C |
C |
C |
N |
V |
W |
W |
0.6 | 0.3 | 0.085 | 2 | 0.6 | 10 | 1 |
This paper introduces a novel optimization algorithm that is based on the basic idea underlying the bisection root-finding method in mathematics. The bisection method is modified for use as an optimizer by weighting each agent or vertex, and the algorithm developed from this process is called the weighted vertices optimizer (WVO). For exploitation and exploration, both swarm intelligence and evolution strategy are used to improve the accuracy and speed of WVO, which is then compared with six other popular optimization algorithms. Results confirm the superiority of WVO in most of the test functions.
Citation: |
Figure 14. block diagram of AVR along with PID controller [17]
Table 1. parameters of WVO
C |
C |
C |
N |
V |
W |
W |
0.6 | 0.3 | 0.085 | 2 | 0.6 | 10 | 1 |
Table 2. benchmark optimization functions
ID | Name | Function | Bound | Global Min |
F1 | Shubert | -186.7309 | ||
F2 | Six-hump camel back | -1.0316285 | ||
F3 | Sphere | 0 | ||
F4 | Ackley | 0 | ||
F5 | Griewank | 0 | ||
F6 | Rastrigin | 0 |
Table 3. cost value of each optimization algorithm in 78th iteration
Method | Cost value |
WVO | 1.78 E-15 |
PSO | 7.36 E-4 |
GA | 9.17 E-5 |
IWO | 2.157 |
HS | 2.56E-3 |
CA | 5.93E-6 |
mIWO | 1.23 E-5 |
mHS | 8.69 E-9 |
mCA | 5.12 E-10 |
Table 4. the performance of each optimization method
WVO | PSO | GA | IWO | HS | CA | mIWO | mHS | mCA | ||
F1 | N1 | 13 | 45 | 27 | 184 | 88 | 47 | 132 | 44 | 23 |
B2 | -176.7309 | -176.7309 | -176.7309 | -176.7309 | -176.7309 | -176.7309 | -176.7309 | -176.7309 | -176.7309 | |
R3 | 1 | 5 | 3 | 9 | 7 | 6 | 8 | 4 | 2 | |
F2 | N | 19 | 50 | 24 | 54 | 60 | 24 | 39 | 45 | 20 |
B | -1.03163 | -1.03163 | -1.03163 | -1.03162 | -1.03162 | -1.03162 | -1.03162 | -1.03162 | -1.03162 | |
R | 1 | 7 | 3 | 8 | 9 | 3 | 5 | 6 | 2 | |
F3 | N | 77 | 1158 | 1500 | 1498 | 1501 | 1500 | 1382 | 1500 | 1500 |
B | 1.71 E-58 | 0 | 7.64 E-128 | 2.43 E-6 | 2 E-10 | 1.89 E-143 | 1.13 E-12 | 3.12E-8 | 0 | |
R | 5 | 1 | 4 | 9 | 7 | 3 | 6 | 8 | 2 | |
F4 | N | 67 | 233 | 199 | 198 | 1501 | 1336 | 173 | 1500 | 1363 |
B | 8.88 E-16 | 4.44 E-15 | 20 | 20 | 6.2 E-5 | 20.29 | 6.13 E-3 | 2.52 E-8 | 1.23 E-4 | |
R | 1 | 2 | 7 | 8 | 4 | 9 | 6 | 3 | 5 | |
F5 | N | 41 | 116 | 251 | 744 | 1501 | 468 | 632 | 432 | 321 |
B | 0 | 0.09747 | 0 | 0.90271 | 1.52 E-8 | 20.25 | 2.38 E-3 | 1.58 E-32 | 9.78 E-2 | |
R | 1 | 7 | 2 | 8 | 4 | 9 | 5 | 3 | 6 | |
F6 | N | 30 | 119 | 418 | 200 | 1501 | 1130 | 123 | 1245 | 1351 |
B | 0 | 5.9697 | 0 | 8.9552 | 4.06 E-10 | 22.94947 | 3.25 E-9 | 3.15 E-21 | 4.65 E-3 | |
R | 1 | 7 | 2 | 8 | 4 | 9 | 5 | 3 | 6 | |
R | 1 | 5 | 2 | 9 | 6 | 8 | 6 | 4 | 3 | |
1N:Number of iteration - 2B:Best cost value - 3R:Rank |
Table 5. the understudy composition functions [12]
CF1 | CF2 | CF3 |
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Table 6. results of optimization algorithms for three CFs
PSO [12] | DE [12] | GA | WVO | ||
CF1 | Mean | 1.7203 E2 | 1.4441 E2 | 1.3451 E2 | 1.1121 E2 |
Std. deviation | 3.2869 E1 | 1.9401 E1 | 1.9142 E1 | 1.4232 E1 | |
CF2 | Mean | 3.1430 E2 | 3.2486 E2 | 3.2314 E2 | 3.0021 E2 |
Std. deviation | 2.0006 E1 | 1.4784 E1 | 1.8154 E1 | 1.6823 E1 | |
CF3 | Mean | 8.3450 E1 | 1.0789 E1 | 7.5421 E1 | 3.8124 E1 |
Std. deviation | 1.0111 E2 | 2.6040 E0 | 1.0512 E1 | 8.5412 E1 |
Table 7. value of AVR's parameters [17]
Parameter | value |
K |
10 |
0.1 | |
K |
1 |
0.4 | |
K |
1 |
1 | |
K |
1 |
0.01 |
Table 8. obtained values and the result for each optimization method
KP | KI | KD | RT | ST (sec) |
OS (%) |
Final error (%) |
Cost value | |
WVO | 0.600518 | 0.41376 | 0.20136 | 0.3101 | 0.5013 | 0.0003 | 0 | 3.11706 |
PSO | 0.600532 | 0.41386 | 0.20137 | 0.3141 | 0.5013 | 0.0017 | 0 | 3.11752 |
GA | 0.610065 | 0.42965 | 0.20784 | 0.3226 | 0.5005 | 0.1522 | 0.018 | 3.17785 |
Table 9.
effect of C
Function | C |
The best cost | Iteration | C |
The best cost | Iteration | C |
The best cost | Iteration | W |
The best cost | Iteration | W |
The best cost | Iteration |
F5 | 0.2 | 3.12E-12 | 46 | 0.2 | 0 | 43 | 0.02 | 0 | 73 | 2 | 0 | 47 | 2 | 0 | 41 |
0.4 | 2.02E-19 | 42 | 0.4 | 0 | 41 | 0.04 | 0 | 45 | 5 | 0 | 43 | 5 | 0 | 45 | |
0.6 | 0 | 42 | 0.6 | 0 | 42 | 0.06 | 0 | 45 | 10 | 0 | 42 | 10 | 0 | 47 | |
0.8 | 0 | 42 | 0.8 | 2.31E-26 | 43 | 0.08 | 0 | 41 | 15 | 0 | 42 | 15 | 0 | 47 | |
1 | 3.12E-30 | 44 | 1 | 8.64E-23 | 45 | 0.1 | 0 | 53 | 20 | 0 | 43 | 20 | 0 | 48 | |
F6 | 0.2 | 2.31E-6 | 45 | 0.2 | 0 | 41 | 0.02 | 2.31E-26 | 45 | 2 | 0 | 45 | 2 | 0 | 43 |
0.4 | 0 | 42 | 0.4 | 0 | 42 | 0.04 | 0 | 43 | 5 | 0 | 43 | 5 | 0 | 43 | |
0.6 | 0 | 42 | 0.6 | 1.12E-28 | 45 | 0.06 | 0 | 43 | 10 | 0 | 43 | 10 | 0 | 43 | |
0.8 | 0 | 45 | 0.8 | 6.78E-25 | 49 | 0.08 | 0 | 41 | 15 | 0 | 44 | 15 | 0 | 45 | |
1 | 0 | 45 | 1 | 1.32E-24 | 51 | 0.1 | 0 | 44 | 20 | 0 | 44 | 20 | 0 | 48 |
Table 10.
the effect of N
Function | N |
The best cost | Iteration |
F5 | 2 | 0 | 47 |
3 | 0 | 43 | |
4 | 0 | 42 | |
5 | 0 | 41 | |
10 | 0 | 43 | |
15 | 4.55E-8 | 116 | |
F5 | 2 | 0 | 45 |
3 | 0 | 43 | |
4 | 0 | 43 | |
5 | 0 | 44 | |
10 | 0 | 44 | |
15 | 0.406497 | 116 |
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