Parameter estimation for low-order models ofcomplex buildings

Anita Martincevic ∗, Antonio Starcic †, Mario Vasak ‡University of Zagreb, Faculty of Electrical Engineering and Computing

Department of Control and Computer Engineering∗ anita.martincevic@fer.hr, † antonio.starcic@fer,hr, ‡ mario.vasak@fer,hr

Abstract—The improvement of the building sector energyefficiency becomes crucially important to attain a balance inmany sectors. Reduction of the energy consumption in buildingsby using model predictive control strategies is recognized as oneof the essential solutions to achieve considerable energy savings.Due to the nature of thermodynamic processes in buildings theunderlying models are mostly nonlinear and of high order. In thiswork Constrained Unscented Kalman Filter is employed to obtaina linear low order model of a large public building applicable forthe predictive control. Through the comparison of results with thedata generated by highly accurate building simulation softwareIDA Indoor Climate and Energy (IDA-ICE), it has been shownthat the first order linear model for each zone, with separatednonlinearities related to the solar radiation effects, is sufficientto capture the main dynamics of the observed building.

Keywords—Constrained Unscented Kalman Filter (UKF),Building temperature prediction, Model Predictive Control(MPC), Online parameter identification, Parameter-Adaptivebuilding model

I. INTRODUCTION

In the last few decades the global warming has becomean important problem that influences the world energymarket. Aware of this, the scientific, economic and politiccommunities started to advocate and encourage rationalconsumption of the energy resources. The building sector iscrucial to the European Union (EU) environment and energypolicies as buildings consume 40% of the total EU energyand generate about the same percentage of greenhouse gasses[1]. The heating, ventilation and air conditioning (HVAC)systems are responsible for majority of the overall energyconsumed in buildings [1]. Use of advanced smart controlmethods to operate the HVAC systems is therefore cruciallyimportant to improve the energy efficiency in the buildingsector.

Model Predictive Control (MPC) framework has attractedconsiderable attention as a promising algorithm for smartenergy efficient buildings. Its clear outstanding advantagesmake it suitable for the Building Energy Management System(BEMS) design. Traditional control algorithms mostly rely onthe calibration of algorithms designed for a typical buildingaccording to approximate rules of thumb or trial and errormethods. While this is sufficient to satisfy the environmentalcondition constraints set by the end users, it is not optimal ina sense of users comfort and power consumption. MPC is anadvanced control strategy that relies on dynamic model of theprocess. In this way the control algorithm is strictly designed

for the particular building. Potential building energy savingscompared to the traditional rule based HVAC controllersare in the range from 16% to 41% [2]–[4]. Mathematicalmodel of the building is a basis for the MPC implementation.Numerous software packages specialized for modeling ofthe building thermal behavior exist on the market. Althoughthose programs can provide very accurate building models,the underlying models are mostly highly nonlinear and of ahigh order. The most popular building modeling frameworkconsists of using the resistor-capacitor (RC) network tomodel thermodynamic processes in buildings. The RCnetwork models are established as simple, computationallyefficient and enough accurate models. The problem of RCrepresentation is a fast increase in the model complexity withan increase of building zones. For large buildings the numberof the system states can be over a couple of thousands. Tobe applicable for the control system design generally, themodel of the process should be simple and yet accurateenough. While nonlinear and higher order models providebetter accuracy, they are computationally too intensive for thecontrol system design. Therefore, from the control viewpoint,the goal is to get a linear model of the lowest possible orderin a way that the model uncertainty is lowest possible.

Besides physical modeling, building model can bedetermined by the use of advanced estimation techniques.Unscented Kalman Filter (UKF) is an algorithm that uses aseries of noisy and inaccurate measurements observed overtime to estimate unknown system states, parameters or evenboth. Use of UKF for estimation of building parameters wasalready reported in [5], [6]. Just like in the RC approach, thenumber of building parameters to estimate can be extremelyhigh, which can further lead to inaccurate estimation ofparameters. In this paper the strengths of both mentionedapproaches are consolidated into a single algorithm in orderto get the model of a large public building suitable for thecontrol system design.

This paper is structured as follows: Section II in shortdescribes the basic principles of the UKF and reasons for itsuse. In Section III the RC modeling framework is describedand, based on it, the modeling structure is proposed. Resultsof the offline and online identification are given in SectionIV. Finally, Section V concludes the paper.

II. UNSCENTED KALMAN FILTER

Kalman filter is an algorithm that estimates values ofunknown states and/or parameters of a system from a seriesof noisy and inaccurate measurements taken over time on it.Early development of the Kalman filter dates back to the earlysixties [7]. Since then the Kalman filter found its place in manyapplications: guidance, robotics, control, signal processing,etc. Over time many extensions and generalizations of theKalman filter have been developed in order to adjust the filterto the specific types of systems. The most known extensions ofthe Kalman filter which can be applied to nonlinear systemsare Extended Kalman Filter (EKF) and Unscented KalmanFilter (UKF). Although both mentioned filters are suitablefor nonlinear systems, it has been shown that UKF in mostapplications outperforms EKF [5], [8]. Instead of the analyticallinearization of nonlinear functions, UKF uses a deterministicsampling approach to capture mean and covariance of theestimates with a minimal set of sample points. Since the spreadof the variables is also taken into account, UKF can be accurateup to a second order in estimating the mean and covariance.

The following discrete-time nonlinear system is considered:

xk = f(xk−1, uk, uk−1) + vk−1,

yk = h(xk) + wk,(1)

where x denotes the system state, v is the process noise,w is observation noise, u is the deterministic system input,y is the noisy system observation and f(·) and h(·) arenonlinear functions. For the identification of a continuoustime system f(·) represents integration of the continous timefunction over a unit-sample time interval. For the case ofsimultaneous estimation of parameters and states, unknownparameters are treated as additional states. Even for the linearmodel case, simultaneous states and parameters estimationmakes the resulting identification nonlinear, i.e. performed ona nonlinear state-update model. Detailed description of theUKF algorithm can be found in [9].

For the models based on physical laws, due to the largenumber of estimation parameters and/or unobservable modelform, the model (1) may result in nonnegative parameters tobe negative or vice versa, whilst this is not physical. UKF canovercome this problem by introducing the constraints [8], [10].The candidates to be constrained are a-priori and a-posterioristate estimates, sigma points, sigma points propagated throughthe nonlinear function, etc. [10]. The main idea of theconstrained UKF is to select the point z from the set definedby constraints that is closest to z by solving the followingconvex problem:

minz

(z − z)T Wk (z − z) ,

s.t. Dkz ≤ dk.(2)

It can be shown that for Wk = I and Dk = I , where Iis the identity matrix, the solution of (2), in the case when zviolates the constraints, is zi = dk,i for those constraints i thatare violated [10]. More generally for constraints introduced as

box constraints, solving (2) gives the same solution as clipping,i.e. setting all points outside the allowed set to the boundaries.

III. MATHEMATICAL MODEL

The model studied in this paper is a model of the 9th and10th floor of the University of Zagreb Faculty of ElectricalEngineering and Computing (FER) building. These two floorsconsist of 50 zones, mainly classrooms, laboratories andoffices. Overall studied building area is about 3000 m2. Figure1 shows the 3D model of the 9th floor of the FER building.First, a detailed simulation model of the considered area isdeveloped with highly accurate building simulation softwareIDA-ICE [11]. This high order nonlinear model is then usedas a source of the identification data and benchmark modelfor the proposed modeling structure.

Fig. 1: 3D model of the 9th floor of FER building.

A. RC modeling framework

RC network models of a building are established as simple,computationally efficient and enough accurate models. TheRC method began to develop in the early eighties [12]. Thebasic strategy of this methodology is to represent buildingelements (or complete zones) with as few thermal circuitelements as possible (lumping). Accuracy of the modeldepends on the number of capacitors used to describe the heattransmission through a wall. The analogy is as follows: heatflow is represented by current, temperatures are representedby voltages, thermal resistances are represented by resistorsand thermal capacitances by capacitors.

Thermal resistance is defined as the temperature differenceacross the structure when a unit of heat energy flows throughit in a time unit. Thermal resistance of convection between afluid and a solid body is defined as:

Rconv =1

h ·A, (3)

where h is convective heat transfer coefficient and A is the areaof the wall exposed to the heat transfer. Thermal resistance ofconduction within a solid body is defined as:

Rcond =

n∑i=1

Li

ki ·A, (4)

where n is the number of material layers that form a wall, Li

is thickness of the ith layer and ki is its thermal conductivity.In order to analyze dynamic behavior of the building it isnecessary to introduce dynamic elements. Dynamic behaviorof the building can be described with a heat capacity concept.

Heat capacity is the property of a body to store the heatenergy.

In buildings with a large share of external windows itis necessary to pay a special attention to the modeling ofsolar radiation influence. Solar radiation can be absorbed bysolid external surfaces or absorbed and transmitted into thezone through external windows. The rate of absorbed solarradiation depends on the material properties. Overall solarradiation transmitted into the zone through windows can becalculated from the known window property called Solar HeatGain Coefficient (SHGC) (Hemisphericaly averaged SHGCfor diffuse solar radiation) [13]. SHGC is the fraction of solarradiation admitted through a window - either transmitteddirectly and/or absorbed, and subsequently released asheat inside a zone. It is a nonlinear function of the solarincidence angle. There are two ways to obtain SHGC and itsdependence on the solar inclination angle for the specifiedtype of window. Tabular dependence of SHGC on the solarincidence angle for most common types of window systemscan be found in ASHRAE handbook [13]. Second way is toobtain SHGC with free software tool, WINDOW from theLawrence Berkeley National Laboratory [14]. Although thesolar radiation model is strictly nonlinear [15], it belongs to aspecial class of nonlinearities called separable nonlinearities.This property allows extraction of the nonlinearity outside amodel and use of its outputs as inputs into the linear modelas shown in Figure 2.

RC model of the FER building, with external wallsrepresented with 2 capacitors and zones and internal wallsrepresented with one capacitor, has order over 500. Oftenit is very difficult to determine the thermal properties ofwalls and the exact values of convection coefficients requiredto accurately determine the parameters of the RC network.In such cases the application of identification techniquesbecomes necessary.

Boundary conditions

Processing of meteorological data(nonlinear calculations)

Direct and diffuse radiations per unit area to the vertical

surfaces(all orientations)

Direct and diffuse radiation per unit area transferred directly

into the zone through window for all orientations

Linear system

Meteorological data

Fig. 2: Structure of the building model.

B. Proposed modeling structure

The objective of this paper is to present the modelingstructure which consolidates the strengths of both the RCapproach and identification procedures and as a result givesthe model applicable for the control system design. In order tofulfill this objective, the following state space model structureis proposed:

x(t) = Atx(t) +Btu(t),

At ∈ Rn×n, Bt ∈ Rn×m,(5)

where n is the number of building zones, m is the numberof the system inputs (outdoor temperature, pre-calculatedabsorbed and transmitted solar radiation, boundary conditions,etc.). Just like in the RC approach, without a-priori knowledgeon the model structure, the number of parameters to estimatecan be extremely high. Large number of parameters canresult in computational inefficiency of the algorithm andinaccurate estimation of parameters. The number of estimationparameters can be reduced if the model structure to beidentified is known (known positions of nonzero elements insystem matrices). In the proposed structure (5) it is assumedthat the dynamics of each zone can be approximated with firstorder differential equation:

dT j

dt= αjT j +

∑i∈N j

βijT i + γjTout + σj1I

ja + σj

2Ijt , (6)

where Tj is the temperature of the jth zone, Tout is theoutdoor air temperature, Ija is the overall solar radiation to theexternal walls belonging to the jth zone and Ijt is the overallsolar radiation transmitted into the zone through the windows.The set N j consists of the indices of zones adjacent to thejth zone. Greek letters α, β, γ and σ denote the estimationparameters. For zones without external walls or without theexternal windows the corresponding terms in (6) are omitted.

Known model structure can significantly reduce thenumber of identification parameters, but for large buildingsthe number of parameters can still be too large for accurateidentification. The main idea of this paper is to take advantageof the known physical parameters of the building such as wallsmaterials and thicknesses. Construction properties of externalwalls are much more complicated than the constructionproperties of the internal walls. Internal walls of FERbuilding are mostly composed from standard full brick andcement plaster. Thermal properties of those materials are welldocumented. For well-known thermal properties of internalwalls the number of independent estimation parameters canbe further reduced by introducing the basic principles of theRC framework into the proposed modeling structure. In asteady state the distribution of temperature through a solidwall is linear, i.e. the heat transfer between two adjacentzones is proportional to the temperature difference betweenthose zones, where factor of proportionality is the overall heatresistance of a wall that separates them, Rij . Thus for known

parameters of internal walls, structure (6) can be rewritten as:

dT j

dt= αjT j + βj

∑i∈N j

T i

Rij+ γjTout + σj

1Ija + σj

2Ijt . (7)

The overall resistance related to the particular internal wall isdefined as a sum of convective heat resistances between theouter and inner face of a wall and air in the correspondingzone and thermal resistance of conduction through thewall. If thermal resistance of some internal wall is initiallywrongly determined, structure (7) assumes that the error actsproportionally to the accompanying thermal resistances onall internal walls related to the same zone. Model structure(7), with Rij calculated using equations (3) and (4), for FERbuilding has twice less parameters than structure (6).

IV. RESULTS

Two approaches to the identification of building parametersare tested: online and offline. Quality of the identificationlargely depends on quality of the data used for identification.The timeframe of the identification data which ensures theexcitation of all system modes depends on climate of thelocation at which the building is placed. For the identificationof FER building, placed in Croatia, the best identification dataare those recorded during the spring or fall. Detailed buildingsimulation model developed in IDA-ICE simulation software

represents a benchmark model for the identified models.Also its zone temperature logs will be used as measurementsrequired for identification. Data used for external conditions(outdoor temperature, global and diffuse solar radiation,relative humidity, wind speed and direction) in IDA-ICEmodel and for the purpose of identification are historicaldata from the nearest meteorological station to the FERbuilding. Sampling time of all data used for identification isone hour. Dynamics of the building excited only by externalenvironmental conditions is very slow which justifies theselection of the sampling time. The main advantage of thecontinuous-time identification approach, applied in this work,is the independence of identified model parameters on thesampling time, so the identified model can be used withan arbitrary sampling time. To ensure the physicality andstability of the proposed model the following constraints areintegrated into the UKF algorithm:

αj < 0, βj , γj , σj1, σ

j2 > 0, ∀j = 1, 2, ..., n. (8)

The comparison of the time-responses of the identified modelsand a benchmark model is presented through the comparisonof the time-response of a single zone. The selected zone issouth oriented, with a large share of external windows area.Such characteristics make it more exposed to the externalconditions (more pronounced dynamics).

10

20

30

25

20

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan−10

0

10

20

30

40

Month of the year

Date Date

Tem

pera

ture

− T

[°C]

Tem

pera

ture

− T

[°C]

Te

mpe

ratu

re −

T [°

C]

01.04 05.04 10.04 15.04 20.04 25.04 30.04 01.07 05.07 10.07 15.07 20.07 25.07 31.07

Outdoor temperature Identified model IDA-ICE model

30

15

25

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec JanMonth of the year

-2

0

2

Mod

el e

rror

− e

[°C]

Fig. 3: The comparison of the time responses of the IDA-ICE model and model identified with UKF (one year predictionhorizon).

A. Offline identification

Data used for offline identification cover the period from thebeginning of March to the end of May. Performance of bothmodels, IDA-ICE model and the identified model, are tested onthe same input data over a whole year. The results are shownin Figure 3. For large buildings dynamics of the zones cansignificantly differ. This difference in dynamics is especiallyevident between zones facing south and zones facing north.TheRoot Mean Squared Error (RMSE) between the identifiedmodel and IDA-ICE benchmark model calculated for wholebuilding (all 50 zones) is shown in Figure 4. Please notethat the best model performance is exhibited in the monthsfollowing the end of the identification period, which clearlymotivates the permanent, on-line model adaptations. Figure5 shows the comparison of the 24 hour prediction of theidentified model and IDA-ICE value of zone temperature(“actual” zone temperature) for two days at the end of March.The dots on each line represent (from the left to the right)prediction of the model from 1h to 24h in advance. As it canbe seen in Figure 5, the prediction error is below 1oC.

One year0

0.2

0.4

0.6

0.8

1

1.2

1.4

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth of the year

RM

SE p

er m

onth

[°C]

RM

SE p

er y

ear

[°C]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 4: RMSE between the identified model and IDA-ICEmodel through months for one year prediction horizon.

0 6 12 18 24 30 36 42 4811.5

12

12.5

13

13.5

14

14.5

Tem

pera

ture

− T

[°C]

Time - t [h]

RMSE=0.52°C

RMSE=0.53°CRMSE=0.30°CRMSE=0.30°C

RMSE=0.62°C

Identified modelIDA-ICE model

Fig. 5: Comparison of 24h prediction of identified model andtrue zone temperature.

B. Online identification

Online identification of the model parameters results inParameter-Adaptive (PA) building model. This model is able tosufficiently capture all building operation modes and to adaptthe model parameters according to the current operation modeor current external conditions. Generally, buildings dynamicsis highly nonlinear and time-varying. For instance, convectiveheat transfer coefficients for external walls highly depend onthe wind speed and direction, radiation heat flux between wallsis nonlinear. This nonlinear dependence can be captured withthe adaptive parameters of the proposed linear model withrespect to the current building operating conditions. Figure6 represent a schematic view of the PA model included intothe BEMS. Figure 7 shows the comparison of the 24 hourprediction of the PA model and the actual zone temperature.At each identification step the new PA model is generatedusing the latest set of identified parameters and exploited tocalculate the 24 hour prediction of zone temperatures.

Expectedly online application of the UKF gives betterresults. Model parameters are tuned in every time step so the

Measurements(measurable system states xk|k and applied control actions uk|k)

External conditions w

System model

Cost functionand

constraints

Dynamicoptimization

Model predictive controlPrediction of external

conditions

User setpoints

Estimated modelparameters

θ Unscented Kalman Filter

Data centerHistorical data

Optimal control actions

k+1|k

k|k[w ,w , . . . , w ]k+1|k k+N-1|k

k|k

Fig. 6: Online identification of a building model used withinthe MPC framework.

0 6 12 18 24 30 36 42 4811.5

12

12.5

13

13.5

14

14.5

Tem

pera

ture

− T

[°C]

Time - t [h]

RMSE=0.12°C

RMSE=0.25°CRMSE=0.13°CRMSE=0.12°C

RMSE=0.23°C

PA modelIDA-ICE model

Fig. 7: Comparison of 24h prediction of PA model and truezone temperature.

resulting PA building model captures the building dynamicsmore accurately than offline identified model. Despite the verylow order of the proposed modeling structure, the RMSE ofboth tested approaches is below 1oC. Shown accuracy and loworder make presented model structure applicable for the designof the MPC.

The future work will include an extension of the proposedstructure with actuators and additional thermal loads, suchas occupancy or equipment loads. By separation of theidentification into several procedures, one during night, whenzones are unoccupied and not influenced by solar radiation,equipment or actuators, and others during day, when zonesare affected by the mentioned gains, the additional heatloads will be satisfactorily estimated [6]. The continuous-timeidentification approach allows easy change of the samplingtime; one hour sampling time is too large given the dynamicsof the zones heated/cooled with the actuators. Independencyof the identified model parameters on the sampling time is ofa high importance in the hierarchical model predictive control,where control is distributed between two or more control levelswith different sampling times.

V. CONCLUSION

Advanced control algorithms such as MPC are appropriatefor the significant energy reduction in buildings. To tailor theMPC towards the building, its mathematical model is required.Knowledge of the mathematical model of the building allowsprediction of the future building behavior. Buildings arecomplex nonlinear dynamic systems with uncertain and time-varying dynamics, therefore the development of the accurateand reliable building model is a challenging task. Whilenonlinear high order models provide better accuracy, they arecomputationally too intensive for the application in controlsystems. Nonlinearities, such as the calculation of the overallsolar radiation to the external walls or overall radiationtransmitted into the zone through window, can be separatedfrom the model, calculated independently from the modeland forwarded to the linear building model as system inputs.The proposed linear first order zone model, with exploitedknowledge from the RC approach to reduce the number ofindependent estimation parameters, proved to be sufficient tocapture the main dynamics of the building. The continuous-time identification approach ensures the independence of theidentified model parameters on the sampling time whichensures applicability of the obtained model with varioussampling times. Constrained UKF identification algorithm,

due to its scalability, easy constraint handling and accuracycan be considered as a promising algorithm for the realimplementation of building energy management system.

ACKNOWLEDGMENT

This work has been financially supported by the EuropeanUnion through project ENHEMS-Buildings – Enhancement ofResearch, Development and Technology Transfer Capacitiesin Energy Management Systems for Buildings, grant No.IPA2007/HR/16IPO/001-040510.

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