Research Article
A Master-Slave Salp Swarm Algorithm Optimizer for Hybrid Energy Storage System Control Strategy in Electric Vehicles
Table 1
Benchmark test function results.
| | MSSSA | SSA | WOA | GA | MFO | PSO | DA |
| | Sphere | f1 | Best score | 4.98-09 | 4.89-09 | 1.23-189 | 1.80+01 | 9.48-06 | 4.30-02 | 1.46-02 | | Worst score | 1.05-08 | 1.31-08 | 2.79-180 | 2.00+01 | 1.00+04 | 4.79-01 | 1.01+02 | | Mean | 8.22-09 | 7.64-09 | 2.92-181 | 1.92+01 | 2.00+03 | 1.52-01 | 1.45+01 | | Standard deviation | 1.64-09 | 2.34-09 | 0.00+00 | 6.32-01 | 4.22+03 | 1.31-01 | 3.14+01 |
| | Schwefel’s problem 2.22 | f2 | Best score | 4.77-06 | 4.26-06 | 5.93-123 | 1.40+01 | 3.50-20 | 1.81-01 | 4.08-01 | | Worst score | 7.54-06 | 1.25-05 | 2.20-114 | 1.90+01 | 2.55-18 | 7.39-01 | 3.92+00 | | Mean | 6.10-06 | 6.95-06 | 4.42-115 | 1.64+01 | 4.57-19 | 5.03-01 | 1.92+00 | | Standard deviation | 9.15-07 | 2.26-06 | 9.18-115 | 1.84+00 | 7.66-19 | 1.43-01 | 1.33+00 |
| | Schwefel’s problem 1.2 | f3 | Best score | 9.93-10 | 5.90-10 | 2.40-09 | 2.20+01 | 2.24-13 | 3.65+01 | 3.40-03 | | Worst score | 5.47-10 | 1.98-09 | 4.34-01 | 2.50+01 | 1.41-09 | 8.70+01 | 1.20+00 | | Mean | 8.68-10 | 1.04-09 | 1.04-01 | 2.35+01 | 2.51-10 | 5.09+01 | 1.98-01 | | Standard deviation | 3.92-10 | 4.01-10 | 1.59-01 | 9.72-01 | 4.78-10 | 1.46+01 | 3.62-01 |
| | Schwefel’s problem 2.21 | f4 | Best score | 9.92-06 | 8.94-06 | 0.00+00 | 1.40+01 | 1.61-10 | 1.06+00 | 1.97-01 | | Worst score | 1.92-05 | 1.93-05 | 3.00+01 | 2.00+01 | 1.48-01 | 1.85+00 | 7.95+00 | | Mean | 1.51-05 | 1.32-05 | 3.06+00 | 1.57+01 | 1.48-02 | 1.34+00 | 3.58+00 | | Standard deviation | 3.14-06 | 2.97-06 | 9.46+00 | 1.89+00 | 4.67-02 | 2.68-01 | 2.36+00 |
| | Generalized Rosenbrock’s function | f5 | Best score | 3.94+00 | 3.45+00 | 5.72+00 | 0.00+00 | 4.39+00 | 9.88+01 | 7.57+00 | | Worst score | 1.49+03 | 1.43+03 | 6.59+00 | 0.00+00 | 9.00+04 | 4.82+02 | 8.41+02 | | Mean | 1.78+02 | 2.01+02 | 6.01+00 | 0.00+00 | 9.61+03 | 2.37+02 | 1.92+02 | | Standard deviation | 4.65+02 | 4.52+02 | 2.60-01 | 0.00+00 | 2.83+04 | 1.34+02 | 2.76+02 |
| | Step function | f6 | Best score | 3.67-10 | 2.96-10 | 1.12-04 | 4.35+01 | 0.00+00 | 3.98-02 | 8.00-04 | | Worst score | 8.84-10 | 8.77-10 | 9.30-06 | 5.55+01 | 7.61-31 | 3.51-01 | 7.83+00 | | Mean | 5.97-10 | 5.92-10 | 5.20-05 | 5.09+01 | 1.18-31 | 1.46-01 | 8.79-01 | | Standard deviation | 1.69-10 | 1.84-10 | 3.50-05 | 4.12+00 | 2.34-31 | 8.25-02 | 2.45+00 |
| | Quadrtic function with noise | f7 | Best score | 1.70-03 | 1.10-03 | 0.00+00 | 2.10+01 | 1.06-03 | 6.20-01 | 2.60-03 | | Worst score | 9.10-03 | 9.50-03 | 3.70-03 | 2.58+01 | 7.04-03 | 2.49+00 | 2.42-02 | | Mean | 4.07-03 | 4.60-03 | 9.30-04 | 2.31+01 | 2.23-03 | 1.10+00 | 1.03-02 | | Standard deviation | 2.28-03 | 3.27-03 | 1.13-03 | 1.54+00 | 1.85-03 | 5.47-01 | 6.07-03 |
| | Generalized Schwefel’s problem 2.26 | f8 | Best score | -3.02+03 | -3.42+03 | -4.19+03 | -9.26+00 | -3.71+03 | -7.73+03 | -3.37+00 | | Worst score | -2.27+03 | -2.61+03 | -2.50+03 | -5.05+00 | -2.76+03 | -5.03+03 | -2.13+00 | | Mean | -2.72+03 | -2.97+03 | -3.39+03 | -6.65+00 | -3.36+03 | -6.51+03 | -2.70+00 | | Standard deviation | 2.29+02 | 2.27+02 | 5.39+02 | 1.56+00 | 3.23+02 | 8.66+02 | 3.40-01 |
| | Generalized Rastrigin’s function | f9 | Best score | 3.98+00 | 2.98+00 | NA | 2.65+02 | 6.96+00 | 5.83+01 | 5.61+00 | | Worst score | 2.19+01 | 3.18+01 | NA | 2.65+02 | 4.28+01 | 1.35+02 | 4.68+01 | | Mean | 1.53+01 | 1.65+01 | NA | 2.65+02 | 2.06+01 | 9.15+01 | 2.60+01 | | Standard deviation | 5.57+00 | 9.22+00 | NA | 0.00+00 | 1.11+01 | 2.42+01 | 1.37+01 |
| | Generalized Griewank function | f10 | Best score | 8.60-06 | 7.57-06 | 8.88-16 | 0.00+00 | 4.44-15 | 2.23-01 | 1.65+00 | | Worst score | 1.30-05 | 2.32+00 | 4.44-15 | 0.00+00 | 4.44-15 | 1.62+00 | 9.36+00 | | Mean | 1.11-05 | 3.47-01 | 3.02-15 | 0.00+00 | 4.44-15 | 7.62-01 | 3.66+00 | | Standard deviation | 1.21-06 | 7.81-01 | 1.83-15 | 0.00+00 | 8.32-31 | 4.72-01 | 2.34+00 |
| | Generalized penalized function | f11 | Best score | 7.13-02 | 9.85-02 | 0.00+00 | 1.01+00 | 3.45-02 | 1.12-02 | 8.62-02 | | Worst score | 4.90-01 | 6.27-01 | 2.45-01 | 1.01+00 | 2.83-01 | 6.17-02 | 9.00-01 | | Mean | 2.33-01 | 2.50-01 | 5.97-02 | 1.01+00 | 1.19-01 | 2.97-02 | 4.73-01 | | Standard deviation | 1.19-01 | 1.64-01 | 9.24-02 | 2.53-04 | 8.09-02 | 1.49-02 | 2.55-01 |
| | Generalized penalized function | f12 | Best score | 4.17-12 | 3.75-12 | 1.39-06 | 3.22+01 | 4.71-32 | 2.00-04 | 3.00-04 | | Worst score | 3.11-01 | 6.22-01 | 2.46-05 | 3.22+01 | 3.11-01 | 1.70-03 | 6.19+00 | | Mean | 3.11-02 | 9.33-02 | 9.79-06 | 3.22+01 | 6.22-02 | 8.20-04 | 1.56+00 | | Standard deviation | 9.83-02 | 2.10-01 | 7.30-06 | 0.00+00 | 1.31-01 | 4.76-04 | 1.79+00 |
| | Generalized penalized function | f13 | Best score | 1.82-11 | 2.88-11 | 0.00+00 | 1.00-01 | 1.35-32 | 5.30-03 | 2.48-01 | | Worst score | 6.11-11 | 1.10-02 | 1.10-02 | 1.00-01 | 1.10-02 | 1.16-01 | 8.36+00 | | Mean | 3.71-11 | 2.20-03 | 1.13-03 | 1.00-01 | 3.30-03 | 3.62-02 | 2.06+00 | | Standard deviation | 1.59-11 | 4.63-03 | 3.47-03 | 1.46-17 | 5.31-03 | 3.27-02 | 2.72+00 |
| | Shekel’s foxhole function | f14 | Best score | 9.98-01 | 9.98-01 | 9.98-01 | NA | 9.98-01 | 9.98-01 | 9.98-01 | | Worst score | 9.98-01 | 9.98-01 | 2.98+00 | NA | 1.99+00 | 5.93+00 | 9.98-01 | | Mean | 9.98-01 | 9.98-01 | 1.20+00 | #DIV/0! | 1.10+00 | 1.69+00 | 9.98-01 | | Standard deviation | 0.00+00 | 0.00+00 | 6.27-01 | #DIV/0! | 3.14-01 | 1.55+00 | 0.00+00 |
| | Shekel’s foxhole function | f15 | Best score | 3.84-04 | 5.40-04 | 3.00-04 | 1.48-01 | 7.11-04 | 5.00-04 | 7.00-04 | | Worst score | 8.07-04 | 1.32-03 | 7.00-04 | 9.68+02 | 7.83-04 | 3.20-03 | 8.30-03 | | Mean | 1.22-03 | 1.76-03 | 1.20-03 | 9.68+02 | 1.49-03 | 9.80-04 | 8.30-03 | | Standard deviation | 2.26-04 | 3.57-04 | 2.77-04 | 4.07+02 | 2.33-04 | 8.00-04 | 2.96-03 |
| | Six-hump camel back function | f16 | Best score | -1.03+00 | -1.03+00 | -1.03+00 | 0.00E+00 | -1.03+00 | -1.03+00 | -1.03+00 | | Worst score | -1.03+00 | -1.03+00 | -1.03+00 | 3.23+00 | -1.03+00 | -1.03+00 | -1.03+00 | | Mean | -1.03+00 | -1.03+00 | -1.03+00 | 1.64+00 | -1.03+00 | -1.03+00 | -1.03+00 | | Standard deviation | 2.34-16 | 2.34-16 | 2.34-16 | 1.47+00 | 2.34-16 | 2.34-16 | 2.34-16 |
| | Branin function | f17 | Best score | NA | NA | 3.98-01 | 3.58+01 | 3.98-01 | 3.98-01 | 3.98-01 | | Worst score | NA | NA | 3.98-01 | 5.56+01 | 3.98-01 | 3.98-01 | 3.98-01 | | Mean | NA | NA | 3.98-01 | 4.53+01 | 3.98-01 | 3.98-01 | 3.98-01 | | Standard deviation | NA | NA | 0.00+00 | 8.11+00 | 0.00+00 | 0.00+00 | 0.00+00 |
| | Goldstein-Price function | f18 | Best score | 3.00+00 | 3.00+00 | 3.00+00 | 6.00+02 | 3.00+00 | 3.00+00 | 3.00+00 | | Worst score | 2.35+01 | 3.00+00 | 3.00+00 | 2.86+04 | 3.00+00 | 3.00+00 | 3.00+00 | | Mean | 8.05+00 | 3.00+00 | 3.00+00 | 9.42+03 | 3.00+00 | 3.00+00 | 3.00+00 | | Standard deviation | 7.94+00 | 0.00+00 | 0.00+00 | 1.33+04 | 0.00+00 | 0.00+00 | 0.00+00 |
| | Hartman’s family | f19 | Best score | -3.86+00 | -3.86+00 | -3.86+00 | NA | -3.86+00 | -3.86+00 | -3.86+00 | | Worst score | -3.29+00 | -3.86+00 | -3.86+00 | NA | -3.86+00 | -3.86+00 | -3.86+00 | | Mean | -3.61+00 | -3.86+00 | -3.86+00 | NA | -3.86+00 | -3.86+00 | -3.86+00 | | Standard deviation | 1.82-01 | 0.00+00 | 6.73-04 | NA | 0.00+00 | 0.00+00 | 0.00+00 |
| | Hartman’s family | f20 | Best score | -3.32+00 | -3.32+00 | -3.32+00 | NA | -3.20+00 | -3.32+00 | -3.32+00 | | Worst score | -2.28+00 | -3.20+00 | -3.15+00 | NA | -3.14+00 | -3.20+00 | -3.05+00 | | Mean | -3.02+00 | -3.21+00 | -3.27+00 | NA | -3.19+00 | -3.27+00 | -3.23+00 | | Standard deviation | 3.88-01 | 3.77-02 | 7.06-02 | NA | 2.65-02 | 6.14-02 | 1.01-01 |
| | Shekel’s family | f21 | Best score | -1.02+01 | -1.02+01 | -1.02+01 | NA | -1.02+01 | -1.02+01 | -1.02+01 | | Worst score | -5.10+00 | -2.63+00 | -1.02+01 | NA | -2.68+00 | -5.06+00 | -5.10+00 | | Mean | -9.14+00 | -9.40+00 | -1.02+01 | NA | -6.87+00 | -8.13+00 | -9.65+00 | | Standard deviation | 2.13+00 | 2.38+00 | 1.71-04 | NA | 2.92+00 | 2.61+00 | 1.60+00 |
| | Shekel’s family | f22 | Best score | -1.04+01 | -1.04+01 | -1.04+01 | NA | -1.04+01 | -1.04+01 | -1.04+01 | | Worst score | -5.09+00 | -5.13+00 | -5.09+00 | NA | -2.77+00 | -5.13+00 | -2.77+00 | | Mean | -9.87+00 | -9.88+00 | -9.34+00 | NA | -9.11+00 | -9.88+00 | -9.11+00 | | Standard deviation | 1.68+00 | 1.67+00 | 2.24+00 | NA | 2.78+00 | 1.67+00 | 2.78+00 |
| | Shekel’s family | f23 | Best score | -1.05+01 | -1.05+01 | -1.05+01 | NA | -1.05+01 | -1.05+01 | 0.00+00 | | Worst score | -1.05+01 | -1.05+01 | -1.05+01 | NA | -2.87+00 | -5.13+00 | 0.00+00 | | Mean | -1.05+01 | -1.05+01 | -1.05+01 | NA | -7.80+00 | -1.00+01 | 0.00+00 | | Standard deviation | 0.00+00 | 0.00+00 | 2.80-04 | NA | 3.59+00 | 1.71+00 | 0.00+00 |
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