# Energy Management through Cost Forecasting for Residential Buildings in New Zealand

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Influencing Factors of Residential Energy Use

#### 2.2. Cost Forecasting Methods

#### 2.3. Energy Consumption Forecasting

## 3. Research Methods

#### 3.1. Data

#### 3.2. Correlation Analysis

#### 3.3. Exponential Smoothing

#### 3.3.1. Holt-Winters Method

_{1}is the growth rate; S

_{t}is a seasonal pattern; ${\u03f5}_{t}$ is the error term.

#### 3.3.2. Multiplicative Holt-Winters Method

#### 3.4. Autoregressive Integrated Moving Average (ARIMA)

_{L}is introduced, where P represents seasonal autoregressive orders, D indicates seasonal differencing orders, Q represents seasonal moving average orders, and L indicates the number of seasons. A seasonal ARIMA model can be shown in Equation (14).

**L**= 4 for quarterly data and

**L**= 12 for monthly data); $\mathsf{\delta}$ is a constant term; ${a}_{t},{a}_{t-1},\cdots $ are random shocks; ${\varnothing}_{1},{\varnothing}_{2},\cdots ,{\varnothing}_{p}$ are non-seasonal autoregressive parameters; ${\phi}_{1,L},{\phi}_{2,L},\cdots ,{\phi}_{P,L}$ are seasonal autoregressive parameters; ${\theta}_{1},{\theta}_{2},\cdots ,{\theta}_{q}$ are non-seasonal moving average parameters, ${\vartheta}_{1,L},{\vartheta}_{2,L},\cdots ,{\vartheta}_{Q,L}$ are seasonal moving average parameters.

#### 3.5. Error Measure for Model Comparison

_{i}) and the forecasting values for the same series by (${\widehat{y}}_{i}$). The mean absolute percentage error (MAPE) can be computed in Equation (16).

#### 3.6. Multilayer Artificial Neutral Networks

#### 3.7. t-Test

## 4. Data Analysis

#### 4.1. Correlation Analysis Results

#### 4.2. Exponential Models for Building Cost

#### 4.3. Seasonal ARIMA Models

#### 4.4. ANN Models

## 5. Results Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACF | Auto-Correlation Function |

ANNs | Artificial Neutral Networks |

AR | Autoregressive |

AR1 | One-storey House |

AR2 | Two-storey House |

AR3 | Townhouse |

AR4 | Apartment |

AR5 | Retirement Village |

ARIMA | Autoregressive Integrated Moving Average |

BIC | Bayesian Information Criterion |

ES | Exponential Smoothing Method |

HVAC | Heating, Ventilation and Air Conditioning |

HW | Holt-Winter Method |

MA | Moving Average |

MAE | Mean Absolute Error |

MAPE | Mean Absolute Percentage Error |

MHW | Multiplicative Holt-Winter Method |

PACF | Partial Auto-Correlation Function |

RMSE | Root Mean Square Error |

SAC | Sample Auto-Correlation Function |

SAR | Seasonal Autoregressive |

SE | Standard Error |

SSE | Sum of Squared Error |

SMA | Seasonal Moving Average |

SPAC | Sample Partial Auto-Correlation Function |

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**Figure 5.**Sample autocorrelation function (ACF) (left panels) and sample partial autocorrelation (PA).

**Figure 6.**ACFs (left panels) and partial autocorrelation functions (PACFs) (right panels) of the differenced data series.

Residential Building Cost | |||||
---|---|---|---|---|---|

Energies | AR1 | AR2 | AR3 | AR4 | AR5 |

Electricity | 0.974 ** | 0.977 ** | 0.984 ** | 0.782 ** | 0.833 ** |

Gas | 0.976 ** | 0.994 ** | 0.968 ** | 0.966 ** | 0.898 ** |

Petrol | 0.919 ** | 0.916 ** | 0.937 ** | 0.884 ** | 0.640 ** |

Series | Exponential Smoothing Model | Parameter | Estimate | SE | p-Value |
---|---|---|---|---|---|

α | 0.370 | 0.116 | 0.002 | ||

ES(AHW) | β | 0.634 | 0.287 | 0.032 ** | |

AR1 | γ | 0 | 0.112 | 0.993 | |

α | 0.379 | 0.112 | 0.001 ** | ||

ES(MHW) | β | 0.537 | 0.251 | 0.037 ** | |

γ | 0.528 | 0.171 | 0.003 ** | ||

α | 0.899 | 0.150 | *** | ||

ES(AHW) | β | 0 | 0.047 | 1 | |

AR2 | γ | 0 | 0.696 | 1 | |

α | 0.846 | 0.145 | *** | ||

ES(MHW) | β | 0.001 | 0.045 | 0.983 | |

γ | 0.028 | 0.298 | 0.925 | ||

α | 0.683 | 0.135 | *** | ||

ES(AHW) | β | 0.218 | 0.113 | 0.059 * | |

AR3 | γ | 0.001 | 0.171 | 0.995 | |

α | 0.578 | 0.128 | *** | ||

ES(MHW) | β | 0.269 | 0.128 | 0.042 ** | |

γ | 0.020 | 0.091 | 0.830 | ||

α | 0.200 | 0.079 | 0.014 ** | ||

ES(AHW) | β | 1.000 | 0.467 | 0.037 ** | |

AR4 | γ | 0 | 0.091 | 1 | |

α | 0.198 | 0.083 | 0.021 ** | ||

ES(MHW) | β | 1.000 | 0.503 | 0.052 * | |

γ | 0.040 | 0.062 | 0.524 | ||

α | 0.469 | 0.118 | *** | ||

ES(AHW) | β | 0.441 | 0.194 | 0.027 ** | |

AR5 | γ | 0.014 | 0.096 | 0.883 | |

α | 0.361 | 0.096 | *** | ||

ES(MHW) | β | 0.511 | 0.225 | 0.028 ** | |

γ | 0.479 | 0.151 | 0.003 ** |

Series | Model | R-Square | RMSE | MAPE | MAE | BIC | Ljung-Box | Shapiro-Wilk |
---|---|---|---|---|---|---|---|---|

AR1 | ES(AHW) | 0.976 | 33.205 | 1.655 | 24.52 | 7.233 | 0.238 | 0.203 |

ES(MHW) | 0.974 | 34.311 | 1.581 | 23.722 | 7.299 | 0.805 | 0.102 | |

AR2 | ES(AHW) | 0.947 | 62.506 | 2.066 | 35.292 | 8.498 | 0.891 | 0.153 |

ES(MHW) | 0.943 | 64.989 | 2.159 | 37.134 | 8.576 | 0.904 | 0.133 | |

AR3 | ES(AHW) | 0.964 | 52.405 | 1.953 | 36.27 | 8.146 | 0.13 | 0.103 |

ES(MHW) | 0.959 | 55.473 | 2.054 | 38.568 | 8.26 | 0.106 | 0.173 | |

AR4 | ES(AHW) | 0.989 | 45.577 | 1.32 | 31.825 | 7.867 | 0.593 | 0.127 |

ES(MHW) | 0.988 | 47.836 | 1.385 | 34.24 | 7.964 | 0.333 | 0.104 | |

AR5 | ES(AHW) | 0.991 | 36.083 | 1.312 | 27.824 | 7.4 | 0.477 | 0.393 |

ES(MHW) | 0.989 | 39.374 | 1.356 | 28.715 | 7.574 | 0.415 | 0.647 |

Series | Model | AR | MA | SAR | SMA |
---|---|---|---|---|---|

AR1 | ARIMA(0,1,3)(0,1,1)_{4}ARIMA(0,1,1)(0,1,1) _{4} | MA(l) = 0.34l MA(2) = −0.l0l MA(3) = −0.295 MA(l) = 0.3l7 | SMA(l) = 0.447 SMA(l) = 0.290 | ||

AR2 | ARIMA(0,1,0)(2,0,0)_{4}ARIMA(0,1,0)(0,0,2) _{4} | SAR(1) = 0.038 SAR(2) = 0.348 | SMA(l) = −0.005 SMA(2) = −0.368 | ||

AR3 | ARIMA(0,1,0)(1,0,0)_{4}ARIMA(0,1,0)(0,1,0) _{4} | SAR(l) = 0.562 | |||

AR4 | ARIMA(1,1,0)(0,1,0)_{4}ARIMA(0,1,1)(0,1,0) _{4} | AR(l) = −0.4l9 | MA(l) = 0.404 | ||

AR5 | ARIMA(0,1,0)(0,1,1)_{4}ARIMA(0,1,0)(0,1,0) _{4} | SMA(l) = 0.554 |

Series | Model | R-Square | RMSE | MAPE | MAE | BIC | Ljung-Box | Shapiro-Wilk |
---|---|---|---|---|---|---|---|---|

AR1 | ARIMA(0,1,3)(0,1,1)_{4} | 0.959 | 37.234 | 1.716 | 25.866 | 7.644 | 0.873 | 0.461 |

ARIMA(0,1,1)(0,1,1)_{4} | 0.953 | 38.665 | 1.856 | 27.895 | 7.556 | 0.519 | 0.158 | |

AR2 | ARIMA(0,1,0)(2,0,0)_{4} | 0.942 | 63.899 | 2.221 | 38.057 | 8.546 | 0.877 | 0.184 |

ARIMA(0,1,0)(0,0,2)_{4} | 0.942 | 63.865 | 2.256 | 38.766 | 8.545 | 0.898 | 0.136 | |

AR3 | ARIMA(0,1,0)(0,1,0)_{4} | 0.944 | 54.277 | 1.913 | 35.836 | 8.07 | 0.855 | 0.153 |

ARIMA(0,1,0)(4,1,0)_{4} | 0.95 | 53.423 | 1.823 | 33.945 | 8.366 | 0.956 | 0.122 | |

AR4 | ARIMA(1,1,0)(0,1,0)_{4} | 0.981 | 51.468 | 1.53 | 37.383 | 8.046 | 0.657 | 0.391 |

ARIMA(0,1,1)(0,1,0)_{4} | 0.981 | 51.889 | 1.505 | 36.868 | 8.062 | 0.628 | 0.24 | |

AR5 | ARIMA(0,1,0)(0,1,1)_{4} | 0.986 | 40.162 | 1.325 | 27.306 | 7.55 | 0.489 | 0.127 |

ARIMA(0,1,0)(0,1,0)_{4} | 0.983 | 43.299 | 1.52 | 31.335 | 7.618 | 0.141 | 0.103 |

Predictors | Parameter Estimates | |||||
---|---|---|---|---|---|---|

Hidden Layer 1 | Output Layer | |||||

H (1:1) | H (1:2) | H (1:3) | H (1:4) | Electricity | ||

Input layer | AR1 | −0.119 | −0.094 | 1.067 | 0.304 | |

AR2 | −0.151 | −0.256 | −0.930 | 0.019 | ||

AR3 | −0.039 | −0.068 | 0.420 | −0.118 | ||

AR4 | 0.178 | −0.219 | 0.414 | 0.245 | ||

AR5 | −0.511 | −0.292 | −0.132 | 0.409 | ||

Bias | −0.339 | −1.477 | −0.409 | −0.215 | ||

Hidden layer 1 | H (1:1) | −0.291 | ||||

H (1:2) | −1.799 | |||||

H (1:3) | −1.209 | |||||

H (1:4) | 0.458 | |||||

Bias | −1.612 | |||||

Model training | SSE = 4.117 | |||||

Model testing | SSE = 1.359 | |||||

t-value | t = −0.841 | |||||

p-value | p = 0.403 |

Predictors | Parameter Estimates | ||||||
---|---|---|---|---|---|---|---|

Hidden Layer 1 | Hidden Layer 2 | Output Layer | |||||

H (1:1) | H (1:2) | H (1:3) | H (2:1) | H (2:2) | Gas | ||

Input layer | AR1 | −0.266 | 0.101 | 0.417 | |||

AR2 | 0.301 | −0.130 | 0.397 | ||||

AR3 | 0.339 | −0.372 | 0.211 | ||||

AR4 | 0.662 | 0.626 | 0.406 | ||||

AR5 | 0.504 | 0.732 | 0.548 | ||||

Bias | −0.394 | −0.456 | −0.029 | ||||

Hidden layer 1 | H (1:1) | −0.476 | −0.196 | ||||

H (1:2) | −0.916 | 0.282 | |||||

H (1:3) | −0.284 | −0.384 | |||||

Bias | 0.332 | 0.301 | |||||

Hidden layer 2 | H (2:1) | −0.852 | |||||

H (2:2) | 0.135 | ||||||

Bias | 0.230 | ||||||

Model training | SSE = 2.437 | ||||||

Model testing | SSE = 0.831 | ||||||

t-value | t = −0.491 | ||||||

p-value | p = 0.625 |

Predictors | Parameter Estimates | ||||
---|---|---|---|---|---|

Hidden Layer 1 | Output Layer | ||||

H (1:1) | H (1:2) | H (1:3) | Petrol | ||

Input layer | AR1 | 0.205 | −1.019 | −0.673 | |

AR2 | 0.091 | 0.756 | −0.086 | ||

AR3 | 0.085 | −0.557 | 0.219 | ||

AR4 | 0.354 | −0.757 | −0.773 | ||

AR5 | 0.018 | −0.636 | −0.441 | ||

Bias | −0.069 | 1.759 | 1.757 | ||

Hidden layer 1 | H (1:1) | −0.287 | |||

H (1:2) | −1.593 | ||||

H (1:3) | −0.338 | ||||

Bias | 1.015 | ||||

Model training | SSE = 1.644 | ||||

Model testing | SSE = 0.422 | ||||

t-value | t = 0.917 | ||||

p-value | p = 0.362 |

Series | Model | MAPE |
---|---|---|

AR1 | ARIMA(0,1,3)(0,1,1)_{4} | 1.813 |

ARIMA(0,1,1)(0,1,1)_{4} | 1.651 | |

ES(AHW) | 1.260 | |

ES(MHW) | 1.556 | |

AR2 | ARIMA(0,1,0)(2,0,0)_{4} | 0.922 |

ARIMA(0,1,0)(0,0,2)_{4} | 0.794 | |

ES(AHW) | 0.395 | |

ES(MHW) | 0.650 | |

AR3 | ARIMA(0,1,0)(0,1,0)_{4} | 1.020 |

ARIMA(0,1,0)(4,1,0)_{4} | 2.318 | |

ES(AHW) | 1.211 | |

ES(MHW) | 1.130 | |

AR4 | ARIMA(1,1,0)(0,1,0)_{4} | 0.501 |

ARIMA(0,1,1)(0,1,0)_{4} | 0.446 | |

ES(AHW) | 0.748 | |

ES(MHW) | 0.809 | |

AR5 | ARIMA(0,1,0)(0,1,1)_{4} | 0.853 |

ARIMA(0,1,0)(0,1,0)_{4} | 1.213 | |

ES(AHW) | 0.917 | |

ES(MHW) | 0.949 |

© 2019 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**

Zhao, L.; Liu, Z.; Mbachu, J. Energy Management through Cost Forecasting for Residential Buildings in New Zealand. *Energies* **2019**, *12*, 2888.
https://doi.org/10.3390/en12152888

**AMA Style**

Zhao L, Liu Z, Mbachu J. Energy Management through Cost Forecasting for Residential Buildings in New Zealand. *Energies*. 2019; 12(15):2888.
https://doi.org/10.3390/en12152888

**Chicago/Turabian Style**

Zhao, Linlin, Zhansheng Liu, and Jasper Mbachu. 2019. "Energy Management through Cost Forecasting for Residential Buildings in New Zealand" *Energies* 12, no. 15: 2888.
https://doi.org/10.3390/en12152888