# Robust Optimization for Household Load Scheduling with Uncertain Parameters

^{1}

^{2}

^{3}

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## 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

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**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 |

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**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