# Performance Analysis of Short-Term Electricity Demand with Atmospheric Variables

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## Abstract

**:**

## 1. Introduction

## 2. Description of Data

#### 2.1. Electricity Demand Data

#### 2.2. Atmospheric Variables

#### 2.3. Weather Station Selection

## 3. Related Works

## 4. Prototype Modeling

#### 4.1. Modeling Trend

#### 4.2. Modeling Cyclicality and Seasonality

#### 4.3. Mathematical Model

- Model A: This model consists of the variables from deterministic terms, historical demand and historical demand-related interaction (e.g., $DH{I}_{h,d}$) term. Therefore, Model A is the sum of Equations (5), (9) and (10) and consists of a total of 47 variables including six correlated error terms (${v}_{h,d}$). The electricity demand from Model A is denoted by $D{A}_{h,d}$ and can be generalized as,$$\begin{array}{cc}\hfill D{A}_{h,d}& =De{t}_{h,d}+DHi{s}_{h,d}+DH{I}_{h,d}+{v}_{h,d}\hfill \end{array}$$
- Model B: Model B consists of Model A (Equation (12)), temperature variables (Equation (7)) and interaction terms with temperature (Equation (11)). Therefore, Model B consists of 83 variables. The electricity demand from Model B is denoted by $D{B}_{h,d}$ and can be generalized as,$$\begin{array}{cc}\hfill D{B}_{h,d}& =D{A}_{h,d}+AtmV{T}_{h,d}+TIn{t}_{h,d}\hfill \end{array}$$
- Model C: Model C consists of all the variables from Model B and atmospheric variables (Equation (8)). Therefore, Model C consists of 94 variables. The electricity demand from Model C is represented by $Dem{C}_{h,d}$ and can be generalized as,$$D{C}_{h,d}=D{B}_{h,d}+Atm{V}_{h,d}$$

## 5. Estimation and Forecasting

#### Algorithm Setup

- Set informative priors,
- Starting values, $p\left(\mathit{\beta}\right)\sim N({\mathit{\beta}}_{ols},{\sum}_{\mathit{ols}})$. The suffix ‘$ols$’ in the symbol means OLS estimation.
- Set a normal prior for serial correlated coefficient $\mathit{\rho}$ as $p\left(\mathit{\rho}\right)\sim N({\mathit{\rho}}^{0},{\sum}_{\rho})$, with starting value $\rho =0$.
- Set an inverse Gamma prior for ${\mathit{\sigma}}^{2}\sim {\Gamma}^{-1}(\frac{{T}_{0}}{2},\frac{{\theta}_{0}}{2})$ where ${T}_{0}$ and ${\theta}_{0}$ represent the degree of freedom and scale factor, respectively.

- Draw the conditional posterior distribution,
- For $\mathit{\beta}$, $P\left(\mathit{\beta}\right|{\mathit{\sigma}}^{2},\mathit{\rho},{\mathit{Y}}_{h})\sim N({\mathit{\mu}}^{*},{\mathit{\zeta}}^{*})$, then $\mathit{\beta}={\mathit{\mu}}^{*}+{[\widehat{\mathit{\mu}}\times {\left({\mathit{\zeta}}^{*}\right)}^{-\frac{1}{2}}]}^{\prime}$ (Appendix C, Theorem A1)
- For correlated error $\mathit{\rho}$, $P\left(\mathit{\rho}\right|{\mathit{\sigma}}^{2},\mathit{\beta},{\mathit{Y}}_{h})\sim N({\mathit{\rho}}^{*},{\mathit{\xi}}^{*})$, then $\mathit{\rho}={\mathit{\rho}}^{*}+{[\widehat{\mathit{\rho}}\times {\left({\mathit{\xi}}^{*}\right)}^{-\frac{1}{2}}]}^{\prime}$ (Appendix C, Theorem A1)

- Given a draw from $\mathit{\beta}$ and $\mathit{\rho}$, draw ${\mathit{\sigma}}^{2}$ from its conditional posterior distribution, $P\left({\mathit{\sigma}}^{2}\right|\mathit{\beta},\mathit{\rho},{\mathit{Y}}_{h})\sim \phantom{\rule{3.33333pt}{0ex}}{\Gamma}^{-1}(\frac{{T}_{1}}{2},\frac{{\theta}_{1}}{2})$. (Appendix C, Theorem A2)
- Repeat Steps 2 and 3 M times to obtain ${\beta}^{1},\dots ,{\beta}^{M}$ and ${\left({\sigma}^{2}\right)}^{1},\dots ,{\left({\sigma}^{2}\right)}^{M}$.

## 6. Results and Discussion

#### 6.1. Atmospheric Variables

#### 6.2. Temperature Variables

#### 6.3. Performance Analysis

#### 6.4. Hypothesis Testing

#### 6.5. Computation Time

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ARMAX | Auto-regressive moving average with exogenous variable |

MAPE | Mean absolute percentage error |

IPCC | Intergovernmental Panel on Climate Change |

MLR | Multiple linear regression |

ANN | Artificial neural network |

SVM | Support vector machine |

GEFCom2012 | Global Energy Forecasting Competition 2012 |

CDD | Cooling degree day |

HDD | Heating degree day |

HEPCO | Hokkaido Electrical Power Company |

JMA | Japan Meteorological Agency |

MW | Megawatt |

OLS | Ordinary least square |

MCMC | Markov chain Monte Carlo |

## Appendix A. Weather Station Selection

#### Appendix A.1. Methodology

#### Appendix A.2. Algorithm Setup

- Denote the temperature variables (daily average, maximum and minimum temperature) of each stations as ${T}_{i}$, $i=1,\dots ,N$.
- Develop the forecasting model (Equation (A1)) where electricity demand is a function of the temperature and calendar variables.
- For speed and simplicity, use OLS estimation and forecasting for a year out of the sample data.
- Calculate the forecasting error and MAPE for all the weather stations separately.
- Sort the resulting error measures in ascending order to find the best individual’s (weather stations) impact on demand.
- Combine (average and weighted average with population) the temperature data of the top k weather stations to create a new temperature series and fit all these combinations to the same forecasting model.
- Calculate the forecasting error and find the combinations of weather stations that give the best (smallest) MAPE.

**Table A1.**Top eight weather stations based on populations and their forecasting performance on electricity demand for Hokkaido territory. Note: figures in parenthesis (8th and 11th column) correspond to the variation in MAPE values.

Weather Station | Forecasting Results of Weather Stations When Their Data Combined as: | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Simple Mean | Weighted by Population | |||||||||||

Sub-Prefecture | Area (m^{2}) | Population | Name | Index | MAPE (%) | Combination | MAPE (%) Variance | R^{2} | Adjusted-R^{2} | MAPE (%) Variance | R^{2} | Adjusted-R^{2} |

Ishikari | 3539.86 | 2,324,878 | Sapporo | 1 | 2.959 | 1, and 2 | 3.012_{(1.702)} | 0.943 | 0.938 | 2.945_{(1.70)} | 0.965 | 0.962 |

Kamikawa | 10,619.20 | 527,575 | Hakodate | 2 | 3.042 | 1, 2 and 7 | 2.894_{(1.667)} | 0.946 | 0.941 | 2.909_{(1.689)} | 0.968 | 0.965 |

Oshima | 3936.46 | 433,475 | Kitami | 3 | 3.318 | 1, 2, 7 and 8 | 2.967_{(1.841)} | 0.943 | 0.938 | 2.942_{(1.744)} | 0.966 | 0.963 |

Iburi | 3698 | 419,115 | Abashiri | 4 | 3.323 | 1, 2, 7, 8 and 6 | 3.324_{(1.909)} | 0.930 | 0.924 | 3.324_{(1.90)} | 0.962 | 0.959 |

Ashahikawa | 5 | 3.174 | 1, 2, 7, 8, 6 and 5 | 2.970_{(1.792)} | 0.943 | 0.937 | 2.946_{(1.725)} | 0.965 | 0.962 | |||

Tokachi | 10,831.24 | 353,291 | Obihiro | 6 | 3.169 | 1, 2, 7, 8, 6, 5 and 3 | 2.899_{(1.730)} | 0.945 | 0.940 | 2.905_{(1.710)} | 0.967 | 0.964 |

Okhotsk | 10,690.62 | 309487 | Moruran | 7 | 3.010 | 1, 2, 7, 8, 6, 5, 3 and 4 | 2.925_{(1.670)} | 0.945 | 0.940 | 2.927_{(1.675)} | 0.968 | 0.965 |

Tomakomai | 8 | 3.156 |

## Appendix B. Forecasting Example

## Appendix C. Theorems

#### Appendix C.1. Theorem A1

**Theorem**

**A1.**

#### Appendix C.2. Theorem A2

**Theorem**

**A2.**

## Appendix D. Figures and Tables

**Figure A1.**(

**a**) Variation of forecasting error for 2015 (Model A); (

**b**) variation of forecasting error for 2015 (Model B).

**Table A2.**Hourly MAPE variation for various day types: working days including holidays (WDH), workings day not including holidays (WDNH), weekends including holidays (WEH), weekends not including holidays (WENH), holidays on working days (HWD), holidays on weekends (HWE).

Hour | Model | Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | WDH | WDNH | WEH | WENH | HWD | HWE | Holiday | NH |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | Model A | 1.6726 | 1.7407 | 1.7456 | 1.8309 | 1.5394 | 1.4631 | 1.4922 | 1.6639 | 0.7753 | 1.5824 | 1.6213 | 1.9812 | 0.8121 | 1.7474 | 1.6325 |

Model B | 1.1504 | 1.4382 | 1.2741 | 1.4257 | 1.2117 | 1.4709 | 1.2896 | 1.3642 | 0.7606 | 1.2200 | 1.2131 | 1.4685 | 1.3576 | 1.4463 | 1.3136 | |

Model C | 1.1919 | 1.4357 | 1.3395 | 1.4745 | 1.1992 | 1.5799 | 1.2373 | 1.4057 | 0.8193 | 1.1919 | 1.2274 | 1.6611 | 0.9614 | 1.5212 | 1.3382 | |

2 | Model A | 1.8561 | 1.7041 | 1.6715 | 1.8159 | 1.5774 | 1.5581 | 1.5430 | 1.6654 | 0.8006 | 1.6996 | 1.7156 | 2.1112 | 1.3818 | 1.9653 | 1.6535 |

Model B | 1.2177 | 1.4480 | 1.1882 | 1.4230 | 1.2913 | 1.5818 | 1.4109 | 1.3865 | 0.7887 | 1.3143 | 1.3056 | 1.3492 | 1.4865 | 1.3767 | 1.3648 | |

Model C | 1.2970 | 1.4602 | 1.1697 | 1.4413 | 1.2717 | 1.6941 | 1.3749 | 1.4074 | 0.8426 | 1.2970 | 1.3407 | 1.5256 | 1.2430 | 1.4691 | 1.3806 | |

3 | Model A | 1.8299 | 1.6551 | 1.7280 | 1.8782 | 1.5712 | 1.3902 | 1.3610 | 1.6445 | 0.6908 | 1.5954 | 1.6039 | 2.2782 | 1.4283 | 2.1082 | 1.5952 |

Model B | 1.2882 | 1.3785 | 1.1462 | 1.4060 | 1.1784 | 1.3799 | 1.3110 | 1.2978 | 0.7097 | 1.2996 | 1.3006 | 1.2319 | 1.2804 | 1.2416 | 1.3021 | |

Model C | 1.3373 | 1.3129 | 1.1836 | 1.3597 | 1.1587 | 1.4631 | 1.3420 | 1.2956 | 0.7436 | 1.3373 | 1.3548 | 1.3017 | 1.0399 | 1.2494 | 1.3121 | |

4 | Model A | 1.7652 | 1.6379 | 1.8458 | 1.8160 | 1.6098 | 1.3923 | 1.3475 | 1.6604 | 0.6981 | 1.5563 | 1.5657 | 2.2030 | 1.3701 | 2.0364 | 1.6007 |

Model B | 1.1658 | 1.2604 | 1.1607 | 1.3288 | 1.2750 | 1.4508 | 1.3543 | 1.2952 | 0.7567 | 1.2600 | 1.2595 | 1.1060 | 1.2704 | 1.1389 | 1.2958 | |

Model C | 1.1385 | 1.2825 | 1.1582 | 1.3299 | 1.1545 | 1.4228 | 1.3618 | 1.2696 | 0.7364 | 1.1385 | 1.2624 | 1.0859 | 1.0082 | 1.0704 | 1.2779 | |

5 | Model A | 1.8329 | 1.6645 | 1.9004 | 1.9173 | 1.5649 | 1.5715 | 1.3441 | 1.7237 | 0.7901 | 1.5885 | 1.5814 | 2.3748 | 1.7288 | 2.2456 | 1.6435 |

Model B | 1.3387 | 1.3066 | 1.3126 | 1.3861 | 1.2598 | 1.4386 | 1.3605 | 1.3407 | 0.7509 | 1.3496 | 1.3297 | 1.4274 | 1.7425 | 1.4904 | 1.3322 | |

Model C | 1.3420 | 1.3241 | 1.2364 | 1.4013 | 1.1265 | 1.4575 | 1.2308 | 1.3092 | 0.7773 | 1.3420 | 1.2897 | 1.3027 | 1.2207 | 1.2863 | 1.3033 | |

6 | Model A | 1.9198 | 1.9072 | 2.0596 | 1.9769 | 1.5467 | 1.5389 | 1.6301 | 1.8059 | 0.7545 | 1.7750 | 1.8003 | 2.5605 | 1.2725 | 2.3029 | 1.7591 |

Model B | 1.6503 | 1.5450 | 1.6708 | 1.5754 | 1.1965 | 1.4065 | 1.5892 | 1.4789 | 0.7142 | 1.6197 | 1.5820 | 1.6762 | 2.3669 | 1.8144 | 1.4965 | |

Model C | 1.6451 | 1.5607 | 1.5380 | 1.5917 | 1.1067 | 1.3907 | 1.4814 | 1.4376 | 0.7524 | 1.6451 | 1.5595 | 1.4951 | 1.6376 | 1.5236 | 1.4687 | |

7 | Model A | 2.0690 | 2.1267 | 2.2502 | 2.2338 | 1.8833 | 1.6164 | 1.9340 | 2.0221 | 0.7776 | 2.0015 | 2.0574 | 2.9955 | 0.8951 | 2.5754 | 1.9747 |

Model B | 1.7640 | 1.7209 | 1.7645 | 1.6313 | 1.4183 | 1.4893 | 2.0553 | 1.6049 | 0.7315 | 1.9097 | 1.9323 | 2.4035 | 1.4605 | 2.2149 | 1.6527 | |

Model C | 1.6466 | 1.7467 | 1.5595 | 1.5090 | 1.2270 | 1.4631 | 1.7972 | 1.5011 | 0.7673 | 1.6466 | 1.7573 | 2.0983 | 1.0206 | 1.8828 | 1.5397 | |

8 | Model A | 2.2325 | 2.3071 | 2.2233 | 2.5286 | 2.4832 | 2.1212 | 2.4363 | 2.3327 | 1.0441 | 2.3344 | 2.3803 | 3.4155 | 1.4263 | 3.0177 | 2.2833 |

Model B | 1.8220 | 1.8314 | 1.9199 | 1.9725 | 1.8258 | 1.7969 | 2.5240 | 1.8693 | 0.8740 | 2.1730 | 2.1622 | 2.8535 | 2.3874 | 2.7603 | 1.8966 | |

Model C | 1.5913 | 1.5995 | 1.5147 | 1.6427 | 1.4574 | 1.5827 | 2.1622 | 1.5594 | 0.8269 | 1.5913 | 1.8738 | 2.2654 | 1.9355 | 2.1994 | 1.6091 | |

9 | Model A | 2.6461 | 2.4470 | 2.7248 | 2.7950 | 2.8503 | 2.4008 | 3.1912 | 2.6436 | 1.1834 | 2.9187 | 2.8304 | 4.3900 | 4.6659 | 4.4452 | 2.5959 |

Model B | 2.1982 | 2.0883 | 2.2785 | 2.2846 | 2.4666 | 2.1374 | 3.1797 | 2.2511 | 1.0574 | 2.6890 | 2.5874 | 3.8711 | 4.6995 | 4.0367 | 2.2544 | |

Model C | 1.9150 | 1.7530 | 1.7447 | 1.9969 | 1.9034 | 1.8084 | 2.8321 | 1.8413 | 0.9390 | 1.9150 | 2.2789 | 3.4363 | 4.2477 | 3.5986 | 1.8751 | |

10 | Model A | 2.9442 | 2.6022 | 3.2205 | 3.0529 | 3.1272 | 2.6851 | 3.6793 | 2.9376 | 1.3814 | 3.3117 | 3.1540 | 4.5038 | 6.4350 | 4.8900 | 2.9090 |

Model B | 2.4234 | 2.3307 | 2.6872 | 2.3936 | 2.7461 | 2.7183 | 3.5322 | 2.5752 | 1.4121 | 2.9778 | 2.8153 | 4.0603 | 6.1947 | 4.4871 | 2.5583 | |

Model C | 2.1795 | 2.0551 | 2.1036 | 1.9529 | 2.1555 | 2.0213 | 3.0005 | 2.0577 | 1.1097 | 2.1795 | 2.4479 | 3.4744 | 5.4029 | 3.8601 | 2.0883 | |

11 | Model A | 3.2398 | 2.6796 | 3.3393 | 3.2835 | 3.2169 | 2.9312 | 3.9262 | 3.0901 | 1.5203 | 3.5830 | 3.3519 | 4.4814 | 8.1586 | 5.2168 | 3.0849 |

Model B | 2.5062 | 2.3502 | 2.7765 | 2.5841 | 2.6018 | 2.9451 | 3.6375 | 2.6516 | 1.5283 | 3.0718 | 2.8516 | 4.2028 | 7.4327 | 4.8488 | 2.6184 | |

Model C | 2.3283 | 1.9570 | 1.9768 | 2.2351 | 2.1442 | 2.2174 | 3.0316 | 2.1061 | 1.2075 | 2.3283 | 2.4878 | 3.2880 | 6.4853 | 3.9274 | 2.1478 | |

12 | Model A | 3.1075 | 2.8537 | 3.4770 | 3.4629 | 3.4499 | 3.1050 | 3.9691 | 3.2697 | 1.6391 | 3.5383 | 3.3037 | 4.9317 | 8.1837 | 5.5821 | 3.1824 |

Model B | 2.5170 | 2.5425 | 2.9768 | 2.7074 | 2.8006 | 3.0701 | 3.5237 | 2.8195 | 1.6153 | 3.0203 | 2.8234 | 4.9172 | 6.9190 | 5.3175 | 2.6972 | |

Model C | 2.3383 | 2.1527 | 2.0263 | 2.2816 | 2.2020 | 2.3525 | 2.9905 | 2.2030 | 1.2876 | 2.3383 | 2.4675 | 3.6795 | 6.5631 | 4.2562 | 2.1932 | |

13 | Model A | 3.3161 | 2.8978 | 4.0335 | 3.7530 | 3.3656 | 3.5846 | 3.8910 | 3.5269 | 1.8386 | 3.6036 | 3.3948 | 4.6860 | 7.7364 | 5.2961 | 3.4198 |

Model B | 2.4320 | 2.6308 | 3.2304 | 2.9576 | 2.7321 | 3.1860 | 3.3674 | 2.9474 | 1.6902 | 2.8997 | 2.7367 | 3.8506 | 6.1283 | 4.3061 | 2.8323 | |

Model C | 2.2623 | 2.1179 | 2.1186 | 2.3961 | 2.2355 | 2.5430 | 2.7583 | 2.2822 | 1.3922 | 2.2623 | 2.3490 | 3.0000 | 5.7045 | 3.5409 | 2.2593 | |

14 | Model A | 3.6238 | 3.0059 | 4.0271 | 3.8494 | 3.3133 | 3.9163 | 4.3122 | 3.6224 | 1.9851 | 3.9680 | 3.7592 | 5.5966 | 8.1018 | 6.0977 | 3.5452 |

Model B | 2.5357 | 2.6686 | 3.3307 | 3.2034 | 2.7270 | 3.4654 | 3.7935 | 3.0790 | 1.8339 | 3.1646 | 2.9757 | 4.8298 | 6.9040 | 5.2447 | 2.9449 | |

Model C | 2.2401 | 2.2966 | 2.0473 | 2.6013 | 2.1359 | 2.5819 | 3.0670 | 2.3326 | 1.3972 | 2.2401 | 2.4697 | 3.7217 | 6.2939 | 4.2362 | 2.2902 | |

15 | Model A | 3.6812 | 3.0742 | 3.9871 | 3.7488 | 3.0118 | 3.9572 | 4.2087 | 3.5558 | 2.0035 | 3.9449 | 3.7478 | 5.2494 | 7.8475 | 5.7690 | 3.5105 |

Model B | 2.7848 | 2.6771 | 3.1999 | 3.3373 | 2.5674 | 3.2893 | 3.4836 | 3.0142 | 1.7151 | 3.1342 | 2.9288 | 4.9817 | 7.2022 | 5.4258 | 2.8723 | |

Model C | 2.3425 | 2.2175 | 2.2240 | 2.4928 | 2.1483 | 2.6142 | 2.9446 | 2.3394 | 1.3864 | 2.3425 | 2.4668 | 4.1316 | 6.1436 | 4.5340 | 2.2705 | |

16 | Model A | 3.5173 | 3.0099 | 3.5821 | 3.5607 | 2.7995 | 3.2589 | 3.8668 | 3.2422 | 1.6404 | 3.6920 | 3.6108 | 4.8270 | 5.2998 | 4.9216 | 3.2550 |

Model B | 2.7110 | 2.6691 | 2.7547 | 3.0173 | 2.3450 | 2.8460 | 3.2405 | 2.7264 | 1.4517 | 2.9758 | 2.8676 | 4.1300 | 5.1174 | 4.3275 | 2.6838 | |

Model C | 2.2337 | 2.1280 | 2.0626 | 2.3022 | 1.9723 | 2.1713 | 2.6555 | 2.1273 | 1.1495 | 2.2337 | 2.3272 | 3.4206 | 4.7688 | 3.6903 | 2.1090 | |

17 | Model A | 2.8113 | 2.2691 | 2.6527 | 2.8911 | 2.4075 | 2.5460 | 3.7121 | 2.5533 | 1.2941 | 3.2617 | 3.1543 | 3.3655 | 5.3881 | 3.7700 | 2.6801 |

Model B | 2.3903 | 2.2728 | 2.0630 | 2.4652 | 1.9515 | 2.1243 | 3.0331 | 2.1754 | 1.1075 | 2.7117 | 2.6350 | 3.1321 | 4.2301 | 3.3517 | 2.2523 | |

Model C | 2.1376 | 1.9282 | 1.6517 | 1.8570 | 1.8850 | 1.7124 | 2.4634 | 1.8069 | 0.9375 | 2.1376 | 2.2313 | 2.5008 | 3.6711 | 2.7349 | 1.8899 | |

18 | Model A | 2.2690 | 2.1932 | 2.2601 | 2.4965 | 2.2489 | 2.2088 | 3.0329 | 2.2815 | 1.1026 | 2.6509 | 2.6016 | 3.4705 | 3.6273 | 3.5019 | 2.3047 |

Model B | 1.8041 | 2.0382 | 1.7465 | 2.0586 | 1.7992 | 1.8877 | 2.4489 | 1.9061 | 0.9407 | 2.1265 | 2.0982 | 2.6811 | 2.6862 | 2.6822 | 1.9161 | |

Model C | 1.8099 | 1.6286 | 1.3981 | 1.7202 | 1.7559 | 1.5168 | 1.8188 | 1.6039 | 0.7631 | 1.8099 | 1.7890 | 2.0452 | 2.3158 | 2.0993 | 1.6323 | |

19 | Model A | 1.9400 | 2.1032 | 2.4597 | 1.7876 | 2.0171 | 2.0373 | 2.6626 | 2.0810 | 1.0018 | 2.3013 | 2.2479 | 3.1036 | 3.3588 | 3.1546 | 2.0693 |

Model B | 1.5986 | 1.9853 | 1.5508 | 1.5214 | 1.5875 | 1.7703 | 2.1094 | 1.6831 | 0.8720 | 1.8540 | 1.8295 | 2.9165 | 2.3402 | 2.8012 | 1.6528 | |

Model C | 1.5358 | 1.7506 | 1.3152 | 1.1828 | 1.6486 | 1.3373 | 1.6731 | 1.4469 | 0.6953 | 1.5358 | 1.5810 | 2.3915 | 2.0686 | 2.3269 | 1.4310 | |

20 | Model A | 1.7961 | 1.8596 | 1.7109 | 1.6868 | 1.8195 | 1.6626 | 2.3081 | 1.7479 | 0.8015 | 2.0521 | 1.9965 | 2.8977 | 3.1537 | 2.9489 | 1.7528 |

Model B | 1.3819 | 1.3910 | 0.9991 | 1.2920 | 1.2984 | 1.1858 | 1.6894 | 1.2333 | 0.5791 | 1.5357 | 1.5211 | 2.4375 | 1.8249 | 2.3150 | 1.2464 | |

Model C | 1.3681 | 1.2429 | 0.9897 | 1.2038 | 1.2219 | 1.0033 | 1.4228 | 1.1323 | 0.5094 | 1.3681 | 1.3759 | 2.1167 | 1.7822 | 2.0498 | 1.1456 | |

21 | Model A | 1.9851 | 1.8714 | 1.9007 | 1.7697 | 1.8257 | 1.7998 | 2.1877 | 1.8334 | 0.8843 | 2.0864 | 2.0404 | 2.9103 | 2.9981 | 2.9278 | 1.8303 |

Model B | 1.4202 | 1.4070 | 1.0795 | 1.1981 | 1.2286 | 1.1109 | 1.6524 | 1.2048 | 0.5500 | 1.5363 | 1.5231 | 2.2684 | 1.7980 | 2.1743 | 1.2350 | |

Model C | 1.4664 | 1.2926 | 0.9844 | 1.0369 | 1.2318 | 1.0341 | 1.3425 | 1.1159 | 0.5441 | 1.4664 | 1.3868 | 2.1136 | 1.7543 | 2.0417 | 1.1364 | |

22 | Model A | 2.1472 | 2.0246 | 1.9386 | 2.0494 | 1.8935 | 1.8998 | 2.1253 | 1.9612 | 0.9233 | 2.1363 | 2.0900 | 2.8940 | 3.0530 | 2.9258 | 1.9436 |

Model B | 1.4406 | 1.4008 | 1.3538 | 1.4628 | 1.3684 | 1.3452 | 1.6044 | 1.3862 | 0.6642 | 1.5225 | 1.5329 | 2.3268 | 1.3174 | 2.1249 | 1.3735 | |

Model C | 1.4462 | 1.4146 | 1.2637 | 1.2127 | 1.3508 | 1.1255 | 1.3280 | 1.2734 | 0.6027 | 1.4462 | 1.3866 | 2.0830 | 1.3962 | 1.9457 | 1.2590 | |

23 | Model A | 2.2922 | 1.9065 | 2.1283 | 2.2634 | 2.0251 | 2.0859 | 2.0248 | 2.0819 | 1.0168 | 2.1585 | 2.1233 | 2.6635 | 2.8549 | 2.7018 | 2.0595 |

Model B | 1.5580 | 1.5780 | 1.4537 | 1.5983 | 1.6215 | 1.5366 | 1.3977 | 1.5576 | 0.7889 | 1.4779 | 1.4828 | 1.7395 | 1.3809 | 1.6678 | 1.5253 | |

Model C | 1.4426 | 1.4823 | 1.3921 | 1.4256 | 1.5368 | 1.4081 | 1.2703 | 1.4490 | 0.7731 | 1.4426 | 1.3404 | 1.5004 | 1.6744 | 1.5352 | 1.4146 | |

24 | Model A | 2.2812 | 1.8947 | 2.3251 | 2.3162 | 1.9561 | 2.3125 | 1.9756 | 2.1609 | 1.1198 | 2.1284 | 2.0778 | 2.1986 | 3.1313 | 2.3852 | 2.1339 |

Model B | 1.7227 | 1.5818 | 1.6329 | 1.5357 | 1.8184 | 1.5679 | 1.5059 | 1.6273 | 0.8113 | 1.6143 | 1.6001 | 1.6980 | 1.8959 | 1.7376 | 1.6158 | |

Model C | 1.6399 | 1.4829 | 1.4895 | 1.3902 | 1.5656 | 1.4349 | 1.3684 | 1.4726 | 0.7713 | 1.6399 | 1.4723 | 1.5999 | 2.1355 | 1.7070 | 1.4653 |

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**Figure 1.**(

**a**) Trend of the electricity demand profile: indicating the constant decreasing trend of demand. This is the averaged data sample of 2013–2015; (

**b**) variation of electricity demand for day types in Hokkaido Prefecture from 2013–2015.

**Figure 2.**(

**a**) Contour plot for electricity demand variation from 2013–2015; (

**b**) daily mean plot of the seasonal variation of electricity demand and temperature in 2013.

**Figure 4.**(

**a**) Non-linear relationship of electricity demand with temperature; (

**b**) piecewise linear approximation for the base temperature of CDD and HDD.

**Figure 6.**Mean, median, and 60 percentile of the forecasts using Model C versus the actual load, for the first week of January 2015.

**Figure 7.**(

**a**) Mean, median, and 60 percentile of the forecasts using Model B versus the actual load, for the first week of January 2015; (

**b**) mean, median, and 60 percentile of the forecasts using Model A versus the actual load, for the first week of January 2015.

**Figure 8.**Mean, median, and 60 percentile of the forecasts using Model C versus the actual load, for the first week of July 2015.

**Figure 9.**(

**a**) Mean, median, and 60 percentile of the forecasts using Model B versus the actual load, for the first week of July 2015; (

**b**) mean, median, and 60 percentile of the forecasts using Model A versus the actual load, for the first week of July 2015.

**Figure 11.**(

**a**) Variation of forecasting error at different hours for 2015 (Model C); (

**b**) percentile distribution of absolute forecasting error for 2015 (Model C).

**Figure 12.**(

**a**) Variation of forecasting error at different hours for 2015 (Model C); (

**b**) MAPE variation at different hours for 2015 (Model C).

**Figure 13.**(

**a**) Seasonal comparison of MAPE among three models; (

**b**) performance improvement of Model C with respect to Model A and Model B.

**Figure 14.**(

**a**) The performance of Models A, B, and C for June 2015; (

**b**) the performance of Models A, B, and C for September 2015.

**Figure 15.**(

**a**) Fluctuation of temperature in June 2015; (

**b**) fluctuation of weather variables in September 2015.

Variables | Winter (January) | Summer (August) |
---|---|---|

Temperature | −0.3495 | 0.5302 |

Rain | 0.0727 | −0.0078 |

Humid | 0.1385 | −0.4348 |

Radiation | −0.2321 | 0.4141 |

Snow | 0.1013 | No snowfall data |

Region | Electrical Data | Base Temperature (°C) | Relationship | Reference |
---|---|---|---|---|

USA | Weekly | 18.3 | Linear with CDD | [32] |

Spain | Daily (1983–1999) | 18.5 | Non-linear | [33] |

Italy | Hourly (2002–September 2003) | 18.7 (HDD), 22 (CDD) | Linear with CDD, HDD | [34] |

London | Hourly (1997–2001) | 16 | CDD HDD | [21] |

South Korea | Monthly (2001–2010) | 16.2–19.4 | CDD HDD | [35] |

Tokyo | Hourly | 15 (HDD), 21.3 (CDD) | Piecewise | [36] |

Tokyo | 2 p.m. data | 17.25 | Piecewise | [37] |

Brisbane | Half-hourly | 18.6 | Linear with CDD, HDD | [38] |

Sydney | Weekly (1999–2000) | 17.5 | Linear with CDD, HDD | [38] |

Melbourne | Weekly (1999–2000) | 16.9 | Linear with CDD, HDD | [38] |

Adelaide | Weekly (1999–2000) | 16.8 | Linear with CDD, HDD | [38] |

Hokkaido | Hourly (2013–2015) | 15.65 (HDD), 21.53 (CDD), 17.1 (min. demand) (2 p.m. data) | Piecewise | This study |

−10.2 (min. exhaustion), 33.18 (max. exhaustion), 16.28 (min. demand) | Polynomial | This study |

Temperature °C | Hour 1 | Hour 14 | Hour 19 | ||||||
---|---|---|---|---|---|---|---|---|---|

CDD_val | Deviation | $\mathit{\delta}\mathit{Demand}$ | CDD_val | Deviation | $\mathit{\delta}\mathit{Demand}$ | CDD_val | Deviation | $\mathit{\delta}\mathit{Demand}$ | |

19.000 | −5.3611 | −0.0178 | −1.7623 | −3.0000 | −0.0194 | −1.9206 | −1.1465 | 0.0068 | 0.6854 |

20.000 | −4.3611 | −0.0119 | −1.1783 | −2.0000 | −0.0134 | −1.3303 | −0.1465 | 0.0165 | 1.6629 |

21.000 | −3.3611 | −0.0059 | −0.5909 | −1.0000 | −0.0074 | −0.7365 | 0.8535 | 0.0262 | 2.6498 |

22.000 | −2.3611 | 0.0000 | 0.0000 | 0.0000 | −0.0014 | −0.1392 | 1.8535 | 0.0358 | 3.6463 |

23.000 | −1.3611 | 0.0059 | 0.5944 | 1.0000 | 0.0046 | 0.4618 | 2.8535 | 0.0455 | 4.6525 |

24.000 | −0.3611 | 0.0119 | 1.1924 | 2.0000 | 0.0106 | 1.0664 | 3.8535 | 0.0551 | 5.6684 |

25.000 | 0.6389 | 0.0178 | 1.7939 | 3.0000 | 0.0166 | 1.6746 | 4.8535 | 0.0648 | 6.6942 |

26.000 | 1.6389 | 0.0237 | 2.3990 | 4.0000 | 0.0226 | 2.2865 | 5.8535 | 0.0745 | 7.7299 |

27.000 | 2.6389 | 0.0296 | 3.0077 | 5.0000 | 0.0286 | 2.9021 | 6.8535 | 0.0841 | 8.7758 |

28.000 | 3.6389 | 0.0356 | 3.6200 | 6.0000 | 0.0346 | 3.5213 | 7.8535 | 0.0938 | 9.8317 |

29.000 | 4.6389 | 0.0415 | 4.2359 | 7.0000 | 0.0406 | 4.1443 | 8.8535 | 0.1034 | 10.8979 |

30.000 | 5.6389 | 0.0474 | 4.8555 | 8.0000 | 0.0466 | 4.7711 | 9.8535 | 0.1131 | 11.9745 |

31.000 | 6.6389 | 0.0533 | 5.4788 | 9.0000 | 0.0526 | 5.4016 | 10.8535 | 0.1228 | 13.0615 |

32.000 | 7.6389 | 0.0593 | 6.1058 | 10.0000 | 0.0586 | 6.0359 | 11.8535 | 0.1324 | 14.1591 |

Model Types | Overall MAPE (%) | Maximum MAPE (%) | MAPE ≤ 2% | MAPE > 2% | MAPE ≥ 5% | MAPE ≥ 10% | MAPE ≥ 15% | Observations | |
---|---|---|---|---|---|---|---|---|---|

Overall | Model A | 2.43 | 19.32 | 53.5 | 46.5 | 11.56 | 1.09 | 0.19 | 8760 |

Model B | 1.98 | 15.41 | 62.49 | 37.51 | 6.97 | 0.53 | 0.01 | ||

Model C | 1.72 | 14.06 | 67.93 | 32.07 | 4.05 | 0.23 | 0 | ||

Working days | Model A | 1.15 | 19.04 | 54.78 | 45.22 | 9.71 | 0.69 | 0.12 | 5784 |

Model B | 1.03 | 13.8 | 64.93 | 35.07 | 5.53 | 0.32 | 0 | ||

Model C | 0.90 | 13.35 | 70.35 | 29.65 | 2.71 | 0.172 | 0 | ||

Weekends | Model A | 2.49 | 11.93 | 52.87 | 47.13 | 12.66 | 0.92 | 0 | 2376 |

Model B | 2.03 | 11.75 | 60.56 | 39.44 | 7.82 | 0.21 | 0 | ||

Model C | 1.81 | 9.2746 | 66.28 | 33.71 | 5.57 | 0 | 0 | ||

Holidays | Model A | 3.52 | 19.33 | 43.5 | 56.5 | 25 | 5.66 | 1.66 | 600 |

Model B | 2.93 | 15.42 | 46.5 | 53.5 | 17.5 | 3.83 | 0.16 | ||

Model C | 2.51 | 14.06 | 51.17 | 48.83 | 11.33 | 1.66 | 0 |

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**MDPI and ACS Style**

Chapagain, K.; Kittipiyakul, S. Performance Analysis of Short-Term Electricity Demand with Atmospheric Variables. *Energies* **2018**, *11*, 818.
https://doi.org/10.3390/en11040818

**AMA Style**

Chapagain K, Kittipiyakul S. Performance Analysis of Short-Term Electricity Demand with Atmospheric Variables. *Energies*. 2018; 11(4):818.
https://doi.org/10.3390/en11040818

**Chicago/Turabian Style**

Chapagain, Kamal, and Somsak Kittipiyakul. 2018. "Performance Analysis of Short-Term Electricity Demand with Atmospheric Variables" *Energies* 11, no. 4: 818.
https://doi.org/10.3390/en11040818