# Novel Genetic Algorithm-Based Energy Management in a Factory Power System Considering Uncertain Photovoltaic Energies

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The problem is solved by a two-level method: the master level determines the optimal states (0/1) of the generators and the elastic loads using a novel genetic algorithm; and the slave level deals with optimal real power scheduling and power purchase/sale from/to the utility, subject to the system operating constraints, using the interior point algorithm.
- (2)
- Two novel encoding schemes associated with new crossover and mutation operations in genetic algorithms are presented. These new operations make this novel GA more efficient to solve the optimal UC and DR in a factory’s power system.
- (3)
- The uncertainty in the PV power generation is studied by using the point estimate method that integrates the master level with the slave level to gain an optimal stochastic mixed-integer solution.
- (4)
- Not only the states of micro-turbine generators in UC but also the states of elastic loads at the production lines in DR are addressed at the same time.

## 2. Problem Formulation

#### Objective Function

#### Equality Constraint

#### Inequality Constraints

## 3. Proposed Method

_{1}+ L) unknown binary variables and T × (2 + G

_{1}) unknown random variables. In this paper, G

_{1}= 4, L = 5 and T = 24. This problem may be solved by binary linear programming or dynamic programming [33,34,35] in the case that no probability density function is involved. The genetic algorithm (GA), on the other hand, randomly produces many chromosomes, which represent solutions, and selects the fittest one. However, the computational burden of the GA becomes large if the binary bit length of a chromosome and the number of functional constraints is large [36,37,38]. Thus, this work proposes an enhanced GA to deal with UC and DR.

#### 3.1. Novel Encoding of Generator States

#### Crossover Operation

#### Mutation Operation

#### 3.2. Novel Encoding of States of Elastic Load

#### Crossover Operation

#### Mutation Operation

#### 3.3. Point Estimation Method

_{2}.

_{1}.

#### 3.4. Overall Flowchart of Algorithmic Steps

_{1}= 4, L = 5 and T = 24. In a day-ahead scheduling problem, T = 24 is always true. Moreover, the number (30 herein) of buses in a factory, which is addressed to consider both UC and DR, is actually large. Considering G

_{1}= 4 and L = 5 is reasonable in a large factory power system. This studied problem is not the same as the traditional UC problem, which is defined in the transmission system and may include many buses and generators. Moreover, traditional DR is concerned in the distribution system or at home; however, the DR is emphasized herein in the end-user’s factory power system. The proposed GA is very efficient to deal with the operational constraints, such as Equation (7), for the studied problem.

## 4. Simulation Results

#### 4.1. Optimal Solutions Obtained by Proposed Method

- (a)
- A positive expected value of power injection at the swing bus ($\tilde{{P}_{sw}}\left(t\right)$) indicates that the factory purchases power ($\tilde{{P}_{p}}\left(t\right)$) while a negative sign implies that the factory sells power ($-\tilde{{P}_{s}}\left(t\right))$ to the utility. The factory sells power to the utility at hours 8:00 a.m.–2:00 p.m., 8:00 p.m. and 9:00 p.m. because the tariff during these periods is high (131$/MWh). In total, 46.1 MWh from the factory is sold to the utility.
- (b)
- Most of the generated PV power is consumed in the factory rather than imported into the utility power system.
- (c)
- Since the tariff for purchasing power from the utility is low during 1:00–8:00 a.m., the elastic loads consume almost all of energy during this period. To fulfill the total demand constraints (75 MWh), the production line at bus 6 also consumes energy during 3:00–7:00 p.m.

#### 4.2. Optimal Solutions without Considering PV Arrays

#### 4.3. Impacts of Different Factors on Total Cost

#### 4.4. Comparisons between Traditional Method and Proposed Method

#### 4.5. Statistics of Convergence Performance of the Proposed Method

#### 4.6. Comparison of Results Considering Different Standard Deviations of PV Power Generations

## 5. Conclusions

- The problem concerning optimal DR and UC, considering uncertain PV power generation, in a factory power system, rather than the UC in the bulk power system or DR at home, is formulated and studied.
- The method based on novel genetic algorithms that are associated with the point estimation and interior point methods is proposed to determine the UC and DR in the factory power system.
- The proposed string encoding in genetic algorithms efficiently performs both crossover and mutation operations for the UC together with DR. This proposed method ensures that feasible chromosomes can evolve to the fittest solution.
- Impacts of different parameters (such as PV generations, electricity tariffs, minimum on/down times, ramp rates and must-run hours) were completely investigated on the optimal solutions.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

- Bertsimas, D.; Litvinov, E.; Sun, X.A.; Zhao, J.; Zheng, T. Adaptive robust optimization for the security constrained unit commitment problem. IEEE Trans. Power Syst.
**2013**, 28, 52–63. [Google Scholar] [CrossRef] - Kalantari, A.; Restrepo, J.F.; Galiana, F.D. Security-constrained unit commitment with uncertain wind generation: The loadability set approach. IEEE Trans. Power Syst.
**2013**, 28, 1787–1796. [Google Scholar] [CrossRef] - Bakirtzis, E.A.; Biskas, P.N.; Labridis, D.P.; Bakirtzis, A.G. Multiple time resolution unit commitment for short-term operations scheduling under high renewable penetration. IEEE Trans. Power Syst.
**2014**, 29, 149–159. [Google Scholar] [CrossRef] - Federal Energy Regulatory Commission. Assessment of Demand Response & Advanced Metering: Staff Report; US Department of Energy: Washington, DC, USA, 2006.
- Federal Energy Regulatory Commission. Benefit of Demand Response in Electric Markets and Recommendations for Achieving Them; U.S. Department of Energy: Washington, DC, USA, 2006.
- Alexandria, V. Demand Responses II-Building the DSR Road Map. In Proceedings of the PJM Symposium on Demand Responses II-Building the DSR Road Map, Baltimore, MD, USA, 12–13 May 2008. [Google Scholar]
- Ott, A.L. Experience with PJM Market Operation, System Design, and Implementation. IEEE Trans. Power Syst.
**2003**, 18, 528–534. [Google Scholar] [CrossRef] - Lijesen, M.G. The real-time price elasticity of electricity. Energy Econ.
**2007**, 29, 249–258. [Google Scholar] [CrossRef] - Huisman, R.; Huurman, C.; Mahieu, R. Hourly electricity prices in day-ahead markets. Energy Econ.
**2007**, 29, 240–248. [Google Scholar] [CrossRef] - Yoon, J.H.; Baldick, R.; Novoselac, A. Dynamic demand response controller based on real-time retail price for residential buildings. IEEE Trans. Smart Grid
**2014**, 5, 121–129. [Google Scholar] [CrossRef] - Tsui, K.M.; Chan, S.C. Demand response optimization for smart home scheduling under real-time pricing. IEEE Trans. Smart Grid
**2012**, 3, 1812–1821. [Google Scholar] [CrossRef] - Pipattanasomporn, M.; Kuzlu, M.; Rahman, S. An algorithm for intelligent home energy management and demand response analysis. IEEE Trans. Smart Grid
**2012**, 3, 2166–2173. [Google Scholar] [CrossRef] - Chavali, P.; Yang, P.; Nehorai, A. A distributed algorithm of appliance scheduling for home energy management system. IEEE Trans. Smart Grid
**2014**, 5, 282–290. [Google Scholar] [CrossRef] - Pourmousavi, S.A.; Patrick, S.N.; Nehrir, M.H. Real-time demand response through aggregate electric water heaters for load shifting and balancing wind generation. IEEE Trans. Smart Grid
**2014**, 5, 769–778. [Google Scholar] [CrossRef] - Salinas, S.; Li, M.; Li, P. Multi-objective optimal energy consumption scheduling in smart grids. IEEE Trans. Smart Grid
**2013**, 4, 341–348. [Google Scholar] [CrossRef] - Tan, Z.; Yang, P.; Nehorai, A. An optimal and distributed demand response strategy with electric vehicles in the smart grid. IEEE Trans. Smart Grid
**2014**, 5, 861–869. [Google Scholar] [CrossRef] - Chang-Chien, L.R.; An, L.N.; Lin, T.W.; Lee, W.J. Incorporating demand response with spinning reserve to realize an adaptive frequency restoration plan for system contingencies. IEEE Trans. Smart Grid
**2012**, 3, 1145–1153. [Google Scholar] [CrossRef] - Abdollahi, A.; Moghaddam, M.P.; Rashidinejad, M.; Sheikh-El-Eslami, M.K. Investigation of economic and environmental-driven demand response measures incorporating UC. IEEE Trans. Smart Grid
**2012**, 3, 12–25. [Google Scholar] [CrossRef] - Zhao, C.Y.; Wang, J.H.; Watson, J.P.; Guan, Y. Multi-stage robust unit commitment considering wind and demand response uncertainties. IEEE Trans. Power Syst.
**2013**, 28, 2708–2717. [Google Scholar] [CrossRef] - Zhao, C.Y.; Guan, Y.P. Unified stochastic and robust unit commitment. IEEE Trans. Power Syst.
**2013**, 28, 3353–3361. [Google Scholar] [CrossRef] - Kuznetsova, E.; Li, Y.F.; Ruiz, C.; Zio, E. An integrated framework of agent-based modelling and robust optimization for microgrid energy management. Appl. Energy
**2014**, 129, 70–88. [Google Scholar] [CrossRef] - Mena, R.; Hennebel, M.; Li, Y.F.; Zio, E. Self-adaptable hierarchical clustering analysis and differential evolution for optimal integration of renewable distributed generation. Appl. Energy
**2014**, 133, 388–402. [Google Scholar] [CrossRef] - Hossain, M.J.; Saha, T.K.; Mithulananthan, N.; Pota, H.R. Robust control strategy for PV system integration in distribution systems. Appl. Energy
**2012**, 99, 355–362. [Google Scholar] [CrossRef] - Wu, Z.; Tazvinga, H.; Xi, X. Demand side management of photovoltaic-battery hybrid system. Appl. Energy
**2015**, 148, 294–304. [Google Scholar] [CrossRef] - Azizipanah-Abarghooee, R.; Niknam, T.; Bina, M.A.; Zare, M. Coordination of combined heat and power-thermal-wind photovoltaic units in economic load dispatch using chance constrained and jointly distributed random variables methods. Energy
**2015**, 79, 50–67. [Google Scholar] [CrossRef] - Niknam, T.; Golestaneh, F.; Malekpour, A. Probabilistic energy and operation management of a microgrid containing wind/photovoltaic/fuel cell generation and energy storage devices based on point estimate method and self-adaptive gravitational search algorithm. Energy
**2012**, 43, 427–437. [Google Scholar] [CrossRef] - Mohammadi, S.; Mozafari, B.; Solimani, S.; Niknam, T. An adaptive modified fire fly optimisation algorithm based on Hong’s point estimate method to optimal operation management in a microgrid with consideration of uncertainties. Energy
**2013**, 51, 339–348. [Google Scholar] [CrossRef] - Niknam, T.; Golestaneh, F.; Shafiei, M. Probabilistic energy management of a renewable microgrid with hydrogen storage using self-adaptive charge search algorithm. Energy
**2013**, 49, 252–267. [Google Scholar] [CrossRef] - Baziar, A.; Kavousi-Fard, A. Considering uncertainty in the optimal energy management of renewable micro-grids including storage devices. Renew. Energy
**2013**, 59, 158–166. [Google Scholar] [CrossRef] - Adi, V.S.K.; Chang, C.T. Development of flexible designs for PVFC hybrid power systems. Renew. Energy
**2015**, 74, 176–186. [Google Scholar] [CrossRef] - Alsayed, M.; Cacciato, M.; Scarcella, G.; Scelba, G. Design of hybrid power generation systems based on multi criteria decision analysis. Sol. Energy
**2014**, 105, 548–560. [Google Scholar] [CrossRef] - Dufo-López, R.; Bernal-Agustín, J.L. Design and control strategies of PV-Diesel systems using genetic algorithms. Sol. Energy
**2005**, 79, 33–46. [Google Scholar] [CrossRef] - Wood, A.J.; Wollenberg, B.F. Power Generation, Operation and Control, 2nd ed.; John Wiley & Son, Inc.: New York, NY, USA, 1996. [Google Scholar]
- Taha, H.A. Integer Programming: Theory, Applications, and Computations; Academic Press: New York, NY, USA, 1975. [Google Scholar]
- Swarup, K.S.; Yamashiro, S. Unit commitment solution methodology using genetic algorithm. IEEE Trans. Power Syst.
**2002**, 17, 87–91. [Google Scholar] [CrossRef] - Gen, M.; Cheng, R. Genetic Algorithms and Engineering Design; John Wiley & Sons: New York, NY, USA, 1997. [Google Scholar]
- Suresh, S.; Huang, H.; Kim, H.J. Hybrid real-coded genetic algorithm for data partitioning in multi-round load distribution and scheduling in heterogeneous systems. Appl. Soft Comput.
**2014**, 24, 500–510. [Google Scholar] [CrossRef] - Karakatič, S.; Podgorelec, V. A survey of genetic algorithms for solving multi depot vehicle routing problem. Appl. Soft Comput.
**2015**, 27, 519–532. [Google Scholar] [CrossRef] - Malekpour, A.R.; Niknam, T. A probabilistic multi-objective daily Volt/Var control at distribution networks including renewable energy sources. Energy
**2011**, 36, 3477–3488. [Google Scholar] [CrossRef] - Aien, M.; Fotuhi-Firuzabad, M.; Rashidinejad, M. Probabilistic optimal power flow in correlated hybrid wind–photovoltaic power systems. IEEE Trans. Smart Grid
**2014**, 5, 130–138. [Google Scholar] [CrossRef] - The Math Works. Optimization Toolbox-Fmincon; MATLAB, The Math Works: Natick, MA, USA, 2009. [Google Scholar]
- Ela, E.; Diakov, V.; Ibanez, E.; Heaney, M. Impacts of Variability and Uncertainty in Solar Photovoltaic Generation at Multiple Timescales; National Renewable Energy Laboratory: Golden, CO, USA, 2013; TP-5500-58274.

**Figure 2.**Identification of overlapping hours in which states of both chromosomes are on and resulting proto-offspring.

**Figure 3.**(

**a**) Insertion of two on-periods and resulting proto-offspring; and (

**b**) trimming redundant slot.

**Figure 5.**(

**a**) Hourly on (grey) and off (white) states of an elastic load over 24 h; and (

**b**) new encoding corresponding to that in 5a.

**Figure 11.**Mean values and standard deviations of PV power at different buses: (

**a**) bus 7; (

**b**) bus 8; (

**c**) bus 9; and (

**d**) bus 10.

**Figure 12.**(

**a**) Probability; and (

**b**) cumulative probability with respect to the percentage of difference between the best and worst costs.

Bus No. | ${\mathbf{P}}_{\mathbf{m}\mathbf{i}\mathbf{n}}$ ($\mathbf{M}\mathbf{W}$) | ${\mathbf{P}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ ($\mathbf{M}\mathbf{W}$) | Cost Coefficients | ||
---|---|---|---|---|---|

${\mathbf{a}}_{\mathit{i}}$ ($\mathbf{\$}/\mathbf{h}$) | ${\mathbf{b}}_{\mathit{i}}$ ($\mathbf{\$}/\mathbf{M}\mathbf{W}\mathbf{h}$) | ${\mathbf{c}}_{\mathit{i}}$ ($/$\mathbf{M}\mathbf{W}{\mathbf{h}}^{2}$) | |||

3 | 0.5 | 4 | 151.28 | 87.87 | 0.14 |

4 | 0.6 | 3.5 | 125.21 | 67.82 | 0.65 |

5 | 0.9 | 4.5 | 89.21 | 31.37 | 1.1 |

6 | 1.1 | 3 | 35.48 | 17.6 | 0.1416 |

Bus No. | Ramp Rate (MW/h) | Start Up Cost ($) | Minimum Up Time, ${\mathit{M}}_{\mathit{i}}^{\mathit{o}\mathit{n}}$ (h) | Minimum Down Time, ${\mathit{M}}_{\mathit{i}}^{\mathit{o}\mathit{f}\mathit{f}}$ (h) |
---|---|---|---|---|

3 | 2.5 | 161.84 | 4 | 2 |

4 | 2.0 | 122.53 | 3 | 2 |

5 | 3.5 | 175.34 | 3 | 1 |

6 | 2.0 | 148.63 | 4 | 2 |

Periods | Prices $\left(\mathbf{\$}/\mathbf{M}\mathbf{W}\mathbf{h}\right)$ |
---|---|

0:00 a.m.–8:59 a.m. | 65 |

9:00 a.m.–8:59 p.m. | 131 |

9:00 p.m.–11:59 p.m. | 65 |

$\mathbf{B}\mathbf{u}\mathbf{s}$ | Must-Run Hours | Maximum Loads (MW) |
---|---|---|

6 | 10 | 7.5 |

16 | 6 | 2.0 |

18 | 7 | 2.5 |

20 | 6 | 2.5 |

26 | 5 | 2.5 |

**Table 5.**(

**a**) Expected values of real power injections (MW) during 1:00 a.m.–12:00 p.m. (

**b**) Expected values of real power injections (MW) during 13:00 p.m.–24:00 a.m.

(a) | ||||||||||||

t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

${P}_{sw}\left(t\right)$ | 15.38 | 13.44 | 14.19 | 12.57 | 12.33 | 5.31 | 1.10 | −1.62 | −6.05 | −6.89 | −6.41 | −6.32 |

${P}_{3}\left(t\right)$ | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 3.00 | 4.00 | 4.00 | 4.00 |

${P}_{4}\left(t\right)$ | 1.15 | 3.15 | 2.05 | 3.50 | 3.50 | 1.50 | 0.60 | 0.60 | 2.60 | 3.50 | 3.50 | 3.50 |

${P}_{5}\left(t\right)$ | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 |

${P}_{6}\left(t\right)$ | 1.59 | 1.10 | 1.16 | 1.10 | 1.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

(b) | ||||||||||||

t | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

${P}_{sw}\left(t\right)$ | −6.45 | −6.53 | 1.73 | 2.04 | 2.23 | 2.25 | 2.27 | −5.29 | −0.54 | 5.10 | 5.01 | 4.88 |

${P}_{3}\left(t\right)$ | 4.00 | 4.00 | 4.00 | 4.00 | 4.00 | 4.00 | 4.00 | 4.00 | 1.50 | 0.50 | 0.50 | 0.50 |

${P}_{4}\left(t\right)$ | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 1.50 | 1.27 | 1.17 | 1.05 |

${P}_{5}\left(t\right)$ | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 0.00 | 0.00 | 0.00 |

${P}_{6}\left(t\right)$ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8–14 | 15 | 16 | 17 | 18 | 19 | 20–24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${P}_{el}^{6}\left(t\right)$ | 7.5 | 7.5 | 7.5 | 7.5 | 7.5 | 0.0 | 0.0 | 0.0 | 7.5 | 7.5 | 7.5 | 7.5 | 7.5 | 0.0 |

${P}_{el}^{16}\left(t\right)$ | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${P}_{el}^{18}\left(t\right)$ | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${P}_{el}^{20}\left(t\right)$ | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${P}_{el}^{26}\left(t\right)$ | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

**Table 7.**(

**a**) Real power injections (MW) during 1:00 a.m.−12:00 p.m. (

**b**) Real power injections (MW) during 13:00 p.m.−24:00 a.m.

(a) | ||||||||||||

t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

${P}_{sw}\left(t\right)$ | 16.37 | 15.39 | 14.50 | 13.67 | 13.90 | 1.06 | −1.89 | −3.25 | −3.92 | −3.18 | −2.92 | −2.81 |

${P}_{3}\left(t\right)$ | 0.50 | 0.50 | 0.50 | 0.50 | 0.50 | 1.58 | 1.08 | 0.58 | 1.02 | 1.51 | 1.97 | 2.47 |

${P}_{4}\left(t\right)$ | 0.06 | 1.20 | 1.80 | 2.40 | 3.00 | 3.50 | 3.48 | 2.88 | 3.48 | 3.50 | 3.42 | 3.50 |

${P}_{5}\left(t\right)$ | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.02 |

${P}_{6}\left(t\right)$ | 1.14 | 1.10 | 1.10 | 1.10 | 0.0 | 1.16 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

(b) | ||||||||||||

t | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

${P}_{sw}\left(t\right)$ | −4.21 | −5.01 | −4.35 | −4.28 | 3.12 | 2.83 | 2.27 | 6.21 | 6.47 | −0.51 | −0.10 | 0.25 |

${P}_{3}\left(t\right)$ | 3.31 | 4.00 | 4.00 | 4.00 | 4.00 | 4.00 | 4.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

${P}_{4}\left(t\right)$ | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 3.50 | 3.48 | 2.88 | 2.28 | 1.68 |

${P}_{5}\left(t\right)$ | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 | 4.50 |

${P}_{6}\left(t\right)$ | 1.10 | 1.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8–16 | 17 | 18 | 19 | 20 | 21 | 22–24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${P}_{el}^{6}\left(t\right)$ | 7.5 | 7.5 | 7.5 | 7.5 | 7.5 | 0.0 | 0.0 | 0.0 | 7.5 | 7.5 | 7.5 | 7.5 | 7.5 | 0.0 |

${P}_{el}^{16}\left(t\right)$ | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${P}_{el}^{18}\left(t\right)$ | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${P}_{el}^{20}\left(t\right)$ | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${P}_{el}^{26}\left(t\right)$ | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Impact Factors | Descriptions | Expected Values of Total Cost ($) |
---|---|---|

Electricity Tariffs | Peak hour: 131$/MWh; off-peak hour: 30$/MWh | 19,028.80 |

As shown in Table 5 | 23,737.68 | |

Peak hour: 131$/MWh; off-peak hour: 100$/MWh | 25,637.07 | |

Minimum Up/Minimum Down Times $({M}_{i}^{on},\text{}{M}_{i}^{off})$ | All are 1 h at all buses | 21,879.39 |

As shown in Table 2 | 23,737.68 | |

(8,4), (8,3), (7,3), (7,4) hours at buses 3, 4, 5, 6. | 25,845.03 | |

Ramp Rates $({R}_{i}^{up}$ = ${R}_{i}^{down}$, MW/h) | 0.5, 0.6, 0.7, 0.8 MW/h at buses 3, 4, 5, 6. | 23,781.71 |

As shown in Table 2 | 23,737.68 | |

4, 3.5, 4.5, 3 MW/h at buses 3, 4, 5, 6. | 23,346.11 | |

Must-run Hours in Production Lines | 7, 3, 4, 3, 2 h at buses 6, 16, 18, 20, 26 | 18,254.53 |

As shown in Table 6 | 23,737.68 | |

15, 11, 12, 11, 10 h at buses 6, 16, 18, 20, 26 | 33,448.68 |

Population Size | Expected Cost ($) | CPU Time (s) | No. of Iterations |
---|---|---|---|

50 | 31345.96 | 25.72 | 15 |

75 | 31300.69 | 54.32 | 23 |

100 | 31321.59 | 56.07 | 20 |

Standard Deviation | Expected Cost ($) | CPU Time (s) | No. of Iterations |
---|---|---|---|

3% | 23,737.68 | 106.99 | 46 |

8% | 23,028.07 | 119.70 | 46 |

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## Share and Cite

**MDPI and ACS Style**

Hong, Y.-Y.; Yo, P.-S.
Novel Genetic Algorithm-Based Energy Management in a Factory Power System Considering Uncertain Photovoltaic Energies. *Appl. Sci.* **2017**, *7*, 438.
https://doi.org/10.3390/app7050438

**AMA Style**

Hong Y-Y, Yo P-S.
Novel Genetic Algorithm-Based Energy Management in a Factory Power System Considering Uncertain Photovoltaic Energies. *Applied Sciences*. 2017; 7(5):438.
https://doi.org/10.3390/app7050438

**Chicago/Turabian Style**

Hong, Ying-Yi, and Po-Sheng Yo.
2017. "Novel Genetic Algorithm-Based Energy Management in a Factory Power System Considering Uncertain Photovoltaic Energies" *Applied Sciences* 7, no. 5: 438.
https://doi.org/10.3390/app7050438