Assessing the relevance of reduced order models for building envelop

ASSESSING THE RELEVANCE OF REDUCED ORDER MODELSFOR BUILDING ENVELOP

Siyamak Sarabi1, Stephane Ploix1, Minh Hoang Le1, Hoang-Anh Dang2 and Frederic Wurtz21G-SCOP lab, CNRS UMR 5272, Grenoble Institute of Technology, France

2G2Elab, CNRS UMR 5269, Grenoble Institute of Technology, France

ABSTRACTThe paper focuses on parameter estimation processesfor physically meaningful models tuned online and de-fine a process to determine whether a model is rele-vant or not for GMBA-BEMS tuning purpose. Theproposed approach relies on the data coming fromthe PREDIS/MHI platform. The first step is to cal-culate realistic parameters with possible intervals be-cause nonlinear optimization, required for physicallyexplicit models, implies initial parameters. The nextstep is to find the best reduced order model structureusing an iterative nonlinear optimization approach us-ing recorded data that leads to parameter estimation.It is based on randomize initial values for parametersto measure the convexity of the search space in study-ing the convergence. Finally, the last step consists inenhancing the time zones where reduced order modeldoes not fit well with the available data. It pointsout some non-modeled phenomena. It is based on aweighted iterative estimation method where weightsdepend on the estimation errors obtain at the previousstep.

INTRODUCTIONThe energy issue is one of the major challenges ofthe 21st century. Building related energy consump-tion accounts for a large part of the total energy bill.Researchers are therefore developing continuous per-formance monitoring, automatic diagnoses and homeenergy management systems to improve building con-sumption. Nevertheless, all these upcoming applica-tions require reduced order models of the building en-velop that can be tuned online. Many models are pro-posed in literature but models are related to specificgoals, with specific time scales. Consequently, assess-ing the relevance of a reduced order model for a spe-cific goal is a key issue. This paper proposes an ap-proach making it possible to determine whether a re-duced order model fits a specific goal or not. The paperfocuses on models for Global Model Based Anticipa-tive - Building Energy Management System (GMBA-BEMS) such as G-homeTech (Ha et al., 2012) but re-sults can be extended to any usage that requires pa-rameter estimation procedure for physical models bycontrast with black box model not directly related tophysics.

STATE OF THE ARTWhen interacting with a system, knowledge about howits variables are related, is needed. With a broad defi-nition, these relationships among observed signals re-

lated to a system is called a model of the system(Ljung, 1999). However a model has to be useful i.e.to fit a specific goals. For instance, very detailed mod-els are not useful for a GMBA-BEMS because theycontain too much parameters that cannot be properlyestimated with a parameter estimation approach. Dataare indeed not sufficient and, when using a 1 hour sam-pling time, it is not meaningful to represent fast dy-namics.The paper focuses on parameter estimation processesfor physically meaningful models tuned online and de-fine a process to determine when a model is relevantor not for GMBA-BEMS tuning purpose. Roughlyspeaking, the simpler the model is, the more impre-cise it could becomes and the more complex it is, themore difficult it is to get physically meaning param-eters. The target is to find a criterion that points outwhen a model structure is suitable taking into accountphysical insights of the system plus identifiability ofthe dedicated parameters.Many models have been proposed in scientific liter-ature to represent the thermal behavior of the build-ings. Linear regressive models such as ARX (AutoRegressive model with exogenous inputs) have beencompared with time scaled identification methods(Malisani et al., 2010) and ARMA (Auto RegressiveMoving Average) models for fault detection purposes(Chowdhury and Carrier, 2000). The structures ofthese models are very general and take into accountneither the actual physical relations between variables,nor the existing links between parameters. It makes itdifficult to extrapolate to other contexts: for instance,finding a ARX model representing the thermal be-havior of a thermal zone is easy when the ventilationflow is constant but extrapolating this model to otherlevels of ventilation is not possible because physicalparameters are distributed into several ARX parame-ters. These kinds of models may be used in real-timefor a given context, using no other information thaninput-output data, considering the system as a blackbox. Based on all previous claims, using black boxapproaches is relevant for GMBA-BEMS purpose.A physical analogy of thermic with electric circuits hasbeen widely used in literature (G. G. J. Achterbosch,1985; Hudson and Underwood, 1999; N. Mendes,2001; G. Fraisse, 2002; M. M. Gouda, 2002; S. Wang,2006). Most of these building models are based ona heat balance equation. By means of this equation,building thermal parameters such as thermal resistanceand thermal capacitance plus indoor/outdoor and ad-jacent zones temperatures, metabolic heat of occu-

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pancies and electric appliances can be adapted to theelectric circuit components such as resistor, capacitor,voltage and current source. These models may be usedto estimate the internal temperature and the heating/-cooling energy demand of buildings (Park et al., 2011;Deng et al., 2010).The work of M. M. Gouda (2002) proved that a sec-ond order RC network with 3 resistors and 2 capaci-tors (3R2C) is sufficient to capture the fine conductivedynamic interaction between two spaces connectedthrough a single wall (Deng et al., 2010) for simula-tion purpose. Deng et al. (2010) suggested a 1R1Clumped parameter circuit which presents a buildingthermal model using thermal-electric analogy. In or-der to avoid opinions about model structures and toget tangible conclusions dependent of model usage, amodeling process is going to be defined.

PROBLEM STATEMENTThe PREDIS/MHI (Monitoring et Habitat Intelligent)platform, located at the ENSE3 school of GrenobleInstitute of Technology, is used as reference build-ing in this paper. This platform is a tertiary low en-ergy building that is highly instrumented where mostof the energy flows are measured using different sen-sor technologies. The studied thermal zone is a class-room surrounded by 5 adjacent thermal zones (one ad-jacent thermal zone has been neglected because of itssmall impact). The classroom is equipped with a CMV(controlled mechanical ventilation) with a static air/airheat exchanger. This CMV may provide heat throughair/water exchanger thanks to a fuel boiler. Data setand model quality assessment methodology are pro-posed in this paper to assess the obtained reduced or-der model.The proposed approach relies on the data coming fromthe PREDIS/MHI platform. The first step is to cal-culate realistic parameters with possible intervals be-cause nonlinear optimization, required for physicallyexplicit models, implies initial parameters. The nextstep is to find the best reduced order model structureusing an iterative nonlinear optimization approach us-ing recorded data that leads to parameter estimation.It is based on randomize initial values for parametersto measure the convexity of the search space in study-ing the convergence. Finally, the last step consists inenhancing the time zones where reduced order modeldoes not fit well with the available data. It pointsout some non-modeled phenomena. It is based on aweighted iterative estimation method where weightsdepend on the estimation errors obtain at the previousstep.

THE PREDIS/MHI PLATFORMA 3D representation of the PREDIS/MHI platform isgiven by figure 1. PREDIS/MHI platform is a lowconsumption building inside another building, whichis a kind of warehouse. The typical year consump-tion are given by(regarding electricity, primary energy

is obtained by multiplying by 2.58 the electric kWh,according to French standards):

• ventilation: 43kWhpe/m2/year (pe=primary en-ergy)

• hot water/air exchanger: 16kWhep/m2/year• lighting: 35kWhep/m2/year• other usage of electricity (computers):

56kWhep/m2/year

In this paper, we focus on the thermal zone namedclassroom. It is surrounded by the following thermalzones:

• a space over the ceiling but also on one side ofclassroom, at temperature T

space

• a corridor at temperature T

corridor

• a downstair thermal zone at temperature T

down

• a adjacent thermal zone corresponding to anopen space with offices at temperature T

offices

• a technical area named control panels in figure1, which is not considered in the thermal mod-eling because the temperature inside this area isalmost the same that in classroom but also be-cause the exchange interface (a wall) is small.

The HVAC system is composed of:

• a double flux ventilation system with a rotativeheat exchanger, whose measured efficiency is50%. It renews air of both zones classroom andoffices. 61% of the injected air is going to class-room and 39% is going to offices.

• 2 hot water/air heat exchanger, one for eachzone of the PREDIS/MHI platform. Hot wateris produced by a site fuel oil boiler.

Thermal Zone

Roof

1st Floor

Down Stair

Space

Office

Down

Classroom

Control Panels

Corridor

Space

Figure 1: 3D overview of PREDIS/MHI platform

The relation between the rotation speed of a ventila-tion drive and the electric power consumption has beenmeasured; results are given by figure 2.


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Figure 2: Characteristics of the air treatment unit

Because there are two drives for ventilation, the re-lation between consumed electric power P

elec

andair flow Q

tot

air

has been interpolated using a quadraticfunction:

P

elec

= 1.728 10

�4(⌦� 500)

2+ 65 (1)

Q

tot

air

= 0.56⌦ (2)

where ⌦ stands for the speed of the two drives.PREDIS/MHI platform is much more than a low con-sumption building: it is a tool for research. Indeed,it contains lots of sensors to measure all the energyflows, including energy transported by air and hot wa-ter. It contains different technologies of sensors: allthe measurements can be accessed thanks to a uniqueRESTful web service connected to many different sen-sor protocols such as OPC for sensors/actuators con-trolled by a SCADA, Zigbee, X10, Oregon Scientificprotocol, 8-20mA analogic protocol,...

CALCULATION OF PARAMETERSCalculation of parameters based on physical con-siderationsIn order to calculate parameter values based onphysics, 4 interfaces have been defined, each interfaceis decomposed of parts. Thermal conduction, convec-tion and radiation have been taken into account foreach layer of a part within an interface but thermalbridges have been neglected. Because of the uncer-tainties about the materials and some dimensions, in-stead of searching average values, surrounding valueshave also been searched taking into account the min-imum and maximum possible values for dimensionsand physical characteristics of materials:

1. the equivalent thermal resistor for the class-room/offices interfaceR

classroom

offices

= 7.12 10

�3K/W 2

[2.79 10

�3, 14 10

�3]

(a) wall part with pvc layer, air layer and pvclayer

(b) glass part with glass layer, air layer and glasslayer

2. the equivalent thermal resistor for the classroom/-corridor interfaceR

classroom

corridor

= 23.2 10

�3K/W 2

[7.91 10

�3, 52.7 10

�3]

(a) wall part with plaster sheet layer, rock-woollayer and wood layer


(c) door part with wood layer, air layer andwood layer

3. the equivalent thermal resistor for the classroom/s-pace interfaceR

classroom

space

= 7.12 10

�3K/W 2

[2.79 10

�3, 14 10

�3]

(a) wall part with plaster sheet layer, rock-woollayer and wood layer


(c) ceiling part with plaster sheet layer and rock-wool layer

4. the equivalent thermal resistor for the classroom/-down interfaceR

classroom

down

= 5.66 10

�3K/W 2

[1.91 10

�3, 22 10

�3]

(a) floor part with concrete layer, air layer andplastic layer

In order to evaluate the capacity, a cross correlation be-tween the air temperature of the classroom T

classroom

and the outdoor temperature Tout

has been done usinga 18 days dataset. Figure 3 points out a time lag com-prised between 1h and 2h. This result will be used todetermined the equivalent capacity in the next section.

Figure 3: Cross correlation between T

classroom

andT

out

Obviously, all these values are approximation andshould be adjusted in order to fit measurements.

Tuning of the calculated parametersAdjusting parameters by identification requires amodel. Because our aim is to configure a GMBA-


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BEMS whose sampling time is 1 hour, dynamics lowerthan one hour would not appear. The model repre-sented in figure 4 has been selected but this choice willbe discussed in the next section. Reduced order phys-ically explicit model is given by:

d

dt

T

classroom

wall

= A(t)T

classroom

wall

+B(t)u(t)

T

classroom

= C(t)T

classroom

wall

+D(t)u(t)

with:

A(t) = �

1R

classroom

down

+

1R

classroom

office

+ . . .

1R

classroom

space

+

1R

classroom

corridor

+ . . .

1R

classroom

wall

+R

classroom

out

(t)

C

classroom

B(t) =

2

666666664

1R

classroom

down

1R

classroom

office

1R

classroom

space

1R

classroom

corridor

1R

classroom

wall

+R

classroom

out

(t)

1

3

777777775

T

C

classroom

C(t) =

R

classroom

out

(t)

R

classroom

wall

+R

classroom

out

(t)

D(t) =

2

66666664

0

0

0

0

R

classroom

out

(t)R

classroom

wall

+R

classroom

out

(t)

0

3

77777775

T

u(t) =

T

down

T

offices

T

space

. . .

. . . T

corridor

T

out

�

classroom

�T

and:• T

classroom

wall

, the average temperature of theclassroom walls

• T

down

, the temperature of the ’down’ zone• T

offices

, the temperature of the ’offices’ zone• T

space

, the temperature of the ’space’ zone• T

corridor

, the temperature of the ’corridor’ zone• �

classroom

, the heat injected into the ’class-room’ zone

• Q

air

, the air flow provided by the CMV• Q

leaks

, the leakage constant air flow

• R

classroom

out

, the equivalent thermal resistor rep-resenting the air flow exchanged with outide

The discrete time model for Ts

= 3600s, is then givenby:

T

classroom

wall

= e

A(t)Ts

T

classroom

wall

+ . . .

(e

A(t)Ts � 1)A(t)

�1Bu(t)

Tout

Tcorridor

Tspace

Tdown

Toffices

Twall/classroom

!classroom

Tclassroom

Rclassroomout

Rclassroomcorridor

Rclassroomspace

Rclassroomdown

Rclassroomoffices

Rclassroomwall

Cclassroom

Figure 4: Selected model for PREDIS/MHI classroom

Nevertheless, calculated parameters cannot be directlyused in the selected model. The transformation illus-trated by figure 5 has been used to obtain model pa-rameters. This transformation relies on the idea thatheat fluxes have to remain identical if T

int

and T

out

are identical. Rwall1

Rwall2

Cwall

Tout1

Tout2

Tin

Twall

0

Tin Twall

Cwall

0

Tout1

Tout2

Rwall ! Rwall1 Rwall

! Rwall2 Rwall

!1

!1

!2

!2

Figure 5: Model transformation used to adapt calcu-lated parameter values

Consequently, the resistor Rwall

has to be determined.Assuming symmetricity, we used:

R

wall

=

1

2R

corridor

+

2R

space

+

2R

down

+

2R

offices

= 1.23 10

�3K/W

It yields:

R

classroom

corridor

= 22 10

�3K/W 2 [6.7 10

�3, 51.5 10

�3]

R

classroom

space

= 5.89 10

�3K/W 2 [1.5 10

�3, 12.8 10

�3]

R

classroom

down

= 4.4 10

�3K/W 2 [0.7 10

�3, 20.8 10

�3]

R

classroom

offices

= 20.1 10

�3K/W 2 [5.2 10

�3, 137 10

�3]

The resistor R

classroom

out

has actually a time varyingvalue because it depends on the ventilation air flow.


��

Because the air flow is known, this resistance can eas-ily be calculated thanks to physics. It yields:

R

classroom

out

=

1

0.61(1� ⇣)c

air

⇢

air

(Q

air

+Q

leaks

)

with

⇣ = 0.5, efficiency of the heat exchangerc

air

= 1006J/kg.K

⇢

air

= 1.204kg/m

3

Q

leaks

= 10.03m

3/s

Q

air

in m

3/S

In the previous section, a time lag of 1 to 2 hours hasbeen observed between T

classroom

and T

out

. In or-der to estimate the value of the equivalent capacity,let’s study the theoretical time lag �. Considering thefirst harmonic whose period in 24h, the delay betweencurves can be calculated because the model is 1st or-der. It yields:

� =

tan

�1�2⇡⌧T

�

360

T

where T = 24h and:

⌧ =

C

classroom

1R

classroom

down

+

1R

classroom

office

+

1R

classroom

space

. . .

+

1R

classroom

corridor

+

1R

classroom

wall

+R

classroom

out

(t)

It can be reformulated as:

C

classroom

=

T ⇥ tan

�360�T

�

2⇡

⇥ . . .

0

BBBBBB@

1R

classroom

down

+ . . .

1R

classroom

office

. . .

+

1R

classroom

space

+ . . .

1R

classroom

corridor

+ . . .

1R

classroom

wall

+R

classroom

out

(t)

1

CCCCCCA

It leads to C = 7.1 10

6[3 10

6, 17.8 10

6].

Using a 18 days dataset, a interior point optimizationalgorithm has then been used to adjust the parametersaccording to their respective intervals. However it ap-pears that the results where very sensitive to the pro-posed initial parameters. Moreover, most of the time,results are not physically meaningful. The phenomenais due to the fact that some parameters are not identifi-able because of the dataset. It can be understood witha sensitivity analysis.The measurements of a typical winter day have beenused and the energy needs in kWh have been estimatedaccording to different simulations when changing in-dependently each parameter to its minimum and max-imum values (average value is 14.2kWh):

parameter min max variationR

classroom

corridor

12.8 18.3 39%R

classroom

space

14.2 38 168%R

classroom

down

8 193 1302%R

classroom

offices

12.7 18 37%R

classroom

wall

14.2 39 175%C

classroom 12.9 18.6 40%It appears that the impact of Rclassroom

corridor

, Rclassroom

offices

and C

classroom is little and that these parameter val-ues will be more difficult to identify. Therefore, toavoid weird values, the following optimization crite-rion has been used because norm 1 does not give moreimportance to larger error:

J =

X

i

|Tmeasured

classroom,i

� Tclassroom, i

model|+ . . .

· · ·+ ⇠|✓calculated

� ✓|

where sample time is one hour.

Figure 6: measured and simulated temperature forTclassroom

Thanks to this objective, the optimization will modifya parameter value only if it reduces the optimizationerror. Consequently, if a parameter is not identifiable,it will be kept at its calculated value.Based on the measurements for 18 days, the optimiza-tion process led to the following values:

R

classroom

corridor

= 22 10

�3K/W ! 22 10

�3K/W

R

classroom

space

= 5.89 10

�3K/W ! 6.4 10

�3K/W

R

classroom

down

= 4.4 10

�3K/W ! 4.4 10

�3K/W

R

classroom

offices

= 20.1 10

�3K/W ! 20.1 10

�3K/W

R

classroom

wall

= 1.23 10

�3K/W ! 1.2 10

�3

C

classroom

= 7.1 10

6K/W ! 15.3 10

6K/W

The error between the measured temperature forT

classroom

and the one deduced from the model withoptimal parameters is given by figure 6.

ASSESSMENT OF THE MODELBefore assessing whether a model is suitable or not fora given goal, the quality of a model has to be defined:


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• a good model has to explain relations betweenobserved phenomena according to expected pre-cision i.e. estimation error has to be smallenough, especially during the validation process

• for BEMS, a good model has links with physicsin order to be able extrapolate behaviors likemodification of the ventilation

• the parameters of a good model has to be iden-tifiable

The first point is easy to check but the second point ismuch more difficult. Sensitivity analysis may be usedbut the link is not direct with identifiability of physicalmodels. In this section, a process is proposed to assessthe quality of a model.

The second point leads to the approach that has beenfollowed in this paper regarding the physics basedmodeling.

Regarding the third point, the basic idea is to checkwhether identification processes are ergodic. There-fore, the proposed process is to draw randomly initialparameters within their possible value set and checkwhether identification process leads to the same bestparameters.

The curve of figure 7 is the main indicator to assessthe quality of a model. It has to be as little as possibleto yield good explanation capacity but also as flat aspossible to guarantee ergodicity. In the PREDIS/MHImodel, same value of criterion is obtained for 60% ofthe optimizations. It increases a lot for 30% of the op-timizations.

Figure 8 points out how the parameters are distributed.If the model is good and if the dataset used for pa-rameter estimation is rich enough, identified param-eters should be gathered around a single best value.If the variance is important, it means that the pa-rameter cannot be found either because the modelcontains to many parameters or because the data setis too poor. Regarding PREDIS/MHI, the parame-ters are from top to bottom: R

classroom

wall

, Cclassroom

,R

classroom

down

, R

classroom

space

, R

classroom

offices

, R

classroom

corridor

, ⇣

and T

classroom

(0). Figure 8 stresses that parametersC

classroom

and R

classroom

space

are difficult to identify.Sensitivity analysis already pointed out that responseis very insensitive to the value of C

classroom

. Nev-ertheless, the result concerning R

classroom

space

points outthat sensitivity analysis is not relevant to evaluate thequality of a model.

Figure 7: Ascendent values of obtained criteria for 20optimizations

Figure 8: Distribution of parameters for 20 optimiza-tions

CONSIDERING UNKNOWN PHENOMENAIn practical situations, adjusting parameters of a modelin order to fit measurements is difficult because ofthe presence of unknown phenomena. Indeed, in theavailable data for PREDIS/MHI, there is no informa-tion about occupancy, neither about the solar gain orabout the position of doors. It means that in practicalsituations, we have to cope with unknown phenomena.The problem to be solved is to give more important toperiod where there is not much unknown phenomena.It amounts to enhance the time periods where reducedorder model does not fit well with the available data.This mismatch points out some non-modeled phe-nomena. An algorithm based on a weighted iterativeestimation method where weights depend on the esti-mation errors obtain at the previous step is introduced.This weighting function is supposed to magnify theimportance of well-matched intervals and minimizethe importance of poor-matched intervals during op-timization process of parameter estimation. If theestimation error becomes high, it means that there aresome phenomena that have not been taken into con-sideration. Here is the function definition:

!(t) = |e(t)| (3)

W (t) = 1� !(t)�min !(t)

max !(t)�min !(t)

(4)


��

where !(t) is the absolute estimation error and W (t)

is the normalized reversed estimation error which isthe weighting function in this case. The procedure ofapplying this weighting factor is shown in figure 9.

Parameter Estimation Weighting Function

𝒆(𝒕)

𝑾(𝒕)

Figure 9: Weighting function algorithm

The idea is to repeat this loop for a certain amount oftime (in our case 20 times) to finally assess the errorsensitivity and analyze the model’s behavior.Figure 10 is the results of 20 weighted iterative estima-tions for a set of 6 days captured data. A good modelshould be able to reduce the error by the help of thisweighting function. Studying step by step the resultsindicates some periods where there is neither signifi-cant change in error nor reduction and even some timesthe errors has increased in further estimations. In thisexperiment, by the help of two other available infor-mation the non-modeled phenomena are introducing.

Figure 10: Results of 20 weighted iterative estimationwith two samples of non-modeled phenomena

Estimated numbers of occupants from emitted CO2and airflow rate of ventilation system have been cor-related by the results of estimations in figure 11. Fivepeaks over midnights are correlated at the same time ofmaximum airflow during midnight. Indeed, two nega-tive peaks occur at the same time of high occupancy infourth and fifth days. Consequently, it is feasible thatthe model does not take into consideration the pres-ence of occupants and also air flow variation of venti-lation system.

Figure 11: Upper plot: airflow of ventilation systemcaptured by sensors, Middle plot: weighted estimationerrors, Lower plot: number of estimated occupancy

CONCLUSIONThis paper proposes a methodology to assess reducedorder physically explicit models. It relies differentsteps. First step consists in calculating parameters withbounds from physics and to proposed a relevant modelstructure according to the expected usage (paper fo-cuses on models for GMBA-BEMS). The second steprelies on a nonlinear optimization algorithm that bothminimize the error between estimations and measure-ments but also keeps parameters close to the valuescalculated using physics in order to avoid weird valuesfor non identifiable parameters. The third step con-sists in the assessment of the model quality using 2curves: one points out whether best parameters canbe found whatever the initial parameters are (ergodic-ity) and the other represents the parameter distributionin order to appreciate parameter identifiability whichdepends both on the selected model and on the useddataset. This procedure has been applied to the PRE-DIS/MHI platform. Physically explicit parameter val-ues have been found but it turns out that it is still dif-ficult to identify some parameters. Finally, a proce-dure based on a recursive weighted parameter estima-tion procedure has been proposed. It makes it possibleto automatically give more important to time periodswhere there are few unknown phenomena.

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Assessing the relevance of reduced order models for building envelop

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