# Modeling and Simulation of Household Appliances Power Consumption

^{*}

## Abstract

**:**

## Featured Application

**The method presented in this paper serves to predict the consumption of household appliances by modeling their behavior and by simulating accordingly.**

## Abstract

^{2}test). To test the veracity of the evaluations, first, a set of random values was simulated for each hour, and their respective averages were calculated. These were compared with the averages of the real values for each hour. With the exception of HVAC during working days, great results were obtained. For the refrigerator, the maximum error was 3.91%, while for the lighting, it was 4.27%. At the point of consumption, the accuracy was even higher, with an error of 1.17% for the dryer while for the washing machine and dishwasher, their minimum errors were less than 1%. The error results confirm that the applied methodology is perfectly acceptable for modeling household appliance consumption and consequently predicting it. However, these consumptions can be only extrapolated to dwellings with similar surface areas and habitats.

## 1. Introduction

^{2}test. To guarantee a correct evaluation, the averages of sets of random values are compared with the averages of real values for each hour, checking that the difference between them is acceptable.

## 2. Methodology

#### 2.1. Continuous Power Consumption Household Appliances

_{α}, the maximum difference allowed according to the level of significance (α) and the type of distribution. D

_{α}, Equation (5), is calculated by checking Table 1 and Table 2 for the values of ${c}_{\alpha}$ and $k\left(n\right)$.

_{α}, the null hypothesis H

_{0}of Normality is accepted so that the corresponding distribution would follow a Normal distribution. 10-min periods whose distribution rejects the null hypothesis of Normality are assessed in order to check if they follow an Exponential distribution with an analogous process to the previous one. The difference is the cumulative distribution function, expressed in Equation (6), where λ is the rate parameter, as well as the different values of c

_{α}and the way k(n) is calculated according to Table 2.

^{2}test [45]. The process begins with the grouping of the data in a number of classes greater or equal to five, in such a way that they cover the whole possible range of values of the variables, and the expected frequency Oi is calculated for each sample.

_{a}and α

_{d}are the skewness and distribution shape factors, respectively.

^{2}, defined in Equation (15), is calculated and compared with X

^{2}

_{α}(k-r-1), where k is the number of classes and r is the number of parameters on which each distribution depends. If X

^{2}is less than or equal to X

^{2}

_{α}(k-r-1) the null hypothesis H

_{0}of the corresponding distribution to be evaluated is accepted.

#### 2.2. Discontinuous Power Consumption Household Appliances

- Simulation of the number of power consumptions of each element, where the probability of each integer value is based on the ratio of the previous count.
- If the above-simulated value is greater than or equal to 1, the duration of each count and its start time are simulated, also based on the data previously collected.
- Simulation of 300 sets of random consumptions according to the duration of the consumptions and their associated distribution, comparing their average value with the average of the real values.
- Calculation of the percentage of error by Equation (16).

^{2}and three occupants, located in Vancouver. The extrapolation of results can be done in homes with an equivalent surface area, an equal number of inhabitants and a similar climatic zone. If these conditions change, the results would be different. Thus, if the surface area was larger, the consumption would increase. If it were a single dwelling, consumption would be lower, on the other hand. However, the methodology proposed in this paper can be applied if the number of consumption values available are significant and correspond to the same appliances that have been evaluated. Moreover, in some cases, such as the HVAC, the climate data influences the models obtained. This fact is not taken into account in this research work but can be of interest for future work.

## 3. Case study

^{2}and three people living in it. Before starting the statistical evaluation, the consumption was grouped according to a 10-min period of power and then separated into three groups, i.e., working days, Saturdays and Sundays. The last step beforehand was to distinguish between continuous and discontinuous consumption, given that the treatment of the latter is more complex.

## 4. Results

^{2}) and the number of inhabitants is the same (three in this case). If these characteristics change, the results would no longer be applicable. However, the methodology can be applied in the same manner and, after a proper validation, the results obtained would be considered valid.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Symbols and Acronyms

μ | Mean |

σ | Standard deviation |

${F}_{O}\left(x\right)$ | Normal cumulative distribution function |

${\widehat{F}}_{n}\left(x\right)$ | Observed cumulative frequency |

${f}_{o}\left(x\right)$ | Probability density function |

${D}^{+}$ | Upper difference between the observed cumulative frequency and normal cumulative Distribution |

${D}^{-}$ | Lower difference between the observed cumulative frequency and normal cumulative distribution |

$D$ | Maximum difference between the observed cumulative frequency and normal cumulative distribution |

${D}_{\alpha}$ | Maximum tabulated difference between the observed cumulative frequency and normal cumulative distribution |

${c}_{\alpha}$ | Coefficient of significance |

$k\left(n\right)$ | Tabulated expression which determines ${D}_{\alpha}$ |

${E}_{i}$ | Expected frequency |

${O}_{i}$ | Observed frequency |

α | Level of significance |

n | Number of samples |

λ | Rate parameter |

β | Scale factor |

α | Shape factor |

μl | Location factor |

λ_{a} | Skewness shape factor |

α_{d} | Distribution shape factor |

X^{2} | Chi square |

X^{2}_{α}(k-r-1) | Tabulated chi square |

DR | Demand Response |

HVAC | Heating, ventilating and air condition |

SVM | Support-vector machines |

POE | Post Occupancy Evaluation |

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**Figure 7.**Random (blue) and real (orange) average consumption of the refrigerator during working days.

**Figure 13.**Random (blue) and real (orange) average consumption of the dryer whose duration is 40 min during working days.

**Figure 14.**Random (blue and yellow) and real (orange and purple) averages consumption of the washing machine, whose durations are 10 and 70 min during working days.

**Figure 15.**Random (blue and purple) and real (red and orange) averages consumption of the dishwasher whose durations are 30 and 40 min during working days.

${\mathit{c}}_{\mathit{\alpha}}$ | A | ||
---|---|---|---|

Model | 0.1 | 0.05 | 0.01 |

General | 1.224 | 1.358 | 1.628 |

Normal | 0.819 | 0.895 | 1.035 |

Exponential | 0.990 | 1.094 | 1.308 |

Weibull n = 10 | 0.760 | 0.819 | 0.944 |

Weibull n = 20 | 0.779 | 0.843 | 0.973 |

Weibull n = 50 | 0.790 | 0.856 | 0.988 |

$\mathrm{Weibull}\text{}n\text{}\to \text{}\infty $ | 0.803 | 0.874 | 1.007 |

Distribution | k(n) |
---|---|

General | $\sqrt{n}+0.12+\frac{0.11}{\sqrt{n}}$ |

Normal | $\sqrt{n}-0.01+\frac{0.85}{\sqrt{n}}$ |

Exponential | $\sqrt{n}+0.12+\frac{0.11}{\sqrt{n}}$ |

Weibull | $\sqrt{n}$ |

**Table 3.**Summary of the evaluated appliances with their type of consumption, type of days and amount of data.

Appliance | Type of Consumption | Type of Days | Amount of Data |
---|---|---|---|

Lightning | Continuous | Working days | 45 |

Saturdays | 9 | ||

Sundays | 9 | ||

Refrigerator | Continuous | Working days | 45 |

Saturdays | 9 | ||

Sundays | 9 | ||

HVAC | Continuous | Working days | 45 |

Saturdays | 9 | ||

Sundays | 9 | ||

Dryer | Occasional | Working days | 45 |

Washing machine | Occasional | Working days | 45 |

Dishwasher | Occasional | Working days | 45 |

Appliance | Type of Day | Duration | Error |
---|---|---|---|

Lightning | Working day | All day | 3.81% |

Saturdays | All day | 4.26% | |

Sundays | All day | 4.27% | |

Refrigerator | Working day | All day | 3.91% |

Saturdays | All day | 3.33% | |

Sundays | All day | 3.61% | |

HVAC | Working day | All day | 5.55% |

Saturdays | All day | 4.72% | |

Sundays | All day | 4.03% | |

Dryer | Working days | 40 min | 1.17% |

Washing machine | Working days | 10 min | 1.04% |

70 min | 0.78% | ||

Dishwasher | Working days | 30 min | 2.33% |

40 min | 0.78% |

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**MDPI and ACS Style**

Villanueva, D.; San-Facundo, D.; Miguez-García, E.; Fernández-Otero, A.
Modeling and Simulation of Household Appliances Power Consumption. *Appl. Sci.* **2022**, *12*, 3689.
https://doi.org/10.3390/app12073689

**AMA Style**

Villanueva D, San-Facundo D, Miguez-García E, Fernández-Otero A.
Modeling and Simulation of Household Appliances Power Consumption. *Applied Sciences*. 2022; 12(7):3689.
https://doi.org/10.3390/app12073689

**Chicago/Turabian Style**

Villanueva, Daniel, Diego San-Facundo, Edelmiro Miguez-García, and Antonio Fernández-Otero.
2022. "Modeling and Simulation of Household Appliances Power Consumption" *Applied Sciences* 12, no. 7: 3689.
https://doi.org/10.3390/app12073689