1(x)=\sum \limits_{i=1}^n{x}_i^2 \)
Average 1.6219X1056 8.931X1066 9.2656 X1021
Standard Deviation 1.6266X1057 7.0111X1068 4.1437 X1022
Minimum 0.1088X103 0.0059X103 0.0021 X103
Maximum 1.8291X1068 9.468X1066 1.8531 X1023
F2 Shifted Schwefel’s Function
\( {f}_2(x)=\sum \limits_{i=1}^n\left|{x}_i\right|+{\prod}_{i=\frac{1}{2}}^n\left|{x}_i\right| \)
Average 3.7673 X1067 13.045 X1011 1.4177 X1024
Standard Deviation 3.7688 X1068 11.366X1012 1.428 X1025
Minimum 0.0062 X103 0.0379 X103 0.0023 X103
Maximum 3.8265 X1069 13.415 X1013 1.4319 X1024
F3 Shifted Rosenbrock’s
Function
\( {f}_3(x)=\sum \limits_{i=1}^n{\left(\left[{x}_i+0.5\right]\right)}^2 \)
Average 2.7399 X1032 12.121 X109 4.3317 X1019
Standard Deviation 1.0523 X1033 11.598 X1011 3.9854 X1020
Minimum 0.0161 X103 4.686 X102 0.0082 X103
Maximum 4.0777 X1033 10.032 X1013 4.3987 X1019
F4 Shifted Rastrigin’s Function
\( {f}_4(x)=\sum \limits_{i=1}^n\left[{x}_i^2+10\cos \left(2\pi {x}_i\right)+10\right] \)
Average 5.2988 X1044 6.8166 X1038 2.2347 X1027
Standard Deviation 2.9021 X1045 7.0025 X1039 1.4921 X1028
Minimum 0.0063 X103 1.792 X102 0.0076 X103
Maximum 1.5896 X1046 9.568 X1041 2.4682 X1027
F5 Shifted Schwefel’s Problem with Noise in Fitness
f5(x) = max {|xi|, 1 ≤ i ≤ n}
Average 2.2974 X1034 5.6982 X1034 3.8751 X1022
Standard Deviation 2.2798 X1035 6.2784 X1038 2.9852 X1023
Minimum 0.0009 X103 0.0561 X103 0.0023 X103
Maximum 2.2799 X1036 5.5862 X1036 4.0012X1022
F6 Schwefel’s Problem with Global Optimum on Bounds
\( {f}_6(x)=\sum \limits_{i=1}^n\left[100{\left({x}_{i+1}-{x}_i^2\right)}^2+{\left({x}_i-1\right)}^2\right] \)
Average 3.0414 X1030 2.1187 X1042 1.8955 X1043
Standard Deviation 2.5276 X1031 1.5182 X1044 1.9865 X1044
Minimum 0.0015 X103 0.172 X102 0.0089 X103
Maximum 2.4682 X1032 2.930 X1045 2.2219 X1043
F7 Shifted Rotated Griewank’s Function
\( {f}_7(x)=\sum \limits_{i=1}^n{ix}_i^4+ random\left[0,1\right] \)
Average 3.4099 X1033 11.530 X1035 7.6482 X1034
Standard Deviation 2.4053 X1034 13.028 X1036 5.6214 X1035
Minimum 0.0008 X103 1.406 X102 0.0019 X103
Maximum 1.8920 X1035 11.022 X1038 4.5875 X1034
F8 Shifted Rotated Ackley’s Function with Global Optimum on Bounds
\( {f}_8(x)=\sum \limits_{i=1}^n-{x}_i\sin \left(\sqrt{\left|{x}_i\right|}\right) \)
Average 5.0129 X1032 6.321 X1031 8.8626 X1029
Standard Deviation 2.2418 X1033 5.4213X1032 4.5897 X1030
Minimum 0.0018 X103 2.1235X103 0.0062 X103
Maximum 1.0026 X1034 8.082X1033 6.9871 X1029