# Chaotic Evolutionary Programming for an Engineering Optimization Problem

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## 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:$$\sum _{j=1}^{{N}_{g}}{P}_{j}-({P}_{D}+{P}_{L})=0$$
- (ii)
- The generator operating limits:$${P}_{j}^{min}\le {P}_{j}\le {P}_{j}^{max}\phantom{\rule{2.em}{0ex}}(j=1,2,\dots ,{N}_{g})$$
- (iii)
- The ramp rate limit.
- As generation increases:$${P}_{j}-{P}_{j}^{0}\le U{R}_{j}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{2.em}{0ex}}\left(j=1,2,\dots ,{N}_{g}\right)$$
- As generation decreases:$${P}_{j}^{0}-{P}_{j}\le D{R}_{j}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{2.em}{0ex}}\left(j=1,2,\dots ,{N}_{g}\right)$$

- (iv)
- Prohibited operating zone constraint:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {P}_{j}^{min}\le {P}_{j}\le {P}_{j,1}^{L}\phantom{\rule{1.em}{0ex}}\left(j=1,2,\dots ,{N}_{g}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {P}_{j,i-1}^{U}\le {P}_{j}\le {P}_{j,i}^{L}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(i=1,2,\dots ,{N}_{zj};j=1,2,\dots ,{N}_{g}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {P}_{j,Nzj}^{U}\le {P}_{j}\le {P}_{j}^{max}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(j=1,2,\dots ,{N}_{g}\right)\hfill \end{array}$$

## 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:$$\begin{array}{c}\hfill {F}_{1}\left(x\right)=\sum _{i=1}^{n}\frac{{x}_{i}^{2}}{4000}-\prod cos\left(\frac{{x}_{i}}{\sqrt{i}}\right)\end{array}$$

- 2.
- Rastrigin’s function: This is described mathematically as:$$\begin{array}{c}\hfill {F}_{2}\left(x\right)=\sum _{i=1}^{n}[{x}_{i}^{2}-10cos\left(2\pi {x}_{i}\right)]\end{array}$$

- 3.
- Rosenbrock’s function: This is described mathematically as:$$\begin{array}{c}\hfill {F}_{3}\left(x\right)=\sum _{i=1}^{n-1}[{({x}_{i}-1)}^{2}+100{({x}_{i+1}-{x}_{i}^{2})}^{2}]\end{array}$$

- 4.
- Schwefel 2.22 function: This is described mathematically as:$$\begin{array}{c}\hfill {F}_{4}\left(x\right)=\sum _{i=1}^{n}|{x}_{i}|+\prod _{i=1}^{n}\left|{x}_{i}\right|\end{array}$$

- 5.
- Sphere function: This is one of the simplest of De Jong’s functions. It is described mathematically as:$$\begin{array}{c}\hfill {F}_{5}\left(x\right)=\sum _{i=1}^{n}{x}_{i}^{2}\end{array}$$

- 6.
- Step function: This is described mathematically as:$$\begin{array}{c}\hfill {F}_{6}\left(x\right)=\sum _{i=1}^{n}\lfloor |{x}_{i}|\rfloor \end{array}$$

- 7.
- Step 2 function: This is described mathematically as:$$\begin{array}{c}\hfill {F}_{7}\left(x\right)=\sum _{i=1}^{n}\lfloor |{x}_{i}+0.5|\rfloor \end{array}$$

#### 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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**Figure 1.**Time series plot of the tent and Gauss maps, respectively [20].

Case | Valve Point Loading | Ramp Rate | Prohibited Operating Zone | Transmission Loss |
---|---|---|---|---|

1 | × | × | × | × |

2 | ✓ | × | × | × |

3 | × | × | ✓ | × |

4 | × | × | × | ✓ |

5 | ✓ | × | ✓ | ✓ |

6 | × | × | ✓ | ✓ |

**Table 2.**Performance analysis of the fitness value of generalized benchmark test functions [29].

Test Function | Fitness | CEP-1 | CEP-2 | CEPPS-1 | CEPPS-2 |
---|---|---|---|---|---|

Worst | 4.61 | 1.08 | 11.05 | 9.29 × ${10}^{-1}$ | |

Griewank function | Average | 4.61 | 1.08 | 11.05 | 9.29 × ${10}^{-1}$ |

Best | 4.61 | 1.08 | 5.48 | 0.01 × ${10}^{-1}$ | |

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 × ${10}^{8}$ | 222.46 | 1.00 × ${10}^{10}$ | 43,304.03 | |

Rosenbrock function | Average | 7.54 × ${10}^{8}$ | 22.36 | 1.00 × ${10}^{10}$ | 7300.95 |

Best | 7.54 × ${10}^{8}$ | 22.36 | 2.91 × ${10}^{8}$ | 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 × ${10}^{-19}$ | |

Sphere function | Average | 13,094.38 | 8.23 | 50471.00 | 9.40 × ${10}^{-20}$ |

Best | 13,094.38 | 8.23 | 18,642.14 | 9.40 × ${10}^{-20}$ | |

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 |

**Table 3.**Test Power System 1, comparison of economic load dispatch (ELD) (${P}_{D}=2700$ MW). BBO, biogeography based optimization; DE, differential evolution; ELHN, enhanced augmented Hopfield neural network; IGA, improved gravitational search algorithm; KHA, krill herd optimization; QP-ALHN, quadratic programming augmented Hopfield neural network.

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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Singh, 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