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Modeling a Multi-Purpose Public Building with Stochastic Gains and Occupancy Schedules.


The representation of occupancy and gains is the most prominent source of error in building models. Error in building energy modeling (BEM) can be seen as coming from four sources: the mathematical models of heat and mass transfer and their solutions, imperfections in the building and HVAC system definitions, weather variations, and user behavior and gains. The treatment of infiltration and building pressurization falls under the first category, and is an under-recognized source of error (Ng, 2012). It is not the focus of this study, and the building in question is considered hermetically. While improvements have been made in reducing the first three error sources through more physically accurate modeling, little progress has been made in the fourth (Ryan, 2012).

When little is known about the usage, occupancy and gains of the building, as in modeling in the design stage, the uncertainty in energy consumption for predictive modeling can be as high as 40% at an 80% confidence interval (Wang, 2012). When more is known about the future or current use of the building, the level of uncertainty is drastically reduced. Even knowing the pattern of usage, such as the difference between the operation of an office, museum or auditorium, is an important distinction to make in representing the occupancy and gains in a model (Hoes, 2009). It can be helpful, when there is no available data on the occupancy and gains levels, to perform a sensitivity analysis to get a range of results.

Models of existing buildings for predictive control, retro-commissioning or energy conservation measure evaluation are also subject to uncertainty due to the unpredictability of occupancy and the difficulty in obtaining reliable data. This paper undertakes to study the uncertainty in building energy modeling and prediction for an existing building when there is a moderate degree of gains and occupancy data available. It uses a calibration case study of a university building to examine the uncertainty created through the use of diversity schedules and explores the possibility of representing the occupancy and gains with stochastic methods.


The standard for simple calculation of occupancy and gains in BEM is the diversity schedule method. In this method, a daily profile is created for each space type in the building. The profile may vary depending on the day type, so, for example, a Sunday generally has lower levels than a weekday. Modeling programs often have a variety of diversity schedules available for generic space types (Design Building, 2012). When more data is available, the recorded internal gains data available for each space may be averaged over each hour to create custom diversity schedules. The same profile is repeated for each day, depending only on the day type.

Early in the process of calibrating a detailed EnergyPlus model of the Mascaro Center for Sustainable Innovation (MCSI) at the University of Pittsburgh, the authors observed that there might be error caused by this simplistic representation of a complex, unpredictable process. The building has a widely varying set of uses with day-to-day variability in occupancy and gains levels. It contains offices and wet and dry labs, but it also forms a pedestrian walkway between two connecting buildings and holds classes, meetings and tours. In addition, much of the activity follows the academic schedule, while many full-time occupants work there throughout the year. The LEED Gold building has three floors and 3,400 [m.sup.2] (36,600 [ft.sup.2]) of conditioned space. An atrium-type stairwell with vertical glazing faces southwest, while a sloped glazed facade faces roughly northeast.


The modeling and calibration of the MCSI model followed the evidence-based procedure described by Bertagnolio et al. (2012). The model was started in DesignBuilder to create the geometry, zones, and constructions. The detailed HVAC model was made in EnergyPlus IDF Editor. Following the calibration of the model, studies were carried out using Monte-Carlo simulation. Monte-Carlo simulation uses the probabilistic distribution of uncertain inputs to determine the distribution of the output of a complex function or model. Compared to other methods, it requires relatively few evaluations to resolve the output distribution even when there are many input variables. For a given number of runs, the convergence ratio of a Monte-Carlo simulation remains constant as the number of input variables increases. A parametric simulation that covers the input space deterministically must increase the number of runs exponentially to maintain convergence. To apply Monte-Carlo simulation, stochastic scheduling methods were developed as model inputs by scaling the daily diversity schedules and using random sampling from the recorded gains data. The Monte-Carlo approach was used to study the effect of uncertainty and natural variation in gains levels on the model output.

Building Energy Modeling of the MCSI

The MCSI building model consisted of 31 zones, with two shading buildings nearby. Connections to adjacent buildings were treated as adiabatic. A dedicated outside air system with energy recovery and VAV fans served the wet lab space, which took up the majority of the second floor of a larger tower. The second air handler, with VAV fans and economization, served the remainder of the 2nd floor of the tower, and the three floors of the MCSI wing. The wing had offices and a reception area on the first floor, a dry lab and open office spaces on the second, and open office space on the third. There were conference rooms on all three floors. The wing also served as a bridge connecting the tower to an adjacent classroom building, with an atrium spanning all three floors and a stairway between them.

The HVAC system operated largely at minimum flow rates set by facilities staff, regardless of the occupancy levels. A building with demand-controlled ventilation (DCV) might be significantly more sensitive to changes in occupancy, so a parallel model was created with ideal DCV control and outside air ventilation set points according to ASHRAE 62.1 (ASHRAE, 2010).

Available Data. The level of accuracy in the creation of internal gains schedules, as well as the validation of the calibrated model, was limited by the data available from the building management system (BMS). Flow rates and temperatures were available at the AHUs and VAV boxes. There were meters on the hot and cold water supplies and returns for each air loop, and 16 carbon dioxide (C[O.sub.2]) sensors spread throughout the building. Eight electric meters were installed on the circuit boxes for lighting and plug loads, and the system recorded each fan's electricity consumption.

The metering of carbon dioxide and electricity consumption was coarser than the zoning in the model, so schedules overlapped several zoned spaces. There were gaps in the data due to changes in the larger building system and discrepancies between meter readings, which created uncertainty in calibration. In the creation of occupancy schedules, carbon dioxide levels and flow rates were used to create a balance and calculate the number of persons present. The difference in C[O.sub.2] ventilated in and out of the space, less the C[O.sub.2] stored in the space air volume, was divided by the average C[O.sub.2] exhaled per person. This method provided a rough estimate for occupancy, but was the best possible method with the available data.

Calibration. The calibration of the model was largely successful, with the caveats that the assumption of adiabatic surfaces between the MCSI and connecting buildings necessarily added significant modeling error, and calibration was completed using heating and cooling loads rather than plant energy consumption, because no data was available for the district plant. Constant COP and efficiencies were assumed when calculating the total point of use energy consumption. Calibration was entirely evidence-based, with no tuning independent of reliable component performance data.

The model calibration verification includes 10 months and approximately 7000 hours for which there was BMS data for energy use. For periods with no gains data, diversity schedules were used rather than the actual gains and occupancy. The model exceeded the calibration standards set by ASHRAE Guideline 14, as can be seen in Table 1 (ASHRAE, 2002).

Schedules. The schedules for occupancy and electric gains are listed in Table 2. There were plug load electric schedules divided between the wing and the tower. The wing had open office, dry lab, reception, conference, and pedestrian areas. The tower had wet lab, student gathering and partitioned office spaces. In addition, there were schedules for the wing server rooms, and the tower fume hoods. Lighting was divided in a similar fashion. There were six occupancy schedules. They represented two conference rooms, a dry lab and open office combined space, an open office space, the wet lab and the partitioned offices with student study areas outside.

Sensitivity to Gains

As a first step in the study of occupancy and gains in the model, a sensitivity analysis was performed by uniformly scaling the schedules from 50% to 150% of their nominal values. As can be seen in Figure 1, an increase in gains results in reduced heating load, but increased overall energy consumption. An increase of 50% in overall gains results in approximately a 5% increase in energy consumption in the calibrated model, or more than 10% in the DCV model.

Schedule Statistics

Hourly calibrated schedules were created directly from the BMS data for each hour. Diversity schedules were used to fill the gaps in the data, which amounted to approximately 2,000 hours of the year. The diversity schedules were created for six day types: academic year weekdays, Saturdays, Sundays and holidays, and summer weekdays and weekends/holidays. The day types were chosen so that there were at least 20 complete days of data for each.

A closer study of the diversity schedules was necessary in order to determine the statistics relevant to a stochastic study of internal gains. The results are shown in Table 2. The best-fit distribution for electric gains was normal, while occupancy more closely resembled lognormal distributions. The means and scaling factors for each distribution are also shown in Table 2.

The gains data are highly correlated. The levels of correlation between occupancy schedules are up to 0.7, and the level of correlation between electric use and occupancy in separate zones is even as high as 0.4. A 12x12 correlation matrix was created for each day type to represent this interdependence.

Stochastic Methods

Sample Mean Uncertainty. Sample mean uncertainty was studied with the first set of Monte-Carlo simulations. A sample of thirty schedules was created with the statistics of the mean of the observed gains levels. For each day type, a vector of twelve standard normal random numbers was created, r. Then, the Cholesky decomposition, U, of the correlation matrix C, was found such that [U.sup.T]U = C The random numbers were correlated with Equation 1. The correlated vector of standard normal numbers was used to create diversity schedules with Equation 2. Normality is assumed for the sample mean distribution because the sample size is greater than 20 for each day type and schedule. In Equations 1 and 2, i is the hour, while j is the day type and k is the schedule, with k = 1 to 12 representing each of the schedules in Table 2. [bar.X] is the sample mean, S is the sample standard deviation, and n is the number of days in the sample.

[mathematical expression not reproducible] (1)

[mathematical expression not reproducible] (2)

Population Variance Diversity Schedules. For this set of simulations, the effect of uncertainty in the diversity schedule levels was expanded by using a sample size n = 1. This could be seen as the level of uncertainty that would result from using a one-day spot measurement as the diversity schedule level.

Stochastic Daily Schedule. This set of correlated stochastic schedules used the diversity schedule profile, but allowed each day's mean level to vary according to the population statistics by scaling the each day's schedules with a different vector of correlated random numbers, v. For the gains, with normal distributions, the hourly levels were calculated with Equation 2 and n = 1 to represent the population statistics. The daily means and standard deviations are shown in the first six rows of Table 2. For the occupancy schedules, the lognormal distribution was used with Equation 3.

The scaling, [sigma], and location, [mu], factors are shown in rows 7-12 of Table 2.

Random Daily Schedule. The second method for representing the daily variation in gains levels was the random selection of profiles from the population of recorded days with complete data. This method required no calculations other than the random selection from the database of days. With this process, the correlation between schedules was naturally preserved. The distribution of the gains levels was also preserved. The profile of the schedule over the day, including arrival and departure times, became randomized according to the observed data rather than smoothed as in the case of a diversity schedule.

Weather. For a comparison between the effects on the model of variations in weather and gains, thirty years of weather data in the Pittsburgh region were simulated.

A set of thirty .idf EnergyPlus input files were created with hourly internal gains schedules for each of the above. The results of each batch of simulations show the degree of variation in model outputs that can be attributed to each variation in inputs.


Figure 2 below shows the results for a summer day. The solid lines represent the calibrated gains taken directly from the data, the mean results for diversity schedules, the mean results for the random day method, and the mean results for the stochastic day method. The dotted lines represent the maximum and minimum outputs for each of the stochastic input sets. It can be seen that the variation in results due to variation in gains is significant. Each of the mean levels is close to the diversity schedule level, as could be expected because the diversity schedule is based on a mean of the data itself. The calibrated model, based on the actual data for the day, varies from the diversity schedule significantly, as do the maximum and minimum levels for the stochastic methods.

This shows that, as expected, the gains patterns can have a significant impact on the model output even when they are statistically representative of the data. Figure 3 shows the same day for the demand controlled ventilation case. For some time steps the model with data calibrated schedules falls outside the maximum and minimum for the stochastic methods, because the actual data deviated from the mean more than any of the schedules stochastically generated for this day.

Table 3 is a comparison of the results of the yearly simulation between the diversity schedule method and each of the stochastic methods. The mean bias error is the difference in total yearly energy use. These average values are small, which should be expected because the expected value of the magnitude of each input schedule is equal. The effects of randomness are demonstrated in the coefficient of variation of the root mean square error (CVRMSE) and the maximum and minimum bias errors. The input set with the highest uncertainty, population variance diversity schedules, shows yearly energy use from -0.45% to 0.85% from the mean. The CVRMSE represents the hour-by-hour difference between the outputs. The values range from 2 to 4%, a relatively small contribution to error but enough to make a difference in the calibration of the model. The minimum and maximum CVRMSE between the stochastic runs and the diversity schedule run are close together, indicating that variations in gains patterns have a uniform effect on this measure of error for this model. The DCV version of the model is about twice as sensitive to variations in gains as the standard version.

The model is much more sensitive to weather data. The 2012 weather year was the most extreme simulated, and the difference between years was up to 3.6% in the base model and 5.3% in the DCV model. The variations in weather were enough to throw either model out of calibration -10% in the base model and 40% for DCV.

Figure 4 compares the CVRMSE between the stochastic methods and the data calibrated model, with error bars for 95% confidence. These levels represent the hourly difference due to daily variations in gains levels not accounted for by day-type schedules. The levels range from 1.5% to 4%. There is a relatively small confidence interval for each method.


When calibrating a building model of a multipurpose green university building, it was shown that having the actual gains and weather data corresponding to the calibration period helped validate calibration. However, this research showed that, for most purposes of predictive modeling, simply obtaining the mean levels will be sufficient. There was a several percent difference in hourly calibration CVRMSE by using the actual data. The impact on long-term energy consumption was not significantly affected by variations in the gains data. This is the more important output when evaluating energy conservation measures, but for controls optimization, a close calibration to operations profiles is important. It was even more important to use the corresponding weather data to the specific time period that was modeled. The variations between weather years caused significant long-term variation in energy consumption, and verification of hourly calibration would have been very difficult without the actual weather data.


This material is based upon work supported by the National Science Foundation under EFRI-SEED Grant No. 1038139, an ASHRAE Graduate Grant-In-Aid, and the Mascaro Center for Sustainable Innovation at the University of Pittsburgh.

S        =  sample standard deviation
U        =  Cholesky decomposition of the correlation matrix
[bar.X]  =  daily sample mean for a given schedule and day type
n        =  the number in the sample population
r        =  vector of standard normal random numbers
v        =  vector of correlated normal random numbers
x        =  the schedule value for a given hour, day and schedule
[sigma]  =  scaling factor for lognormal distributions
[mu]     =  location factor for lognormal distributions
i        =  hour
j        =  day type
k        =  schedule number


Ng, Lisa C., Persily, Andrew K., Emmerich, Steven J., (2012). NIST Technical Note 1734.Airflow and Indoor Air Quality Models of DOE Reference Commercial Buildings.

NREL: OpenStudio. (2012). Retrieved 8/13/2012, 2012, from

Design Building Made Easy. (2012). DesignBuilder Retrieved 8/13/12, 2012, from

ASHRAE. (2002). Measurement of Energy and Demand Savings, ASHRAE Guideline 14-2002.

Bertagnolio, S., & Lemort, V. (2012). Simulation assisted audit & Evidence based calibrated methodology.

American Society of Heating Refrigeration and Air Conditioning Engineers. (2010). ASHRAE 62.1 Ventilation for Indoor Air Quality: ASHRAE.

Hoes, P., Hensen, J. L. M., Loomans, M. G. L. C., de Vries, B., & Bourgeois, D. (2009). User behavior in whole building simulation. [doi: 10.1016/j.enbuild.2008.09.008]. Energy and Buildings, 41(3), 295-302.

Ryan, E. M., & Sanquist, T. F. (2012). Validation of building energy modeling tools under idealized and realistic conditions. [doi: 10.1016/j.enbuild.2011.12.020]. Energy and Buildings, 47(0), 375-382.

Wang, L., Mathew, P., & Pang, X. (2012). Uncertainties in Energy Consumption Introduced by Building Operations and Weather for a Medium-Size Office Building. [doi: 10.1016/j.enbuild.2012.06.017]. Energy and Buildings(0).

Justin C DeBlois

Student Member ASHRAE

Alex K Jones, PhD

William O Collinge, PhD

Laura A Schaefer, PhD


Melissa M Bilec, PhD

Justin C DeBlois is a PhD Candidate in the Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA. William O Collinge is a postdoctoral researcher in the Department of Civil and Environmental Engineering, University of Pittsburgh.

Melissa M Bilec is an Assistant Professor in the Department of Civil and Environmental Engineering, University of Pittsburgh. Alex K Jones is an Associate Professor in the Department of Electrical and Computer Engineering, University of Pittsburgh. Laura A Schaefer is a Professor in the Department of Mechanical Engineering and Materials Science, University of Pittsburgh.
Table 1. Model Calibration Results

   Calibration Method      Normalized Mean  Coefficient of Variation
                              Bias Error             of the
                                             Root Mean Square Error

         Monthly               1.30%               7.10%
ASHRAE Guideline 14 Limit      5.00%              15.00%
         Hourly                1.20%              15.20%
ASHRAE Guideline 14 Limit     10.00%              30.00%

Table 2. Recorded Data and Schedule Statistical Characteristics for
Academic Year Day Type

Name                         Best-Fit    Area, [m.sup.2] ([ft.sup.2])

Wing Electric                 Normal            1,249 (13,444)
Tower Electric                Normal            1,683 (18,116)
Fume Hood                     Normal              150 (1,615)
Server Rooms                  Normal               30 (323)
Wing Lighting                 Normal            1,492 (16,060)
Tower Lighting                Normal            1,683 (18,116)
2nd Floor Wing Occ.         Lognormal             581 (6,254)
2nd Floor Conference Occ.   Lognormal              35 (377)
3rd Floor Wing Occ.         Lognormal             500 (5,382)
3rd Floor Conference Occ.   Lognormal              30 (323)
Tower Offices Occ.          Lognormal             616 (6,631)
Tower Wet Lab Occ.          Lognormal            1114 (11,991)

Name                       Daily Mean or    Daily Standard Deviation
                           Location Factor     or Scaling Factor

Wing Electric                   7.62                  0.68
Tower Electric                  7.67                  0.35
Fume Hood                      39.35                  1.23
Server Rooms                   46.59                  5.72
Wing Lighting                   6.21                  0.46
Tower Lighting                 11.29                  0.66
2nd Floor Wing Occ.             0.06                  0.31
2nd Floor Conference Occ.       0.11                  0.75
3rd Floor Wing Occ.             0.06                  0.30
3rd Floor Conference Occ.       0.02                  0.83
Tower Offices Occ.              0.11                  0.73
Tower Wet Lab Occ.              0.09                  0.40

Table 3. Total Energy Consumption: Comparison Between Stochastic
Simulations of Gains and Weather Representations, and Deterministic
Diversity Schedule Method

    Stochastic       Statistic  Baseline Hourly  Baseline Hourly
  Representation                     NMBE            CVRMSE

 Uncertain Sample     Average        0.05%            1.47%
  Mean Diversity        Min          0.00%            1.45%
                        Max          0.14%            1.51%
 Stochastic Daily     Average        0.33%            2.10%
     Schedule           Min          0.30%            2.04%
                        Max          0.36%            2.17%
   Random Daily       Average        0.02%            2.10%
     Schedule           Min         -0.08%            1.90%
                        Max          0.09%            2.29%
Population Variance   Average        0.29%            2.09%
     Diversity          Min         -0.45%            1.75%
                        Max          0.85%            2.42%
    Thirty Year       Average       -1.43%            9.73%
      Weather           Min         -3.67%            0.00%
                        Max          0.00%            9.73%

    Stochastic       DCV Hourly  DCV Hourly
  Representation        NMBE       CVRMSE

 Uncertain Sample       0.13%      2.94%
  Mean Diversity       -0.05%      2.90%
                        0.29%      2.99%
 Stochastic Daily       0.01%      3.75%
     Schedule          -0.04%      3.66%
                        0.08%      3.87%
   Random Daily         0.08%      4.11%
     Schedule          -0.06%      3.80%
                        0.28%      4.46%
Population Variance     0.01%      3.73%
     Diversity         -1.11%      3.21%
                        1.09%      4.53%
    Thirty Year        -0.41%     37.07%
      Weather          -4.58%      0.00%
                        5.29%     40.84%
COPYRIGHT 2014 American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc. (ASHRAE)
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Author:DeBlois, Justin C.; Jones, Alex K.; Collinge, William O.; Schaefer, Laura A.; Bilec, Melissa M.
Publication:ASHRAE Conference Papers
Date:Dec 22, 2014
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