# Development of Regression Models considering Time-Lag and Aerosols for Predicting Heating Loads in Buildings

Authors: Hong Soo Lim; Gon Kim
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

Building automation systems is becoming more vital, especially in regard to reduced building energy consumption. However, the accuracy of such systems in calculating building thermal loads is limited as they are unable to predict future thermal loads based on prevailing environmental factors. The current paper therefore seeks to improve the understanding of the interactions between outdoor meteorological data and building energy consumption through a statistical analysis. Using weather data collected by the Korean Meteorological Agency (KMA) over a period of three years (2011–2014), prediction models that are able to predict heating thermal loads considering the time-lag phenomenon are developed. In addition, the study develops different prediction models for buildings of different sizes. The results confirm the existence of the time-lag phenomenon: the heating load experienced by a building at a given time is better explained by a regression model developed using the climatic conditions that existed two hours before. As such, conventional building simulation programs must endeavor to include time-lag as well as Aerosol Optical Depth (AOD) data as important factors in the prediction of building heating loads.


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## Body

## 1. Introduction

Energy efficiency is of particular importance, especially in the building sector. The building sector consumes a significant amount of energy in comparison with other sectors [1]. As such, reducing the energy consumed by buildings has been at the core interest of building-related research.Over the years, various energy-saving systems have been developed and implemented. Research from multiple disciplines, such as IT (information technology) and mechanical engineering, are being utilized to improve the energy performance of buildings through building automation. For simplification, building automation can be divided into three categories: the building management system (BMS), the security system (SS), and the energy management system (EMS). The EMS is more closely associated with the energy-consuming units of the building, such as the heating, ventilating, and air conditioning (HVAC) systems and systems that require mechanical operations, such as elevators, automated doors, and so forth. The likes of such systems are essential in predicting the energy consumed by buildings in that they collect the energy usage patterns of a given building, which can be used later in the calculation of the total energy consumed by the building [2]. These systems are also helpful in understanding complex relationships between building occupants and their environments [3].Although systems that assist in the prediction of building energy performance already exist, such as the EMS and other related building simulation software, the process of accurately predicting the amount of energy consumed by a given space is rather complex [4]. It requires an extensive understanding of certain factors that influence how a building behaves. An example of such a factor is the outdoor climatic conditions. There is a plethora of studies that deal with the complex relationships between buildings and physical factors [5]. However, studies that deal with outdoor climatic conditions and their influence on building energy consumption are rather insufficient. The current study seeks to fill this gap and improve the understanding of the interaction between the prevailing outdoor elements and building energy consumption. In addition, this study seeks to present ways in which the newly found relationship between outdoor climatic conditions and building energy performance can be used to predict building heating thermal loads. Furthermore, this study considers aerosol weather data that were not previously taken into account when predicting building energy using computer simulation programs.


## 2. Energy Prediction in Buildings

Achieving significant reductions in the energy consumed by buildings requires knowledge about the thermal behavior of a given building. With the knowledge regarding building behavior and how a building interacts with its surroundings, the energy consumed by a building can then be predicted and subsequently reduced. Traditionally, building energy consumption could be predicted through mathematical equations. To do this, factors related to building thermal loads and their influence on the thermal performance of the building had to be extensively studied. Nowadays, new and simpler methods are constantly being developed to assist building professionals in their endeavors to optimize building designs and improve building energy performance [6]. Additionally, the estimation of building energy consumption has become a key factor in documenting energy consumed by buildings. However, it is not a simple task to estimate building energy consumption, as it involves various factors that are associated with the characteristics of buildings, such as the climatic conditions, occupant schedule, HVAC systems, and so forth [7].Recently, scientific studies have been focusing on human behavior and its influence on energy consumption by buildings. Erickson et al. alluded to the issue of modern buildings being conditioned by relying on the assumption of maximum occupancy and, thus, conditioning spaces unnecessarily. Subsequently, energy is utilized where it is not needed. In addition, the study presented a method by which energy occupancy models can be developed using a wireless sensor network integrated into building HVAC systems to ensure that spaces are not conditioned without occupancy [8]. Furthermore, Ahn and Park [9] explained how poor consideration of occupant behavior might lead to inconsistencies between the actual building energy data and the modeled building energy data. In the study, multiple approaches to optimize the occupant schedule data in building modelling were also discussed. Virote [10] also stresses the importance of considering occupant behavior and space occupancy in building energy modelling. Different occupant schedules are likely to lead to different energy usage patterns. In addition, Virote demonstrates the applicability of stochastic models to estimate occupant behavior and space occupancy.Various statistical modelling techniques are also playing a significant role in building modelling. A number of studies have developed models that either predict the amount of energy consumed by buildings or models that improve the accuracy of the building energy prediction software that already exists. For instance, a study by Ahmad et al. [11] has demonstrated how building electric energy can be predicted using artificial intelligence methods. His study discusses two common approaches used in deep learning-related predictions: support vector machine (SVM) and artificial neural network (ANN). In addition, the study shines light on new data-handling techniques being developed to improve building modelling and building behavior prediction. Some of these methods include least square support vector machine (LSSVM) and group method of data handling (GMDH). Ahmad concludes that there are advantages and disadvantages associated with each model, and it is thus difficult to pinpoint which of the modelling techniques is superior. Deep learning techniques have also been used to study other aspects of building modelling such as occupant behavior. Zhao et al. [12] developed a data-mining model that learns the behavior of occupants based on the office appliance usage. Grey-box models have also been predominantly used in studying short-term behavior of buildings [13].Li et al. [14] also introduced an advanced deep learning approach to increase accuracy in building energy consumption predictions. The introduced method is a combination of two approaches: stacked autoencoders (SAEs) and extreme learning machine (ELM). The SAE is used in studying the building’s energy usage patterns whereas the ELM is used as the prediction learning tool. The proposed approach shows better performance when compared with other deep learning techniques such as support vector regression (SVR), generalized radial basis function neural network (GRBFNN) and multiple linear regression (MLR). Sensor-based machine learning techniques are also widely applied in forecasting building energy consumption. Such techniques require less input data than conventional statistical methods and are thus less complex [15]. Fan et al. [16] discuss the application of data-mining techniques in developing a set of predictive models that estimate next-day energy consumption and peak power demand. The developed ensemble models show a much higher performance in comparison with single-base models.Among statistical approaches used in building energy estimation, regression analysis has shown significant outcomes. Regression analysis is widely adopted because it is easy to use and provides reasonably accurate results. Braun et al. used regression models to predict the future electricity and gas consumption of a supermarket in northern England, obtaining theoretically accurate results [17]. Similarly, Tso and Yau also developed regression models to predict the electricity consumed by buildings [18]. Lam et al. further highlighted the usefulness of regression analysis to predict building behavior in the early design stage of buildings [19]. The accuracy of regression analysis in predicting the behavior of buildings has been so far outstanding. Regression models have been used to estimate the heating energy demand in buildings, obtaining accurate results with R-values as high as 0.987 [20] and 0.90 [21]. The accuracy of regression models to predict building energy consumption is constantly being improved through continuous research. For instance, Fumo and Rafe Biswas [22] point out the importance of the time interval of measured data in regard to the quality of produced regression models. Similarly, the current paper seeks to utilize regression models to improve accuracy in the prediction of building thermal heating loads by considering the time-lag phenomenon and a new climatic factor, aerosols. Both of these factors have not been studied extensively in regard to building energy modelling through regression analysis.


## 3. Methodology

### 3.1. Weather Data and Energy Predictions in Buildings

Predicting the amount of energy used by a given space requires a number of factors. One of the most important factors is the weather data. Previous studies [23] have shown indoor climatic conditions and outdoor climatic conditions to be correlated in a directly proportional manner. As such, existing outdoor climatic conditions have a significant influence on how a building occupant behaves; for instance, if it is cold outside, the occupant is more likely to activate the heating system. Many energy prediction software programs have methods in which information about the climatic conditions of an area is entered into the program for analysis. In cases where real-time data from weather stations are unavailable, weather data can be obtained from online databases. Some types of weather data are even derived from developed mathematical equations. However, weather data obtained using methods other than actual data from weather stations have been known to cause inconsistencies in the predicted energy results. In fact, the Lawrence Berkeley National Laboratory (LBNL) attributes the differences between the actual amount of energy consumed by a building and the amount of energy predicted by simulation software to be due to inaccuracies in the weather data used [24]. Due to the amount of climatic conditions used in simulation software to predict energy used by a building, there have been attempts [25] to use statistical methods such as the Taguchi method to sieve out the least important climatic factors and thus reduce the number of input parameters that describe weather conditions. In a similar study, Kapetanakis et al. [26] used data from Building Energy Management System (BEMS) to examine and subsequently provide appropriate input weather valuables necessary for predicting building heating thermal loads. Different energy prediction software also uses different formats of weather data [27]. This is mainly because such software was developed using different algorithms and might not be able to process a weather data file with a different format. Consequently, the current study converted the format of the original data (data obtained from KMA) into a format that can be read by EnergyPlus, “epw.” Table 1 shows the weather data used within this study.Table 1
Weather data and aerosol data considered in this study.
Name
Units
Used by EnergyPlus
Dry bulb temperature
°C
Provided by KMA
Wet bulb temperature
°C
Calculated by equation
Dew point temperature
°C
Provided by KMA
Humidity ratio
g/m3
Provided by KMA
Relative humidity
%
Provided by KMA
Atmospheric pressure
Pa
Provided by KMA
Extraterrestrial horizontal radiation
Wh/m2
Calculated by equation
Extraterrestrial direct normal radiation
Wh/m2
Calculated by equation
Solar altitude
Degrees
Calculated by equation
Horizontal infrared radiation intensity from the sky
Wh/m2
Calculated by equation
Global horizontal radiation
Wh/m2
Provided by KMA
Direct normal radiation
Wh/m2
Calculated by equation
Diffuse horizontal radiation
Wh/m2
Calculated by equation
Wind direction
Degrees
Provided by KMA
Wind speed
m/s
Provided by KMA
Total sky cover
Tenths
Provided by KMA
Visibility
km
Provided by KMA
Precipitable water
mm
Measured by AERONET
Aerosol optical depth
Thousandths
Measured by AERONET


### 3.2. The Time-Lag Phenomenon

The time-lag phenomenon is being studied in the current paper to address the inconsistencies brought upon by the usage of weather data in building energy predictions. Theoretically, it is usually assumed that indoor climatic conditions are a direct representation of outdoor climatic conditions, especially in passive buildings that lack mechanical systems to adjust indoor environmental conditions. In reality, however, outdoor climatic conditions at a given time are not accurately matched by indoor climatic conditions at that same time. The term “time-lag” is being used to represent the delay that occurs between peak energy usage by a given building and peak outdoor climatic conditions. Figure1 gives an example of the time-lag phenomenon. For May 13, 2011, the peak radiation from the sun, peak temperature, and peak cooling load occurred at approximately 1 pm, 3 pm, and 4 pm, respectively. Theoretically, the peak cooling load should occur at the same time as the peak outdoor temperature, but in reality, there was a delay of 1 hour between the time the peak temperature occurred and the time the peak cooling load occurred. Moon and Kim [28] discussed in detail the issue of time-lag in buildings.Figure 1
Time-lag phenomenon.


### 3.3. Reference Building

The current study used the U.S. Department of Energy (DOE) commercial buildings as case models for analysis. The DOE reference models are standard models used for energy performance assessments in building-related studies [29]. The reference building models incorporate a total of 16 building types and 16 different geographical locations in the U.S. They represent approximately 60% of the characteristics usually found within commercial buildings [30]. Table 2 shows the DOE reference buildings used.Table 2
DOE commercial reference buildings.
U.S. Department of Energy commercial reference building models of the national building stock
Small office
Medium office
Large officeFurthermore, ASHRAE specifies building material according to different climatic conditions. South Korea, which is the base area of this study, falls under ASHRAE climatic zones 4A and 4B; the building materials used are in line with ASHRAE standard 90.1–2004 to match the climatic conditions of the location under study. It is worth noting that the regression equations determined through this study might need revisions for buildings made of different materials.


### 3.4. Research Process

As mentioned in the previous sections of this study, the current paper seeks to develop heating load prediction models while considering the time-lag phenomenon. Multiple regression modelling is used to study the relationship between the outdoor climatic conditions and heating load in buildings of three different sizes while considering time-lag. Ideally, the purpose is to find which weather data best describes the heating load at a given time. To do that, EnergyPlus is made to output hourly data in three phases. First, when the time-lag is zero (time-lag0), meaning that the output results from EnergyPlus are for a time that is in sync with the time of the weather data used. Second, when the time-lag is one (time-lag1), meaning that the output results from EnergyPlus are for a time that is one hour ahead of the time of the weather data used. Finally, when the time-lag is two (time-lag2), meaning that the output results from EnergyPlus are for a time that is two hours ahead of the time of the weather data used. This procedure is done for each building size (small, medium, and large DOE reference buildings). Figure2 below explains the study procedure.Figure 2
Study methodology workflow.


## 3.1. Weather Data and Energy Predictions in Buildings

Predicting the amount of energy used by a given space requires a number of factors. One of the most important factors is the weather data. Previous studies [23] have shown indoor climatic conditions and outdoor climatic conditions to be correlated in a directly proportional manner. As such, existing outdoor climatic conditions have a significant influence on how a building occupant behaves; for instance, if it is cold outside, the occupant is more likely to activate the heating system. Many energy prediction software programs have methods in which information about the climatic conditions of an area is entered into the program for analysis. In cases where real-time data from weather stations are unavailable, weather data can be obtained from online databases. Some types of weather data are even derived from developed mathematical equations. However, weather data obtained using methods other than actual data from weather stations have been known to cause inconsistencies in the predicted energy results. In fact, the Lawrence Berkeley National Laboratory (LBNL) attributes the differences between the actual amount of energy consumed by a building and the amount of energy predicted by simulation software to be due to inaccuracies in the weather data used [24]. Due to the amount of climatic conditions used in simulation software to predict energy used by a building, there have been attempts [25] to use statistical methods such as the Taguchi method to sieve out the least important climatic factors and thus reduce the number of input parameters that describe weather conditions. In a similar study, Kapetanakis et al. [26] used data from Building Energy Management System (BEMS) to examine and subsequently provide appropriate input weather valuables necessary for predicting building heating thermal loads. Different energy prediction software also uses different formats of weather data [27]. This is mainly because such software was developed using different algorithms and might not be able to process a weather data file with a different format. Consequently, the current study converted the format of the original data (data obtained from KMA) into a format that can be read by EnergyPlus, “epw.” Table 1 shows the weather data used within this study.Table 1
Weather data and aerosol data considered in this study.
Name
Units
Used by EnergyPlus
Dry bulb temperature
°C
Provided by KMA
Wet bulb temperature
°C
Calculated by equation
Dew point temperature
°C
Provided by KMA
Humidity ratio
g/m3
Provided by KMA
Relative humidity
%
Provided by KMA
Atmospheric pressure
Pa
Provided by KMA
Extraterrestrial horizontal radiation
Wh/m2
Calculated by equation
Extraterrestrial direct normal radiation
Wh/m2
Calculated by equation
Solar altitude
Degrees
Calculated by equation
Horizontal infrared radiation intensity from the sky
Wh/m2
Calculated by equation
Global horizontal radiation
Wh/m2
Provided by KMA
Direct normal radiation
Wh/m2
Calculated by equation
Diffuse horizontal radiation
Wh/m2
Calculated by equation
Wind direction
Degrees
Provided by KMA
Wind speed
m/s
Provided by KMA
Total sky cover
Tenths
Provided by KMA
Visibility
km
Provided by KMA
Precipitable water
mm
Measured by AERONET
Aerosol optical depth
Thousandths
Measured by AERONET


## 3.2. The Time-Lag Phenomenon

The time-lag phenomenon is being studied in the current paper to address the inconsistencies brought upon by the usage of weather data in building energy predictions. Theoretically, it is usually assumed that indoor climatic conditions are a direct representation of outdoor climatic conditions, especially in passive buildings that lack mechanical systems to adjust indoor environmental conditions. In reality, however, outdoor climatic conditions at a given time are not accurately matched by indoor climatic conditions at that same time. The term “time-lag” is being used to represent the delay that occurs between peak energy usage by a given building and peak outdoor climatic conditions. Figure1 gives an example of the time-lag phenomenon. For May 13, 2011, the peak radiation from the sun, peak temperature, and peak cooling load occurred at approximately 1 pm, 3 pm, and 4 pm, respectively. Theoretically, the peak cooling load should occur at the same time as the peak outdoor temperature, but in reality, there was a delay of 1 hour between the time the peak temperature occurred and the time the peak cooling load occurred. Moon and Kim [28] discussed in detail the issue of time-lag in buildings.Figure 1
Time-lag phenomenon.


## 3.3. Reference Building

The current study used the U.S. Department of Energy (DOE) commercial buildings as case models for analysis. The DOE reference models are standard models used for energy performance assessments in building-related studies [29]. The reference building models incorporate a total of 16 building types and 16 different geographical locations in the U.S. They represent approximately 60% of the characteristics usually found within commercial buildings [30]. Table 2 shows the DOE reference buildings used.Table 2
DOE commercial reference buildings.
U.S. Department of Energy commercial reference building models of the national building stock
Small office
Medium office
Large officeFurthermore, ASHRAE specifies building material according to different climatic conditions. South Korea, which is the base area of this study, falls under ASHRAE climatic zones 4A and 4B; the building materials used are in line with ASHRAE standard 90.1–2004 to match the climatic conditions of the location under study. It is worth noting that the regression equations determined through this study might need revisions for buildings made of different materials.


## 3.4. Research Process

As mentioned in the previous sections of this study, the current paper seeks to develop heating load prediction models while considering the time-lag phenomenon. Multiple regression modelling is used to study the relationship between the outdoor climatic conditions and heating load in buildings of three different sizes while considering time-lag. Ideally, the purpose is to find which weather data best describes the heating load at a given time. To do that, EnergyPlus is made to output hourly data in three phases. First, when the time-lag is zero (time-lag0), meaning that the output results from EnergyPlus are for a time that is in sync with the time of the weather data used. Second, when the time-lag is one (time-lag1), meaning that the output results from EnergyPlus are for a time that is one hour ahead of the time of the weather data used. Finally, when the time-lag is two (time-lag2), meaning that the output results from EnergyPlus are for a time that is two hours ahead of the time of the weather data used. This procedure is done for each building size (small, medium, and large DOE reference buildings). Figure2 below explains the study procedure.Figure 2
Study methodology workflow.


## 4. Results

### 4.1. Multiple Regression Analysis for the Prediction of the Heating Load for a Small Office considering Time-Lag

To estimate the heating load of a small office building, 20,352 weather data points were used. In addition, the influence of outdoor climatic conditions on the heating load was assessed through multiple regression analysis. Without considering time-lag, the heating load values extracted from EnergyPlus after one hour and two hours were matched separately by hour during the analysis. The results show R-values of the time-lag0 model, time-lag1 model, and time-lag2 model to be 0.534, 0.542, and 0.592, respectively. In addition, the R-square values of time-lag0, time-lag1, and time-lag2 were 0.285, 0.294, and 0.350, respectively; this implies that the regression model for the DOE small office has a higher explanatory or predictive power when using a time-lag of two hours. Table3 shows the results of the small office model summary considering time-lag.Table 3
DOE small office model summary considering time-lag.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Time-lag0
0.534b
0.285
0.285
7463851.345
0.285
737.844
11
20,340
0.000
0.799
Time-lag1
0.542b
0.294
0.294
7417381.535
0.294
770.449
11
20,339
0.000
0.834
Time-lag2
0.592b
0.350
0.350
7116200.437
0.350
996.953
11
20,338
0.000
0.778
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.TheF-variations for all the developed models were higher than 700, and the p values were all less than 0.001, thus confirming the statistical significance of the models. Furthermore, the Durbin–Watson test was conducted, and the results were 0.799, 0.834, and 0.778 for the time-lag0, time-lag1, and time-lag2 models, respectively; this means that among the three developed models, the time-lag1 model has a higher estimating power as it has a higher R-value and Durbin–Watson value.The results of the ANOVA analysis shown in Table4 indicate a p value less than 0.05 for all the models; thus, all the models are assumed to be statistically significant. The time-lag2 model showed the largest variance, whereas the time-lag0 model showed the largest variance of residue. Additionally, the F-value belonging to the time-lag2 model was the highest among the three developed models, thus indicating that the time-lag2 model has the best fit.Table 4
ANOVA for a DOE small building depending on time-lag.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Time-lag0
Regression
452150586586186300.000
11
41104598780562392.000
737.844
0.000b
Residual
1133122624135237120.000
20,340
55709076899470.850
—
—
Total
1585273210721423360.000
20,351
—
—
—
Time-lag1
Regression
466270598690733180.000
11
42388236244612096.000
770.449
0.000b
Residual
1119001925868610430.000
20,339
55017548840582.650
—
—
Total
1585272524559343620.000
20,350
—
—
—
Time-lag2
Regression
555346313497669060.000
11
50486028499788096.000
996.953
0.000b
Residual
1029922597401856130.000
20,338
50640308653842.860
—
—
Total
1585268910899525120.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table5 shows the tolerance limit, significance of probability, and t-value of the variables for each model. For accuracy, if a high goodness-of-fit is found for the independent variables, the t-value and p value should meet certain criteria: t-value ≥ 1.96 and p<0.05. In addition, to avoid issues related to multicollinearity, the tolerance limit should be 0.1 or higher; variables that do not satisfy these conditions are usually omitted from the analysis to improve the accuracy of the developed models.Table 5
Regression coefficients for a DOE small office building depending on time-lag.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Time-lag0
(Constant)
−173421890.552
11828102.396
—
−14.662
0.000
—
—
Dry bulb temperatures
−577018.579
43,428.001
−0.540
−13.287
0.000
0.021
47.057
Dew point temperatures
−185925.394
46,080.321
−0.195
−4.035
0.000
0.015
66.370
Relative humidity
27,650.459
12,842.837
0.061
2.153
0.031
0.044
22.624
Atmospheric station pressure
1668.517
114.145
0.117
14.618
0.000
0.553
1.809
Horizontal infrared radiation
35,821.811
1622.471
1.031
22.079
0.000
0.016
62.031
Global horizontal radiation
−-6515.011
1386.208
−0.144
−4.700
0.000
0.038
26.628
Direct normal radiation
4723.975
747.304
0.113
6.321
0.000
0.109
9.146
Diffuse radiation
11,820.058
1778.320
0.128
6.647
0.000
0.095
10.484
Wind speed
−294501.046
40,293.443
−0.050
−7.309
0.000
0.753
1.328
Total sky cover
−2154444.294
95,068.697
−0.957
−22.662
0.000
0.020
50.797
Visibility
123613.977
102564.923
0.009
1.205
0.228
0.669
1.495
Time-lag1
(Constant)
−181234643.381
11755825.707
—
−15.417
0.000
—
—
Dry bulb temperatures
−614138.021
43,157.633
−0.575
−14.230
0.000
0.021
47.050
Dew point temperatures
−68945.791
45,794.132
−0.072
−1.506
0.132
0.015
66.368
Relative humidity
−1938.780
12,763.123
−0.004
−0.152
0.879
0.044
22.625
Atmospheric station pressure
1780.718
113.449
0.124
15.696
0.000
0.553
1.809
Horizontal infrared radiation
29,704.944
1612.435
0.855
18.422
0.000
0.016
62.032
Global horizontal radiation
−24780.315
1377.584
−0.547
−17.988
0.000
0.038
26.627
Direct normal radiation
17,040.575
742.657
0.409
22.945
0.000
0.109
9.146
Diffuse radiation
32,792.426
1767.248
0.354
18.556
0.000
0.095
10.484
Wind speed
−348289.011
40,047.290
−0.059
−8.697
0.000
0.753
1.328
Total sky cover
−1728577.821
94,481.810
−0.768
−18.295
0.000
0.020
50.800
Visibility
−64173.273
101930.905
−0.005
−0.630
0.529
0.669
1.495
Time-lag2
(Constant)
−179888432.974
11279313.576
—
−15.949
0.000
—
—
Dry bulb temperatures
−685748.939
41,405.258
−0.642
−16.562
0.000
0.021
47.042
Dew point temperatures
86,402.645
43,935.195
0.091
1.967
0.049
0.015
66.365
Relative humidity
−47343.593
12,245.089
−0.104
−3.866
0.000
0.044
22.625
Atmospheric station pressure
1811.848
108.852
0.127
16.645
0.000
0.553
1.810
Horizontal infrared radiation
24,711.629
1547.029
0.711
15.974
0.000
0.016
62.033
Global horizontal radiation
−49815.842
1321.653
−1.099
−37.692
0.000
0.038
26.627
Direct normal radiation
34,307.818
712.510
0.823
48.151
0.000
0.109
9.146
Diffuse radiation
61,552.875
1695.489
0.664
36.304
0.000
0.095
10.484
Wind speed
−371456.480
38,421.767
−0.063
−9.668
0.000
0.753
1.328
Total sky cover
−1367478.673
90,650.584
−0.608
−15.085
0.000
0.020
50.803
Visibility
−544550.045
97,794.601
−0.038
−5.568
0.000
0.669
1.495
aDependent variable: heating load of a DOE small office building.Consequently, visibility was omitted from the time-lag0 model, whereas the dew point temperature, relative humidity, and visibility were also removed from the time-lag1 model; these variables are thus absent from the final models.As indicated by the results after taking into account thet-value, significance of the probability, and multicollinearity issues, the time-lag2 model was found to be the most suitable for a DOE small office building; this model is defined as the “Modified Model” and is presented in Table 6. Secondly, another model is developed based on all the variables considered in the time-lag2 model, AOD, and precipitable water values; this model is defined as the “New Model,” and it is also presented in Table 6. Additionally, a third model based on only those variables received from the Korea Meteorological Administration (KMA) and which can be input by the user into EnergyPlus was developed considering a time-lag of 2 hours; this third model is defined as the “Limited Model” and is shown in Table 6.Table 6
Model summary for a DOE small office building depending on different models.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Modified modelb
0.592b
0.350
0.350
7116200.437
0.350
996.953
11
20,338
0.000
0.778
New modelc
0.422c
0.178
0.176
3742573.281
0.178
130.924
8
4845
0.000
1.509
Limited modeld
0.513d
0.264
0.263
7575212.077
0.264
1213.784
6
20,343
0.000
0.767
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.A multiple regression analysis was performed to predict the heating thermal load. As indicated by the obtained results, the “New Model” showed the minimumR-squared (0.422) value and the highest Durbin–Watson value (1.509); in this case, a Durbin–Watson value above 1.3 indicates a weak correlation between the variables. In addition, although the R-squared values for the “Modified Model” and “Limited Model” are higher than that of the “New Model,” multicollinearity issues were found for the two models.TheF-variation of the “New Model” was shown to be higher than 170 at p<0.001. Furthermore, the variance of the regression model and residual were smaller than those of both the “Modified Model” and “Limited Model.” The F-value was shown to be highest in the “Limited Model.” Additionally, the results predicted by the “New Model” exhibited the smallest mean and standard deviation, as well as the smallest difference between the maximum and minimum predictions (Tables 7–9). A regression equation for the “New Model” can be drafted based on the final variables listed in Table 6.Table 7
ANOVA for a DOE small office building depending on different models.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Modified modelb
Regression
555346313497669060.000
11
50486028499788096.000
996.953
0.000b
Residual
1029922597401856130.000
20,338
50640308653842.860
—
—
Total
1585268910899525120.000
20,349
—
—
—
New modelc
Regression
14670632319938900.000
8
1833829039992363.000
130.924
0.000c
Residual
67863211318381720.000
4845
14006854761275.896
—
—
Total
82533843638320624.000
4853
—
—
—
Limited modeld
Regression
417909494247049020.000
6
69651582374508168.000
1213.784
0.000d
Residual
1167359416652476160.000
20,343
57383838010739.625
—
—
Total
1585268910899525120.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, relative humidity, atmospheric station pressure, diffuse radiation, dew point temperatures, global horizontal radiation, and horizontal infrared radiation; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.Table 8
Residual statistics for a DOE small office building depending on different models.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Modified modelb
Predicted value
−16410367.00
28590780.00
3786390.10
5224087.118
20,350
Residual
−20649606.000
64699612.000
0.000
7114276.785
20,350
Std. predicted value
−3.866
4.748
0.000
1.000
20,350
Std. residual
−2.902
9.092
0.000
1.000
20,350
New modelc
Predicted value
−2313629.00
13152829.00
1123572.72
1738678.448
4854
Residual
−10689199.000
46347212.000
0.000
3739487.258
4854
Std. predicted value
−1.977
6.919
0.000
1.000
4854
Std. residual
−2.856
12.384
0.000
0.999
4854
Limited modeld
Predicted value
−8359116.00
17571524.00
3786390.10
4531787.978
20,350
Residual
−15785886.000
70738864.000
0.000
7574095.201
20,350
Std. predicted value
−2.680
3.042
0.000
1.000
20,350
Std. residual
−2.084
9.338
0.000
1.000
20,350
aDependent variable: heating load of a DOE small office building. bPredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, relative humidity, atmospheric station pressure, diffuse radiation, dew point temperatures, global horizontal radiation, and horizontal infrared radiation; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.Table 9
Regression coefficients for a DOE small office building depending on different models.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Modified model
(Constant)
−179888432.974
11279313.576
—
−15.949
0.000
—
—
Dry bulb temperatures
−685748.939
41,405.258
−0.642
−16.562
0.000
0.021
47.042
Dew point temperatures
86,402.645
43,935.195
0.091
1.967
0.049
0.015
66.365
Relative humidity
−47343.593
12,245.089
−0.104
−3.866
0.000
0.044
22.625
Atmospheric station pressure
1811.848
108.852
0.127
16.645
0.000
0.553
1.810
Horizontal infrared radiation
24,711.629
1547.029
0.711
15.974
0.000
0.016
62.033
Global horizontal radiation
−49815.842
1321.653
−1.099
−37.692
0.000
0.038
26.627
Direct normal radiation
34,307.818
712.510
0.823
48.151
0.000
0.109
9.146
Diffuse radiation
61,552.875
1695.489
0.664
36.304
0.000
0.095
10.484
Wind speed
−371456.480
38,421.767
−0.063
−9.668
0.000
0.753
1.328
Total sky cover
−1367478.673
90,650.584
−0.608
−15.085
0.000
0.020
50.803
Visibility
−544550.045
97,794.601
−0.038
−5.568
0.000
0.669
1.495
New model
(Constant)
−84468362.750
11178380.426
—
−7.556
0.000
—
—
Dew point temperatures
−153900.636
11,782.451
−0.332
−13.062
0.000
0.263
3.800
Atmospheric station pressure
820.654
109.775
0.123
7.476
0.000
0.626
1.599
Horizontal infrared radiation
7302.493
1652.577
0.423
4.419
0.000
0.019
54.035
Global horizontal radiation
−28329.843
5547.379
−0.340
−5.107
0.000
0.038
26.094
Direct normal radiation
22,573.966
1829.080
0.430
12.342
0.000
0.140
7.149
Diffuse radiation
31,803.161
6859.159
0.208
4.637
0.000
0.084
11.876
Total sky cover
−380750.599
94,783.568
−0.349
−4.017
0.000
0.023
44.375
AOD
−215562.688
100939.898
−0.029
−2.136
0.033
0.945
1.059
Limited model
(Constant)
−198730470.861
11744538.154
—
−16.921
0.000
—
—
Dry bulb temperatures
−717138.604
42,245.667
−0.671
−16.975
0.000
0.023
43.216
Dew point temperatures
296105.476
45,462.975
0.310
6.513
0.000
0.016
62.710
Relative humidity
−90189.068
12,260.853
−0.198
−7.356
0.000
0.050
20.018
Atmospheric station pressure
2096.977
113.658
0.146
18.450
0.000
0.574
1.741
Global horizontal radiation
8298.520
340.007
0.183
24.407
0.000
0.643
1.555
Wind speed
−241986.775
40,515.794
−0.041
−5.973
0.000
0.767
1.303
aDependent variable: heating load of a DOE small office building.


### 4.2. Multiple Regression Analysis for the Prediction of Heating Load for a Medium Office considering Time-Lag

For a DOE medium office building, 20,352 weather data points were used to estimate the heating load. In addition, the influence of the outdoor climatic conditions on the heating load was assessed through multiple regression analysis. Without considering time-lag, heating load values extracted from EnergyPlus after one hour and two hours were matched separately by hour during the analysis. The results of the analysis together with the input variables describing the climatic conditions are shown in Tables10–13. The results show the R-values of the time-lag0 model, time-lag1 model, and time-lag2 model to be 0.537, 0.537, and 0.579, respectively. In addition, the R-square values of time-lag0, time-lag1, and time-lag2 were 0.289, 0.289, and 0.335, respectively; this implies that the regression model for the DOE medium office has a higher explanatory or predictive power when using a time-lag of two hours. Table 10 shows the model summaries of a DOE medium office building considering time-lag.Table 10
Model summary for a DOE medium office building depending on time-lag.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Time-lag0
0.537b
0.289
0.288
12945018.511
0.289
750.955
11
20,340
0.000
0.706
Time-lag1
0.537b
0.289
0.288
12947428.028
0.289
750.070
11
20,339
0.000
0.731
Time-lag2
0.579b
0.335
0.335
12518253.739
0.335
931.428
11
20,338
0.000
0.677
aDependent variable: heating load (medium); bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 11
Residual summary for a DOE medium office building depending on time-lag.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Time-lag0
Regression
1384242517670826240.000
11
125840228879166016.000
750.955
0.000b
Residual
3408445076552884200.000
20,340
167573504255304.030
—
—
Total
4792687594223710200.000
20,351
—
—
—
Time-lag1
Regression
1383124549966582780.000
11
125738595451507520.000
750.070
0.000b
Residual
3409546418166475800.000
20,339
167635892529941.300
—
—
Total
4792670968133058600.000
20,350
—
—
—
Time-lag2
Regression
1605570375500446210.000
11
145960943227313184.000
931.428
0.000b
Residual
3187100390118337000.000
20,338
156706676670190.620
—
—
Total
4792670765618783200.000
20,349
—
—
—
aDependent variable: heating load of DOE medium office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 12
Model summary for a DOE medium office building depending on time-lag.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. Error
Beta
Tolerance
VIF
Time-lag0
(Constant)
−232365034.136
20514208.737
—
−11.327
0.000
—
—
Dry bulb temperatures
−1239414.048
75,319.865
−0.667
−16.455
0.000
0.021
47.057
Dew point temperatures
−354626.266
79,919.948
−0.214
−4.437
0.000
0.015
66.370
Relative humidity
66,371.259
22,274.126
0.084
2.980
0.003
0.044
22.624
Atmospheric station pressure
2116.094
197.968
0.085
10.689
0.000
0.553
1.809
Horizontal infrared radiation
99,198.849
2813.952
1.642
35.252
0.000
0.016
62.031
Global horizontal radiation
−3376.020
2404.186
−0.043
−1.404
0.160
−0.038
26.628
Direct normal radiation
788.013
1296.095
0.011
0.608
0.543
0.109
9.146
Diffuse radiation
13,477.568
3084.249
0.084
4.370
0.000
0.095
10.484
Wind speed
−479633.798
69,883.408
−0.047
−6.863
0.000
0.753
1.328
Total sky cover
−6135755.540
164883.515
−1.568
−37.213
0.000
0.020
50.797
Visibility
408798.727
177884.682
0.017
2.298
0.022
0.669
1.495
Time-lag1
(Constant)
−242363953.699
20520409.598
—
−11.811
0.000
—
—
Dry bulb temperatures
−1289408.649
75,333.909
−0.694
−17.116
0.000
0.021
47.050
Dew point temperatures
−155279.237
79,936.057
−0.094
−1.943
0.052
0.015
66.368
Relative humidity
16,763.249
22,278.700
0.021
0.752
0.452
0.044
22.625
Atmospheric station pressure
2277.502
198.032
0.091
11.501
0.000
0.553
1.809
Horizontal infrared radiation
87,070.308
2814.591
1.441
30.935
0.000
0.016
62.032
Global horizontal radiation
−39853.596
2404.644
−0.506
−16.574
0.000
0.038
26.627
Direct normal radiation
25,968.009
1296.347
0.358
20.032
0.000
0.109
9.146
Diffuse radiation
54,012.154
3084.824
0.335
17.509
0.000
0.095
10.484
Wind speed
−616322.819
69,904.642
−0.060
−8.817
0.000
0.753
1.328
Total sky cover
−5285555.158
164922.949
−1.351
−32.049
0.000
0.020
50.800
Visibility
35,564.622
177925.734
0.001
0.200
0.842
0.669
1.495
Time-lag2
(Constant)
−242862773.124
19841671.213
—
−12.240
0.000
—
—
Dew point temperatures
102726.330
77,287.300
0.062
1.329
0.184
0.015
66.365
Relative humidity
−57775.571
21,540.586
−0.073
−2.682
0.007
0.044
22.625
Atmospheric station pressure
2360.095
191.484
0.095
12.325
0.000
0.553
1.810
Horizontal infrared radiation
76,361.042
2721.411
1.264
28.059
0.000
0.016
62.033
Global horizontal radiation
−82461.104
2324.947
−1.047
−35.468
0.000
0.038
26.627
Direct normal radiation
56,837.692
1253.391
0.784
45.347
0.000
0.109
9.146
Diffuse radiation
101106.744
2982.570
0.628
33.899
0.000
0.095
10.484
Total sky cover
−4510822.624
159465.297
−1.153
−28.287
0.000
0.020
50.803
Dry bulb temperatures
−1387805.281
72,836.836
−0.747
−19.054
0.000
0.021
47.042
Wind speed
−677958.805
67,588.517
−0.066
−10.031
0.000
0.753
1.328
Visibility
−799488.056
172032.483
−0.032
−4.647
0.000
0.669
1.495
aDependent variable: heating load of a DOE medium office building.Table 13
Residual statistics for a DOE medium office building depending on time-lag.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Time-lag0b
Predicted value
−59638412.00
34621280.00
5299548.66
8247327.016
20,352
Residual
−31375230.000
142983696.000
0.000
12941519.557
20,352
Std. predicted value
−7.874
3.555
0.000
1.000
20,352
Std. residual
−2.424
11.045
0.000
1.000
20,352
Time-lag1b
Predicted value
−53244212.00
40414716.00
5299348.31
8244198.466
20,351
Residual
−28412308.000
145224064.000
0.000
12943928.250
20,351
Std. predicted value
−7.101
4.259
0.000
1.000
20,351
Std. residual
−2.194
11.216
0.000
1.000
20,351
Time-lag2b
Predicted value
−47442988.00
48511468.00
5299326.20
8882662.010
20,350
Residual
−35604016.000
136248016.000
0.000
12514869.803
20,350
Std. predicted value
−5.938
4.865
0.000
1.000
20,350
Std. residual
−2.844
10.884
0.000
1.000
20,350
aDependent variable: heating load of a DOE medium office building. bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.TheF-variation for all the developed models was higher than 750, and the p values were all less than 0.001, thus confirming the statistical significance of the models. Furthermore, the Durbin–Watson test was conducted, and the results were 0.706, 0.731, and 0.677 for the time-lag0 model, time-lag1 model, and time-lag2 model, respectively; this means that among the three developed models, the time-lag1 model has the highest estimating power as it had the highest R-value and Durbin–Watson value, as shown in Table 11.The results of the ANOVA analysis indicate ap value less than 0.05 for all models, and thus, all the models are assumed to be statistically significant. The time-lag2 model showed the largest variance with the smallest variance in the residuals. In addition, the F-value belonging to the time-lag2 model was the highest among the three developed models and thus indicated that the time-lag2 model has the best fit.Table12 shows the tolerance limit, significance of the probability, and t-value of the variables for each model. For accuracy, if a high goodness-of-fit is found for the independent variables, the t-value and p value should meet certain criteria: t-value ≥ 1.96 and p<0.05. In addition, to avoid issues related to multicollinearity, the tolerance limit should be 0.1 or higher; variables that do not satisfy these conditions are usually omitted from the analysis to improve the accuracy of the developed models.Since the time-lag phenomenon was not taken into consideration in the time-lag0 model, the most important variables when it comes to thermal conditions, such as global horizontal radiation and direct normal radiation, failed to satisfy the conditions. The dew point temperature and relative humidity in the time-lag1 model and the dew point temperature in the time-lag2 model were the variables found to be rejected from each model. These variables are thus absent from the final models. Table13 shows residual statistics for a DOE medium office building considering time-lag.As indicated by the results after taking into account thet-value, significance of the probability, and multicollinearity issues, the time-lag2 model was found to be the most suitable for a DOE medium office building; this model is defined as the “Modified Model” and is presented in Table 14. Secondly, another model is developed based on all the variables considered in the time-lag2 model, AOD, and precipitable water values; this model is defined as the “New Model,” and it is also presented in Table 14. Additionally, a third model based on only those variables received from the Korea Meteorological Administration (KMA) and which can be input by the user into EnergyPlus was developed considering a time-lag of 2 hours; this third model is defined as the “Limited Model.”Table 14
Model summary for a DOE medium office building depending on different models.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin-Watson
R square change
F change
df1
df2
Sig.F change
Modified modelb
0.568b
0.323
0.323
12629176.660
0.323
970.983
10
20,339
0.000
0.671
New modelc
0.458c
0.210
0.209
6510053.800
0.210
159.315
8
4798
0.000
1.227
Limited modeld
0.485d
0.235
0.235
13421600.463
0.235
894.764
7
20,342
0.000
0.655
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dew point temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Multiple regression analysis was performed to predict the heating thermal load. As indicated by the obtained results, the “New Model” showed the minimumR-squared (0.422) value and the highest Durbin–Watson value (1.509); in this case, a Durbin–Watson value above 1.3 indicates a weak correlation between the variables. In addition, although the R-squared values for the “Modified Model” and “Limited Model” are higher than that of the “New Model,” multicollinearity issues were found for the two models.The “Modified Model” is a good explanation of the dependent variable, as it exhibits the highestR-values. The variables considered in the “Modified Model” also exhibit positive correlations, as indicated by the Durbin–Watson value (0.671). The “Limited Model” indicated R-values lower than those of the “Modified Model” but higher than those of the “New Model” by a value of 0.03. The “New Model” showed the lowest R-value (0.458) and thus is an indication of the lowest explanatory power.Tables15–17 show the statistical results of the new regression model dealing with AOD. As shown in the table, the F-variation of the model was higher than 159 at p<0.001. Its variance and residual are slightly lower than that of the “Modified Model.” In addition, the F-values of the “Modified Model” were shown to be the highest of all three models. The “Limited Model” indicated the largest difference between the maximum and minimum predictions, as well as the largest mean.Table 15
ANOVA for a DOE medium office building depending on different models.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Modified modelb
Regression
1548679524700758530.000
10
154867952470075808.000
970.983
0.000b
Residual
3243991240918024700.000
20,339
159496103098383.620
—
—
Total
4792670765618783200.000
20,349
—
—
—
New modelc
Regression
54015103645277112.000
8
6751887955659638.000
159.315
0.000c
Residual
203343080683039488.000
4798
42380800475831.500
—
—
Total
257358184328316608.000
4806
—
—
—
Limited modeld
Regression
1128275925296639100.000
7
161182275042376992.000
894.764
0.000d
Residual
3664394840322144300.000
20,342
180139358977590.400
—
—
Total
4792670765618783200.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dew point temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Table 16
Residual statistics for a DOE medium office building depending on different models.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Modified modelb
Predicted value
−41989976.00
47291244.00
5299326.20
8723871.147
20,350
Residual
−34259568.000
135437520.000
0.000
12626073.134
20,350
Std. predicted value
−5.421
4.813
0.000
1.000
20,350
Std. residual
−2.713
10.724
0.000
1.000
20,350
New modelc
Predicted value
−5426438.00
19135408.00
1919966.69
3352476.356
4807
Residual
−19135408.000
73064632.000
0.000
6504633.271
4807
Std. predicted value
−2.191
5.135
0.000
1.000
4807
Std. residual
−2.939
11.223
0.000
0.999
4807
Limited modeld
Predicted value
−15162106.00
27670688.00
5299326.20
7446224.483
20,350
Residual
−22803048.000
150226688.000
0.000
13419291.767
20,350
Std. predicted value
−2.748
3.004
0.000
1.000
20,350
Std. residual
−1.699
11.193
0.000
1.000
20,350
aDependent variable: heating load of a DOE medium office building. bPredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dew point temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, dew point temperatures.Table 17
Regression coefficients for a DOE medium office building depending on different models.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Modified model
(Constant)
−332178078.900
19450841.631
—
−17.078
0.000
—
—
Dew point temperatures
−1278243.171
27,075.391
−0.770
−47.211
0.000
0.125
8.002
Relative humidity
318029.341
8735.744
0.402
36.406
0.000
0.274
3.656
Atmospheric station pressure
2937.887
190.743
0.118
15.402
0.000
0.567
1.764
Horizontal infrared radiation
66,114.532
2691.386
1.094
24.565
0.000
0.017
59.610
Global horizontal radiation
−90530.015
2306.310
−1.149
−39.253
0.000
0.039
25.744
Direct normal radiation
59,029.740
1259.159
0.814
46.880
0.000
0.110
9.069
Diffuse radiation
109882.074
2972.907
0.682
36.961
0.000
0.098
10.234
Total sky cover
−3968922.089
158298.925
−1.014
−25.072
0.000
0.020
49.187
Wind speed
−640957.692
68,159.260
−0.062
−9.404
0.000
0.754
1.327
Visibility
−340103.073
171843.921
−0.014
−1.979
0.048
0.682
1.466
New model
(Constant)
−137086417.229
19542748.049
—
−7.015
0.000
—
—
Dew point temperatures
−112436.116
27,307.607
−0.136
−4.117
0.000
0.151
6.625
Atmospheric station pressure
1264.005
192.177
0.107
6.577
0.000
0.623
1.604
Horizontal infrared radiation
47,644.241
3375.835
1.556
14.113
0.000
0.014
73.809
Direct normal radiation
16,462.981
1213.513
0.178
13.566
0.000
0.962
1.040
Total sky cover
−2787320.969
196312.532
−1.437
−14.198
0.000
0.016
62.171
Dry bulb temperatures
−581774.230
37,805.690
−0.586
−15.389
0.000
0.114
8.794
AOD
−566944.126
182564.249
−0.042
−3.105
0.002
0.879
1.137
Precipitable water
1076401.110
240985.736
0.080
4.467
0.000
0.512
1.955
Limited model
(Constant)
−263437804.945
21012890.768
—
−12.537
0.000
—
—
Dry bulb temperatures
−1285938.756
75,197.392
−0.692
−17.101
0.000
0.023
43.618
Dew point temperatures
582520.586
80,798.698
0.351
7.210
0.000
0.016
63.097
Relative humidity
−152490.019
22,022.183
−0.193
−6.924
0.000
0.049
20.572
Atmospheric station pressure
2831.572
203.680
0.114
13.902
0.000
0.561
1.781
Global horizontal radiation
13,177.043
602.849
0.167
21.858
0.000
0.642
1.557
Wind speed
−445559.152
71,983.529
−0.043
−6.190
0.000
0.763
1.310
Total sky cover
−153324.249
27,559.024
−0.039
−5.563
0.000
0.758
1.320
aDependent variable: heating load of a DOE medium office building.


### 4.3. Multiple Regression Analysis for the Prediction of the Heating Load for a Large Office considering Time-Lag

For a DOE large office building, 20,352 weather data points were used to estimate the heating load. In addition, the influence of the outdoor climatic conditions on the heating load was assessed through multiple regression analysis. Without considering time-lag, heating load values extracted from EnergyPlus after one hour and two hours were matched separately by hour during the analysis. The results of the analysis together with the input variables describing the climatic conditions are shown in Tables18–21. The results show R-values of the time-lag0 model, time-lag1 model, and time-lag2 model to be 0.514, 0.516, and 0.579, respectively. In addition, the R-square values of time-lag0, time-lag1, and time-lag2 were 0.264, 0.266, and 0.335, respectively; this implies that the regression model for the DOE large office has a higher explanatory or predictive power when using a time-lag of two hours.Table 18
Model summary for a DOE large office building depending on time-lag.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Time-lag0
0.514b
0.264
0.264
172903173.811
0.264
663.260
11
20,340
0.000
0.732
Time-lag1
0.516b
0.266
0.266
172668699.008
0.266
670.178
11
20,339
0.000
0.761
Time-lag2
0.579b
0.335
0.335
12518253.739
0.335
931.428
11
20,338
0.000
0.677
aDependent variable: heating load of a DOE large office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 19
ANOVA for a DOE large building depending on time-lag.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Time-lag0b
Regression
218113322176116700000.000
11
19828483834192430000.000
663.260
0.000b
Residual
608074622833789500000.000
20,340
29895507513952288.000
—
—
Total
826187945009906200000.000
20,351
—
—
—
Time-lag1b
Regression
219791177256880500000.000
11
19981016114261864000.000
670.178
0.000b
Residual
606396700934593400000.000
20,339
29814479617217832.000
—
—
Total
826187878191473900000.000
20,350
—
—
—
Time-lag2b
Regression
1605570375500446210.000
11
145960943227313184.000
931.428
0.000b
Residual
3187100390118337000.000
20,338
156706676670190.620
—
—
Total
4792670765618783200.000
20,349
—
—
—
aDependent variable: heating load of DOE large office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 20
Regression coefficients for a DOE large office building depending on time-lag.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Time-lag0
(Constant)
−2733111537.015
274002837.132
—
−9.975
0.000
—
—
Dry bulb temperatures
−15671733.697
1006027.437
−0.643
−15.578
0.000
0.021
47.057
Dew point temperatures
−4753374.820
1067469.519
−0.218
−4.453
0.000
0.015
66.370
Relative humidity
990564.362
297509.579
0.095
3.330
0.001
0.044
22.624
Atmospheric station pressure
24,491.862
2644.212
0.075
9.262
0.000
0.553
1.809
Horizontal infrared radiation
1277515.990
37,585.214
1.610
33.990
0.000
0.016
62.031
Global horizontal radiation
−33027.580
32,112.069
−0.032
−1.029
0.304
0.038
26.628
Direct normal radiation
9862.429
17,311.599
0.010
0.570
0.569
0.109
9.146
Diffuse radiation
163741.885
41,195.501
0.077
3.975
0.000
0.095
10.484
Wind speed
−5511093.363
933414.112
−0.041
−5.904
0.000
0.753
1.328
Total sky cover
−78732270.417
2202305.310
−1.533
-35.750
0.000
0.020
50.797
Visibility
6728944.425
2375958.447
0.021
2.832
0.005
0.669
1.495
Time-lag1
(Constant)
−2823375262.525
273663033.374
—
−10.317
0.000
—
—
Dry bulb temperatures
−16209199.274
1004663.472
−0.665
−16.134
0.000
0.021
47.050
Dew point temperatures
−2245663.075
1066038.360
−0.103
−2.107
0.035
0.015
66.368
Relative humidity
362476.854
297111.842
0.035
1.220
0.222
0.044
22.625
Atmospheric station pressure
26,181.850
2640.978
0.080
9.914
0.000
0.553
1.809
Horizontal infrared radiation
1119195.915
37,535.772
1.411
29.817
0.000
0.016
62.032
Global horizontal radiation
−538804.813
32,068.672
−0.521
−16.802
0.000
0.038
26.627
Direct normal radiation
355624.778
17,288.268
0.374
20.570
0.000
0.109
9.146
Diffuse radiation
718224.310
41,139.637
0.340
17.458
0.000
0.095
10.484
Wind speed
−7333997.059
932258.024
−0.054
−7.867
0.000
0.753
1.328
Total sky cover
−67661397.473
2199435.361
−1.317
−30.763
0.000
0.020
50.800
Visibility
1918837.055
2372842.304
0.006
0.809
0.419
0.669
1.495
Time-lag2
(Constant)
−242862773.124
19841671.213
—
−12.240
0.000
—
—
Dew point temperatures
102726.330
77,287.300
0.062
1.329
0.184
0.015
66.365
Atmospheric station pressure
2360.095
191.484
0.095
12.325
0.000
0.553
1.810
Horizontal infrared radiation
76,361.042
2721.411
1.264
28.059
0.000
0.016
62.033
Direct normal radiation
56,837.692
1253.391
0.784
45.347
0.000
0.109
9.146
Total sky cover
−4510822.624
159465.297
−1.153
−28.287
0.000
0.020
50.803
Dry bulb temperatures
−1387805.281
72,836.836
−0.747
−19.054
0.000
0.021
47.042
Relative humidity
−57775.571
21,540.586
−0.073
−2.682
0.007
0.044
22.625
Global horizontal radiation
−82461.104
2324.947
−1.047
−35.468
0.000
0.038
26.627
Diffuse radiation
101106.744
2982.570
0.628
33.899
0.000
0.095
10.484
Wind speed
−677958.805
67,588.517
−0.066
−10.031
0.000
0.753
1.328
Visibility
−799488.056
172032.483
−0.032
−4.647
0.000
0.669
1.495
aDependent variable: heating load of a DOE large office building.Table 21
Residual statistics for a DOE large office building depending on time-lag.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Time-lag0
Predicted value
−767601856.00
432252768.00
63956011.14
103525710.371
20,352
Residual
−392874016.000
1979168896.000
0.000
172856439.204
20,352
Std. predicted value
−8.032
3.558
0.000
1.000
20,352
Std. residual
−2.272
11.447
0.000
1.000
20,352
Time-lag1
Predicted value
−684187840.00
516718464.00
63956412.79
103925691.005
20,351
Residual
−356752608.000
2007745792.000
0.000
172622025.484
20,351
Std. predicted value
−7.199
4.357
0.000
1.000
20,351
Std. residual
−2.066
11.628
0.000
1.000
20,351
Time-lag2
Predicted value
−47442988.00
48511468.00
5299326.20
8882662.010
20,350
Residual
−35604016.000
136248016.000
0.000
12514869.803
20,350
Std. predicted value
−5.938
4.865
0.000
1.000
20,350
Std. residual
−2.844
10.884
0.000
1.000
20,350
aDependent variable: heating load of a DOE large office building.All regression models developed in the study showF-variations higher than 660 at p<0.001. In addition, the time-lag2 model indicated the highest R-value (0.931) amongst the three models. The Durbin–Watson test showed values of 0.732, 0.761, and 0.677 for the time-lag1, time-lag2, and time-lag3 models, respectively.The results of the ANOVA analysis indicate ap value less than 0.05 for all the models, and thus, all the models are assumed to be statistically significant. The time-lag2 model showed the largest variance with the smallest variance in the residuals. In addition, the F-value belonging to the time-lag2 model was the highest among the three developed models and thus indicated that the time-lag2 model has the best fit.Table20 shows the tolerance limit, significance of the probability, and t-value of the variables for each model. For accuracy, if a high goodness-of-fit is found for the independent variables, the t-value and p value should meet certain criteria: t-value ≥ 1.96 and p<0.05. In addition, to avoid issues related to multicollinearity, the tolerance limit should be 0.1 or higher; variables that do not satisfy these conditions are usually omitted from the analysis to improve the accuracy of the developed models.Since the time-lag phenomenon was not taken into consideration in the time-lag0 model, the most important variables when it comes to thermal conditions, such as global horizontal radiation and direct normal radiation, failed to satisfy the criteria. In addition, the dew point temperature and relative humidity in the time-lag1 model and the dew point temperature in the time-lag2 model did not meet the criteria and thus were omitted from the final model.As indicated through the obtained results, the time-lag2 model was shown to be the most suitable for heating load prediction of the DOE reference for a large office building. After taking into consideration the criteria related to the t-value, significance probability, and multicollinearity, the dew point temperature was excluded from the analysis; this model is defined as the “Modified Model” and is presented in Table22. Secondly, another model is developed based on all the variables considered in the time-lag2 model, AOD, and precipitable water values; this model is defined as the “New Model” and it is also presented in Table 22. Additionally, a third model based on only those variables received from the Korea Meteorological Administration (KMA) and which can be input by the user into EnergyPlus was developed considering a time-lag of 2 hours; this third model is defined as the “Limited Model” and is shown in Table 22. In other words, the third model considers all real data with the exception of the data obtained from equations. The data considered for each model are described as prediction a, prediction b, and prediction c and are shown in Table 22.Table 22
Model summary for a DOE large office building depending on different models.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Modified modelb
0.579b
0.335
0.335
12518489.662
0.335
1024.355
10
20,339
0.000
0.678
New modelc
0.458c
0.210
0.209
6510053.800
0.210
159.315
8
4798
0.000
1.227
Limited modeld
0.462d
0.214
0.213
178702667.526
0.214
789.878
7
20,342
0.000
0.684
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Tables23–25 show the statistical results of the new regression model dealing with AOD. As shown in the table, the F-variation of the model was higher than 159 at p<0.001. The variance and the residuals of the “New Model” are slightly lower than those of the “Modified Model.” In addition, the F-values of the “Modified Model” were shown to be the highest of all three models. The “Limited Model” indicated the largest difference between the maximum and minimum predictions, as well as the largest mean.Table 23
ANOVA for a DOE large office building depending on different models.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Modified modelb
Regression
1605293531717734140.000
10
160529353171773408.000
1024.355
0.000b
Residual
3187377233901048800.000
20,339
156712583406315.400
—
—
Total
4792670765618783200.000
20,349
—
—
—
New modelc
Regression
54015103645277112.000
8
6751887955659638.000
159.315
0.000c
Residual
203343080683039488.000
4798
42380800475831.500
—
—
Total
257358184328316608.000
4806
—
—
—
Limited modeld
Regression
176571232917599750000.000
7
25224461845371390000.000
789.878
0.000d
Residual
649614515656888500000.000
20,342
31934643381028828.000
—
—
Total
826185748574488200000.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Table 24
Residual statistics for a DOE large office building depending on different models.
Residual statisticsa
Minimum
Maximum
Mean
Std. Deviation
N
Modified modelb
Predicted value
−47685040.00
48544072.00
5299326.20
8881896.171
20,350
Residual
−35572376.000
136128544.000
0.000
12515413.336
20,350
Std. predicted value
−5.965
4.869
0.000
1.000
20,350
Std. residual
−2.842
10.874
0.000
1.000
20,350
New modelc
Predicted value
−5426438.00
19135408.00
1919966.69
3352476.356
4807
Residual
−19135408.000
73064632.000
0.000
6504633.271
4807
Std. predicted value
−2.191
5.135
0.000
1.000
4807
Std. residual
−2.939
11.223
0.000
0.999
4807
Limited modeld
Predicted value
−198986816.00
346896704.00
63958680.44
93151196.759
20,350
Residual
−280478208.000
2066232704.000
0.000
178671928.270
20,350
Std. predicted value
−2.823
3.037
0.000
1.000
20,350
Std. residual
−1.570
11.562
0.000
1.000
20,350
aDependent variable: heating load of a DOE large office building. bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, relative humidity, atmospheric station pressure, diffuse radiation, dew point temperatures, global horizantal radiation, and horizantal infrared radiation; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.Table 25
Regression coefficients for a DOE large office building depending on different models.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Modified Model
(Constant)
−247363320.740
19550986.572
—
−12.652
0.000
—
—
Atmospheric station pressure
2380.032
190.899
0.096
12.467
0.000
0.556
1.799
Horizontal infrared radiation
76,728.709
2707.367
1.270
28.341
0.000
0.016
61.392
Direct normal radiation
56,924.641
1251.706
0.785
45.478
0.000
0.110
9.121
Total sky cover
−4535356.153
158396.421
−1.159
−28.633
0.000
0.020
50.122
Dry bulb temperatures
−1297018.379
25,292.663
−0.698
−51.280
0.000
0.176
5.672
Relative humidity
−30575.346
6723.627
−0.039
−4.547
0.000
0.454
2.204
Global horizontal radiation
−82908.168
2300.532
−1.052
−36.039
0.000
0.038
26.070
Diffuse radiation
101684.144
2950.819
0.631
34.460
0.000
0.097
10.261
Wind speed
−678986.494
67,585.368
−0.066
−10.046
0.000
0.753
1.328
Visibility
−773429.426
170914.893
−0.031
−4.525
0.000
0.678
1.476
New Model
(Constant)
−137086417.229
19542748.049
—
−7.015
0.000
—
—
Atmospheric station pressure
1264.005
192.177
0.107
6.577
0.000
0.623
1.604
Horizontal infrared radiation
47,644.241
3375.835
1.556
14.113
0.000
0.014
73.809
Direct normal radiation
16,462.981
1213.513
0.178
13.566
0.000
0.962
1.040
Total sky cover
−2787320.969
196312.532
−1.437
−14.198
0.000
0.016
62.171
Dry bulb temperatures
−581774.230
37,805.690
−0.586
−15.389
0.000
0.114
8.794
Dew point temperatures
−112436.116
27,307.607
−0.136
−4.117
0.000
0.151
6.625
AOD
−566944.126
182564.249
−0.042
−3.105
0.002
0.879
1.137
Precipitable water
1076401.110
240985.736
0.080
4.467
0.000
0.512
1.955
Limited Model
(Constant)
−3092395121.054
279777336.777
—
−11.053
0.000
—
—
Dry bulb temperatures
−16496447.837
1001219.983
−0.677
−16.476
0.000
0.023
43.618
Dew point temperatures
7565123.922
1075798.887
0.347
7.032
0.000
0.016
63.097
Relative humidity
−1931850.757
293215.611
−0.186
−6.588
0.000
0.049
20.572
Atmospheric station pressure
33,401.664
2711.907
0.102
12.317
0.000
0.561
1.781
Global horizontal radiation
172387.218
8026.661
0.167
21.477
0.000
0.642
1.557
Wind speed
−4842190.418
958428.818
−0.036
−5.052
0.000
0.763
1.310
Total sky cover
−1842155.224
366936.200
−0.036
−5.020
0.000
0.758
1.320
aDependent variable: heating load of a DOE large office building.


## 4.1. Multiple Regression Analysis for the Prediction of the Heating Load for a Small Office considering Time-Lag

To estimate the heating load of a small office building, 20,352 weather data points were used. In addition, the influence of outdoor climatic conditions on the heating load was assessed through multiple regression analysis. Without considering time-lag, the heating load values extracted from EnergyPlus after one hour and two hours were matched separately by hour during the analysis. The results show R-values of the time-lag0 model, time-lag1 model, and time-lag2 model to be 0.534, 0.542, and 0.592, respectively. In addition, the R-square values of time-lag0, time-lag1, and time-lag2 were 0.285, 0.294, and 0.350, respectively; this implies that the regression model for the DOE small office has a higher explanatory or predictive power when using a time-lag of two hours. Table3 shows the results of the small office model summary considering time-lag.Table 3
DOE small office model summary considering time-lag.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Time-lag0
0.534b
0.285
0.285
7463851.345
0.285
737.844
11
20,340
0.000
0.799
Time-lag1
0.542b
0.294
0.294
7417381.535
0.294
770.449
11
20,339
0.000
0.834
Time-lag2
0.592b
0.350
0.350
7116200.437
0.350
996.953
11
20,338
0.000
0.778
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.TheF-variations for all the developed models were higher than 700, and the p values were all less than 0.001, thus confirming the statistical significance of the models. Furthermore, the Durbin–Watson test was conducted, and the results were 0.799, 0.834, and 0.778 for the time-lag0, time-lag1, and time-lag2 models, respectively; this means that among the three developed models, the time-lag1 model has a higher estimating power as it has a higher R-value and Durbin–Watson value.The results of the ANOVA analysis shown in Table4 indicate a p value less than 0.05 for all the models; thus, all the models are assumed to be statistically significant. The time-lag2 model showed the largest variance, whereas the time-lag0 model showed the largest variance of residue. Additionally, the F-value belonging to the time-lag2 model was the highest among the three developed models, thus indicating that the time-lag2 model has the best fit.Table 4
ANOVA for a DOE small building depending on time-lag.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Time-lag0
Regression
452150586586186300.000
11
41104598780562392.000
737.844
0.000b
Residual
1133122624135237120.000
20,340
55709076899470.850
—
—
Total
1585273210721423360.000
20,351
—
—
—
Time-lag1
Regression
466270598690733180.000
11
42388236244612096.000
770.449
0.000b
Residual
1119001925868610430.000
20,339
55017548840582.650
—
—
Total
1585272524559343620.000
20,350
—
—
—
Time-lag2
Regression
555346313497669060.000
11
50486028499788096.000
996.953
0.000b
Residual
1029922597401856130.000
20,338
50640308653842.860
—
—
Total
1585268910899525120.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table5 shows the tolerance limit, significance of probability, and t-value of the variables for each model. For accuracy, if a high goodness-of-fit is found for the independent variables, the t-value and p value should meet certain criteria: t-value ≥ 1.96 and p<0.05. In addition, to avoid issues related to multicollinearity, the tolerance limit should be 0.1 or higher; variables that do not satisfy these conditions are usually omitted from the analysis to improve the accuracy of the developed models.Table 5
Regression coefficients for a DOE small office building depending on time-lag.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Time-lag0
(Constant)
−173421890.552
11828102.396
—
−14.662
0.000
—
—
Dry bulb temperatures
−577018.579
43,428.001
−0.540
−13.287
0.000
0.021
47.057
Dew point temperatures
−185925.394
46,080.321
−0.195
−4.035
0.000
0.015
66.370
Relative humidity
27,650.459
12,842.837
0.061
2.153
0.031
0.044
22.624
Atmospheric station pressure
1668.517
114.145
0.117
14.618
0.000
0.553
1.809
Horizontal infrared radiation
35,821.811
1622.471
1.031
22.079
0.000
0.016
62.031
Global horizontal radiation
−-6515.011
1386.208
−0.144
−4.700
0.000
0.038
26.628
Direct normal radiation
4723.975
747.304
0.113
6.321
0.000
0.109
9.146
Diffuse radiation
11,820.058
1778.320
0.128
6.647
0.000
0.095
10.484
Wind speed
−294501.046
40,293.443
−0.050
−7.309
0.000
0.753
1.328
Total sky cover
−2154444.294
95,068.697
−0.957
−22.662
0.000
0.020
50.797
Visibility
123613.977
102564.923
0.009
1.205
0.228
0.669
1.495
Time-lag1
(Constant)
−181234643.381
11755825.707
—
−15.417
0.000
—
—
Dry bulb temperatures
−614138.021
43,157.633
−0.575
−14.230
0.000
0.021
47.050
Dew point temperatures
−68945.791
45,794.132
−0.072
−1.506
0.132
0.015
66.368
Relative humidity
−1938.780
12,763.123
−0.004
−0.152
0.879
0.044
22.625
Atmospheric station pressure
1780.718
113.449
0.124
15.696
0.000
0.553
1.809
Horizontal infrared radiation
29,704.944
1612.435
0.855
18.422
0.000
0.016
62.032
Global horizontal radiation
−24780.315
1377.584
−0.547
−17.988
0.000
0.038
26.627
Direct normal radiation
17,040.575
742.657
0.409
22.945
0.000
0.109
9.146
Diffuse radiation
32,792.426
1767.248
0.354
18.556
0.000
0.095
10.484
Wind speed
−348289.011
40,047.290
−0.059
−8.697
0.000
0.753
1.328
Total sky cover
−1728577.821
94,481.810
−0.768
−18.295
0.000
0.020
50.800
Visibility
−64173.273
101930.905
−0.005
−0.630
0.529
0.669
1.495
Time-lag2
(Constant)
−179888432.974
11279313.576
—
−15.949
0.000
—
—
Dry bulb temperatures
−685748.939
41,405.258
−0.642
−16.562
0.000
0.021
47.042
Dew point temperatures
86,402.645
43,935.195
0.091
1.967
0.049
0.015
66.365
Relative humidity
−47343.593
12,245.089
−0.104
−3.866
0.000
0.044
22.625
Atmospheric station pressure
1811.848
108.852
0.127
16.645
0.000
0.553
1.810
Horizontal infrared radiation
24,711.629
1547.029
0.711
15.974
0.000
0.016
62.033
Global horizontal radiation
−49815.842
1321.653
−1.099
−37.692
0.000
0.038
26.627
Direct normal radiation
34,307.818
712.510
0.823
48.151
0.000
0.109
9.146
Diffuse radiation
61,552.875
1695.489
0.664
36.304
0.000
0.095
10.484
Wind speed
−371456.480
38,421.767
−0.063
−9.668
0.000
0.753
1.328
Total sky cover
−1367478.673
90,650.584
−0.608
−15.085
0.000
0.020
50.803
Visibility
−544550.045
97,794.601
−0.038
−5.568
0.000
0.669
1.495
aDependent variable: heating load of a DOE small office building.Consequently, visibility was omitted from the time-lag0 model, whereas the dew point temperature, relative humidity, and visibility were also removed from the time-lag1 model; these variables are thus absent from the final models.As indicated by the results after taking into account thet-value, significance of the probability, and multicollinearity issues, the time-lag2 model was found to be the most suitable for a DOE small office building; this model is defined as the “Modified Model” and is presented in Table 6. Secondly, another model is developed based on all the variables considered in the time-lag2 model, AOD, and precipitable water values; this model is defined as the “New Model,” and it is also presented in Table 6. Additionally, a third model based on only those variables received from the Korea Meteorological Administration (KMA) and which can be input by the user into EnergyPlus was developed considering a time-lag of 2 hours; this third model is defined as the “Limited Model” and is shown in Table 6.Table 6
Model summary for a DOE small office building depending on different models.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Modified modelb
0.592b
0.350
0.350
7116200.437
0.350
996.953
11
20,338
0.000
0.778
New modelc
0.422c
0.178
0.176
3742573.281
0.178
130.924
8
4845
0.000
1.509
Limited modeld
0.513d
0.264
0.263
7575212.077
0.264
1213.784
6
20,343
0.000
0.767
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.A multiple regression analysis was performed to predict the heating thermal load. As indicated by the obtained results, the “New Model” showed the minimumR-squared (0.422) value and the highest Durbin–Watson value (1.509); in this case, a Durbin–Watson value above 1.3 indicates a weak correlation between the variables. In addition, although the R-squared values for the “Modified Model” and “Limited Model” are higher than that of the “New Model,” multicollinearity issues were found for the two models.TheF-variation of the “New Model” was shown to be higher than 170 at p<0.001. Furthermore, the variance of the regression model and residual were smaller than those of both the “Modified Model” and “Limited Model.” The F-value was shown to be highest in the “Limited Model.” Additionally, the results predicted by the “New Model” exhibited the smallest mean and standard deviation, as well as the smallest difference between the maximum and minimum predictions (Tables 7–9). A regression equation for the “New Model” can be drafted based on the final variables listed in Table 6.Table 7
ANOVA for a DOE small office building depending on different models.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Modified modelb
Regression
555346313497669060.000
11
50486028499788096.000
996.953
0.000b
Residual
1029922597401856130.000
20,338
50640308653842.860
—
—
Total
1585268910899525120.000
20,349
—
—
—
New modelc
Regression
14670632319938900.000
8
1833829039992363.000
130.924
0.000c
Residual
67863211318381720.000
4845
14006854761275.896
—
—
Total
82533843638320624.000
4853
—
—
—
Limited modeld
Regression
417909494247049020.000
6
69651582374508168.000
1213.784
0.000d
Residual
1167359416652476160.000
20,343
57383838010739.625
—
—
Total
1585268910899525120.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, relative humidity, atmospheric station pressure, diffuse radiation, dew point temperatures, global horizontal radiation, and horizontal infrared radiation; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.Table 8
Residual statistics for a DOE small office building depending on different models.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Modified modelb
Predicted value
−16410367.00
28590780.00
3786390.10
5224087.118
20,350
Residual
−20649606.000
64699612.000
0.000
7114276.785
20,350
Std. predicted value
−3.866
4.748
0.000
1.000
20,350
Std. residual
−2.902
9.092
0.000
1.000
20,350
New modelc
Predicted value
−2313629.00
13152829.00
1123572.72
1738678.448
4854
Residual
−10689199.000
46347212.000
0.000
3739487.258
4854
Std. predicted value
−1.977
6.919
0.000
1.000
4854
Std. residual
−2.856
12.384
0.000
0.999
4854
Limited modeld
Predicted value
−8359116.00
17571524.00
3786390.10
4531787.978
20,350
Residual
−15785886.000
70738864.000
0.000
7574095.201
20,350
Std. predicted value
−2.680
3.042
0.000
1.000
20,350
Std. residual
−2.084
9.338
0.000
1.000
20,350
aDependent variable: heating load of a DOE small office building. bPredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, relative humidity, atmospheric station pressure, diffuse radiation, dew point temperatures, global horizontal radiation, and horizontal infrared radiation; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.Table 9
Regression coefficients for a DOE small office building depending on different models.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Modified model
(Constant)
−179888432.974
11279313.576
—
−15.949
0.000
—
—
Dry bulb temperatures
−685748.939
41,405.258
−0.642
−16.562
0.000
0.021
47.042
Dew point temperatures
86,402.645
43,935.195
0.091
1.967
0.049
0.015
66.365
Relative humidity
−47343.593
12,245.089
−0.104
−3.866
0.000
0.044
22.625
Atmospheric station pressure
1811.848
108.852
0.127
16.645
0.000
0.553
1.810
Horizontal infrared radiation
24,711.629
1547.029
0.711
15.974
0.000
0.016
62.033
Global horizontal radiation
−49815.842
1321.653
−1.099
−37.692
0.000
0.038
26.627
Direct normal radiation
34,307.818
712.510
0.823
48.151
0.000
0.109
9.146
Diffuse radiation
61,552.875
1695.489
0.664
36.304
0.000
0.095
10.484
Wind speed
−371456.480
38,421.767
−0.063
−9.668
0.000
0.753
1.328
Total sky cover
−1367478.673
90,650.584
−0.608
−15.085
0.000
0.020
50.803
Visibility
−544550.045
97,794.601
−0.038
−5.568
0.000
0.669
1.495
New model
(Constant)
−84468362.750
11178380.426
—
−7.556
0.000
—
—
Dew point temperatures
−153900.636
11,782.451
−0.332
−13.062
0.000
0.263
3.800
Atmospheric station pressure
820.654
109.775
0.123
7.476
0.000
0.626
1.599
Horizontal infrared radiation
7302.493
1652.577
0.423
4.419
0.000
0.019
54.035
Global horizontal radiation
−28329.843
5547.379
−0.340
−5.107
0.000
0.038
26.094
Direct normal radiation
22,573.966
1829.080
0.430
12.342
0.000
0.140
7.149
Diffuse radiation
31,803.161
6859.159
0.208
4.637
0.000
0.084
11.876
Total sky cover
−380750.599
94,783.568
−0.349
−4.017
0.000
0.023
44.375
AOD
−215562.688
100939.898
−0.029
−2.136
0.033
0.945
1.059
Limited model
(Constant)
−198730470.861
11744538.154
—
−16.921
0.000
—
—
Dry bulb temperatures
−717138.604
42,245.667
−0.671
−16.975
0.000
0.023
43.216
Dew point temperatures
296105.476
45,462.975
0.310
6.513
0.000
0.016
62.710
Relative humidity
−90189.068
12,260.853
−0.198
−7.356
0.000
0.050
20.018
Atmospheric station pressure
2096.977
113.658
0.146
18.450
0.000
0.574
1.741
Global horizontal radiation
8298.520
340.007
0.183
24.407
0.000
0.643
1.555
Wind speed
−241986.775
40,515.794
−0.041
−5.973
0.000
0.767
1.303
aDependent variable: heating load of a DOE small office building.


## 4.2. Multiple Regression Analysis for the Prediction of Heating Load for a Medium Office considering Time-Lag

For a DOE medium office building, 20,352 weather data points were used to estimate the heating load. In addition, the influence of the outdoor climatic conditions on the heating load was assessed through multiple regression analysis. Without considering time-lag, heating load values extracted from EnergyPlus after one hour and two hours were matched separately by hour during the analysis. The results of the analysis together with the input variables describing the climatic conditions are shown in Tables10–13. The results show the R-values of the time-lag0 model, time-lag1 model, and time-lag2 model to be 0.537, 0.537, and 0.579, respectively. In addition, the R-square values of time-lag0, time-lag1, and time-lag2 were 0.289, 0.289, and 0.335, respectively; this implies that the regression model for the DOE medium office has a higher explanatory or predictive power when using a time-lag of two hours. Table 10 shows the model summaries of a DOE medium office building considering time-lag.Table 10
Model summary for a DOE medium office building depending on time-lag.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Time-lag0
0.537b
0.289
0.288
12945018.511
0.289
750.955
11
20,340
0.000
0.706
Time-lag1
0.537b
0.289
0.288
12947428.028
0.289
750.070
11
20,339
0.000
0.731
Time-lag2
0.579b
0.335
0.335
12518253.739
0.335
931.428
11
20,338
0.000
0.677
aDependent variable: heating load (medium); bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 11
Residual summary for a DOE medium office building depending on time-lag.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Time-lag0
Regression
1384242517670826240.000
11
125840228879166016.000
750.955
0.000b
Residual
3408445076552884200.000
20,340
167573504255304.030
—
—
Total
4792687594223710200.000
20,351
—
—
—
Time-lag1
Regression
1383124549966582780.000
11
125738595451507520.000
750.070
0.000b
Residual
3409546418166475800.000
20,339
167635892529941.300
—
—
Total
4792670968133058600.000
20,350
—
—
—
Time-lag2
Regression
1605570375500446210.000
11
145960943227313184.000
931.428
0.000b
Residual
3187100390118337000.000
20,338
156706676670190.620
—
—
Total
4792670765618783200.000
20,349
—
—
—
aDependent variable: heating load of DOE medium office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 12
Model summary for a DOE medium office building depending on time-lag.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. Error
Beta
Tolerance
VIF
Time-lag0
(Constant)
−232365034.136
20514208.737
—
−11.327
0.000
—
—
Dry bulb temperatures
−1239414.048
75,319.865
−0.667
−16.455
0.000
0.021
47.057
Dew point temperatures
−354626.266
79,919.948
−0.214
−4.437
0.000
0.015
66.370
Relative humidity
66,371.259
22,274.126
0.084
2.980
0.003
0.044
22.624
Atmospheric station pressure
2116.094
197.968
0.085
10.689
0.000
0.553
1.809
Horizontal infrared radiation
99,198.849
2813.952
1.642
35.252
0.000
0.016
62.031
Global horizontal radiation
−3376.020
2404.186
−0.043
−1.404
0.160
−0.038
26.628
Direct normal radiation
788.013
1296.095
0.011
0.608
0.543
0.109
9.146
Diffuse radiation
13,477.568
3084.249
0.084
4.370
0.000
0.095
10.484
Wind speed
−479633.798
69,883.408
−0.047
−6.863
0.000
0.753
1.328
Total sky cover
−6135755.540
164883.515
−1.568
−37.213
0.000
0.020
50.797
Visibility
408798.727
177884.682
0.017
2.298
0.022
0.669
1.495
Time-lag1
(Constant)
−242363953.699
20520409.598
—
−11.811
0.000
—
—
Dry bulb temperatures
−1289408.649
75,333.909
−0.694
−17.116
0.000
0.021
47.050
Dew point temperatures
−155279.237
79,936.057
−0.094
−1.943
0.052
0.015
66.368
Relative humidity
16,763.249
22,278.700
0.021
0.752
0.452
0.044
22.625
Atmospheric station pressure
2277.502
198.032
0.091
11.501
0.000
0.553
1.809
Horizontal infrared radiation
87,070.308
2814.591
1.441
30.935
0.000
0.016
62.032
Global horizontal radiation
−39853.596
2404.644
−0.506
−16.574
0.000
0.038
26.627
Direct normal radiation
25,968.009
1296.347
0.358
20.032
0.000
0.109
9.146
Diffuse radiation
54,012.154
3084.824
0.335
17.509
0.000
0.095
10.484
Wind speed
−616322.819
69,904.642
−0.060
−8.817
0.000
0.753
1.328
Total sky cover
−5285555.158
164922.949
−1.351
−32.049
0.000
0.020
50.800
Visibility
35,564.622
177925.734
0.001
0.200
0.842
0.669
1.495
Time-lag2
(Constant)
−242862773.124
19841671.213
—
−12.240
0.000
—
—
Dew point temperatures
102726.330
77,287.300
0.062
1.329
0.184
0.015
66.365
Relative humidity
−57775.571
21,540.586
−0.073
−2.682
0.007
0.044
22.625
Atmospheric station pressure
2360.095
191.484
0.095
12.325
0.000
0.553
1.810
Horizontal infrared radiation
76,361.042
2721.411
1.264
28.059
0.000
0.016
62.033
Global horizontal radiation
−82461.104
2324.947
−1.047
−35.468
0.000
0.038
26.627
Direct normal radiation
56,837.692
1253.391
0.784
45.347
0.000
0.109
9.146
Diffuse radiation
101106.744
2982.570
0.628
33.899
0.000
0.095
10.484
Total sky cover
−4510822.624
159465.297
−1.153
−28.287
0.000
0.020
50.803
Dry bulb temperatures
−1387805.281
72,836.836
−0.747
−19.054
0.000
0.021
47.042
Wind speed
−677958.805
67,588.517
−0.066
−10.031
0.000
0.753
1.328
Visibility
−799488.056
172032.483
−0.032
−4.647
0.000
0.669
1.495
aDependent variable: heating load of a DOE medium office building.Table 13
Residual statistics for a DOE medium office building depending on time-lag.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Time-lag0b
Predicted value
−59638412.00
34621280.00
5299548.66
8247327.016
20,352
Residual
−31375230.000
142983696.000
0.000
12941519.557
20,352
Std. predicted value
−7.874
3.555
0.000
1.000
20,352
Std. residual
−2.424
11.045
0.000
1.000
20,352
Time-lag1b
Predicted value
−53244212.00
40414716.00
5299348.31
8244198.466
20,351
Residual
−28412308.000
145224064.000
0.000
12943928.250
20,351
Std. predicted value
−7.101
4.259
0.000
1.000
20,351
Std. residual
−2.194
11.216
0.000
1.000
20,351
Time-lag2b
Predicted value
−47442988.00
48511468.00
5299326.20
8882662.010
20,350
Residual
−35604016.000
136248016.000
0.000
12514869.803
20,350
Std. predicted value
−5.938
4.865
0.000
1.000
20,350
Std. residual
−2.844
10.884
0.000
1.000
20,350
aDependent variable: heating load of a DOE medium office building. bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.TheF-variation for all the developed models was higher than 750, and the p values were all less than 0.001, thus confirming the statistical significance of the models. Furthermore, the Durbin–Watson test was conducted, and the results were 0.706, 0.731, and 0.677 for the time-lag0 model, time-lag1 model, and time-lag2 model, respectively; this means that among the three developed models, the time-lag1 model has the highest estimating power as it had the highest R-value and Durbin–Watson value, as shown in Table 11.The results of the ANOVA analysis indicate ap value less than 0.05 for all models, and thus, all the models are assumed to be statistically significant. The time-lag2 model showed the largest variance with the smallest variance in the residuals. In addition, the F-value belonging to the time-lag2 model was the highest among the three developed models and thus indicated that the time-lag2 model has the best fit.Table12 shows the tolerance limit, significance of the probability, and t-value of the variables for each model. For accuracy, if a high goodness-of-fit is found for the independent variables, the t-value and p value should meet certain criteria: t-value ≥ 1.96 and p<0.05. In addition, to avoid issues related to multicollinearity, the tolerance limit should be 0.1 or higher; variables that do not satisfy these conditions are usually omitted from the analysis to improve the accuracy of the developed models.Since the time-lag phenomenon was not taken into consideration in the time-lag0 model, the most important variables when it comes to thermal conditions, such as global horizontal radiation and direct normal radiation, failed to satisfy the conditions. The dew point temperature and relative humidity in the time-lag1 model and the dew point temperature in the time-lag2 model were the variables found to be rejected from each model. These variables are thus absent from the final models. Table13 shows residual statistics for a DOE medium office building considering time-lag.As indicated by the results after taking into account thet-value, significance of the probability, and multicollinearity issues, the time-lag2 model was found to be the most suitable for a DOE medium office building; this model is defined as the “Modified Model” and is presented in Table 14. Secondly, another model is developed based on all the variables considered in the time-lag2 model, AOD, and precipitable water values; this model is defined as the “New Model,” and it is also presented in Table 14. Additionally, a third model based on only those variables received from the Korea Meteorological Administration (KMA) and which can be input by the user into EnergyPlus was developed considering a time-lag of 2 hours; this third model is defined as the “Limited Model.”Table 14
Model summary for a DOE medium office building depending on different models.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin-Watson
R square change
F change
df1
df2
Sig.F change
Modified modelb
0.568b
0.323
0.323
12629176.660
0.323
970.983
10
20,339
0.000
0.671
New modelc
0.458c
0.210
0.209
6510053.800
0.210
159.315
8
4798
0.000
1.227
Limited modeld
0.485d
0.235
0.235
13421600.463
0.235
894.764
7
20,342
0.000
0.655
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dew point temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Multiple regression analysis was performed to predict the heating thermal load. As indicated by the obtained results, the “New Model” showed the minimumR-squared (0.422) value and the highest Durbin–Watson value (1.509); in this case, a Durbin–Watson value above 1.3 indicates a weak correlation between the variables. In addition, although the R-squared values for the “Modified Model” and “Limited Model” are higher than that of the “New Model,” multicollinearity issues were found for the two models.The “Modified Model” is a good explanation of the dependent variable, as it exhibits the highestR-values. The variables considered in the “Modified Model” also exhibit positive correlations, as indicated by the Durbin–Watson value (0.671). The “Limited Model” indicated R-values lower than those of the “Modified Model” but higher than those of the “New Model” by a value of 0.03. The “New Model” showed the lowest R-value (0.458) and thus is an indication of the lowest explanatory power.Tables15–17 show the statistical results of the new regression model dealing with AOD. As shown in the table, the F-variation of the model was higher than 159 at p<0.001. Its variance and residual are slightly lower than that of the “Modified Model.” In addition, the F-values of the “Modified Model” were shown to be the highest of all three models. The “Limited Model” indicated the largest difference between the maximum and minimum predictions, as well as the largest mean.Table 15
ANOVA for a DOE medium office building depending on different models.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Modified modelb
Regression
1548679524700758530.000
10
154867952470075808.000
970.983
0.000b
Residual
3243991240918024700.000
20,339
159496103098383.620
—
—
Total
4792670765618783200.000
20,349
—
—
—
New modelc
Regression
54015103645277112.000
8
6751887955659638.000
159.315
0.000c
Residual
203343080683039488.000
4798
42380800475831.500
—
—
Total
257358184328316608.000
4806
—
—
—
Limited modeld
Regression
1128275925296639100.000
7
161182275042376992.000
894.764
0.000d
Residual
3664394840322144300.000
20,342
180139358977590.400
—
—
Total
4792670765618783200.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dew point temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Table 16
Residual statistics for a DOE medium office building depending on different models.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Modified modelb
Predicted value
−41989976.00
47291244.00
5299326.20
8723871.147
20,350
Residual
−34259568.000
135437520.000
0.000
12626073.134
20,350
Std. predicted value
−5.421
4.813
0.000
1.000
20,350
Std. residual
−2.713
10.724
0.000
1.000
20,350
New modelc
Predicted value
−5426438.00
19135408.00
1919966.69
3352476.356
4807
Residual
−19135408.000
73064632.000
0.000
6504633.271
4807
Std. predicted value
−2.191
5.135
0.000
1.000
4807
Std. residual
−2.939
11.223
0.000
0.999
4807
Limited modeld
Predicted value
−15162106.00
27670688.00
5299326.20
7446224.483
20,350
Residual
−22803048.000
150226688.000
0.000
13419291.767
20,350
Std. predicted value
−2.748
3.004
0.000
1.000
20,350
Std. residual
−1.699
11.193
0.000
1.000
20,350
aDependent variable: heating load of a DOE medium office building. bPredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dew point temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, dew point temperatures.Table 17
Regression coefficients for a DOE medium office building depending on different models.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Modified model
(Constant)
−332178078.900
19450841.631
—
−17.078
0.000
—
—
Dew point temperatures
−1278243.171
27,075.391
−0.770
−47.211
0.000
0.125
8.002
Relative humidity
318029.341
8735.744
0.402
36.406
0.000
0.274
3.656
Atmospheric station pressure
2937.887
190.743
0.118
15.402
0.000
0.567
1.764
Horizontal infrared radiation
66,114.532
2691.386
1.094
24.565
0.000
0.017
59.610
Global horizontal radiation
−90530.015
2306.310
−1.149
−39.253
0.000
0.039
25.744
Direct normal radiation
59,029.740
1259.159
0.814
46.880
0.000
0.110
9.069
Diffuse radiation
109882.074
2972.907
0.682
36.961
0.000
0.098
10.234
Total sky cover
−3968922.089
158298.925
−1.014
−25.072
0.000
0.020
49.187
Wind speed
−640957.692
68,159.260
−0.062
−9.404
0.000
0.754
1.327
Visibility
−340103.073
171843.921
−0.014
−1.979
0.048
0.682
1.466
New model
(Constant)
−137086417.229
19542748.049
—
−7.015
0.000
—
—
Dew point temperatures
−112436.116
27,307.607
−0.136
−4.117
0.000
0.151
6.625
Atmospheric station pressure
1264.005
192.177
0.107
6.577
0.000
0.623
1.604
Horizontal infrared radiation
47,644.241
3375.835
1.556
14.113
0.000
0.014
73.809
Direct normal radiation
16,462.981
1213.513
0.178
13.566
0.000
0.962
1.040
Total sky cover
−2787320.969
196312.532
−1.437
−14.198
0.000
0.016
62.171
Dry bulb temperatures
−581774.230
37,805.690
−0.586
−15.389
0.000
0.114
8.794
AOD
−566944.126
182564.249
−0.042
−3.105
0.002
0.879
1.137
Precipitable water
1076401.110
240985.736
0.080
4.467
0.000
0.512
1.955
Limited model
(Constant)
−263437804.945
21012890.768
—
−12.537
0.000
—
—
Dry bulb temperatures
−1285938.756
75,197.392
−0.692
−17.101
0.000
0.023
43.618
Dew point temperatures
582520.586
80,798.698
0.351
7.210
0.000
0.016
63.097
Relative humidity
−152490.019
22,022.183
−0.193
−6.924
0.000
0.049
20.572
Atmospheric station pressure
2831.572
203.680
0.114
13.902
0.000
0.561
1.781
Global horizontal radiation
13,177.043
602.849
0.167
21.858
0.000
0.642
1.557
Wind speed
−445559.152
71,983.529
−0.043
−6.190
0.000
0.763
1.310
Total sky cover
−153324.249
27,559.024
−0.039
−5.563
0.000
0.758
1.320
aDependent variable: heating load of a DOE medium office building.


## 4.3. Multiple Regression Analysis for the Prediction of the Heating Load for a Large Office considering Time-Lag

For a DOE large office building, 20,352 weather data points were used to estimate the heating load. In addition, the influence of the outdoor climatic conditions on the heating load was assessed through multiple regression analysis. Without considering time-lag, heating load values extracted from EnergyPlus after one hour and two hours were matched separately by hour during the analysis. The results of the analysis together with the input variables describing the climatic conditions are shown in Tables18–21. The results show R-values of the time-lag0 model, time-lag1 model, and time-lag2 model to be 0.514, 0.516, and 0.579, respectively. In addition, the R-square values of time-lag0, time-lag1, and time-lag2 were 0.264, 0.266, and 0.335, respectively; this implies that the regression model for the DOE large office has a higher explanatory or predictive power when using a time-lag of two hours.Table 18
Model summary for a DOE large office building depending on time-lag.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Time-lag0
0.514b
0.264
0.264
172903173.811
0.264
663.260
11
20,340
0.000
0.732
Time-lag1
0.516b
0.266
0.266
172668699.008
0.266
670.178
11
20,339
0.000
0.761
Time-lag2
0.579b
0.335
0.335
12518253.739
0.335
931.428
11
20,338
0.000
0.677
aDependent variable: heating load of a DOE large office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 19
ANOVA for a DOE large building depending on time-lag.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Time-lag0b
Regression
218113322176116700000.000
11
19828483834192430000.000
663.260
0.000b
Residual
608074622833789500000.000
20,340
29895507513952288.000
—
—
Total
826187945009906200000.000
20,351
—
—
—
Time-lag1b
Regression
219791177256880500000.000
11
19981016114261864000.000
670.178
0.000b
Residual
606396700934593400000.000
20,339
29814479617217832.000
—
—
Total
826187878191473900000.000
20,350
—
—
—
Time-lag2b
Regression
1605570375500446210.000
11
145960943227313184.000
931.428
0.000b
Residual
3187100390118337000.000
20,338
156706676670190.620
—
—
Total
4792670765618783200.000
20,349
—
—
—
aDependent variable: heating load of DOE large office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, horizontal infrared radiation, and dew point temperatures.Table 20
Regression coefficients for a DOE large office building depending on time-lag.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Time-lag0
(Constant)
−2733111537.015
274002837.132
—
−9.975
0.000
—
—
Dry bulb temperatures
−15671733.697
1006027.437
−0.643
−15.578
0.000
0.021
47.057
Dew point temperatures
−4753374.820
1067469.519
−0.218
−4.453
0.000
0.015
66.370
Relative humidity
990564.362
297509.579
0.095
3.330
0.001
0.044
22.624
Atmospheric station pressure
24,491.862
2644.212
0.075
9.262
0.000
0.553
1.809
Horizontal infrared radiation
1277515.990
37,585.214
1.610
33.990
0.000
0.016
62.031
Global horizontal radiation
−33027.580
32,112.069
−0.032
−1.029
0.304
0.038
26.628
Direct normal radiation
9862.429
17,311.599
0.010
0.570
0.569
0.109
9.146
Diffuse radiation
163741.885
41,195.501
0.077
3.975
0.000
0.095
10.484
Wind speed
−5511093.363
933414.112
−0.041
−5.904
0.000
0.753
1.328
Total sky cover
−78732270.417
2202305.310
−1.533
-35.750
0.000
0.020
50.797
Visibility
6728944.425
2375958.447
0.021
2.832
0.005
0.669
1.495
Time-lag1
(Constant)
−2823375262.525
273663033.374
—
−10.317
0.000
—
—
Dry bulb temperatures
−16209199.274
1004663.472
−0.665
−16.134
0.000
0.021
47.050
Dew point temperatures
−2245663.075
1066038.360
−0.103
−2.107
0.035
0.015
66.368
Relative humidity
362476.854
297111.842
0.035
1.220
0.222
0.044
22.625
Atmospheric station pressure
26,181.850
2640.978
0.080
9.914
0.000
0.553
1.809
Horizontal infrared radiation
1119195.915
37,535.772
1.411
29.817
0.000
0.016
62.032
Global horizontal radiation
−538804.813
32,068.672
−0.521
−16.802
0.000
0.038
26.627
Direct normal radiation
355624.778
17,288.268
0.374
20.570
0.000
0.109
9.146
Diffuse radiation
718224.310
41,139.637
0.340
17.458
0.000
0.095
10.484
Wind speed
−7333997.059
932258.024
−0.054
−7.867
0.000
0.753
1.328
Total sky cover
−67661397.473
2199435.361
−1.317
−30.763
0.000
0.020
50.800
Visibility
1918837.055
2372842.304
0.006
0.809
0.419
0.669
1.495
Time-lag2
(Constant)
−242862773.124
19841671.213
—
−12.240
0.000
—
—
Dew point temperatures
102726.330
77,287.300
0.062
1.329
0.184
0.015
66.365
Atmospheric station pressure
2360.095
191.484
0.095
12.325
0.000
0.553
1.810
Horizontal infrared radiation
76,361.042
2721.411
1.264
28.059
0.000
0.016
62.033
Direct normal radiation
56,837.692
1253.391
0.784
45.347
0.000
0.109
9.146
Total sky cover
−4510822.624
159465.297
−1.153
−28.287
0.000
0.020
50.803
Dry bulb temperatures
−1387805.281
72,836.836
−0.747
−19.054
0.000
0.021
47.042
Relative humidity
−57775.571
21,540.586
−0.073
−2.682
0.007
0.044
22.625
Global horizontal radiation
−82461.104
2324.947
−1.047
−35.468
0.000
0.038
26.627
Diffuse radiation
101106.744
2982.570
0.628
33.899
0.000
0.095
10.484
Wind speed
−677958.805
67,588.517
−0.066
−10.031
0.000
0.753
1.328
Visibility
−799488.056
172032.483
−0.032
−4.647
0.000
0.669
1.495
aDependent variable: heating load of a DOE large office building.Table 21
Residual statistics for a DOE large office building depending on time-lag.
Residual statisticsa
Minimum
Maximum
Mean
Std. deviation
N
Time-lag0
Predicted value
−767601856.00
432252768.00
63956011.14
103525710.371
20,352
Residual
−392874016.000
1979168896.000
0.000
172856439.204
20,352
Std. predicted value
−8.032
3.558
0.000
1.000
20,352
Std. residual
−2.272
11.447
0.000
1.000
20,352
Time-lag1
Predicted value
−684187840.00
516718464.00
63956412.79
103925691.005
20,351
Residual
−356752608.000
2007745792.000
0.000
172622025.484
20,351
Std. predicted value
−7.199
4.357
0.000
1.000
20,351
Std. residual
−2.066
11.628
0.000
1.000
20,351
Time-lag2
Predicted value
−47442988.00
48511468.00
5299326.20
8882662.010
20,350
Residual
−35604016.000
136248016.000
0.000
12514869.803
20,350
Std. predicted value
−5.938
4.865
0.000
1.000
20,350
Std. residual
−2.844
10.884
0.000
1.000
20,350
aDependent variable: heating load of a DOE large office building.All regression models developed in the study showF-variations higher than 660 at p<0.001. In addition, the time-lag2 model indicated the highest R-value (0.931) amongst the three models. The Durbin–Watson test showed values of 0.732, 0.761, and 0.677 for the time-lag1, time-lag2, and time-lag3 models, respectively.The results of the ANOVA analysis indicate ap value less than 0.05 for all the models, and thus, all the models are assumed to be statistically significant. The time-lag2 model showed the largest variance with the smallest variance in the residuals. In addition, the F-value belonging to the time-lag2 model was the highest among the three developed models and thus indicated that the time-lag2 model has the best fit.Table20 shows the tolerance limit, significance of the probability, and t-value of the variables for each model. For accuracy, if a high goodness-of-fit is found for the independent variables, the t-value and p value should meet certain criteria: t-value ≥ 1.96 and p<0.05. In addition, to avoid issues related to multicollinearity, the tolerance limit should be 0.1 or higher; variables that do not satisfy these conditions are usually omitted from the analysis to improve the accuracy of the developed models.Since the time-lag phenomenon was not taken into consideration in the time-lag0 model, the most important variables when it comes to thermal conditions, such as global horizontal radiation and direct normal radiation, failed to satisfy the criteria. In addition, the dew point temperature and relative humidity in the time-lag1 model and the dew point temperature in the time-lag2 model did not meet the criteria and thus were omitted from the final model.As indicated through the obtained results, the time-lag2 model was shown to be the most suitable for heating load prediction of the DOE reference for a large office building. After taking into consideration the criteria related to the t-value, significance probability, and multicollinearity, the dew point temperature was excluded from the analysis; this model is defined as the “Modified Model” and is presented in Table22. Secondly, another model is developed based on all the variables considered in the time-lag2 model, AOD, and precipitable water values; this model is defined as the “New Model” and it is also presented in Table 22. Additionally, a third model based on only those variables received from the Korea Meteorological Administration (KMA) and which can be input by the user into EnergyPlus was developed considering a time-lag of 2 hours; this third model is defined as the “Limited Model” and is shown in Table 22. In other words, the third model considers all real data with the exception of the data obtained from equations. The data considered for each model are described as prediction a, prediction b, and prediction c and are shown in Table 22.Table 22
Model summary for a DOE large office building depending on different models.
Model summarya
Model
R
R square
AdjustedR square
Std. error of the estimate
Change statistics
Durbin–Watson
R square change
F change
df1
df2
Sig.F change
Modified modelb
0.579b
0.335
0.335
12518489.662
0.335
1024.355
10
20,339
0.000
0.678
New modelc
0.458c
0.210
0.209
6510053.800
0.210
159.315
8
4798
0.000
1.227
Limited modeld
0.462d
0.214
0.213
178702667.526
0.214
789.878
7
20,342
0.000
0.684
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Tables23–25 show the statistical results of the new regression model dealing with AOD. As shown in the table, the F-variation of the model was higher than 159 at p<0.001. The variance and the residuals of the “New Model” are slightly lower than those of the “Modified Model.” In addition, the F-values of the “Modified Model” were shown to be the highest of all three models. The “Limited Model” indicated the largest difference between the maximum and minimum predictions, as well as the largest mean.Table 23
ANOVA for a DOE large office building depending on different models.
ANOVAa
Model
Sum of squares
df
Mean square
F
Sig.
Modified modelb
Regression
1605293531717734140.000
10
160529353171773408.000
1024.355
0.000b
Residual
3187377233901048800.000
20,339
156712583406315.400
—
—
Total
4792670765618783200.000
20,349
—
—
—
New modelc
Regression
54015103645277112.000
8
6751887955659638.000
159.315
0.000c
Residual
203343080683039488.000
4798
42380800475831.500
—
—
Total
257358184328316608.000
4806
—
—
—
Limited modeld
Regression
176571232917599750000.000
7
25224461845371390000.000
789.878
0.000d
Residual
649614515656888500000.000
20,342
31934643381028828.000
—
—
Total
826185748574488200000.000
20,349
—
—
—
aDependent variable: heating load of a DOE small office building; bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, atmospheric station pressure, dew point temperatures, dry bulb temperatures, and horizontal infrared radiation; dpredictors: (constant), total sky cover, wind speed, dry bulb temperatures, global horizontal radiation, relative humidity, atmospheric station pressure, and dew point temperatures.Table 24
Residual statistics for a DOE large office building depending on different models.
Residual statisticsa
Minimum
Maximum
Mean
Std. Deviation
N
Modified modelb
Predicted value
−47685040.00
48544072.00
5299326.20
8881896.171
20,350
Residual
−35572376.000
136128544.000
0.000
12515413.336
20,350
Std. predicted value
−5.965
4.869
0.000
1.000
20,350
Std. residual
−2.842
10.874
0.000
1.000
20,350
New modelc
Predicted value
−5426438.00
19135408.00
1919966.69
3352476.356
4807
Residual
−19135408.000
73064632.000
0.000
6504633.271
4807
Std. predicted value
−2.191
5.135
0.000
1.000
4807
Std. residual
−2.939
11.223
0.000
0.999
4807
Limited modeld
Predicted value
−198986816.00
346896704.00
63958680.44
93151196.759
20,350
Residual
−280478208.000
2066232704.000
0.000
178671928.270
20,350
Std. predicted value
−2.823
3.037
0.000
1.000
20,350
Std. residual
−1.570
11.562
0.000
1.000
20,350
aDependent variable: heating load of a DOE large office building. bpredictors: (constant), visibility, diffuse radiation, atmospheric station pressure, wind speed, total sky cover, dry bulb temperatures, direct normal radiation, relative humidity, global horizontal radiation, and horizontal infrared radiation, and dew point temperatures; cpredictors: (constant), precipitable water, direct normal radiation, total sky cover, AOD, relative humidity, atmospheric station pressure, diffuse radiation, dew point temperatures, global horizantal radiation, and horizantal infrared radiation; dpredictors: (constant), wind speed, dry bulb temperatures, relative humidity, global horizontal radiation, atmospheric station pressure, and dew point temperatures.Table 25
Regression coefficients for a DOE large office building depending on different models.
Coefficientsa
Model
Unstandardized coefficients
Standardized coefficients
t
Sig.
Collinearity statistics
B
Std. error
Beta
Tolerance
VIF
Modified Model
(Constant)
−247363320.740
19550986.572
—
−12.652
0.000
—
—
Atmospheric station pressure
2380.032
190.899
0.096
12.467
0.000
0.556
1.799
Horizontal infrared radiation
76,728.709
2707.367
1.270
28.341
0.000
0.016
61.392
Direct normal radiation
56,924.641
1251.706
0.785
45.478
0.000
0.110
9.121
Total sky cover
−4535356.153
158396.421
−1.159
−28.633
0.000
0.020
50.122
Dry bulb temperatures
−1297018.379
25,292.663
−0.698
−51.280
0.000
0.176
5.672
Relative humidity
−30575.346
6723.627
−0.039
−4.547
0.000
0.454
2.204
Global horizontal radiation
−82908.168
2300.532
−1.052
−36.039
0.000
0.038
26.070
Diffuse radiation
101684.144
2950.819
0.631
34.460
0.000
0.097
10.261
Wind speed
−678986.494
67,585.368
−0.066
−10.046
0.000
0.753
1.328
Visibility
−773429.426
170914.893
−0.031
−4.525
0.000
0.678
1.476
New Model
(Constant)
−137086417.229
19542748.049
—
−7.015
0.000
—
—
Atmospheric station pressure
1264.005
192.177
0.107
6.577
0.000
0.623
1.604
Horizontal infrared radiation
47,644.241
3375.835
1.556
14.113
0.000
0.014
73.809
Direct normal radiation
16,462.981
1213.513
0.178
13.566
0.000
0.962
1.040
Total sky cover
−2787320.969
196312.532
−1.437
−14.198
0.000
0.016
62.171
Dry bulb temperatures
−581774.230
37,805.690
−0.586
−15.389
0.000
0.114
8.794
Dew point temperatures
−112436.116
27,307.607
−0.136
−4.117
0.000
0.151
6.625
AOD
−566944.126
182564.249
−0.042
−3.105
0.002
0.879
1.137
Precipitable water
1076401.110
240985.736
0.080
4.467
0.000
0.512
1.955
Limited Model
(Constant)
−3092395121.054
279777336.777
—
−11.053
0.000
—
—
Dry bulb temperatures
−16496447.837
1001219.983
−0.677
−16.476
0.000
0.023
43.618
Dew point temperatures
7565123.922
1075798.887
0.347
7.032
0.000
0.016
63.097
Relative humidity
−1931850.757
293215.611
−0.186
−6.588
0.000
0.049
20.572
Atmospheric station pressure
33,401.664
2711.907
0.102
12.317
0.000
0.561
1.781
Global horizontal radiation
172387.218
8026.661
0.167
21.477
0.000
0.642
1.557
Wind speed
−4842190.418
958428.818
−0.036
−5.052
0.000
0.763
1.310
Total sky cover
−1842155.224
366936.200
−0.036
−5.020
0.000
0.758
1.320
aDependent variable: heating load of a DOE large office building.


## 5. Conclusion

The current study aimed to understand the effect of time-lag as well as Aerosol Optical Depth on the prediction of the heating load in buildings. In the process, this study also explains how heating load predictions can be improved through the consideration of time-lag. Various studies have already established that outdoor weather conditions play a significant role in determining the thermal behavior of a building; the term time-lag in this sense is used to define the phenomenon in which the time of the effect of the outdoor climatic conditions on the building heating load is misrepresented during prediction. The current study uses regression models to identify which outdoor climatic conditions, in terms of time, best explain the heating load being experienced.Many building energy simulation programs assume that the time at which peak outdoor conditions occur is the same time the effect of these conditions is highest inside the building. However, the results obtained by the current study prove otherwise. For instance, for all the building sizes considered (DOE reference buildings for small, medium, and large offices), it has been shown that the heating load at a given time is best described by the outdoor climatic conditions that occur two hours prior to that time. In this same manner, the developed regression models can be used to predict the heating load of a future time, for example, two hours in advance; a number of factors, such as the building materials used, can explain the reason for this. For example, the outer material of a building might absorb heat during the peak solar hours only to release it later into the building. In this case, a building simulation program would estimate the highest thermal load to be at the time of the peak solar hours, yet the effect is actually experienced later when the heat has been released from the building mass. The regression models developed here, however, can be used accurately to estimate the heating load since they consider the time-lag caused by factors such as the building mass or building size. Table26 is a summary of the regression models that best estimate the heating load per building size. In addition, the table presents the developed models in two sets: the first set considering only the time-lag and the second set considering both time-lag and Aerosol Optical Depth data.Table 26
Multiple regression equations using new variables based on building size.
Building Types
Linear Regression Model
Original variables only
DOE small office (time-lag0)
Y = −179888432.974 + (−685748.939)X1 + (86,402.645)X2 + (−47343.593)X3 + (1811.848)X4 + (24,711.629)X5 + (−49815.842) X6 + (34,307.818)X7 + (61,552.875)X8 + (−371456.480)X9 + (−1367478.673)X10 + (−544550.045) X11
DOE medium office (time-lag1)
Y = −242862773.124 + (−57775.571)X2 + (2360.095)X3 + (76,361.042)X4 + (−82461.104)X5 + (56,837.692)X6 + (101106.744)X7 + (−4510822.624)X8 + (−1387805.281)X9 + (−677958.805)X10 + (−799488.056)X11
DOE large office (time-lag2)
Y = −242862773.124 + (2360.095) X2 + (76,361.042)X3 + (56,837.692)X4 + (−4510822.624)X5 + (−1387805.281)X6 + (−57775.571)X7 + (−82461.104)X8+ (101106.744)X9 + (−677958.805)X10 + (−799488.056)X11
With new variables
DOE small office (new model)
Y = −84468362.750 + (−153900.636)X2 + (820.654)X4 + (7302.493)X5 + (−28329.843)X6 + (22,573.966)X7 + (31,803.161)X8 + (−380750.599)X10 + (−215562.688)X12
DOE medium office (new model)
Y = −137086417.229 + (−581774.230)X1 + (−112436.116)X2 + (1264.005)X4 + (47,644.241)X5 + (16,462.981)X7 + (−2787320.969)X10 + (−566944.126)X12 + (1076401.110)X13
DOE large office (new model)
Y = −137086417.229 + (−581774.230)X1 + (−112436.116)X2 + (1264.005)X4 + (47,644.241)X5 + (16,462.981)X7 + (−2787320.969)X10 + (−566944.126)X12 + (1076401.110)X13
X
1 = dry bulb temperatures, X2 = dew point temperatures, X3 = relative humidity, X4 = atmospheric station pressure, X5 = horizontal infrared radiation, X6 = global horizontal radiation, X7 = direct normal radiation, X8 = diffuse radiation, X9 = wind speed, X10 = total sky cover, X11 = visibility, X12 = AOD, X13 = precipitable water, and Y = heating load.In conclusion, the key finding of this study is that both time-lag and Aerosol Optical Depth (AOD) data are critical factors in the estimation of building energy usage. As such, conventional building simulation programs must endeavor to include time-lag as an important factor in the prediction process. The impact of each input variable on the output can be assessed by the size of the coefficient values. For example, for the DOE small office linear regression model, total sky cover has the biggest impact on building heating loads.



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*Source: 4878021-2018-02-01.xml*