# Optimal Scheduling of Domestic Appliances via MILP

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Demand-Side Consumption Scheduling

#### 2.1. System Description

_{h,i,j,k}= 1, which indicates that the i − th appliance in the h − th dwelling processes its j − th phase in the time slot k). The model constraints are organized into several groups: power constraints, timing constraints and user constraints.

Symbol | Description |
---|---|

Indices and model size | |

h | Dwelling index |

i | Appliance index |

j | Energy phase index |

k | Time slot index |

H | Number of dwellings |

I | Number of appliances |

J | Maximal number of power phases defined for each appliance |

K | Number of scheduled time slots |

Decision variables | |

x_{h,i,j,k} | Binary variable; one if the phase j in the appliance i in the dwelling h is running in the time slot k |

y_{h,i,j,k} | Binary variable; one if the phase j in the appliance i in the dwelling h is already finished in the time slot k |

z_{h,i,j,k} | Binary variable; one if the phase j in the appliance i in the dwelling h is waiting for run in the time slot k |

Constants (let Ψ = H · I) | |

PT | Processing time of all appliance classes and power phases (Ψ × J ) |

PC | Power consumed by all appliance classes during the power phases (Ψ × J ) |

PP | Peak power of all appliance classes during the power phases (Ψ × J ) |

PD | Maximum allowed waiting time for the next phase if the previous one is finished for all appliances and their phases (Ψ × J ) |

AH | Matrix of the presence of appliances in the dwellings ( H × I). Note that this matrix determines the instantiation of the appliance classes to concrete dwellings. |

UP | User preference matrix (Ψ × K). A value of one states that the appliance i in the dwelling h can be scheduled in the time slot k |

UC | Appliances consequence matrix (Ψ × Ψ). A value of one states that the appliance i_{1} in the dwelling h_{1} must finish its cycle before the appliance i_{2} in the dwelling h_{2} starts. |

Θ | Energy price vector |

P_{max} | Peak power |

P_{base} | Vector of the household base load caused by unmanageable devices |

#### 2.2. Consumption Scheduling Optimization

_{max}− P

_{base}value, which represents the non-schedulable consumption. Equation (2) then allows enabling or disabling a certain appliance in a certain dwelling.

_{i}

_{1}

_{,i}

_{2}= 1 for any two appliances i

_{1}and i

_{2}, then (for every time slot k) the appliance i

_{2}cannot perform its first power phase if the last power phase of the appliance i

_{1}has not been finished yet. Moreover, for all i

_{1}and i

_{2}, there must hold UC

_{i}

_{1}

_{,i}

_{2}+ UC

_{i}

_{2}

_{,i}

_{1}≤ 1.

#### 2.3. Objective Function

#### 2.4. Problem Formulation

## 3. Case Study

**Table 2.**Presence of the appliances (washing machines (WM), dishwashers (DW), tumble dryers (TD), electronic water heaters (EWH), electric ovens (EO) and home lighting subsystems (HL)) in the dwellings.

Identifier | Saturation | Dwelling | Cycles per year | |||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |||

WM | 0.97 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | 220 |

DW | 0.66 | ✓ | ✓ | ✓ | ✓ | 240 | ||

TD | 0.42 | ✓ | ✓ | 147 | ||||

EWH | 0.50 | ✓ | ✓ | ✓ | 328 | |||

EO | 0.85 | ✓ | ✓ | ✓ | ✓ | ✓ | 182 | |

HL | 1.00 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | 330 |

ID | Phase length | Energy per phase/kWh | Peak per phase/kW |
---|---|---|---|

WM | 5 × 1 | 0.15, 0.29, 0.03, 0.03, 0.02 | 2.12, 2.12, 0.28, 0.26, 0.18 |

DW | 6 × 1 | 0.38, 0.28, 0.11, 0.43, 0.01, 0.01 | 2.23, 2.12, 2.09, 2.07, 0.01, 0.01 |

TD | 6 × 1 | 0.15, 0.20, 0.20, 0.20, 0.17, 0.01 | 2.20, 2.20, 2.20, 2.20, 2.20, 0.10 |

EWH | 9 × 2 | 2
×
[0.90, 0.90, 0.85, 0.85] ^{1}, 0.81 | 2 × [1.80, 1.80, 1.75, 1.75], 1.80 |

EO | 6 × 1 | 0.44, 0.24, 0.17, 0.15, 0.27, 0.01 | 1.84, 1.88, 1.88, 1.89, 2.18, 0.01 |

HL | 4 × 8, 4 | 0.01, 0.02, 0.05, 0.02, 0.01 | 0.04, 0.08, 0.19, 0.06, 0.04 |

^{1}The section in brackets is repeated twice.

**Figure 1.**(

**a**) Base load; (

**b**) pricing tariffs; (

**c**) real DW and WM profiles; (

**d**) discrete form of DW peak and energy.

## 4. Results

Dwelling | Absolute price (CZK) | Relative cuts (%) | ||||
---|---|---|---|---|---|---|

S1 | S2 | S3 | S1→S2 | S2→S3 | S1→S3 | |

1 | 105.4 | 68.6 | 58.1 | 34.9% | 15.3% | 44.8% |

2 | 33.6 | 32.1 | 29.4 | 4.7% | 8.2% | 12.5% |

3 | 100.5 | 65.0 | 62.4 | 35.3% | 4.0% | 37.9% |

4 | 34.8 | 33.1 | 29.4 | 5.1% | 11.1% | 15.6% |

5 | 91.5 | 57.1 | 55.0 | 37.7% | 3.7% | 39.9% |

6 | 33.3 | 30.7 | 25.9 | 7.9% | 15.6% | 22.2% |

#### 4.1. Scheduling Algorithm Verification

Identifier | Type | W/O scheduling | With scheduling |
---|---|---|---|

WM | Shiftable | 16:00 to 21:00 | 8:00 to 16:00 |

DW | Shiftable | 18:00 to 22:00 | 8:00 to 16:00 |

TD | Shiftable | 18:00 to 22:00 | 8:00 to 16:00, after WM cycle |

EWH | Thermostatic | starts at 20:00 | 0:00 to 23:59 |

EO | Non-shiftable | 19:00 to 20:30 | 18:00 to 19:30 ^{1} |

HL | Non-shiftable | 06:00 to 08:30 | 16:00 to 23:00 ^{2} |

^{1}The preferences of electric oven users allow only a single running time a day;

^{2}To observe the MILP problem formulation, the home lighting system is modeled as an appliance, which starts at 6:00; the two peak periods during the day are modeled as two phases with an exact interphase delay time.

Instance | Tariff | Constraints | Total price | Max. peak | Solving time |
---|---|---|---|---|---|

C1 | Tariff 2 | No | 290 CZK | 34 kW | 18 s |

C2 | Tariff 2 | 7360 Watts | 325 CZK | 14 kW (7 kW) | 80 s |

## 5. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Bradac, Z.; Kaczmarczyk, V.; Fiedler, P. Optimal Scheduling of Domestic Appliances via MILP. *Energies* **2015**, *8*, 217-232.
https://doi.org/10.3390/en8010217

**AMA Style**

Bradac Z, Kaczmarczyk V, Fiedler P. Optimal Scheduling of Domestic Appliances via MILP. *Energies*. 2015; 8(1):217-232.
https://doi.org/10.3390/en8010217

**Chicago/Turabian Style**

Bradac, Zdenek, Vaclav Kaczmarczyk, and Petr Fiedler. 2015. "Optimal Scheduling of Domestic Appliances via MILP" *Energies* 8, no. 1: 217-232.
https://doi.org/10.3390/en8010217