# Robust Optimization for Household Load Scheduling with Uncertain Parameters

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Two typical examples of uncertain parameters, outdoor temperature and hot water demand, are modelled as uncertainty sets, based on which household load scheduling problem with uncertainties is formulated. For researching the uncertainties, the uncertain parameters are presented in the form of interval numbers.
- A robust optimization method is applied to deal with the uncertainties in the comfort constraints. The robust counterpart transformation is the key component of robust optimization. Via deducing the robust counterpart, the original scheduling problem is transformed into a mixed integer linear programming problem, of which the global optimum can be found by mature tools, such as CPLEX. The proposed method avoids the time-consuming iterations in many other uncertain optimization methods.
- An uncertainty analysis that quantifies the violation degree of the comfort constraints is designed and conducted to test the proposed method. The results show that the complete robust schedules is able to guarantee that the comfort constraints will not be violated. Moreover, schedules with different robust levels can be obtained to make a trade-off between the comfort violation and electricity payment.

## 2. Mathematical Model

#### 2.1. Uncertain Parameters

#### 2.2. Objective Function

#### 2.3. Constraints

#### 2.3.1. Loads with Uncertain Parameters

^{6}; the constant C is the specific heat capacity of water, being 4.2 × 10

^{3}; M is the mass of the water in full tank; and ${\theta}_{water,1}$ is the initial water temperature in tank, which is also given before scheduling.

#### 2.3.2. Interruptible Loads

#### 2.3.3. Uninterruptible Loads

#### 2.3.4. Energy Storage Device

## 3. Robust Optimization for Household Load Scheduling

#### 3.1. Robust Optimization

**A**is uncertain, it should be replaced by ${\tilde{a}}_{ij}$ that takes value in $\left[{\overline{a}}_{ij}-\text{\hspace{0.17em}}{\widehat{a}}_{ij},{\overline{a}}_{ij}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\widehat{a}}_{ij}\right]$ of which ${\overline{a}}_{ij}$ and ${\widehat{a}}_{ij}$ are the mean value and range of the coefficient (the elements in $\mathit{b}$ can be seen as ${a}_{ij}$ through some transformations [12,13]). A new parameter is then defined as${z}_{ij}=\left({\tilde{a}}_{ij}-{\overline{a}}_{ij}\right)/{\widehat{a}}_{ij}$, so ${z}_{ij}$ is a bounded random variable (but with unknown distribution) which is within [−1,1].

#### 3.2. Robust Counterpart Transformation

#### 3.2.1. Robust Counterpart Transformation of the Uncertain Outdoor Temperature

**A**. Thus, introducing a new variable ${x}_{AC,N+1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ makes this uncertain parameter classified into matrix

**A**. To simplify the proof, set ${k}_{1}={e}^{-\Delta t/RC},{k}_{2}=(1-{e}^{-\Delta t/RC})$. Then the matrix form for constraints is as follows:

#### 3.2.2. Robust Counterpart Transformation of Uncertain Water Demand

**A**, there is another one uncertain parameter which do not appear in matrix

**A**in each constraint. Thus, introducing the new variable ${x}_{EWH,N+1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ makes this uncertain parameter classified into matrix

**A**. To simplify the proof, set $k=\rho /CM,r=({\theta}_{water,1}-{\theta}_{cold})>0$. Then the matrix form for constraints is as follows:

#### 3.3. Solution to the Robust Counterpart

## 4. Simulation Result

#### 4.1. Simulation Design

#### 4.2. The Impact of the Uncertainties

#### 4.3. Complete Robust Schedules

#### 4.4. Schedules with Different Robust Levels

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Siano, P. Demand response and smart grids—A survey. Renew. Sustain. Energy Rev.
**2014**, 30, 461–478. [Google Scholar] [CrossRef] - Datchanamoorthy, S.; Kumar, S.; Ozturk, Y.; Lee, G. Optimal time-of-use pricing for residential load control. In Proceedings of the 2011 IEEE International Conference on Smart Grid Communications, Brussels, Belgium, 17–20 October 2011; pp. 375–380. [Google Scholar]
- Fallah, S.N.; Deo, R.C.; Shojafar, M.; Conti, M.; Shamshirband, S. Computational Intelligence Approaches for Energy Load Forecasting in Smart Energy Management Grids: State of the Art, Future Challenges, and Research Directions. Energies
**2018**, 11, 596. [Google Scholar] [CrossRef] - Beaudin, M.; Zareipour, H. Home energy management systems: A review of modelling and complexity. Renew. Sustain. Energy Rev.
**2015**, 45, 318–335. [Google Scholar] [CrossRef] - Zhao, Z.; Lee, W.C.; Shin, Y.; Song, K.B. An optimal power scheduling method for demand response in home energy management system. IEEE Trans. Smart Grid
**2013**, 4, 1391–1400. [Google Scholar] [CrossRef] - Sun, H.C.; Huang, Y.C. Optimization of power scheduling for energy management in smart homes. Procedia Eng.
**2012**, 38, 1822–1827. [Google Scholar] [CrossRef] - Paterakis, N.G.; Erdinc, O.; Bakirtzis, A.G.; Catalão, J.P. Optimal household appliances scheduling under day-ahead pricing and load-shaping demand response strategies. IEEE Trans. Ind. Inform.
**2015**, 11, 1509–1519. [Google Scholar] [CrossRef] - Liu, X.; Ivanescu, L.; Kang, R.; Maier, M. Real-time household load priority scheduling algorithm based on prediction of renewable source availability. IEEE Trans. Consum. Electron.
**2012**, 58, 318–326. [Google Scholar] - Soares, A.; Antunes, C.H.; Oliveira, C.; Gomes, Á. A multi-objective genetic approach to domestic load scheduling in an energy management system. Energy
**2014**, 77, 144–152. [Google Scholar] [CrossRef] - Bouzerdoum, M.; Mellit, A.; Pavan, A.M. A hybrid model (SARIMA–SVM) for short-term power forecasting of a small-scale grid-connected photovoltaic plant. Sol. Energy
**2013**, 98, 226–235. [Google Scholar] [CrossRef] - Weron, R. Electricity price forecasting: A review of the state-of-the-art with a look into the future. Int. J. Forecast.
**2014**, 30, 1030–1081. [Google Scholar] [CrossRef] - Chen, X.; Wei, T.; Hu, S. Uncertainty-aware household appliance scheduling considering dynamic electricity pricing in smart home. IEEE Trans. Smart Grid
**2013**, 4, 932–941. [Google Scholar] [CrossRef] - Hong, Y.Y.; Lin, J.K.; Wu, C.P.; Chuang, C.C.C. Multi-objective air-conditioning control considering fuzzy parameters using immune clonal selection programming. IEEE Trans. Smart Grid
**2012**, 3, 1603–1610. [Google Scholar] [CrossRef] - Huang, Y.; Wang, L.; Guo, W.; Kang, Q.; Wu, Q. Chance Constrained Optimization in a Home Energy Management System. IEEE Trans. Smart Grid
**2016**, 99, 1–9. [Google Scholar] [CrossRef] - Soyster, A.L. Technical note-convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res.
**1973**, 21, 1154–1157. [Google Scholar] [CrossRef] - Ben-Tal, A.; Goryashko, A.; Guslitzer, E.; Nemirovski, A. Adjustable robust solutions of uncertain linear programs. Math. Program.
**2004**, 99, 351–376. [Google Scholar] [CrossRef] - Bertsimas, D.; Sim, M. The price of robustness. Oper. Res.
**2004**, 52, 35–53. [Google Scholar] [CrossRef] - Bertsimas, D.; Sim, M. Robust discrete optimization and network flows. Math. Program.
**2003**, 98, 49–71. [Google Scholar] [CrossRef] - Wang, C.; Zhou, Y.; Jiao, B.; Wang, Y.; Liu, W.; Wang, D. Robust optimization for load scheduling of a smart home with photovoltaic system. Energy Convers. Manag.
**2015**, 102, 247–257. [Google Scholar] [CrossRef] - Chen, Z.; Wu, L.; Fu, Y. Real-time price-based demand response management for residential appliances via stochastic optimization and robust optimization. IEEE Trans. Smart Grid
**2012**, 3, 1822–1831. [Google Scholar] [CrossRef] - Conejo, A.J.; Morales, J.M.; Baringo, L. Real-time demand response model. IEEE Trans. Smart Grid
**2010**, 1, 236–242. [Google Scholar] [CrossRef] - Wang, J.; Li, Y.; Zhou, Y. Interval number optimization for household load scheduling with uncertainty. Energy Build.
**2016**, 130, 613–624. [Google Scholar] [CrossRef] - Wang, C.; Zhou, Y.; Wu, J.; Wang, J.; Zhang, Y.; Wang, D. Robust-index method for household load scheduling considering uncertainties of customer behavior. IEEE Trans. Smart Grid
**2015**, 6, 1806–1818. [Google Scholar] [CrossRef] - Pedrasa, M.A.A.; Spooner, T.D.; MacGill, I.F. Coordinated scheduling of residential distributed energy resources to optimize smart home energy services. IEEE Trans. Smart Grid
**2010**, 1, 134–143. [Google Scholar] [CrossRef] - Du, P.; Lu, N. Appliance commitment for household load scheduling. IEEE Trans. Smart Grid
**2011**, 2, 411–419. [Google Scholar] [CrossRef] - Alrumayh, O.; Bhattacharya, K. Model predictive control based home energy management system in smart grid. In Proceedings of the 2015 IEEE Electrical Power and Energy Conference, London, ON, Canada, 26–28 October 2015; pp. 152–157. [Google Scholar]
- 2012 ASHRAE Handbook—HVAC Systems and Equipment Amer. Soc. Heating, Refrigerating and Air-Conditioning Eng., Inc., 2012. Available online: http://app.knovel.com/web/search.v?q=2012%20ASHRA.E%20Handbook&my_subscription=FALSE&search_type=tech-reference (accessed on 10 October 2012).

**Figure 1.**(

**a**) Power output of the photovoltaic generation and power consumption of the uncontrollable loads for the next day; and (

**b**) the real-time price for the next day.

**Figure 2.**(

**a**) The uncertainty set of the outdoor temperature for the next day; and (

**b**) the uncertainty set of the hot water demand for the next day.

**Figure 3.**The schedules of household appliances (without considering the uncertainties). CW: clothes washers; EV: electric vehicles; CD: clothes dryer; DW: dishwasher; WH: water heater; AC: air conditioner.

**Figure 4.**(

**a**) The actual temperature set of the air in room; and (

**b**) the actual temperature set of the hot water in tank.

**Figure 5.**(

**a**) The actual temperature range of the air in room (under the complete robust schedules); and (

**b**) the actual temperature range of the hot water in tank (under the complete robust schedules).

Appliance | ${\mathit{b}}_{\mathit{I}\mathit{L}}$ | ${\mathit{e}}_{\mathit{I}\mathit{L}}$ | ${\mathit{l}}_{\mathit{I}\mathit{L}}$ (h) | ${\mathit{p}}_{\mathit{I}\mathit{L}}^{\mathit{r}}$ (KWh) |
---|---|---|---|---|

Cloth Washer (CW) | 7:00 | 17:00 | 3 | 1 |

Electrical Vehicle (EV) | 0:00 | 8:00 | 4 | 2.5 |

Appliance | ${\mathit{b}}_{\mathit{U}\mathit{I}\mathit{L}}$ | ${\mathit{e}}_{\mathit{U}\mathit{I}\mathit{L}}$ | ${\mathit{l}}_{\mathit{U}\mathit{I}\mathit{L}}$ (h) | ${\mathit{p}}_{\mathit{U}\mathit{I}\mathit{L}}^{\mathit{r}}$ (KWh) |
---|---|---|---|---|

Cloth Dryer (CD) | 12:00 | 22:00 | 2 | 3 |

Dish Washer (DW) | 12:00 | 20:00 | 3 | 0.8 |

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

${\eta}_{ch}$ | 0.95 | ${p}_{\mathrm{max}}^{ch}\text{}\left(\mathrm{kW}\right)$ | 2 | $SO{C}_{\mathrm{min}}$ | 0.1 |

${\eta}_{dch}$ | 0.95 | ${p}_{\mathrm{max}}^{dch}\text{}\left(\mathrm{kW}\right)$ | 2 | $SO{C}_{\mathrm{max}}$ | 0.9 |

$\epsilon \text{}(\mathrm{kWh}/\mathrm{h})$ | 0.004 | $SO{C}_{ini}$ | 0.5 | - | - |

Robust Level $\mathit{\alpha}$ | Violation Rate for the Comfort Constraints of AC | Violation Rate for the Comfort Constraints of WH | Electricity Bill/Economy (Yuan) |
---|---|---|---|

1 | 0 | 0 | 12.75 |

0.8 | 0.003 | 0.008 | 12.48 |

0.6 | 0.024 | 0.075 | 12.20 |

0.4 | 0.062 | 0.113 | 11.70 |

0.2 | 0.181 | 0.643 | 11.11 |

0 | 0.517 | 1.000 | 9.81 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, J.; Li, P.; Fang, K.; Zhou, Y.
Robust Optimization for Household Load Scheduling with Uncertain Parameters. *Appl. Sci.* **2018**, *8*, 575.
https://doi.org/10.3390/app8040575

**AMA Style**

Wang J, Li P, Fang K, Zhou Y.
Robust Optimization for Household Load Scheduling with Uncertain Parameters. *Applied Sciences*. 2018; 8(4):575.
https://doi.org/10.3390/app8040575

**Chicago/Turabian Style**

Wang, Jidong, Peng Li, Kaijie Fang, and Yue Zhou.
2018. "Robust Optimization for Household Load Scheduling with Uncertain Parameters" *Applied Sciences* 8, no. 4: 575.
https://doi.org/10.3390/app8040575