# Economic Dispatch Using Modified Bat Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Problem Objectives

#### 2.2. Problem Constraints

## 3. Bat Algorithm

## 4. Modifications

#### 4.1. Add Bad Experience Component

#### 4.2. Nonlinear Inertia Weight

## 5. Experiments and Results

#### 5.1. Test Case 1—Six-Generator Test System with System Losses

Generator | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Pmin (MW) | 10 | 10 | 35 | 35 | 130 | 125 |

Pmax (MW) | 125 | 150 | 225 | 210 | 325 | 315 |

No. | a | b | c |
---|---|---|---|

1 | 0.15240 | 38.53973 | 756.79886 |

2 | 0.10587 | 46.15916 | 451.32513 |

3 | 0.02803 | 40.39655 | 1049.9977 |

4 | 0.03546 | 38.30553 | 1243.5311 |

5 | 0.02111 | 36.32782 | 1658.5596 |

6 | 0.01799 | 38.27041 | 1356.6592 |

Parameter | Value |
---|---|

${A}_{i}^{\left(0\right)}$ | 0.9 |

${r}_{i}^{\left(0\right)}$ | 0.1 |

$\left[\zeta \text{}\varsigma \right]$ | [0.97 0.95] |

$[{W}_{min}{W}_{max}]$ | [0.4 0.9] |

$[{C}_{1}{C}_{2}{C}_{3}{C}_{4}]$ | $\left[\mathfrak{u}\left(0,3\right)\mathfrak{u}\left(0,2\right)\mathfrak{u}\left(0,1\right)\mathfrak{u}\left(0,1\right)\right]$ |

$p$ | 2000 |

$\left[GH\right]$ | [10 200] |

P1 (MW) | P2 (MW) | P3 (MW) | P4 (MW) | P5 (MW) | P6 (MW) | Loss (MW) | Cost ($/h) | ||
---|---|---|---|---|---|---|---|---|---|

λ Iteration | 28.304 | 10 | 118.897 | 118.733 | 230.733 | 212.831 | 19.433 | 36912.14 | |

GA | Best | 26.79976 | 15.89313 | 107.3073 | 123.932 | 228.3426 | 217.1609 | 19.44 | 36924.15 |

Avg. | 45.57365 | 48.619 | 105.8057 | 106.478 | 211.4508 | 200.6767 | 18.61 | 37505.72 | |

Std | 19.7706 | 28.6733 | 43.3288 | 36.2062 | 45.62 | 45.0436 | 1.325 | 382.88 | |

PSO | Best | 28.30223 | 9.999884 | 118.9522 | 118.6706 | 230.7563 | 212.7375 | 19.431 | 36911.54 |

Avg. | 28.39792 | 10.02338 | 119.0863 | 118.5947 | 230.588 | 212.7238 | 19.4262 | 36911.75 | |

Std | 0.85864 | 0.13943 | 0.83555 | 0.62292 | 1.1889 | 0.4948 | 0.027862 | 1.4869 | |

BA | Best | 28.07394 | 10.05693 | 119.9855 | 117.7729 | 231.1333 | 212.3918 | 19.4238 | 36911.79 |

Avg. | 28.39414 | 10.26771 | 119.159 | 119.0363 | 230.2951 | 212.2449 | 19.4092 | 36912.54 | |

Std | 0.69285 | 0.26761 | 2.2262 | 1.7091 | 2.9539 | 3.8 | 0.059993 | 1.0006 | |

MBA | Best | 28.14831 | 10.03893 | 119.7243 | 118.052 | 231.0219 | 212.4194 | 19.4239 | 36911.27 |

Avg. | 28.28837 | 10.21736 | 119.3942 | 118.6366 | 230.4744 | 212.3904 | 19.4146 | 36912.13 | |

Std | 0.73114 | 0.19558 | 2.485 | 1.8194 | 3.38 | 3.6097 | 0.065668 | 0.84625 |

P1 (MW) | P2 (MW) | P3 (MW) | P4 (MW) | P5 (MW) | P6 (MW) | Loss (MW) | Cost ($/h) | ||
---|---|---|---|---|---|---|---|---|---|

λ Iteration | 32.599 | 14.483 | 141.544 | 136.041 | 257.6588 | 243.003 | 25.330 | 41896.63 | |

GA | Best | 39.63015 | 13.23341 | 170.317 | 155.1286 | 232.4949 | 213.4204 | 24.2359 | 41976.08 |

Avg. | 55.35765 | 54.95395 | 130.4268 | 134.2949 | 230.3903 | 218.5409 | 23.9787 | 42614.68 | |

Std | 25.9155 | 30.1187 | 45.3717 | 39.9879 | 49.7911 | 45.6905 | 1.3105 | 436.61 | |

PSO | Best | 32.59937 | 14.48227 | 141.5412 | 136.0392 | 257.6555 | 242.9997 | 25.3299 | 41895.98 |

Avg. | 32.5959 | 14.51256 | 141.4859 | 135.9388 | 257.6442 | 243.1419 | 25.3322 | 41896.02 | |

Std | 0.19817 | 0.2575 | 0.31681 | 0.66662 | 0.33471 | 0.86126 | 0.020216 | 0.23259 | |

BA | Best | 32.46774 | 14.34427 | 141.9097 | 135.7294 | 257.7276 | 243.1421 | 25.3359 | 41895.88 |

Avg. | 32.58662 | 14.49149 | 141.7122 | 136.2057 | 257.3597 | 242.9548 | 25.3232 | 41896.17 | |

Std | 0.38275 | 0.49502 | 0.97076 | 0.88628 | 1.2144 | 1.3829 | 0.037035 | 0.25826 | |

MBA | Best | 32.49975 | 14.43056 | 141.6805 | 135.9817 | 257.502 | 243.2203 | 25.3329 | 41895.71 |

Avg. | 32.6766 | 14.35507 | 142.1353 | 135.802 | 257.5361 | 242.803 | 25.3222 | 41896.09 | |

Std | 0.12392 | 0.37326 | 0.61012 | 0.59674 | 0.31035 | 1.0403 | 0.03666 | 0.21975 |

#### 5.2. Test Case 2–Five-Generator Test System with System Losses

Generator | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Pmin (MW) | 50 | 20 | 30 | 10 | 40 |

Pmax (MW) | 300 | 125 | 175 | 75 | 250 |

No. | $a$ | $b$ | $c$ | $d$ | $e$ |
---|---|---|---|---|---|

1 | 0.0015 | 1.8 | 40 | 200 | 0.035 |

2 | 0.0030 | 1.8 | 60 | 140 | 0.040 |

3 | 0.0012 | 2.1 | 100 | 160 | 0.038 |

4 | 0.0080 | 2.0 | 25 | 100 | 0.042 |

5 | 0.0010 | 2.0 | 120 | 180 | 0.037 |

P1 (MW) | P2 (MW) | P3 (MW) | P4 (MW) | P5 (MW) | Cost ($/h) | ||
---|---|---|---|---|---|---|---|

GA | Best | 234.82 | 100.92 | 112.80 | 71.46 | 209.99 | 2068.06 |

Avg. | 253.89 | 91.39 | 127.71 | 49.37 | 207.62 | 2357.02 | |

Std | 29.12 | 19.19 | 27.23 | 15.85 | 23.30 | 111.80 | |

PSO | Best | 229.51 | 102.98 | 112.67 | 75.00 | 209.81 | 2029.63 |

Avg. | 248.71 | 94.28 | 126.39 | 55.04 | 205.56 | 2165.08 | |

Std | 30.85 | 19.33 | 27.77 | 23.06 | 26.58 | 104.24 | |

BA | Best | 229.14 | 101.30 | 114.05 | 74.26 | 211.23 | 2042.88 |

Avg. | 250.69 | 90.03 | 133.26 | 53.18 | 202.84 | 2176.06 | |

Std | 38.69 | 26.89 | 32.47 | 27.81 | 28.00 | 106.92 | |

MBA | Best | 231.06 | 99.59 | 113.48 | 74.42 | 211.44 | 2032.23 |

Avg. | 256.94 | 96.53 | 131.55 | 47.81 | 197.14 | 2141.50 | |

Std | 35.60 | 17.06 | 29.09 | 29.65 | 30.24 | 95.39 |

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Saadat, H. Power System Analysis. Available online: http://www.psapublishing.com/ (accessed on 1 July 2014).
- Kothari, D.P. Power system optimization, 2nd ed.Ghosh, A.K., Ed.; PHI Learning Private Limited: New Dehli, India, 2012. [Google Scholar]
- Li, L.; Sun, Z. Dynamic Energy Control for Energy Efficiency Improvement of Sustainable Manufacturing Systems Using Markov Decision Process. Syst. Man Cybern. Syst. IEEE Trans.
**2013**, 43, 1195–1205. [Google Scholar] [CrossRef] - Fletcher, R. Practical Methods of Optimization; John Wiley & Sons: Chichester, SXW, UK, 2013. [Google Scholar]
- Frank, S.; Steponavice, I.; Rebennack, S. Optimal power flow: A bibliographic survey I. Energy Syst.
**2012**, 3, 221–258. [Google Scholar] [CrossRef] - Frank, S.; Steponavice, I.; Rebennack, S. Optimal power flow: A bibliographic survey II. Energy Syst.
**2012**, 3, 259–289. [Google Scholar] [CrossRef] - Abido, M.A. A niched Pareto genetic algorithm for multiobjective environmental/economic dispatch. Int. J. Electr. Power Energy Syst.
**2003**, 25, 97–105. [Google Scholar] [CrossRef] - Subbaraj, P.; Rengaraj, R.; Salivahanan, S. Enhancement of self-adaptive real-coded genetic algorithm using Taguchi method for economic dispatch problem. Appl. Soft Comput.
**2011**, 11, 83–92. [Google Scholar] [CrossRef] - Amjady, N.; Nasiri-Rad, H. Solution of nonconvex and nonsmooth economic dispatch by a new adaptive real coded genetic algorithm. Expert Syst. Appl.
**2010**, 37, 5239–5245. [Google Scholar] [CrossRef] - Selvakumar, A.I.; Thanushkodi, K. A new particle swarm optimization solution to nonconvex economic dispatch problems. Power Syst. IEEE Trans.
**2007**, 22, 42–51. [Google Scholar] [CrossRef] - Selvakumar, A.I.; Thanushkodi, K. Anti-predatory particle swarm optimization: Solution to nonconvex economic dispatch problems. Electr. Power Syst. Res.
**2008**, 78, 2–10. [Google Scholar] [CrossRef] - Gaing, Z.-L. Particle swarm optimization to solving the economic dispatch considering the generator constraints. Power Syst. IEEE Trans.
**2003**, 18, 1187–1195. [Google Scholar] [CrossRef] - Cai, J.; Ma, X.; Li, L.; Haipeng, P. Chaotic particle swarm optimization for economic dispatch considering the generator constraints. Energy Convers. Manag.
**2007**, 48, 645–653. [Google Scholar] [CrossRef] - Yang, X.S. A new metaheuristic bat-inspired algorithm. In Nature Inspired Cooperative Strategies for Optimization (NICSO 2010); Springer: Berlin, Heidelberg, Germany, 2010; pp. 65–74. [Google Scholar]
- Yang, X.S.; He, X. Bat algorithm: Literature review and applications. Int. J. Bioinspired Comput.
**2013**, 5, 141–149. [Google Scholar] [CrossRef] - Sidi-Bel-Abbes, A. Economic dispatch problem using bat algorithm. Leonardo J. Sci.
**2014**, 24, 75–84. [Google Scholar] - Niknam, T.; Azizipanah-Abarghooee, R.; Zare, M.; Bahmani-Firouzi, B. Reserve constrained dynamic environmental/economic dispatch: A new multiobjective self-adaptive learning bat algorithm. Syst. J. IEEE
**2012**, 7, 763–776. [Google Scholar] [CrossRef] - Ramesh, B.; Chandra Jagan Mohan, V.; Veera Reddy, V.C. Application of bat algorithm for combined economic load and emission dispatch. Int. J. Electr. Eng. Telecommun.
**2013**, 2, 1–9. [Google Scholar] - Fister, I., Jr.; Fister, D.; Yang, X.-S. A hybrid bat algorithm. Elektroteh. Vestn.
**2013**, 80, 1–7. [Google Scholar] - Jamil, M.; Zepernic, H.-J.; Yang, X.S. Improved bat algorithm for global optimization. Appl. Soft Comput.
**2013**. submitted for publication. [Google Scholar]

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**MDPI and ACS Style**

Latif, A.; Palensky, P. Economic Dispatch Using Modified Bat Algorithm. *Algorithms* **2014**, *7*, 328-338.
https://doi.org/10.3390/a7030328

**AMA Style**

Latif A, Palensky P. Economic Dispatch Using Modified Bat Algorithm. *Algorithms*. 2014; 7(3):328-338.
https://doi.org/10.3390/a7030328

**Chicago/Turabian Style**

Latif, Aadil, and Peter Palensky. 2014. "Economic Dispatch Using Modified Bat Algorithm" *Algorithms* 7, no. 3: 328-338.
https://doi.org/10.3390/a7030328