BASELINE MODELS OF WHOLE BUILDING ENERGY

Transcription

1 CHAPTER 3 BASELINE MODELS OF WHOLE BUILDING ENERGY CONSUMPTION 3.1 Introduction This chapter describes the research design, methodology, and results for the baseline model development of whole building energy consumption for the Tropics. Classical regression analysis approach is applied in this study. Monthly utility bills are collected for modeling and analysis. Figure 3.1 shows the developmental stages of baseline model development of whole building energy consumption. 3. Mathematical background of regression models 3..1 Regression model selection criterion In this study, regression approach is applied in the baseline model establishing. The reasons have been presented in Chapter. The criterion used to select the most appropriate regression model is to maximize the goodness of fit using the simplest mode or combination of models (Draper and Smith, 1981). According to the literature review (section.,.3), it is believed that the coefficient of determination ( R ) and the coefficient of variance of the root-mean-square error (CV-RMSE) are two major 31

2 Figure 3.1 Developmental stages of baseline model development of whole building energy consumption Available Buildings Possibility of bill reading dates Normalize Energy Use Weather data collection Simple Linear Regression Primary Analysis Primary modeling of whole building energy consumption Selection of Buildings Multiple Regression Approach Improved utility bill analysis Model Develop ment Correlationship analysis Uncertainty analysis Normalize Energy Use Baseline model identification Verify baseline model performance Regression Approach Predicting error analysis Predicted Monthly data Verification Determine prediction uncertainty and provide recommendations of baseline model selection 3

3 measures to evaluate the goodness of fit of a model. The value of R is defined as the Pearson correlation coefficient between the observed and fitted values. The CV-RMSE is a non-dimensional measure that is found by dividing root-mean-square error (RMSE) by the mean value of total energy consumption E. It is usually presented as a percentage. A CV-RMSE value of 1% indicates that the mean variation in E not explained by the regression model is only 1% of the mean value of E (Reddy et al., 1997a). The CV-RMSE defined below: Where, RMSE CV RMSE =.1 (3.1) E RMSE = [ MSE] 1/ = n i= 1 ( E Eˆ ) i n p 1/ (3.) E i is the actual energy consumption of single month/day/hour i (i =1, n). E is the mean value of E. Ê is the value of E predicted by the regression model, n is the number of observations; p is the number of model parameters. For simplicity, the direct deviation between the two types of energy consumption values, actual versus predicted, which is also called mean absolute error (MAE), is defined as follows: E predicted Emeasured MAE (%) = 1 (3.3) E measured 33

4 How to determine the goodness of fit of the model with the two measures, namely, R and CV-RMSE, have been well discussed and examined by researchers. Fels et al. (1993) arbitrarily suggested that monthly models with R.7 and CV-RMSE 7% be deemed good models. Reddy et al. (1997a) pointed out that CV-RMSE of less than 5% are considered excellent models, those less than 1% are considered good models, and those less than % are taken to be mediocre models and those greater than % are considered to be poor model. ASHRAE Guideline 14- states that for the whole building performance path the baseline model shall have a maximum CV-RMSE of % for energy and 3% for the demand quantities when less than 1 months worth of post-retrofit data are available for computing savings; these requirements are 5% and 35%, respectively, when 1 to 6 months of data will be used in computing savings; when more than 6 months of data will be available, these requirements are 3% and 4%, respectively. For this thesis, the values of R and CV-RMSE follow the criteria pointed out by Reddy et al. (1997a) as it applies to the whole building energy consumption analysis, and it may also be deemed to become rigorous than normal industry models. 3.. Errors and bias of baseline model The established baseline model is intended to predict future energy consumption. Hence, the analysis of errors and bias inside the model, and the predicted results is 34

5 extremely important. The first and most obvious error is the sampling error. Sampling error refers to errors resulting from the fact that a sample of units was observed rather than observing the entire set of units under study (ASHRAE, ). In this study, there is no sampling error because all the units in the sample are used and just the entire units. Secondly, besides the sampling error, there are several other sources of uncertainty within the model when dealing with the observed data. Reddy et al. (1998) had identified three categories of regression model error. They are as follows: (a) model misspecification error: This is due to approximation of the true driving function of the response variable; The typical causes of these errors are (i) the inclusion of irrelevant regressor variables or non-inclusion of important regressor variables; (ii) the assumption of a linear relationship, when the physical equations suggest nonlinear interaction among the regressor variables; and (iii) the incorrect order of the model, i.e. either a lower order or a higher order model than that suggested by the physical equations. (b) model prediction errors: This error arises because a model is never perfect, which means that R will never be 1.; invariably, a certain amount of the observed variance in the response variable is unexplained by the model. This variance 35

6 introduces an uncertainty in prediction. This uncertainty may arise even when the exact functional form of the regression model is known. The model parameters are random variables, and as a result of randomness, these are inherent observed errors in the regressor and response variables. (c) model extrapolation error: This fundamental error arises because of the fact that the prediction outside the domain of the original data, particularly for daily data which can represent the annual behavior of the system. Models identified from short data sets, which do not satisfactorily represent the annual behavior of the system, will be subject to this source of error. The prediction of long-term building loads from short-term in-situ tests will also suffer from this type of error. In this study, ordinary least squares (OLS) for regression estimation has been used and subsequently the projected model is used for prediction. Hence, error due to source (b) will be purely random and no bias will be introduced (Reddy et al. 1998). Furthermore, using the OLS as the algorithm for baseline model determination also means that there is no net determination bias which is always generated by the computational accuracy of modern computational tools such as DOE-.1. Finally, the assumptions regarding model residuals are under Gauss-Markov set of assumptions (Beck and Arnold, 1977). They are: (1) zero mean; () constant variances; (3) uncorrelated; (4) from a model where the regressor variables are non-stochastic, i.e. there is no measurement error in the regressor variables. It is under such 36

7 assumptions that the OLS yield the minimum variance estimators, and the true model uncertainty can be estimated properly. 3.3 Generate 9% uncertainty bands The ultimate goal of baseline model is to predict energy use in the post retrofit period where the actual energy use could not be ascertained. The degree of credibility related to the prediction results is statistically expressed in terms of confidence level. Vine and Kushler (1995) claimed that adopting 9% of confidence levels seemed to be the standard in electric utility load research. ASHRAE Guideline 14 () states that the requirements of 1% precision at 9% confidence has been adopted in part by the extension of the Public Utilities Regulatory Policy Act (PURPA) requirements for a class load research sample. Reddy et al. (1997a) pointed out the 9% prediction interval (PI) bounds of the estimates. Physically, if 9% confidence level is adopted, this means that if Ê is the value predicted by the model, then nine out of ten times, the next actual value of Ê will be between ( Ê +PI) and ( Ê -PI). For a simple two parameter model, the uncertainty band at the 9% confidence level for a given x (i.e. a given month) is defined as follows (Reddy et al., 1997a): PI α MSE m = t 1, n p m + m n + m = 1 n i= 1 ( x x) ( x x) i 1/ (3.4) 37

8 Where, t is the t-statistic evaluated at (1- α /, n p) ; α is the significance level (which for 9% confidence bands is equal to.1); MSE is the mean square error; m is the number of month; n is the number of observations; p is the number of parameters in the model; x represents the individual independent variable ( in this study, the outdoor dry-bulb temperature); x i is the independent variables; and x is the mean (annual )value of x i. Equation (3.4) is strictly applicable only to the two-parameter models. It also shows that if there is no big difference between x and x i, and if MSE is constant, the value of PI will change with the t-statistic evaluated at (1- α /, n p). It is consistent with uncertainty analysis, which shows that when the confidence level improves the bound of precision becomes wider to try to include more data. For multiple linear regression models, the PI for one point is defined (Bowerman, 199) as follows: PI 1 { 1+ Χ ( Χ Χ } 1/ α = t 1, n p s ) Χ (3.5) Here, S is the standard error of the estimate as given below: e i S = (3.6) n p 38

9 And, Χ is a row vector containing the values of the independent variables for which we wish to predict E. e i is the residuals; n is the sample number; p is the model parameter. Another important evaluating parameter for model prediction ability is the variance of forecast error (VAR). VAR is defined below (Tan, ): ( ) ( ) 1 x x VAR( Eˆ E) = S (3.7) n n x i x i= 1 Finally, the predicted energy consumption may be expressed as within the 9% PI. E & = E ˆ ± VAR 3.4 Normalized energy use One of the objectives of this study is to know the changes between baseline energy consumption and projected energy consumption. The baseline model can be used to correct for changes in energy use due to changes in weather data from year to year, month to month and day to day. However, another two important factors are changes in conditioned area and population. The impacts of these changes are needed so that 39

10 adjustments may be made to remove such effects from the resultant changes and these by giving an accurate estimate of the improvement attributed to energy efficiency and O&M measures. Most studies in the literature have assumed a proportional relationship between the energy use and changes in the conditioned area (Fels and Keating, 1993). Hence, normalized area-change energy use is merely the annual mean monthly energy use divided by the conditioned area for that particular year. However, until now there is no clear methodology for normalized population-varied energy use. Here, it is assumed that normalizing energy use by conditioned area would also implicitly normalize energy use for population changes, if it is speculated that population could be related to conditioned area, i.e. there could be a tendency to increase the conditioned are, if more people had to be accommodated. (Reddy et al., 1997a) 3.5 Percentage changes based on annul energy use For the baseline model based on monthly data, it can track monthly and annual energy changes. Following Reddy et al. (1997a), the equation for percentage change between actual energy use and projected energy use is given as follows: Emeasured Ebaseline projected E (%) = 1 (3.8) E baseline projected Here, E measured is the annual-based mean monthly energy use determined by simply 4

11 averaging 1 monthly utility bills for projected year. E is the annual baseline projected energy use predicted by the baseline model using corresponding weather data for the projected year. Furthermore, if one wants to compare the utility bills directly, the equation is (Reddy et al., 1997): Emeasured Ebaseline measured E (%) = 1 (3.9) E baseline measured The PI (Prediction Interval) for E(%) is defined below (ASME, 199): PI annual E (% E) = 1 E measured baseline projected α t(1, n p). VAR Model [ Eˆ ] baseline porjected measurm ment + error fraction 1/ (3.1) It is known that the measurement error is only around % for electricity measurement at 9% confidence level. Even when the regression model is very good, the contribution of measurement uncertainty to the total uncertainty tends to be very small, which is less than 3% (Reddy et al., 1997). Hence, the measurement error fraction part is taken as, and the formula is transferred as below: PI annual E (% E) = 1 E measured baseline projected α VAR t(1, n p). Eˆ Model baselinepr ojected (3.11) 41

12 According to equation 3.5 and 3.7, equation 3.11 can be transformed to equation 3.1 below: PI Emeasured (% E ) = 1 PI (3.1) ( E ) annual baseline projected 3.6 Possibilities of utility bill reading dates An important thing in the model development is to verify uncertainty in the bill reading dates. Neither the building owners nor Singapore Power Service Company provide detailed information on utility bill reading dates. In addition, every building may have its own policy for recording electricity use. Hence, it is difficult to verify the specific utility reading period. Here, the methodology was recommended by Reddy et al. (1997a) who pointed out that there were three possibilities in recording building energy use: a. The utility bill period is correspondent with weather data period. It means that the utility bill reading dates are the same dates as weather data dates. b. The utility bill period is one month later than the weather data period. Since utility personnel may read the meter in the first few days of a month, a common oversight of a data-entry clerk who subsequently has to transfer the utility bills 4

13 along with the associated month into a central data base would be to associate the energy use with the end date of utility bill period. Hence, such an oversight could cause a shift of one month. It means that the present utility bill actually represents the previous month s electricity use. c. The utility bill period is 15 days later than the weather data period. It is decided to take the average value of the temperature of the present month and that of the previous month and associate those values to the particular utility bill. If the model turns out to be substantially better than the two pervious cases, this would imply that the utility bill was read sometime around the middle of the month as against the beginning. However, due to the laborious procedure, the scope has limited to the middle corrections only. For each of the baseline model, all these three possibilities are performed. Then, the one which has the best-fit regression is selected. 3.7 Primary modeling of whole building energy consumption The energy consumption of a building is a complex function of climatic conditions such as dry-bulb and dew-point temperatures, building characteristics such as loss coefficients and internal loads, building usage such as for commercial or residential use, system characteristics and type of heating, ventilation and air conditioning equipment used. Since some of these parameters are difficult to estimate and measure 43

14 in an actual building, they are impractical to be considered as model variables. Although the building usage and operation parameters change from hour to hour, they are effectively constant from day to day. Hence, their variations have influence on the monthly and daily time scales. The most important indicator for baselining building consumption in the literature review is outdoor dry-bulb temperature. To ascertain the importance of outdoor temperature in the tropical region, a primary study is carried out through simple linear regression approach. In addition, this primary study was conducted using traditional utility bill analysis method in the literature (Reddy et al., 1997a) Simple linear regression approach As mentioned before, regression analysis is a statistical technique used to relate variables. Its basic aim is to build a mathematical model to relate a dependent variable to independent variables. Regression analysis has been widely applied in building energy research. In the primary modeling stage, simple linear regression method is applied. Simple linear regression here means single-variant linear regression as discussed in Chapter. Single-variant regression has only one independent variable and one dependent variable. The primary study tries to use the outdoor dry-bulb temperature ( T ) to baseline whole building energy consumption ( E ) based on monthly utility bills. The regression expression should be: 44

15 E a + a + ε (3.13) = 1T Where, ε is a normally distributed random number with zero mean and standard deviation. ε is difficult to discover since it changes for each observation, while a and a 1 remain fixed. Hence, the predicted value of E is more interesting thanε. Using ordinary least square (OLS) method, the equation of estimator Ê is expressed as: Eˆ = β + β T (3.14) Data collection Building energy use data Twelve buildings were selected randomly among all the buildings around the Central Business District in Singapore. They are all pure office buildings for commercial use. The utility bills of these twelve buildings were collected through surveys which were carried by the previous research on building efficiency (Lee, 1). In order to retain the individual building anonymity, these twelve buildings are referred to as building A,B,C,D,E,F,G,H,I,J,K and L. Table 1 shows the building size and the annual energy use of these buildings. For building A, B and C, was their modeling year. 45

16 Building D, E, H, I and J take year 1 as their modeling year. The utility bills from building F are only available from September 1999 to September 1 and therefore the baseline year is set to be from October to September 1. So does building K and L where October 1998 to October 1999 and June to April 1 is taken as their modeling years respectively. Building Table 3.1 Size and energy use in twelve buildings Modeling Year Total Bldg. Area(m ) Air Conditioned Area(m ) Total Bldg. Energy Consumption ( MWh/yr) A B C D E F Oct ~ Sept G Aug ~July H I J K Oct 98~ Oct L June ~Apl Weather data The correspondent weather data was obtained from National Environment Agency, Singapore. There are four weather stations in Singapore. They are Tengah, Changi, Seletar and Senbawang. The station in Seletar is selected as it is nearest to the twelve 46

17 buildings among the four stations. This is because onsite weather data are not available. The monthly data is found by averaging the hourly data of the whole month. Actually, there is little difference among these four weather points in terms of monthly mean temperature. Figure 3. shows the monthly mean outdoor dry-bulb temperature from year to 1. The highest temperature appeared in May as 8.6, while the lowest temperature appeared in January as 6.4.Hence, the overall range of temperature is only., which determines the difference of energy consumption among 4 months should be with little change. Figure 3. Temperature profile from January to December 1 Temperature Months Results of primary baseline model development As discussed before, there are three possible ways of utility bill readings dates. For each of the baseline model, all these three possibilities are performed. Then, the model which has the best-fit regression is selected. Table 3. shows the summary of 47

18 primary baseline model identification. Obviously, the three different assumed billing arrangements generate different R values. The best R value and the correspondent CV-RMSE value are easily selected as shown in the shaded box of Table 3. Table 3. Summary of primary baseline model identification Building Value CV-RMSE of Same Month Previous Month Middle Month Best R A B C D E F G H I*...1 N/A J K*.1..1 N/A L R Table 3. shows that all the buildings except for buildings I and K (shown with *) have certain relationships between whole building energy consumption and outdoor dry-bulb temperature. Because of the bad co-relationships in building I and K, where R is. and. respectively, they are both excluded in the future primary study. Table 3. also shows the identification of the three possible baseline models. Building D, F and G have the highest R value, which are.76,.77 and.61 respectively, when the utility bill period is the same month as the weather data, while the R value appears highest in building B, C, E and J, which are.7,.75,.78 and.71 respectively, when the utility bill period is one month later than the temperature data 48

19 file. Building A, H and L have the highest R value, which are.76,.6 and.76 respectively, when bill period is 15 days later than weather data period. In addition, the results in Table 3. also indicate that the way building owner records utility bill could not be arbitrarily justified because any of the methods could lead to the possibility of having the highest R value. Hence, all three possibilities should be considered to establish the best relationship between utility bills and weather data variations. Furthermore, all the best R values are between.6 and.8. The best and worst values appear in building E and G, which are.78 and.61 respectively. The correspondent CV-RMSE values of the best R are also good, which is 3.76 and 3.1. All CV-RMSE are below 5%, which are considered to be excellent models based on previous study. Reddy et al. (1997a) pointed out that CV-RMSE is a more appropriate statistical element for selecting the best among competing models to be used for baselining purposes in terms of three parameter and four parameter models. While in this study, the obvious difference between different projected values of R shows that it is sufficient for selecting baseline models in terms of two-parameter ones for certain buildings Results of primary baseline model performance verification 49

20 Once the baseline models have been established, it may be used for tracking and predicting future or past energy use. However, only six buildings have an additional year of utility bills. Hence, only building A, B, C, D, E and F are processed for baseline model performance verification. As discussed in section 3.3, the 9% prediction interval (PI) bands of the whole year are calculated for each building to evaluate the prediction ability of baseline models. Meanwhile, the 9% PI band on monthly basis is computed for building C only as an example for how baseline year model can be used for tacking energy use. Figure 3.3 shows the correlation between monthly energy consumption and out door dry-bulb temperature in Building C. Figure 3.3 Correlation between temperature and TBEC Building C Total Building Energy Consumption per m (TBEC) y =.3169x R = Out Door Dry-bulb Temperature The correlation equation of the baseline model of Building C is E ˆ =.317 T (3.15) 5

21 The range of T is from 6.4 to 8.6. The P value of T is only.3, which means T is statistically significant. Following the correlation equation, the monitored temperature was substituted in and consequently the predicted energy consumptions were computed. Figure 3.4 shows how the energy consumption changes through month to month in building C. It is assumed that for all PI calculations there is no measurement error as discussed before. For building C, changes of energy use from month to month are small. Figure 3.4 also demonstrates a good tracking ability of baseline model of building C. However, the deviation between predicted and actual energy consumption appears particularly large in January, June and December. The reason could be that during January and December there are Christmas and Chinese New Year holidays. Many office buildings are shut down. Hence, the predicted value will be larger than the actual one. Figure 3.4 Predicted vs. actual monthly energy use with 9% PI band 51

22 Building C Normalized Monthly Energy Consumption (MKh/month/m ) Projected Month Predicted (+)9%PI (-)9%PI Measured Figure 3.5 Residues of estimated whole building energy consumption Building C Differences between Predicted and Measued Monthly Energy Use (kwh/month/m ) Projected Month All the actual energy consumption values are within the 9% prediction interval of predicted energy consumptions. Hence, judging from this point, the estimates of electricity consumption are highly convincing. Figure 3.5 shows the residuals plot of predicted energy consumption. It shows that although the model gives an excellent 5

23 prediction, the distribution of residuals is not constant. It demonstrates a downward trend of the predicted energy consumption. This is against the assumption of OLS method. The abnormal distribution means that actual energy consumptions are lower than those predicted. Furthermore, the last month has a sudden surge in value. Therefore, the energy consumption predicted by the baseline model can be used by building owner to check whether the energy consumption follows a normal development. 53

24 Table 3.3 Summary of identified baseline models and the relevant statistical measures for six buildings Building Baseline Year Final Year Annual Energy Use (kwh/m /month) VAR(Model) (kwh/m /mo nth) MAE (%) 9% PI(kWh/mont h/m ) Percentage Change(from baseline year to final year) 9% PI Error( wit hout measurem ent error) Actual Modeled Actual ** Modeled A % % * -.45% 1.51% B C D E F Oct ~ Sep Note: * Negative value means the predicted energy use is higher than the actual one ** Actual percentage change is the result of direct bill comparing. 53

25 Table 3.3 gives a summary of percentage changes of energy consumption per square meter on monthly basis and statistics analysis from baseline year to final year for all six buildings. For building A, changes are clearly negative, which means the energy consumption shows an increasing trend. However, in terms of actual energy consumption, building D and E have positive percentage change, which means that the energy use is lower than the previous year. Such kind of behavior is contrary to the normal electricity use trends. In addition, building B has a positive value on the modeled percentage change, which indicates the opposite trends of the actual value. Building E is in such condition as well. The other four correctly predict the energy consumption trends. All of the models VARs, which indicate the variance of the forecast error, are small, except for building D. It increases according to the building energy consumption. It appears highest in building D, which is 9.79(kWh/m /month) and lowest in building B, 1.35(kWh/m /month). This means in terms of accuracy, the predicted data in building B is the best one. Furthermore, direct deviations, namely MAE, from modeled and actual energy use are all below 3%, except for building D. The deviation is from the difference of predicted and actual on based on actual one. The lower the deviation, the more accurate the predicted energy consumption is. At 9% confidence level, following equation (3.6), 54

26 one can get a PI (Prediction Interval), which shows the error band of forecast value. For building A, the monthly electricity use in year 1 is predicted as 4.11(±1.71) (kwh/m /month). This represents an increase of about.9% (±1.51). The actual energy use is 3.5(kWh/m /month), which represents an increase of about.45%. Although the difference between the modeled and the actual energy use is only.59(kwh/m /month), the error band is rather wide. For example, the worst prediction results belong to building D. In this case, a modeled energy consumption of 4.33(±5.46) (kwh/m /month) was computed and it decreases its energy use by 18.6%(±48.15%) comparing with the baseline year, while the actual energy use is 8.59(kWh/m /month) and it decreases by 16.58%. The results therefore show a large uncertainty in estimates Discussion Table 3.3 shows that all the error bands of percentage changes are wider than percentage changes, whether they are actual values or modeled. Hence, the percentage changes are not very significant statistically. The reasons are given as follows: (a) Low R value. Although), these models are included among the excellent model (Reddy et al., 1997a), all the R values are lower than.8. This means that more than % of the variation in Y value cannot be explained by these models. This study shows that, in the tropical region, building energy consumption has a 55

27 certain relationship with temperature, but maybe not sufficiently to provide an excellent prediction of energy use as a single parameter. (b) Little changes in monthly mean temperature in tropical region. This leads to small percentage changes both in actual and modeled annual energy use. Furthermore, together with reason (a) and the goal of the prediction interval of baseline-projected energy use to meet 9% confidence level, the width of PI becomes large. (C) Although all the actual and modeled percentages are within 9% uncertainty interval, one shall not only look at such fraction intervals to judge its prediction ability. As discussed before, 9% error band for percentage changes has little statistical meaning and the actual value of PI for modeled energy use is more comparative and objective. For example, in building B, the modeled value is.59 (kwh/m /month) with 1.6 prediction interval at 9% confidence level, while the modeled percentage is.4% with 1.6% prediction interval. Obviously, the former value has more statistical meaning. 3.8 Improved utility bill analysis method Multivariate regression study From the primary evaluating study, which is a popular and standard method in the literature, it seems there is a potential to improve prediction accuracy. Here an 56

28 improved methodology is carried out by adding more independent variables using multiple linear regressions Selection of more weather variables Relative humidity (RH) and global solar radiation (GSR) are chosen as two additional variables relevant to local conditions. The reasons could be from the engineering principles for modeling cooling energy consumption in large commercial buildings. It is well known that the cooling energy consumption accounts the most in the whole commercial building energy consumption. Previous research studies by Knebel (1983), Katipamula and Claridge (1993), and Liu and Claridge (1995) have shown that buildings can be physically modeled as a two zone buildings, one exterior zone and one interior zone, with adequate accuracy. Again, to simplify the analysis, both zones are assumed to have same set point temperature, the total cooling energy consumption of the building with a DDCV system as an example is given by Eˆ c ( Tm Tc ) [ m c ( T T ) + ( U ( T T ) + q + q ) + q ] + m h ( w w ) = & t p h z o o z sol i, s, e i, s, d c v m c ( Th Tc ) (3.16) where, m& means mass flow rate, c p means specific heat of air, U is the overall heat transmission coefficient and hv is the latent heat of vaporization. Equation (3.16) shows that total cooling energy consumption ( E ) has relationships with total global c 57

29 horizontal solar radiation ( q sol ), internal sensible heat gains ( q i, s ), and specific humidity of air ( w ) etc. However, T h, T c, T m, T z, w m and w c are seldom actual on a continuous basis. In most commercial buildings, a major portion of the latent loads is due to ventilation; therefore, the term w w ) in Equation (3.16) can be replaced by ( m c + ( T dp Ts ), where the subscripts m c, dp, and s means mixed air, cooling, dew-point temperature and soil surface respectively, and T means the dry-bulb temperature. This term is set equal to zero when it is negative, as represented by the superscript +. Hence, Equation (3.16) finally changed to Equation below: ˆ + Ec = α + β1t + β I + β 3IT + β 4Tdp + β 5qi + β 6 q sol (3.17) whereα, β 1, β, β 3, β 4, β 5 and β 6 are regression coefficients. With such engineering analysis processes, Katipamula et al. (1998) applied multivariate regression method to model cooling energy consumption in commercial buildings. The results showed that only outdoor dry-bulb temperature and dew point temperature were statistically meaningful and accounted for 9% of the variation in the cooling energy consumption. However, there is no further research in the literature carried out for the whole building level, especially in the tropical area Multiple linear regression analysis After verifying the climate variables, namely, T, RH and GSR, a multiple linear 58

30 regression model is derived as follows: ˆ = β + β T + β RH + β GSR (3.18) E 1 3 As stated in Section 3., R and the coefficient of variance (CV-RMSE) are two major measures to evaluate the goodness of fit of the model. In addition, another important verification parameter in the multiple linear regression (MLR) analysis is the partial correlations. The partial correlations procedure computes partial correlation coefficients that describe the linear relationship between two variables while controlling for the effects of one or more additional variables (SPSS 1999). This parameter clearly shows the correlations between every independent variable with the dependent variable. The whole multiple linear regression analysis is processed using backward elimination method in SPSS. The backward method tries to examine only the best regressions containing a certain number of variables. After the regression model containing all variables has been set up, the partial F-test is calculated for every predictor variable. Based on the critical F-value, which are defined as.1 in this study, the backward procedure removes all unneeded X-variables one by one. This backward method lists all possible R based on three possible variables. Hence, when adding additional variables, it can check its effect to the regression model itself. 59

31 3.8.3 Multicollinearity analysis and Durbin-Watson (DW) test Another important issue in MLR is the multicollinearity. MLR analysis assumes that there is no correlation between independent variables, which is also the limitation of MLR. A rule of thumb is that multicollinearity effects may be important if the simple correlation between two variables is larger than the correlation of one or either variable with a dependent variable (Fels et al., 1993). In addition, previous research carried by Wu et al. (199), Reddy et al.(1997a) and Katipamula et al.(1998) showed that there is no strong collinearity between outdoor dry-bulb temperature, outdoor dew-point temperature, relative humidity, solar radiation and internal heat gains on a daily basis in the temperate zone. In this study, in order to examine the limitation of MLR, a correlation matrix between every additional variable was made using SPSS. This is to assess the degree of collinearities when using the final multiple variant model. If the final baseline model was finally verified to be single variant regression model, a Durbin-Watson (DW) test will be performed. This test is used to detect whether there is auto-correlation in the model error term. Sometimes, the mode residue may be auto-correlated rather than random, which is against the OLS assumption that error terms appear randomly. This phenomenon often exists especially with short-term actual data. 6

32 3.8.4 Data collection The same twelve buildings used for single-parameter regression tests described in Section have been utilized for the Multiple-variant regression study. The magnitude and the energy consumption pattern of twelve buildings were presented in Table 3.1. Here, additional weather data profiles are presented including dry-bulb temperature (T ), relative humidity (RH) and global solar radiation (GSR) as shown in Figure 3.6. Figure 3.6 Weather data from to 1 Monthly Mean Temperature ( ) Relative Humidity (%) Global Solar Radiation/1(cal/cm) Month Figure 3.6 shows the monthly average weather data from year to 1. Due to the big value of GSR, it is plotted as monthly value divided by ten. Figure 3.6 also shows that there is no significant variation over the two years. For example, the highest average temperature appeared in May as 8.6, while the lowest average temperature appeared in January as 6.4.Hence, the overall 61

33 difference of average temperature is only.. The same situation appears in RH and GSR data Results of improved baseline model development The flowchart of the developmental steps for the improved utility bill analysis method is shown in Figure 3.7. The results of possible utility bill reading dates were performed in Table 3.. Hence, for the multiple regression analysis, the same method is processed, following the previous results. The summary of primary MLR results is shown in Table 3.4. It also shows the individual effect of each variable to the whole MLR model. The partial correlation parameter R of.86 for T means that the outdoor dry-bulb temperature explains 86% of the variation in the whole building energy consumption. For most of these twelve buildings, the partial R of T are more than.75, which means that T explains most of the variation in the whole building energy use. For building K and I, GSR explains most of this variation. 6

34 Fig. 3.7 improved utility bill analysis method flowchart Collect utility bills from building owners Collect correspondent weather data from nearest weather station Preliminary Multiple Linear Model Model 1 Model Model 3 Primary Analysis R 1, CV 1 R, CV R 3, CV 3 Highest Model R, lowest CV Backward Elimination Process R 1, CV 1 R, CV R 3, CV 3 t test at 9% confidence level > One One Variable Variable Model Develop ment Multicollinearity test Durbin-Watson test Final Regression Model Input: MMT of projected year Output: Projected Year s Energy Use Verification Generation of 9% PI Band Annual Percentage Change (both projected and actual) Generation of 9% PI for Percentage Change (projected only) 63

35 In addition, the critical value of t-test at 9% confidence level (CL) for n data points (n=1) and p, number of parameters (p=4), is Hence, judging from the t-test, at 9% confidence level, for 1 out of 1 buildings, including Building A, B, C, D, E, F, G, H, J and L, T is statistically significant, where all the t-test values of T is larger than GSR is also statistically significant in building A, H, I and K, which are -.49, -.1, 3.45 and.14 respectively. The RH in building J and I are statistically significant, which are and.4. Table 3.4 shows that, at 9% confidence level, some buildings models are found to be single variant linear regression models. These include Building B, C, D, E, F, G and L. Building A, H, I and J are accepted as multiple linear regression models. Table 3.4 also shows that the R of multiple regressions of most of the buildings are around.8, which is quite good comparing with the results reported in the literature (Sullivan et al., 1985; Reddy et al., 1997b; Sonderegger, 1998; Katipamula et al., 1998; Kissock et al., 1998), which are typically between.74 and.95. The R of building K and I are only.49 and.61 respectively. Hence, we believed that the weather data has little relationships with the whole building energy consumption in these two buildings. These two buildings are excluded in the further studies. 64

36 Table 3.4 Summary of statistical regression results for each variable at 9% CL Vari A B C ables Correlation Correlation Correlation Partial R Model t Partial R R Model t Partial R R Model R t Partial t Partial T RH GSR (Continued) Vari D E F ables Correlation Correlation Correlation Partial Model Model Model t R R R R R R T RH GSR t (Continued) Vari G H I ables Correlation Correlation Correlation Partial R Model t Partial R R Model t Partial R R Model R T t Partial t Partial RH GSR (Continued) Vari J K L ables Correlation Correlation Correlation Partial Model Model Model t R R R R R R T RH GSR t In order to understand whether additional parameters will improve the model R and accuracy, model RMSE and CV were calculated based on different number of model 65

37 variables as shown in Table 3.5. In the backward regression, the effect of adding each additional variable can be assessed by comparing three values of R. The elimination criterion for each step is following the increase of partial R of every independent value. For example, for building B, a R value of.79 fort, RH and GSR means that these three variables can explain 79% of the changes in whole building energy consumption. In the second step, due to the low statistical significant value of GSR, judging from Table 3.4, only T and RH are included. However, T and RH can only explain 76%. Finally, T alone can only explain 73% of such changes. Some of the buildings are following such kind of elimination until onlyt is included. However, for building A and H, GSR is more significant than RH. Hence, in the second step, RH was excluded. Furthermore, for building A, G, J and H, the multiple linear regression models show much better than single variant regression model. Their R increase by more than.1. Both R in building G and H improve from.6 to.81, while for building A, R improves from.76 to.9. For other buildings, with adding more variables, R also improves a little bit, but insignificantly. The average improvement of the value of R among other buildings is only.. In terms of R, MLR is better than simple linear regression which has only with one variable, namely, T. 66

38 Table 3.5 Summary of statistical results for multiple linear regression Variables B C D T,RH,GS R R RMSE CV R RMSE CV R RMSE CV T,RH T Variables E F G T,RH,GS R R RMSE CV R RMSE CV R RMSE CV T,RH T Variables J L T,RH,GS R R RMSE CV R RMSE CV T,RH T Variables H A T,RH,GS R R RMSE CV R RMSE CV T,GSR T An enhanced value of R is not sufficient to conclude that the model is suitable for modeling whole building energy consumption. For building C, D E and F, although R improves with more variables, RMSE and CV values are higher. If increased RMSE and CV are compiled with R, it may mean that the MLR model is not acceptable as a baseline model set up. In order to develop a model as accurate as possible, RMSE and CV should be as low as possible. Figure 3.8 shows the changes of CV with the addition of independent variables to the model. Consequently, in term of accuracy, for building C, D, E and F, simple linear regression model is better than 67

39 multiple linear regressions. For building A, B, G, H, J and L, RMSE and CV decrease by 1% with the increasing value of R, which demonstrates that multiple linear regression model is better than simple linear regression models in these six buildings. Figure 3.8 Additional independent variables to the model A B C D E F G H J L B J H G L A T RH GSR CV(%) Results of error analysis However, at 9% confidence level, as discussed before, only building A, G, H and J are accepted as multiple linear regression models. An example of MLR analysis, taken as building A, is shown below. For building A, RH was rejected, while T and GSR are independent variables. The multiple regression equation for building A is given in Equation 3.19 together with other derived regression equations for rest of the nine building are shown in Table 3.6: Eˆ = T. GSR (3.19) 68

40 Subsequently, a multicollinearity test between T and GSR was done only for Building A. The result of multicollinearity test from SPSS was shown in Table 3.7. TBEC means the total building energy consumption. It shows that T has a positive correlation with GSR at.57, which means there is no significant relationship. The P-value in the significance row, which is.43, also shows the same result. For building G, H and J, the same analysis process was conducted and the results were shown in Table 3.8, 3.9 and 3.1. There is also no significant multicollinearity between the independent variables. Hence, the assumption thatt, RH and GSR are considered as three independent variables without any co-relationships is valid. Table 3.6 Summary of all the regression equations Building A B C D E F G H J L Regression Equation Eˆ = T. GSR E ˆ = T E = T E ˆ = T E ˆ = T E ˆ = T Eˆ = T. 8RH + Eˆ = T. GSR Eˆ = T. 5RH E ˆ = T 69

41 Table 3.7 Result of multicollinearity test for building A Pearson Correlation TBEC T GSR TBEC T GSR Sig. (1-tailed) TBEC...75 T...43 GSR Table 3.8 Result of multicollinearity test for building G TBEC T RH Pearson Correlation TBEC T RH Sig. (1-tailed) TBEC T RH Table 3.9 Result of multicollinearity test for building H TBEC T GSR Pearson Correlation TBEC T GSR Sig. (1-tailed) TBEC...58 T...43 GSR Table 3.1 Result of multicollinearity test for building J TBEC T RH Pearson Correlation TBEC T RH Sig. (1-tailed) TBEC...63 T RH

42 Meanwhile, a Durbin-Watson (DW) test was done for all other six buildings. This test was used to detect whether there is any auto-correlation in the model error term. Sometimes, the model residue may be auto-correlated rather than random. This phenomenon is against the OLS assumption that error terms are random. The results of Durbin-Watson test from SPSS are as shown in Table As for the study, at 95% confidence level, when the observation values are twelve and parameter are two, the upper, denoted by d u, and lower critical, denoted by d l, values of DW test are 1.58 and.81 respectively. All the test statistic values are between d u and 4-d u (building B, D, E, F and L) or 4-d u and 4-d l (building C). Hence, it is believed that there is no auto-correlationship in the model error term. Therefore, the developed linear regression models are excellent and accurate. Table 3.11 Results of Durbin-Watson Test Building B C D E F L DW Value Results of improved baseline model performance verification Finally, to assess the prediction ability of the regression models, the models identified in Table 3.6 were used to predict monthly whole building energy for another year. However, only building A, B, C, D, E and F have one additional year data. As building B, C, D, E and F are accepted as single-variant regression model, the results are the same as presented in Table 3.3. Hence, Table 3.1 only shows the results of 71

43 MLR prediction of building A. Chapter 3 Baseline Models of Whole Building Energy Consumption Table 3.1 Results of regression prediction of Building A Building Baseline Year Final Year Annual Energy Use (kwh/month/m ) VAR (kwh/month/m ) Deviation (%) Actual Modeled A Table 3.1 shows that the variance of forecast error (VAR) is improved comparing with the result in Table 3.3, which is.9, while the deviation is more or less the same. The total improvement in VAR is 137.4%. It indicates that the modeled annual energy use is more accurate than that predicted using the single-variant regression model. Because of the limitation of data obtained, only one building has been verified. This may lead to the bias of the conclusion made before. However, in this section, the proposed methodology is paid more attention than the result itself. It is believed that the MLR method is better than single-variant regression method in the Tropics as well as the temperate zones in the literature (Katipamula et al., 1998) Discussion The results as presented in the improved utility bill analysis stage suggest that baseline modeling of the whole building energy consumption in most of the large pure office buildings can be performed either by multiple linear regression models or single variant regression models depending on different model requirements such as confidence level and prediction accuracy. However, as shown in Table 3.5, the CV of 7

44 some models will increase by adding more independent variables. Hence, for such buildings, the MLR models are neither suitable nor better than using simple linear regression. Furthermore, at 9% confidence level, only building A, G, H and J are accepted using MLR analysis. However, it is still believed that for the suitable buildings, this improved method will increase the credibility and prediction accuracy because of the improved R and less CV. Among MLR regression models, the independent variables are also different. For building A and H, T and GSR are most important, while T and RH are significant in building J and G. The reasons may be that: a) although all these ten buildings are within the same area, namely, central business district, their surroundings are totally different. Some buildings are in the shadow of other buildings all the year. Some buildings are very tall and have no taller buildings around. The location and height of buildings have important effect on solar radiation gains and heat gains. b) different building has different shape, façade materials and orientation, which decide the different OTTV (overall thermal transmit value). c) office arrangement inside these buildings may be different too. Some buildings have office both in the South and North sides, while some may only have in one side. 73