# 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:$${P}_{j}-{P}_{j}^{0}\le U{R}_{j}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{2.em}{0ex}}\left(\right)open="("\; close=")">j=1,2,\dots ,{N}_{g}$$
- As generation decreases:$${P}_{j}^{0}-{P}_{j}\le D{R}_{j}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{2.em}{0ex}}\left(\right)open="("\; close=")">j=1,2,\dots ,{N}_{g}$$

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

## 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

- Dhillon, J.; Kothari, D. Power System Optimization; PHI Learning Pvt. Ltd.: New Delhi, India, 2004. [Google Scholar]
- Singh, N.J.; Dhillon, J.; Kothari, D. Synergic predator-prey optimization for economic thermal power dispatch problem. Appl. Soft Comput.
**2016**, 43, 298–311. [Google Scholar] [CrossRef] - Singh, N.J.; Dhillon, J.; Kothari, D. Surrogate worth trade-off method for multi-objective thermal power load dispatch. Energy
**2017**, 138, 1112–1123. [Google Scholar] [CrossRef] - Jayabarathi, T.; Jayaprakash, K.; Jeyakumar, D.; Raghunathan, T. Evolutionary programming techniques for different kinds of economic dispatch problems. Electr. Power Syst. Res.
**2005**, 73, 169–176. [Google Scholar] [CrossRef] - Gaing, Z.-L. Particle swarm optimization to solving the economic dispatch considering the generator constraints. IEEE Trans. Power Syst.
**2003**, 18, 1187–1195. [Google Scholar] [CrossRef] - Chen, P.-H.; Chang, H.-C. Large-scale economic dispatch by genetic algorithm. IEEE Trans. Power syst.
**1995**, 10, 1919–1926. [Google Scholar] [CrossRef] - Chellapilla, K.; Fogel, D.B. Two new mutation operators for enhanced search and optimization in evolutionary programming. In Optical Science, Engineering and Instrumentation; International Society for Optics and Photonics: Bellingham, WA, USA, 1997; pp. 260–269. [Google Scholar]
- Fogel, L.J.; Fogel, D.B. A preliminary investigation on extending evolutionary programming to include self-adaptation on finite state. Informatica
**1994**, 18, 387–398. [Google Scholar] - Yao, X.; Liu, Y. Fast evolutionary programming. Evol. Program.
**1996**, 3, 451–460. [Google Scholar] - Yang, H.-T.; Yang, P.-C.; Huang, C.-L. Evolutionary programming based economic dispatch for units with non-smooth fuel cost functions. IEEE Trans. Power Syst.
**1996**, 11, 112–118. [Google Scholar] [CrossRef] - Back, T.; Schwefel, H.P. An overview of evolutionary algorithms for parameter optimization. Evolu. Comput.
**1993**, 1, 1–23. [Google Scholar] [CrossRef] - Yao, X.; Liu, Y.; Lin, G. Evolutionary programming made faster. IEEE Trans. Evolu. Comput.
**1999**, 3, 82–102. [Google Scholar] - Liu, Y.; Yao, X.; Zhao, Q.; Higuchi, T. Scaling up fast evolutionary programming with cooperative co evolution. In Proceedings of the Congress on Evolutionary Computation, Seoul, Korea, 27–30 May 2001; Volume 2, pp. 1101–1108. [Google Scholar]
- Lee, C.-Y.; Yao, X. Evolutionary programming using mutations based on the levy probability distribution. IEEE Trans. Evol. Comput.
**2004**, 8, 1–3. [Google Scholar] [CrossRef] [Green Version] - Thangaraj, R.; Pant, M.; Chelliah, T.R.; Abraham, A. Opposition based chaotic differential evolution algorithm for solving global optimization problems. In Proceedings of the Fourth World Congress on Nature and Biologically Inspired Computing, Mexico City, Mexico, 5–9 November 2012; pp. 1–15. [Google Scholar]
- Dos Santos Coelho, L.; Mariani, V.C. Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect. IEEE Trans. Power Syst.
**2006**, 21, 989–996. [Google Scholar] [CrossRef] - He, D.; Dong, G.; Wang, F.; Mao, Z. Optimization of dynamic economic dispatch with valve-point effect using chaotic sequence based differential evolution algorithms. Energy Conver. Manag.
**2011**, 52, 1026–1032. [Google Scholar] [CrossRef] - Bharti, K.K.; Singh, P.K. Chaotic gradient artificial bee colony for text clustering. Soft Comput.
**2016**, 20, 1113–1126. [Google Scholar] [CrossRef] - Zhang, Z.; Wang, T.; Liu, X. Melt index prediction by aggregated rbf neural networks trained with chaotic theory. Neuro Comput.
**2014**, 131, 368–376. [Google Scholar] [CrossRef] - Gandomi, A.H.; Yang, X.-S. Chaotic bat algroithm. J. Compt. Sci.
**2014**, 5, 224–232. [Google Scholar] [CrossRef] - Chuang, L.-Y.; Hsiao, C.-J.; Yang, C.-H. Chaotic particle swarm optimization for data clustering. Expert Syst. Appl.
**2011**, 38, 62–66. [Google Scholar] [CrossRef] - Kumar, S.; Mandal, K.K.; Chakraborty, N. Optimal DG placement by multi-objective opposition based chaotic differential evolution for techno-economic analysis. Appl. Soft Comput.
**2019**, 78, 70–83. [Google Scholar] [CrossRef] - Kaur, M.; Dhillon, J.S.; Kothari, D.P. Crisscross differential evolution algorithm for constrained hydro thermal scheduling. Appl. Soft. Comput.
**2020**, 93, 1–19. [Google Scholar] [CrossRef] - Lu, H.; Wang, X.; Fei, Z.; Qiu, M. The Effects of Using Chaotic Map on Improving the Performance of Multi-objective Evolutionary Algorithms. Math. Probl. Eng.
**2014**, 2014, 924652. [Google Scholar] [CrossRef] [Green Version] - Tutueva, A.V.; Nepomuceno, E.G.; Karimov, A.I.; Andreev, V.S.; Butusov, D.N. Adaptive chaotic maps and their application to pseudo-random numbers generation. Chaos Solitons Fractals
**2020**, 133, 109615. [Google Scholar] [CrossRef] - Nepomuceno, E.G.; Lima, A.M.; Arias-García, J.; Perc, M.; Repnik, R. Minimal digital chaotic system. Chaos Solitons Fractals
**2019**, 120, 62–66. [Google Scholar] [CrossRef] - Chen, F.; Huang, G.; Fan, Y.; Liao, R. A nonlinear fractional programming approach for environmental economic power dispatch. Int. J. Electr. Power Energy Syst.
**2016**, 78, 463–469. [Google Scholar] [CrossRef] - Fogel, D.B. Applying evolutionary programming to selected traveling salesman problems. Cybern. Syst.
**1993**, 24, 27–36. [Google Scholar] [CrossRef] - Jamil, M.; Yang, X.-S. A literature survey of benchmark functions for global optimisation problems. Int. J. Math Model. Numer. Optim.
**2013**, 4, 150–194. [Google Scholar] [CrossRef] [Green Version] - Chiang, C.-L. Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels. IEEE Trans. Power Syst.
**2005**, 20, 1690–1699. [Google Scholar] [CrossRef] - Bhattacharya, A.; Chattopadhyay, P.K. Solving complex economic load dispatch problems using bio geography-based optimization. Expert Syst. Appl.
**2010**, 37, 3605–3615. [Google Scholar] [CrossRef] - Park, J.-B.; Jeong, Y.-W.; Shin, J.-R.; Lee, K.Y. An improved particle swarm optimization for non convex economic dispatch problems. IEEE Trans. Power Syst.
**2010**, 25, 156–166. [Google Scholar] [CrossRef] - Barisal, A. Dynamic search space squeezing strategy based intelligent algorithm solutions to economic dispatch with multiple fuels. Int. J. Electr. Power Energy Syst.
**2013**, 45, 50–59. [Google Scholar] [CrossRef] - Bhattacharya, A.; Chattopadhyay, P.K. Hybrid differential evolution with bio geography-based optimization for solution of economic load dispatch. IEEE Trans. Power Syst.
**2010**, 25, 1955–1964. [Google Scholar] [CrossRef] - Vo, D.N.; Ongsakul, W. Economic dispatch with multiple fuel types by enhanced augmented Lagrange Hopfield network. Appl. Energy
**2012**, 91, 281–289. [Google Scholar] [CrossRef] - Mandal, B.; Roy, P.K.; Mandal, S. Economic load dispatch using krill herd algorithm. Int. J. Electr. Power Energy Syst.
**2014**, 57, 1–10. [Google Scholar] [CrossRef] - Dieu, V.N.; Schegner, P. Augmented lagrange hopfeld network initialized by quadratic programming for economic dispatch with piece wise quadratic cost functions and prohibited zones. Appl. Soft Comput.
**2013**, 13, 292–301. [Google Scholar] [CrossRef]

**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