calibration of envelope parameters using control-based

commands. The combination of all of these variableswas used to ascertain the operation schedules and rel-ative heating/cooling that was provided to the buildingfor each of the two AHUs.

Building Model

An energy model was constructed of the building us-ing EnergyPlus (version 7.2.0). The architectural lay-out was obtained from original drawings while in-formation about updates to the building design weregathered by walk-through discussion with the build-ing manager. The model consists of eight basic con-structions types: exterior walls, doors, windows, roof,flooring, interior walls, doors and furnishings. Eachconstruction is built of at least one layer from 12 differ-ent materials. The materials were taken from the com-mercial reference building model for existing build-ings constructed before 1980 for the outpatient healthcare building type developed by (U.S. Department ofEnergy, 2013). An R16 insulation layer was addedto the envelope of the building as it was found thatthe insulation in the template file was not sufficient.These template materials are only a starting guess, asthe exact properties of the building are not known. Thematerial parameters are later varied in parallel simula-tions to get a model that behaves the same way as thereal building. In total 182 zones were created, whichaccount for every room in the building and sectionedhallways.No HVAC systems or internal loads (i.e. people,equipment, and lighting) were accounted for in thismodel. For this reason we focus on unoccupied hourssuch as nights and weekends - when the building isempty and systems are off. The intent at the initialstages of this project is to identify characteristics ofthe envelope, followed by detailed characterization ofloads in each zone.Sensor installation began in September 2012, whiledata for January 2013 was used for this study.

Developing the controller to determine QID

To perform the identification of the heat balance, theBuilding Controls Virtual Test Bed (BCVTB version1.2.0) was integrated with EnergyPlus and the temper-ature data that was obtained from the wireless sensornetwork. A discrete proportional-integral-derivative(PID) controller which was implemented with PtolmeyII’s graphical modeling environment was used to iden-tify the energy input to the model. The controller andthe model were connected to form the feedback con-trol loop shown in Figure 2. There is one controller foreach of the 182 thermal zones. The measured sensortemperatures is the reference to each PID controller,and the output of the model for these same zones isthe feedback signal. For the rooms where there wasno temperature sensor installed (black rooms in Fig-ure 1), data from an adjacent with similar behavior wasassigned to it. The PID controller then calculates the

PID Energy PlusQIDTs e Tm

Figure 2: Diagram of the heat flow (QID) identifier.The PID block is implemented in the BCVTB and Ts

are measured sensor temperatures, while Tm are mod-eled operative temperatures.

energy QID from Equation 3

QID(t) = KP e(t)+KI

Z t

0e(⌧) d⌧+KD

d

dte(t) (4)

needed to drive this error e to zero. Equation 5 is usedfor the simulations is the discrete form of Equation 4

QID[k] = KP e[k] +KIts

kX

i=0

e[i] +KDe[k]�e[k�1]

ts(5)

with KP , KI , and KD being the PID coefficients andts the sampling time (in this case, the sample time is5 minutes). The controller is implemented in its statespace representation in the controller canonical form.The transfer function of a general infinite impulse re-sponse (IIR) filter in the z-domain

H(z) = Y (z)U(z) =

b0+b1z�1+···+bNb

z�Nb

1+a1z�1+···+aNaz�Na

H(z) = b0 +(b1�b0a1)z�1+···+(bN�b0aN )z�N

1+a1z�1+···+aNaz�Na

H(z) = b0 +�1z

�1+···+�Nz�N

1+a1z�1+···+aNaz�Na (6)

is transformed to the transfer function in state-space

C(zI �A)�1B +D (7)

with the matrices

A =

"�a1 �a2 ··· �aN�1 �aN

1 0 ··· 0 00 1 ··· 0 0

···0 0 ··· 1 0

#B =

"100···0

#

C = [ �1 �2 ··· �N ] D = [ b0 ] .

This applied to a PID controller

H(z) = KP +KIz

z�1 +KDz�1z

H(z) = (KP+KI+KD)z2+(�KP�2KD)z+KD

z(z�1)

H(z) = (KP+KI+KD)+(�KP�2KD)z�1+KDz�2

1�z�1 (8)

leads to the coefficients

a1 = �1 �1 = KI �KD

a2 = 0 �2 = KD

to realize the PID in its controller canonical form

A = [ 1 01 0 ] B = [ 10 ]

C = [KI�KD KD ] D = [KP+KI+KD ] .

Proceedings of BS2013: 13th Conference of International Building Performance Simulation Association, Chambéry, France, August 26-28

- 3829 -

Each of the 182 PID controllers needs to be tuned dueto different dynamics in every room. Because thatwould be too time consuming, for a start one room wasselected and tuned by applying the Ziegler-Nicholsmethod, which is an automated way to obtain PID co-efficients for a particular system (G.F. Franklin, 2006).To gain information about the dynamics of the system,both the integral (KI ) and derivative (KD) gains of thePID were set to zero while increasing the proportional(KP ) gain until the system oscillates with a constantamplitude and frequency as shown in Figure 3. Forthis test, the outdoor temperature was held constantand a step was used as reference temperature becausethe system may or may not be initially stable. Thegain at which the system begins to oscillate is calledthe ultimate gain Ku. This and the period of the oscil-lation Pu are used to identify the PID gains using theZiegler-Nichols’ rules

KP =Ku

1.7, KI =

2Ku

1.7Pu, and KD =

KuPu

1.7 · 8 . (9)

With this, the coefficients for the PID controller forone zone are established. To determine the coeffi-cients for the other zones, these coefficients are scaledbased on relative difference in the volume of each zoneas compared to the zone in which the controller wastuned for. The coefficients increase all by the sameratio with increasing zone volume.

210 220 230 240 250 260 270 280 290 30019

20

21

22

23

Time step

Tem

pera

ture

[°C

]

Reference temperatureModel temperature

Figure 3: Tuning of a PID controller with the Ziegler-Nichols method. The proportional gain is increaseduntil the system oscillates with constant amplitude andperiod. The integral gain is set to zero, hence thesteady state error, and so is the derivative gain. Thefound gain and period are put into Equation 9 to ob-tain the PID coefficients.

The structure of the control system in Figure 2 is asimple way to identify the heat necessary for the zonethermal balance. Other structures were tested includ-ing disturbance rejection acting on the outdoor temper-ature in addition to the reference tracking. This wasfound to not have much influence on the controllingquality, hence unnecessary and was removed for themodel analysis.An example of how well the controller identifies theheat necessary to drive the model temperature to matchthe sensor temperature is illustrated in Figures 4 and 5.Monitoring the norm of the error in Figure 5 for allzones is a quick way to establish how well the PID

Jan Feb10

12

14

16

18

20

22

24

26

Tem

pera

ture

[°C

]

Jan Feb−6

−4

−2

0

2

4

6

8

10

Time

Erro

r [%

]

Sensor temperatureModel temperatureTemperature error

Figure 4: Sensor and model temperature for a typicalzone (practically on top of each other). The PID con-troller manages to identify the heat needed to maintaina model temperature that is very close to the actualtemperature. On the right y-axis is the tracking errorin percent with a standard deviation (from zero error)of only 0.73 pp.

Jan Feb−1

−0.5

0

0.5

1

1.5

2

Time

Tem

pera

ture

[°C

]

Temperature error

Figure 5: The error between reference and model tem-perature is kept between ±1.7 �C for this typical zone,the root mean squared error is only 0.16 �C.

controllers are tuned for each zone without investigat-ing time traces for all 182 zones.

Parameter SamplingParameter sampling was used to characterize the be-havior of the model with parameter values differentthan the nominal values. Because there is no envi-ronmental systems, occupancy, lighting, or equipmentloads in the model, there was only 69 parameters thatwere candidates for envelope calibration. These pa-rameters include material properties for walls, and in-ternal mass, as well as window properties and infiltra-tion rates. Other numerical parameters in the modelthat were not included for calibration were architec-tural dimensions and simulator parameters.Most of the parameters that were selected were variedby ±75 % of their nominal value using a uniform dis-tribution (all parameters were positive and non-zero),the solar heat gain coefficient was varied by ±10000 %because of large uncertainty in the modeling of theshading devices, and the material thicknesses werevaried by ±1000 %. Regardless of the desired pertur-


- 3830 -

bation range, the range was ultimately constrained asnecessary (e.g. [0,1] if fractional, etc.). A matrix of500 values for each of these 69 parameters was cre-ated using a deterministic sampling algorithm (simi-lar to (Burhenne et al., 2011) or (Eisenhower et al.,2012)). Simulations were performed in parallel on a16-core desktop workstation, which took about 8 hoursto complete the entire batch of simulations. A singlesimulation for this month with twelve time steps perhour, i.e. every five minutes, takes about four minuteson an Intel Xenon E5-2650 CPU running at 2 GHz.

Data AnalysisTo calibrate parameters for the envelope, the heat bal-ance is calculated during times when the building isnot occupied and when the equipment is off (QInt andQHVAC from Equation 3 are near zero), and the vari-ance of the remaining identified portion of the heat bal-ance is minimized. As mentioned in the introduction,at any time, the heat identified by the PID controllerincludes heat from internal loads, the HVAC system,and any heat that is not accurately captured in the en-velope model. When the building is not occupied andthe HVAC system is turned off, we expect there to bea relatively constant load from remaining equipment(e.g. idle office equipment or refrigerators, etc.). It isexpected that these loads are constant and do not trackoutdoor conditions.If the envelope model is calibrated correctly, all of theheat transfer between the building and the outdoorswill be accurately captured in QAmb and Qpart in Equa-tion 2. Any identified heat from the PID controllerthat tracks outdoor conditions is expected to be errorsor residuals in this heat flow to the ambient. There-fore, the goal is to minimize any variance or trackingof outdoor conditions when it is expected that there isno variable internal load or HVAC heat addition (smallconstant loads are expected from idle equipment). Aformal metric for how this is being accomplished wasestablished by calculating the variance for the sum ofall QID in the building (summed over all zones for eachtime step) over the hours 20-4 each night, additionally11-20 on Saturdays, and all day on Sundays.With a metric for how accurate the envelope portionof the model is, this variance is calculated for all 500simulations from the sampled set. From this data, ameta-model is calculated using support vector regres-sion which is then used to identify critical parametersof the model using sensitivity analysis. The criticalparameters are parameters that have a strong influenceon the variance discussed above. Figure 6 illustratesthe sensitivities for the 69 parameters with the top 10most critical parameters highlighted in color.Once critical parameters are identified (10 of the to-tal 69) a reduced order meta-model is calculated thatonly accounts for this reduced set of parameters. Thisreduced model is then used to in conjunction with anumerical optimizer to identify which values for theseparameters is needed to calibrate the model based on

0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Parameter number

Sens

itivi

ty

Metal Decking − DensityMetal Decking − Spec. HeatMetal Decking − ThicknessStructural Wood − ConductivityStructural Wood − DensityStructural Wood − Spec. HeatStructural Wood − ThicknessR16 − Thermal ResistanceWindow − U−factorZone Infiltration − ACH

Figure 6: Sensitivities for the 69 parameters that arerelated to the envelope portion of the model. Thesesensitivities indicate the influence on the variance ofthe residual heat calculated during unoccupied hours.

minimizing the variance of QID during unoccupiedhours. An interior point optimizer was used to performthis optimization, and since the optimization processtakes only a fraction of a second, two optimizationswere performed: minimizing the variance of QID andmaximizing the same (which is not desired). Figure7 illustrates QID for the nominal case and these twooptimized cases (Table 1 presents the values for thecritical parameters for these three EnergyPlus simula-tions - all other parameters remained the same as thenominal case).

Table 1: The 10 critical parameters’ values for thereduced order meta-model.

Parameter Nominal Min var Max varMetal Decking- Density [kg/m3] 7680 1932 13176- Spec. Heat [J/kgK] 418 238 733- Thickness [m] 0.0015 0.0082 0.0147Structural Wood- Conductivity [W/mK] 0.1200 0.1705 0.1825- Density [kg/m3] 100 106 175- Spec. Heat [J/kgK] 1210 1204 2118- Thickness [m] 0.0100 0.0001 0.1101R16 insulation- Thermal Resistance [m2K/W] 16 23 20Window- U-factor [W/m2K] 4 1 5Zone Infiltration- ACH [1/h] 0.2100 0.1551 0.2512

In Figure 7, it is evident that the optimization pro-cess reduces the variance of the QID during unoccu-pied hours when this is the goal. It is also evident thatby maximizing the variance, the QID tracks outdoorconditions to a larger degree. This suggests that theenvelope properties of the model are calibrated (in theminimized case). Further work is needed to verify withnew data that the response is the same for new time pe-riods and that these envelope parameters calibrate the


- 3831 -

0

100

200

QID

[kW

]

Maximum variance

−20

0

20

40

60

QID

[kW

]

NominalMinimum variance

Jan Feb0

10

20

Tem

pera

ture

[°C

]

Time

Jan Feb0

500

1000

Rad

iatio

n [W

h/m

2 ]

Outdoor temperatureGlobal Horizontal Radiation

Figure 7: The sum of the identified heat (QID) over allzones for the month of January that is needed to drivethe operative temperatures in each zone to match thesensor values. The first plot shows QID for the nomi-nal and the minimum variance optimized case. In themiddle is the maximum variance case, note that this isplotted in a different scale. The bottom plot shows theoutdoor temperature and the global horizontal radia-tion from the sun for the simulated period.

model for other seasons.

CONCLUSION

In this paper we discussed an approach to calibrateenvelope parameters of a building energy model us-ing cheap thermal sensors, a PID control loop in aco-simulation environment, and uncertainty and sen-sitivity analysis coupled with numerical optimization.By identifying the amount of supplemental heat that isneeded in the energy balance for each zone, we cal-ibrate parameters of the model so that this varianceis minimized. This is only the first step in the for-mal calibration of the building energy model whichwill be used for model-based recommissioning. Thenext steps will be to use the identification of the ele-ments of the heat balance to capture typical time de-pendent loads in the building (from occupants and in-ternal equipment) and investigate how the calibratedmodel performs on new data in a validation step. Withthis information, optimized control strategies will besought to maintain and enhance comfort in the build-ing while minimizing energy used. This work wasperformed using EnergyPlus and the Buildings Con-trol Virtual Test Bed, while the methods can be usedin many other software environments.

ACKNOWLEDGEMENTThis work was partially sponsored by the Universityof California, Santa Barbara Green Initiative Fund andthe Department of Student Affairs. We would also liketo thank Kazimir Gasljevic (facilities manager) for hishelp with access to the building in the case study andZheng O’Neill for discussions on this topic.

NOMENCLATUREACH Air changes per hourAHU Air handling unitAmb AmbientAMY Actual meteorological yearBCVTB Building Controls Virtual Test Bed (software)BMS Building management systemC Thermal capacitanceD Derivative coefficientEnv Envelope" ErrorHVAC Heating, ventilation, and air conditioningI Integral coefficienti Zone number iID IdentifiedIIR Infinite impulse responseInt InternalK GainKu Ultimate gain (Ziegler-Nichols)N Total number (of subscript)P Proportional coefficientpart Internal partitions and thermal mass elementsPID Proportional-integral-derivative (controller)Pu Ultimate oscillation period (Ziegler-Nichols)Q Heat transfer rateTm Model temperaturesTs Sensor temperaturests Sampling time

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calibration of envelope parameters using control-based

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