Chaotic Evolutionary Programming for an Engineering Optimization Problem
Abstract
:1. Introduction
- Introduction of the chaotic sequence based population initialization process.
- A chaotic mutation operator is proposed and employed.
- A chaos guided tournament selection operator is considered to select better candidates.
- The Powell’s pattern search is applied to enhance the exploitation of the proposed algorithm.
2. Economic Load Dispatch Problem
- (i)
- The power balance equality constraint:
- (ii)
- The generator operating limits:
- (iii)
- The ramp rate limit.
- As generation increases:
- As generation decreases:
- (iv)
- Prohibited operating zone constraint:
3. Evolutionary Programming
4. Proposed Algorithm
4.1. Chaotic Evolutionary Programming
4.2. Powell’s Pattern Search Method
5. Simulation Test Problems
5.1. Generalized Test Functions
- 1.
- Griewank function: This is described mathematically as:
- 2.
- Rastrigin’s function: This is described mathematically as:
- 3.
- Rosenbrock’s function: This is described mathematically as:
- 4.
- Schwefel 2.22 function: This is described mathematically as:
- 5.
- Sphere function: This is one of the simplest of De Jong’s functions. It is described mathematically as:
- 6.
- Step function: This is described mathematically as:
- 7.
- Step 2 function: This is described mathematically as:
5.2. Multi-Fuel Economic Load Dispatch Problem
6. Results and Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Case | Valve Point Loading | Ramp Rate | Prohibited Operating Zone | Transmission Loss |
---|---|---|---|---|
1 | × | × | × | × |
2 | ✓ | × | × | × |
3 | × | × | ✓ | × |
4 | × | × | × | ✓ |
5 | ✓ | × | ✓ | ✓ |
6 | × | × | ✓ | ✓ |
Test Function | Fitness | CEP-1 | CEP-2 | CEPPS-1 | CEPPS-2 |
---|---|---|---|---|---|
Worst | 4.61 | 1.08 | 11.05 | 9.29 × | |
Griewank function | Average | 4.61 | 1.08 | 11.05 | 9.29 × |
Best | 4.61 | 1.08 | 5.48 | 0.01 × | |
Worst | 21,893.57 | 305.44 | 40,041.02 | 97.11 | |
Rastrigin function | Average | 21,893.57 | 305.44 | 40,041.02 | 61.70 |
Best | 21,893.57 | 305.44 | 15,691.13 | 61.70 | |
Worst | 7.54 × | 222.46 | 1.00 × | 43,304.03 | |
Rosenbrock function | Average | 7.54 × | 22.36 | 1.00 × | 7300.95 |
Best | 7.54 × | 22.36 | 2.91 × | 7300.95 | |
Worst | 48.16 | 68.45 | 7.57 | 27.71 | |
Schwefel’s 2.22 function | Average | 48.16 | 68.45 | 7.57 | 27.71 |
Best | 48.16 | 68.45 | 7.57 | 27.71 | |
Worst | 13,094.38 | 8.23 | 50,471.00 | 4.79 × | |
Sphere function | Average | 13,094.38 | 8.23 | 50471.00 | 9.40 × |
Best | 13,094.38 | 8.23 | 18,642.14 | 9.40 × | |
Worst | 670.00 | 37.00 | 969.00 | 28.00 | |
Step function | Average | 670.00 | 37.00 | 969.00 | 28.00 |
Best | 670.00 | 15.00 | 532.00 | 15.00 | |
Worst | 15,349.00 | 6.00 | 39,277.00 | 8.00 | |
Step 2 function | Average | 15,349.00 | 6.00 | 39,277.00 | 5.00 |
Best | 15,349.00 | 6.00 | 17,381.00 | 5.00 |
Algorithm | Cost ($/h) | |||||
---|---|---|---|---|---|---|
Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
BBO [31] | 624.51 | – | – | – | – | – |
CPSO [32] | – | 623.82 | – | – | – | – |
CGA-MU [33] | 623.80 | 624.71 | – | – | – | – |
DE [33] | 623.80 | 624.46 | – | – | – | – |
DEBBO [34] | 624.51 | – | – | – | – | – |
ELHN [35] | 624.51 | – | – | – | – | – |
IGA [30] | 624.51 | – | – | – | – | – |
IGA-MU [30] | 623.80 | 624.51 | – | – | – | – |
KHA [36] | 624.51 | – | – | – | – | – |
PSO [33] | 623.80 | 624.24 | – | – | – | – |
QP-ALHN [37] | 623.80 | – | 624.32 | – | – | – |
SPPO | 623.80 | 623.82 | 624.32 | 700.29 | 700.77 | 700.48 |
CEPPS-1 | 623.75 | 623.87 | 623.76 | 699.70 | 699.54 | 704.94 |
CEPPS-2 | 623.75 | 623.88 | 623.77 | 699.77 | 699.73 | 700.60 |
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Singh, N.J.; Singh, S.; Chopra, V.; Aftab, M.A.; Hussain, S.M.S.; Ustun, T.S. Chaotic Evolutionary Programming for an Engineering Optimization Problem. Appl. Sci. 2021, 11, 2717. https://doi.org/10.3390/app11062717
Singh NJ, Singh S, Chopra V, Aftab MA, Hussain SMS, Ustun TS. Chaotic Evolutionary Programming for an Engineering Optimization Problem. Applied Sciences. 2021; 11(6):2717. https://doi.org/10.3390/app11062717
Chicago/Turabian StyleSingh, Nirbhow Jap, Shakti Singh, Vikram Chopra, Mohd Asim Aftab, S. M. Suhail Hussain, and Taha Selim Ustun. 2021. "Chaotic Evolutionary Programming for an Engineering Optimization Problem" Applied Sciences 11, no. 6: 2717. https://doi.org/10.3390/app11062717
APA StyleSingh, N. J., Singh, S., Chopra, V., Aftab, M. A., Hussain, S. M. S., & Ustun, T. S. (2021). Chaotic Evolutionary Programming for an Engineering Optimization Problem. Applied Sciences, 11(6), 2717. https://doi.org/10.3390/app11062717