Modeling Hourly Electric Utility Loads in the Phoenix Area

Transcription

1 3/10/2012 SOS 598: Applied Data Analysis for Energy Scientists 12/06/2011 Modeling Hourly Electric Utility Loads in the Phoenix Area Seth Herron Joseph Fernandi SOS 598: Applied Data Analysis for Energy Scientists 12/06/2011 Objectives Analyze hourly interval data for average residential energy use and total electric load for a large utility in Phoenix. Observe the relationship between dry-bulb temperature and energy use for both cases Identify correlations between energy use and time of use (hour of day and day of the week) 1

2 3/10/2012 Data Set Hourly interval data for: Average residential energy use in utility territory (kw) Total utility load (MW) Dry-bulb temperature ( F) Methodology Energy & Temperature Change-point model (5P) was used to determine the critical temperatures at which energy use becomes heavily influenced by thermal load Energy, Time of Use, & Temperature Fourier series model was used to relate energy use to time of use AR1 model was used to improve Durbin-Watson (DW) statistic & R 2 value 2

3 3/10/2012 Weather Data - Dry-bulb Temperature ( F) CDD50 = 8425 hrs HDD65 = 1110 hrs 1 1 Source: ASHRAE Energy Use vs. Temperature Total Utility Load (MW) 3

4 3/10/2012 Change Point Model - Residential R 2 = 0.83 RMSE = CV-RSME = 19.3% Xcp 1 = F Xcp 2 = F Ycp 1 = kw Average Residential Load = (61.64 T) (T 82.36) Change Point Model Total Utility Load R 2 = 0.88 RMSE = CV-RSME = 11.0% Xcp 1 = F Xcp 2 = F Ycp 1 = 2434 MW Total SRP Load = 2, (52.74 T) (T 79.4) 4

5 3/10/2012 Change Point Model Summary of Results Total Load Average Residential Load Critical Heating Temperature F F Critical Cooling Temperature 79.4 F F Explanation of Results Critical Heating Temperature Commercial/Industrial sector has higher internal gains (more equipment/machinery/people etc.) than residential - therefore less need for heating More likely to find programmed temperature setbacks (EMS/T-stats) t in commercial sector More prevalence of gas heat in commercial/industrial Critical Cooling Temperature Commercial sector more likely to keep cooling setpoint lower to increase comfort Change Point Model Residuals Total Utility Load Fairly even distribution of residuals No observable patterns 5

6 3/10/2012 Energy Use by Month Total Utility Load Larger variance over Summer months (high impact of thermal load) Smaller variance during shoulder months Energy Use vs. Day of Week Total Residential Considerably higher energy use on weekends Total Utility Considerably higher energy use on weekdays 6

7 3/10/2012 Total Energy Use vs. Time of Day Conclusions Observable effects of heat storage Increased usage on weekdays during normal business hours Residential Energy Use vs. Time of Day Conclusions Increased usage as people prepare for work (6-8am) & return from work (5-8pm) Observably higher consumption on weekends 7

8 3/10/2012 Residential Energy Use vs. Time of Day (Weekday) Fourier Series Average Residential Load (kw) = *Avg Res I *Avg Res I *CH *CH *CH *CH *SH *SH *SH *SH *CH1*Avg Res I *SH1*Avg Res I *SH2*Avg Res I *SH4*Avg Res I *CH1*Avg Res I *CH2*Avg Res I *CH3*Avg Res I *CH4*Avg Res I *SH1*Avg Res I *SH2*Avg Res I *SH3*Avg Res I *SH4*Avg Res I2 R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = Residential Energy Use vs. Time of Day (Weekday) AR1 Model YR MO Date Day Hour Total SRP Avg Res Avg Res New Avg Temp Residuals AR(1) Residuals SSR SST Load Load (pred) Res (pred) AR1 Model R-squared = percent Durbin-Watson statistic =

9 3/10/2012 Total Energy Use vs. Time of Day (Weekday) Fourier Series R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = AR1 Model R-squared = percent Durbin-Watson statistic = Residential Energy Use vs. Time of Day (Weekend) Fourier Series R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = AR1 Model R-squared = percent Durbin-Watson statistic =

10 3/10/2012 Total Energy Use vs. Time of Day (Weekend) Fourier Series R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = AR1 Model R-squared = percent Durbin-Watson statistic = Total Utility Load vs. Average Residential Load Utility customer base: 949,388 as of April 30, B-intercept indicates a residential customer base of ~1.2M 1 Source: 10

11 3/10/2012 Questions 11

12 Modeling Hourly Electric Utility Loads in the Phoenix Area Seth Herron Joseph Fernandi SOS 598: Applied Data Analysis for Energy Scientists 12/06/2011 Table of Contents Abstract...2 Description of Problem/Scope...3 Methodology of Analysis/Modeling...3 Analysis of Results...6 Summary & Conclusion...13 References...14 Appendix A...15

13 Abstract Models that can predict electricity use at a specified time, given a particular ambient temperature, are extremely valuable for utility companies. Such models can provide utilities with the ability to better plan demand-response programs, justify the installation of large-scale solar generation, properly sequence gas-fired plants, or even fine-tune rate schedules that offer different billing rates depending on time of use. In this report, we develop such models, one for the total electric utility load and another for only the total residential sector. The models are based on a year s worth of hourly interval data provided by a utility company. The dataset contains dry bulb temperature, total electric load, and the average load of all residential customers. First, using only temperature and load data, a change-point model (5P) was developed to find the critical temperatures at which energy use becomes dominated by thermal load. For the total service territory, the critical heating and cooling temperatures were found to be 52.7 F and 79.4 F respectively, with R 2 = For the residential sector, the critical heating and cooling temperatures were found to be 61.6 F and 82.4 F, with R 2 = Then, a Fourier series model & OLS regression (forward-model) was combined with the change point model to incorporate the impacts of time of use. For total load on weekdays and weekends/holidays, the model fit the data with R 2 terms of 0.95 and 0.93, and Durbin-Watson statistics of 0.31 and 0.26 respectively. For average residential load on weekdays and weekends/holidays, the model fit the data with R 2 terms of 0.93 and 0.92, and Durbin-Watson statistics of 0.30 and 0.26 respectively. The low Durbin-Watson statistics indicate that there remain patterns in the model that are unaccounted for by the parameters. In an effort to improve the model outputs, we used an AR(1) model. For total load on weekdays and weekends/holidays, the model fit the data with an R 2 term of 0.99 for both, and Durbin-Watson statistics of 2.0 and 1.98 respectively. For average residential load on weekdays and weekends/holidays, the model fit the data with an R 2 term of 0.99 for both, and Durbin-Watson statistics of 2.06 and 2.0 respectively. While the AR adjusted model is extremely accurate, it is limited in its predictive ability because it can only model electricity load for a given hour and temperature if it has residuals from previous hours. In the future, we would like to incorporate additional methods in our analysis to identify the shortcomings of the original OLS model. 2

14 Description of Problem/Scope In a particularly hot climate such as Phoenix, AZ, the impacts of weather on energy use can be severe. As a result of the number of cooling degree-days, there is a strong correlation between temperature and electricity use, given the increased need for mechanical cooling to maintain thermal comfort. Additionally, in a largely urban population, energy use can vary significantly based on the time of use (e.g. time of week or time of day) as people commute to and from work. The ability to forecast electricity use at any given time, given a particular ambient temperature and time of day or week is extremely valuable to utility companies who must plan accordingly to generate enough capacity to meet a given load. As such, models for energy load based on temperature and time of day are invaluable tools for energy utility providers. Such models can provide utilities with the ability to better plan demand-response programs, justify the installation of large-scale solar generation, properly sequence gas-fired plants, or even fine-tune rate schedules that offer different billing rates depending on time of use. The aim of this report is to develop such a model. We obtained a year s worth of hourly interval data for both average residential and total utility-scale electrical loads from a utility company. In addition to hourly loads, the dataset also included hourly dry-bulb temperature as measured in Phoenix, AZ. Using this dataset, we formulate models that quantify, with hourly resolution, the relationships of dry-bulb temperature 1 and time of day to both average residential and total utility-scale electrical load. Because of differences in hourly energy use on weekdays versus weekends, separate models for the two types of days needed to be created. Methodology of Analysis/Modeling Prior to performing any analysis on the data that was provided by the utility, the data was processed to remove dry-bulb temperature readings of 0 F, that were likely caused by the temperature monitoring system being shut off for maintenance or interruptions in power supply. This process removed 237 of the 8,760 data points for temperature, total load, and average residential load. There were no interruptions in data for total load or average residential energy use. First, electricity use was plotted against temperature for both the total load and the average residential load to observe the relationship between dry-bulb temperature and energy use. This plot clearly illustrated the effects of heating and cooling load on electricity use, as a fairly linear correlation could be seen between temperature and energy above some critical temperature for cooling and below another critical temperature for heating. A common method for accounting for this behavior is through a change point model (Ali et al, 2011). Based on the 1 Generally, when such energy-use models are produced, other weather parameters are considered humidity being the most common and most significant. However, in the arid climate of Phoenix, AZ, humidity is generally negligible. Thus, its absence from our analysis should not have a noticeable impact on our modeling results. 3

15 distribution of the data points, a 5P change point model was selected. The 5P model accounts for a rise in electricity load due to cooling above one change point temperature and a rise in electricity load due to heating below another change point temperature. Between these points, the plot of electricity load vs. temperature is relatively flat, as there is relatively little thermal load to overcome. The software Energy Explorer was used to fit a 5P model to each set of data by selecting the appropriate change points and performing a linear regression. Because the change point model includes two distinct change points, it takes the following form (Kissock et al, 2003): Or, it can be rewritten in the form: Y 1 2 X X X 2 E T H T I 1 T T C I 2 where E is electricity load (in kwh in the average residential load model, and in MWh in the total load model), T is the temperature variable (in degrees F), T H is the heating change point (the temperature at which heating systems are switched on), T C is the cooling change point (the temperature at which cooling systems are switched on),,, and are coefficients selected by the software Energy Explorer, and I 1 and I 2 are indicator variables such that: I 1 1 if T T H if 0 T T H I 2 1 if T T C if 0 T T C In order to observe the relationship between energy use and time of use, first a plot of energy use vs. day of the week was generated. This plot showed that for total load, energy use was considerably higher during the week, while for average residential load, energy use was considerably higher over the weekend. This data confirmed that there would indeed be significantly different hourly profiles for each total load and average residential load depending on whether it was a weekday or a weekend. This difference in hourly profiles is easily explained: during the week, many adults and children leave the home to go to work or school. Thus, during the weekdays, average residential load is relatively lower while total utility-scale load is higher. The inverse is true during weekends when residents are more likely to stay at home and commercial and industrial buildings are closed. With this information, we separated the data into four sections: Table 1: Organization of data Total load on weekdays Total load on weekend/holidays Average residential load on weekdays Average residential load on weekends/holidays 4

16 In order to account for the daily pattern in hourly energy use, we utilized a time-series model. When data displays a clear periodic cycle, as this does, a Fourier series can model the cycle (Reddy, 2011). For our purpose, the final model we construct must be capable of handling both temperature and time of day as inputs. So, our time series model must be an ordinary least squares (OLS) model so that it can be combined with the change-point temperature model (which is also an OLS model). A Fourier series meets this criterion by converting periodic trends into parameters that can be correlated using OLS techniques. For each set of data, a Fourier series model was generated in Microsoft Excel to relate energy use to time of use using the equation below (Reddy, 2011): b 0 yt n j 1 a j cos j t For our model, we set n=4 with a period of 2 /24: Et b 0 4 j 1 b j sin j t a j cos 2 jt b 24 j sin 2 jt 24 where E(t) is the output of the electricity load model (kwh for the average residential load model, MWh for the total utility-scale load model), t is time during the day (hour), and b 0, a j, b j, etc are coefficients determined by a commercial software package (CSP). In order to integrate the results of the first model (energy use vs. temperature) the indicators for the change point model (with interaction terms) were included in the OLS forwardmodel in CSP. The results showed that the models fit the data rather well based on the R 2 value, however the Durbin-Watson statistic was well under 2 in each case, suggesting that a pattern existed in the residuals, and that further refinement was necessary. In order to improve the model, an autoregressive AR(1) model was used to smooth the data. The AR model can smooth over unaccounted for patterns in the model by incorporating the past time step s residual into the current time step (Reddy, 2011): AR(t) E(t) R t 1 where AR(t) is the output of the AR model for time period t, E(t) is the output of our electricity use model at time period t, R t-1 is the residual of our electricity load model at time t-1, and is the lag coefficient given by CSP. 5

17 Analysis of Results Prior to performing any specific analysis or generating any models, the raw temperature and energy use data was plotted to observe the relationship between the two as shown below in Figures 1 and 2. Figure 1: Average Residential Energy Use vs. Temperature Figure 2: Total Load vs. Temperature From the figures above, it can be seen that there exists a correlation between energy use and temperature above some critical temperature, where mechanical cooling begins, and below some temperature, where heating is required, while the energy use throughout the middle band of temperature remains relatively flat, as no (or relatively little) heating or cooling is required. In order to determine these critical temperatures at which the energy use becomes heavily influenced by thermal load, a change point model (5P) was used for both average residential load and total load. The plots that have been generated using Energy Explorer, along with a statistical summary including change points and model equations are shown below. 6

18 Figure 3:Change Point Model (5P) Average Residential Load Table 2: Change Point Model (5P) Average Residential Load Summary of Results R 2 = 0.83 RMSE = CV-RSME = 19.3% Xcp 1 = F Xcp 2 = F Ycp 1 = kw Figure 4: Change Point Model (5P) Total Load Table 3: Change Point Model (5P) Total Load Summary of Results R 2 = 0.88 RMSE = CV-RSME = 11.0% Xcp 1 = F Xcp 2 = F Ycp 1 = 2434 MW 7

19 In both models, the R 2 value suggests a relatively good model fit, however the model can be improved by incorporating the relationship between energy use and time of use. Using the equations generated by Energy Explorer, residual plots were generated, which can be found in Appendix A. The residual plots show a fairly even distribution of residual points with no observable patterns. The table below summarizes the critical temperatures at which the energy use becomes heavily influenced by thermal load. Table 4: Change Point Model (5P) Critical Cooling & Heating Temperatures Total Load Average Residential Load Critical Heating Temperature F F Critical Cooling Temperature 79.4 F F From the change point model, it can be seen that the critical heating temperature is considerably lower for the total load than for the average residential load. This difference can be attributed to several factors. In general, the commercial/industrial sector (which dominates much of the electricity use outside of the residential sector) has considerably higher internal gains as a result of more equipment, machinery, people, etc. than the residential sector, therefore there is less need to provide additional heating. Another contributing factor is the prevalence of energy management systems and programmable thermostats in the commercial/industrial sector, which utilize temperature setbacks. Since the majority of the colder bin temperatures will occur during the night, or during unoccupied periods, the higher heating setpoint allowed by programmed temperature control avoids the use of additional heating when the building is unoccupied. Lastly, since the building stock in the commercial and industrial sector is much larger on average, it is more likely to find gas heat (i.e. boiler systems) in larger spaces, whereas the residential sector will have a higher prevalence of heat pumps (which use electricity rather than gas). The change point model also shows that the critical cooling temperature is slightly lower for the total load than for the average residential load. This difference can be attributed to the fact that the commercial sector (primarily retail/office) is more likely to keep the cooling setpoint lower to increase occupant comfort, whereas the residential sector might utilize temperature setbacks during the day since they are more likely to be unoccupied during the week. Additionally, homeowners and renters are more likely to be conscious of what their thermostat is set at since they are the ones paying the utility bill, while in the commercial sector, the people who occupy the space are often not tied to the party who pays the utility bills. Other contributing factors include a generally higher window to wall ratio in non-residential spaces, higher internal loads for non-residential spaces, the ability for residences to open windows to maintain desirable comfort levels during moderate outdoor temperatures, and lower temperatures (in general) experienced by the residential sector as a result of being located primarily in suburban areas with more vegetation. 8

20 As mentioned in the methods section, the data for both average residential energy use and total load has been divided into the following two categories: weekdays and weekends/holidays. The two figures below show the relationship between both average residential load and total load and time of day. Figure 5: Temperature & Average Total Load vs. Time of Day Figure 6: Temperature & Average Residential Load vs. Time of Day 9

21 From the Figure 5 above, it can be seen that the total load is observably higher during the week, which justifies the fact that most utilities offer Off-Peak rates over the weekend. Figure 6 illustrates that on average, there is considerably higher electricity consumption on weekends in the residential sector, which makes sense as people are typically home for more hours out of the day on weekends. Additionally, the chart shows increased electricity consumption as people prepare for work (~6-8am) and as people return home from work (~5-8pm). In order to investigate the impacts of time of use in addition to temperature on energy use for both cases, we used a Fourier Series analysis to find the coefficients for the ordinary least squares (OLS) model, which would be analyzed in CSP. Additionally, the change point indicators, as well as their interaction terms, were input as independent variables to the forwardmodel. CSP generated OLS models for each of the four cases. The model for the average residential load during weekdays is as follows: where Avg Res I1 stands in for (T H -T)I 1 or (T H -61.6)I 1, Avg Res I2 stands in for (T-T C )I 2 or (T-82.4)I 2, and CH1 stands for cos(1*2 *t/24), and so on. Below are residual plots for the average residential load during weekdays as well as a summary table of the results. Figure 6: Average Residential Load vs. Time of Day (Weekday) Observed vs. Predicted 10

22 Figure 7: Average Residential Load vs. Time of Day (Weekday) Residuals vs. X Figure 8: Average Residential Load vs. Time of Day (Weekday) Residuals vs. Predicted Table 5: Summary of Results for Average Residential Load vs. Time of Day (Weekday) R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = Upon analyzing the results, the R 2 values for all four sets of data were fairly well-fit, however the Durbin-Watson statistics were ~0.30 for all four cases. This suggests that there exists a pattern in the residuals that the model does not account for, although this pattern was not easily observed in the residual plots generated by CSP. As a result, we have generated the following two plots, which highlight the model s behavior during both a warm weather month, and a cold weather month. 11

23 Figure 9: Predicted Average Residential Load and Residuals for One Week in July (Weekdays) Figure 10: Predicted Average Residential Load and Residuals for a One Week in January (Weekdays) Looking at Figures 9 and 10, we can see that the model more accurately predicts the load for warm-weather than it does for cold-weather. Figure 9 displays no observable pattern in the residuals, however looking at Figure 10, there appears to be a pattern in the residuals during the week in January. During the mornings, the model over-predicts the electricity load fairly consistently. The following table shows the results of our AR(1) model compared to the original OLS model for all four cases. The comprehensive results of both the ordinary least squares models and the AR(1) models for the remaining three scenarios are located in the appendix. 12

24 Table 6: Summary of AR(1) Model Results for All Four Cases OLS R 2 AR(1) R 2 OLS Durbin Watson AR(1) Durbin Watson Avg Residential Weekday Avg Residential Weekend Total Weekday Total Weekend The AR(1) model has significantly improved both the R 2 value, as well as the Durbin- Watson statistic. The AR(1) model presented a sound method for accounting for lags and/or patterns that the original model could not account for. Summary & Conclusion In our effort to model both average residential and total electrical load as function of temperature and time of day, we were fairly successful. We began with a 5P change point model that simply related temperature to electricity use. The results fit reasonably well with the measured values, with R 2 statistics of 0.83 and 0.88 respectively. In formulating the change point models for both average residential and total electrical loads, we found that the critical points at which electricity use becomes dominated by thermal load were different between the two models. The variation in the change point temperatures can be attributed to a number of factors, including the difference in temperature setpoint between residential and commercial buildings. In order to improve the change point model, we explored the impacts of time of use on the electrical loads. First, we separated the data into two bins, weekdays and weekends/holidays, to account for the difference in building operating schedules during these two types of day. Then, we coupled a Fourier series analysis with the previous change point model. This combination of inputs significantly improved the model results. The R 2 statistics for the four versions of the model were between 0.92 and Adding the time series element also made the model more diverse in that it could now account for both temperature and time of use. While, all four models have high R 2 values which indicate good fits to the measured data, they all have low Durbin Watson statistics (between 0.26 and 0.3). The low Durbin-Watson statistics indicate that there remain patterns in the model that are unaccounted for by the parameters. By observing the residuals, we noticed that the model seemed to overpredict morning electricity use during colder months, though we could not account for this phenomenon. In an effort to improve the model outputs, we used an AR(1) model. This approach greatly improved the R 2 and the Durbin Watson statistics. While the AR adjusted model is extremely accurate, it is limited in its predictive ability. The AR adjusted model can only model electricity load for a given hour and temperature if it has residuals from previous hours. In the future, we would like to incorporate additional methods in our analysis to identify the 13

25 shortcomings of the original OLS model. Once the model is improved to account for the hidden patterns that we heretofore could not identify, it could be used by utility companies for its intended purpose accurately predicting average residential or total electrical load given temperature and hour of the day. References Reddy, T. A. (2011). Applied data analysis and modeling for energy engineers and scientists. (1 ed., Vol. 1). New York, NY: Springer. Kissock, J.K., Haberl J. and Claridge, D.E. (2003) Inverse Modeling Toolkit (1050RP): Numerical Algorithms, ASHRAE Transactions, Vol. 109, Part 2, pp Ali, M. T., Mokhtar, M., Chiesa, M., and Armstrong, P. (2011) A cooling change-point model of community-aggregated electrical load. Energy and Buildings 43, pp

26 Appendix A Figure A.1: Hourly Bin Data for Phoenix, AZ Figure A.2: Average Residential Load Change Point Residuals Figure A.3: Total Load Change Point Residuals 15

27 Figure A.4: Average Residential Energy Use by Month Figure A.5: Total Load by Month Figure A.6: Total Load & Average Residential Load vs. Day of Week 16

28 YR MO Date Day Hour Total Load Avg Res Load Temp Avg Res (pred) Residuals AR(1) New Avg Res (pred) Residuals SSR SST Table A.1: AR(1) Model 17

29 Figure A.7: Total Load vs. Time of Day (Weekday) Observed vs. Predicted Figure A.8: Total Load vs. Time of Day (Weekday) Residuals vs. X Figure A.9: Total Load vs. Time of Day (Weekday) Residuals vs. Predicted 18

30 Fourier Series R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = AR(1)1 Model R-squared = percent Durbin-Watson statistic = Table A.2: Summary of Results for Total Load vs. Time of Day (Weekday) Figure A.10: Average Residential Load vs. Time of Day (Weekend) Observed vs. Predicted Figure A.11: Average Residential Load vs. Time of Day (Weekend) Residuals vs. X 19

31 Figure A.12: Average Residential Load vs. Time of Day (Weekend) Residuals vs. Predicted Fourier Series R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = AR(1) Model R-squared = percent Durbin-Watson statistic = Table A.3: Summary of Results for Average Residential Load vs. Time of Day (Weekend) Figure A.13: Total Load vs. Time of Day (Weekend) Observed vs. Predicted 20

32 Figure A.14: Total Load vs. Time of Day (Weekend) Residuals vs. X Figure A.15: Total Load vs. Time of Day (Weekend) Residuals vs. Predicted Fourier Series R-squared = percent Mean absolute error = R-squared (adjusted for d.f.) = percent Durbin-Watson statistic = (P=0.0000) Standard Error of Est. = Lag 1 residual autocorrelation = AR1 Model R-squared = percent Durbin-Watson statistic = Table A.4: Summary of Results for Total Load vs. Time of Day (Weekend) 21

33 YR MO Date Day Hour Total Load Avg Residen tial Load Temp Total Load (pred) Resid uals AR(1) New Total Load (pred) Resid uals SSR (before AR(1)) SST (before AR(1)) SSR (after AR(1)) SST (after AR(1)) Table A.5: Regression Applied to May Data w/ AR(1) Model 22

34 Before AR(1) R 2 DW After AR(1) R 2 DW Table A.6: Summary of Regression Applied to May Data w/ AR(1) Model Figure A.16: Total Load vs. Average Residential Load 23