1

Identification of a Control-oriented Energy Modelfor a System of Fan Coil Units

Anita Martincevica∗, Mario Vasakb and Vinko Lesicc

abc University of Zagreb, Faculty of Electrical Engineering and ComputingLaboratory for Renewable Energy Systems (url: www.lares.fer.hr)

Unska 3, HR-10000 Zagreb, Croatiaaanita.martincevic@fer.hr, bmario.vasak@fer.hr, cvinko.lesic@fer.hr

Abstract—In recent years, application of advanced control,fault detection and diagnosis algorithms for building heating andcooling systems has been intensively investigated with the aim toimprove their energy efficiency and bring the buildings sectorinto the smart city arena. Hindering the trend, hysteresis andproportional-integral-derivative controllers are still a commonpractice for temperature control in buildings with Fan CoilUnits (FCUs). Introduction of more sophisticated controllersfor additional savings requires a cost-effective approach foridentification of an energy model which accurately resemblesthermal and hydraulic performance of a system of FCUs.In the present work, the control-oriented energy model of asystem of FCUs is developed and accompanied with replicable,robust and simple methodologies for its identification derived byconsolidating the advantages of physical modelling, identificationmethods and manufacturer’s catalogue data. The validity ofthe developed approach is tested on the 248-office living-lab.The introduced simple and accurate dynamic characterization ofenergy transmitted from a FCU to zone air fills the gap betweenthermal and energy management for buildings. This enablesimplementation of predictive building controls and unleashessignificant energy and cost-saving potentials of a smart buildingin a smart city.

Index Terms—control-oriented fan coil unit model; hydraulicmodel; electric-hydraulic analogy; thermodynamic fan coil unitmodel; heat capacity estimation

I. INTRODUCTION

Increasing global energy demand and noticeable effects ofirrational energy consumption highlighted the improvementof buildings sector energy efficiency as one of the prioritiesfor ensuring long-term energy security due to its significantshare in total annual energy consumption [1], [2]. The systemswith the largest potential for improvement of buildings sectorenergy efficiency, with estimated nearly 60% of overall energyconsumption in buildings, are Heating, Ventilation, and AirConditioning (HVAC) systems [2]. The inherent complexityof HVAC systems with uncertain and time-varying dynamics,as well as the presence of unmeasurable disturbances, presentserious challenges for the development of corresponding effi-cient control, fault detection and diagnosis algorithms. ModelPredictive Control (MPC) applied in building managementsystems has been recognized as one of the most promisingsolutions to achieve considerable energy savings in buildingswith estimated theoretical energy saving potential up to 70% in

* Corresponding authoremail: anita.martincevic@fer.hr, phone: +38516129893

particular comprehensive applications [3]–[5]. Only recently,MPC has found its place in practice, with experimentally-validated building energy efficiency increase by 15-53% [6]–[10]. The reasons are primarily the difficulties in obtaining asuitable mathematical model of a building and its subsystems.Acquiring of accurate thermal energy data from the control-lable thermal sources in building zones is the main prerequisitefor identification of accurate building thermal models thattransform zone energy input into a temperature output. Someauthors coped with the problem by using electric heating[11], [12] to excite the building temperature. However, this isoften impractical since most of the commercial and residentialbuildings have water-based (hydronic) or air-based heatingand/or cooling [2].

Fan Coil Units (FCUs) are one of the most commonheating/cooling elements found in office buildings today. AFCU consists of a fan and one or more air-water heat ex-changers. Multiple FCUs connected in parallel to a commonsupply line form a system of FCUs. Overall performance ofa FCU as a part of the system is described with a hydraulicmodel characterizing the distribution of the heating/coolingmedium through the system and a thermodynamic modelfor assessment of thermal energy generated by the FCU. Ingeneral, thermodynamic heat exchanger models in literatureare divided into three groups: i) physical models based ona number of transfer units or the logarithmic mean temper-ature difference relations [13]–[17] or relied on fundamentalphysical laws, ii) non-physical models completely relied onthe experimental data and iii) semi-physical models as acompromise between the first two groups. Physical modelsrequire detailed physical properties of a FCU, such as finthickness or tube dimensions, which are often omitted frommanufacturer’s catalogue and are hardly measurable on thefinal on-site product [18]–[22]. Non-physical models, usuallyput in a form of simple linear approximations around anoperating point [6] or neural networks [23]–[26], decreasein accuracy when operating outside the training range. Semi-physical models exploit physical knowledge or some other a-priori information to specify the model structure while theunknown parameters are identified based on the experimental[27]–[29] or manufacturer’s catalogue data [27]. Whilst theexperimental analysis of heat exchangers in general is widelyelaborated, experimental analysis of a FCU is scarcely con-sidered in only few papers that concern with mainly a single

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NOMENCLATURE

Symbolsα exponent of pressure drop charachteristicsβ exponent of Uo function∆h change in pipe elevation [m]∆p pressure drop [Pa]η flow share through an individual FCU [%]µ dynamic viscosity [Pa·s]φ heat transfer coefficient [K−1]εεε set of coefficients of Uo functionρ density [kg·m−3]τ Transport delay [s]a set of coefficients of Uo functionb set of coefficients of Uo functionc specific heat capacity [J·(kg·K)−1]d diameter [m]fD friction coefficientg acceleration of gravity [9.81 m · s−2]i FCU index i = 1, . . . , nl length [m]m mass [kg]N number of zones supplied through the considered ductn number of FCUs connected to the same ductP power [W]q mass flow [kg·s−1]

R hydraulic resistanceRe Reynolds numberT temperature [oC]Uo overall heat transfer coefficient [W·K−1]x fan speed x ∈ {off,L,M,H}H high fan speedL low fan speedM medium fan speedoff fan switched offSuperscriptsa a-prioric calibrated measurementcd manufacturer’s catalogue datain inputm measured valueout outputraw raw measurements supplyx associated with a certain fan speed xSubscriptsa aircal calorimeterfc fan coil unitp pipew water

FCU [6], [30]. The thermodynamic performance of a FCUdepends on thermal properties of the heating/cooling mediumused within the system of FCUs. Most hydronic systems inbuildings use softened tap water or water-glycol mixtures as aworking fluid with rarely known accurate thermal properties.To the best of the authors’ knowledge, this issue is not coveredwhen discussing modelling of heat exchangers.

While thermodynamic model of a single FCU can be easilyassessed if measurements of the medium flow through theunit as well as measurements of supply and return mediumand air temperatures are available, such information is oftenunavailable for a system of FCUs. To be able to determineunknown flow through a certain FCU, based on one centralmeasurement of the flow through the system, a hydraulicmodel of the system is required. Use of the hydraulic modelfor the system design and subsequently its management sig-nificantly reduces operating costs [31], [32]. Inclusion of both,hydraulic and thermodynamic model into the building manage-ment system offers further savings by enabling dynamic flowcontrol with respect to the thermal demands per zones [33].A hydraulic performance of a FCU and sensors-free solutionfor determining the medium flow through the unit is rarelydiscussed. Authors of [34] suggest using pressure drop sensorsto determine the flow through the heat exchanger. However,this tends to be cost-intensive when applied to individualFCUs due to a large number of expensive sensors requiredand corresponding installation costs.

In the present work, a control-oriented energy model of asystem of FCUs is developed and accompanied by its identifi-cation methodology. The presented methodology consolidatesthe advantages of physical modelling, identification methodsand manufacturer’s catalogue data making it robust, simple

and easily replicable. Cost-effectiveness of the methodologyis reflected in the number of required sensors, e.g. singletemperature sensor per each FCU, calorimeters installed onlyon major supply ducts and temperature sensors in every zone,thus minimizing the commissioning cost for the deployment.

The paper is organized as follows. In Section II, an overviewof a control-oriented energy model for a system of FCUs isgiven. In Section III, an algorithm for finding flow distributionin heating/cooling installations, based on electric-hydraulicanalogy is developed. A general thermodynamic model ofa FCU is introduced in Section IV. The methodology foridentification of the energy model is given in Section V. Thetest-site configuration is described in Section VI. In SectionVII, the developed energy model is experimentally validatedon a system of FCUs in the test-site. Section VIII concludesthe paper.

II. A CONTROL-ORIENTED ENERGY MODEL FOR A SYSTEMOF FCUS

Control-oriented energy model for a system of FCUs con-sists of a hydraulic model of the system and thermodynamicmodels of all FCUs, where thermodynamic model is the samefor equal FCU types. The overall scheme of the model is givenin Fig. 1. The model inputs are: i) central measurement ofmedium mass flow through the entire system qm

w,o, ii) individualmeasurements of fan speeds xm

i , iii) zone air temperaturesof every considered zone T in,m

a,i and iv) central or individualmeasurements of supply medium temperature T in,m

w,i . Index i insubscripts denotes measurements related to the ith FCU and nis the overall number of FCUs in the system. Outputs of thehydraulic system model are individual medium mass flowsthrough every FCU in the system qw,i. Individual medium

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qw,i

qmw,o

T in,ma,i

xmi

T in,mw,i i = 1

i = n

Pa,i

Energy model of a system of FCUs

T outw,i

ThermodynamicFCU model

Hydraulic model ofthe system

Fig. 1: Scheme of the control-oriented energy model of a system ofFCUs.

mass flows are subsequently used in the the thermodynamicmodels to calculate the thermal powers affecting the zonesPa,i and simulated return medium temperatures T out

w,i for everyconsidered FCU.

The developed energy model gives a direct relation betweenthe thermal energy exerted to zone, actuation commands andzone temperature. As such, the model is suitable for: i) ac-quiring thermal powers per zone required for identification ofbuilding thermodynamic model, ii) estimation of unmeasuredthermal loads affecting the zone for a very broad use inbuilding monitoring and control, iii) usage of an advancedFCU control algorithm for direct control of thermal energyinputs per zone via fan speeds, which makes it possible to re-alize optimized thermal energy inputs computed via predictiveenergy management schemes for maintaining zones thermalcomfort [35], [36] and iv) development of fault detection anddiagnosis algorithms for FCUs [37]–[39].

III. HYDRAULIC MODEL OF A SYSTEM OF FCUS

The medium flow through a FCU depends on the pressuredrop across the various elements that form up the entiresystem. A practical way of modelling complex hydraulicsystems is the transition to an analogous electrical modelwhere medium mass flow qw, pressure drop ∆p and hydraulicresistance R behave equivalently to electrical current, voltageand electrical resistance, respectively. The equation relatingpressure drop and mass flow through a hydraulic networkelement is equal to:

∆p = R · qαw , (1)

where R is a constant hydraulic resistance. The values of αdepend on the methodology used for calculation of R and theelement type.

Pressure loss in pipes consists of three components: i) hy-drostatic pressure loss ∆ph, ii) frictional pressure loss ∆pfand iii) kinetic pressure loss. For most applications, kineticlosses are minimal and can be ignored. Thus, the equationthat describes the overall pressure loss in pipes is expressedas a sum of two major terms:

∆pp = ∆pf + ∆ph. (2)

The hydrostatic pressure drop occurs only when there aredifferences in elevation from the inlet to the outlet of a pipesegment:

∆ph = ρ · g ·∆h, (3)

where g is acceleration of gravity and ∆h is change in pipeelevation. The frictional pressure drop in a circular pipe withconstant inner diameter d and length l is defined by Darcy-Weisbach equation:

∆pf = fD8 · l

ρ · π2 · d5· q2

w, (4)

where ρ is the density of heating/cooling medium and fD is thefriction factor. For hydraulically smooth pipes, fD is definedby Blasius equation:

fD = 0.3164 ·Re−0.25, (5)

where Re is Reynolds number defined as:

Re =4

µ · d · π · qw, (6)

with µ as dynamic viscosity of the medium. In addition tothe losses due to the friction or elevation difference, there arealso losses associated with flow through valves and fittings.These, so called minor pressure losses, are accounted by usingthe equivalent length method [40]. The method uses empiricaltables to convert each fitting into an equivalent length of thestraight pipe leq which is then added to the pipe length l. Theleq/d ratio for most common types of fittings can be found in[40], [41]. By inserting (6) and (5) into (4) and including theminor losses, the final form of frictional pressure drop acrossthe circular pipe section is defined as:

∆pf = 0.241 · µ0.25 · (l +

∑leq)

d4.75 · ρ · q1.75w . (7)

Hydraulic resistance of the FCU and medium mass flowthrough the unit are fully determined with the pressure dropwithin it:

∆pfc = Rfc · qαfcw , (8)

where Rfc and αfc are parameters to be found based on theexperiments or pressure drop data from the manufacturer’scatalogue.

Based on the electric-hydraulic analogy an equivalent elec-trical model of the system is derived for a most commonheating/cooling network topology (Fig. 2). Supply pipe, returnpipe and FCU hydraulic resistances are denoted as Rs

p, Rrp and

Rfc, respectively. For clarity, hydraulic resistances of pipesin parallel branches are omitted. For every closed loop ofthe circuit, once the hydraulic resistances and mass flows areknown, the pressure drop is defined with Kirchhoff’s circuitlaws ∀j = 1, . . . , n− 1:

∆pj =

∆pj+1 −

j∑i=1

q1.75w,i (Rs

p,j +Rrp,j) for j ≤ k,

∆pj+1 +n∑i=j

q1.75w,i (Rs

p,j +Rrp,j) for j > k,

(9)

where ∆pj is the overall pressure drop in a parallel branchincluding pressure drop through FCU and pressure drop in

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Rfc,n Rfc,n−1 Rfc,k Rfc,1

Rsp,n Rs

p,n−1 Rsp,k Rs

p,1

Rrp,n Rr

p,n−1 Rrp,k Rr

p,1

∆pn ∆pn−1 ∆po ∆pk ∆p1

qw,n qw,n−1 qw,o qw,k qw,1

Zone n Zone n− 1 Zone k Zone 1

+−

Fig. 2: The analogous electrical model of standard heating/cooling installations.

associated vertical supply and return pipes, ∆pk+1 = ∆po isthe overall pressure drop in the entire system, n is the totalnumber of FCUs in the system and qw,i is the medium massflow though the ith FCU. For known overall medium massflow denoted with qm

w,o, the individual FCU mass flows qw,iare found by solving the following optimization problem:

min∆p0

|qmw,o − qw,o|

s.t. qw,o =

n∑i=1

qw,i,

(2), (3), (7), (8), (9).

(10)

The optimization problem (10) belongs to a class of NonLinearPrograms (NLPs) which can be efficiently solved with e.g.genetic algorithms [42]. Flow share through the ith FCU isdefined as ηi = qw,i/qw,o where qw,i is recalculated based on(9) and the optimal ∆p0 obtained as solution of optimizationproblem (10). For installations with operable valves, whereflow distribution is time-variable and based on the valvepositions, the procedure is extended by introducing variablevalves hydraulic resistances in the network.

IV. THERMODYNAMIC FCU MODEL

Heat transfer within a FCU consists of three parts: con-vection of the heating/cooling medium (e.g. water), heatconduction through the heat exchanger and convection of air tobe heated or cooled. For modelling, the following assumptionsare made:• there are four possible fan speeds: off, Low, Medium and

High denoted respectively as off, L, M, H,• air mass flow qa inside the FCU varies with the fan speed

and is assumed to be constant for each speed,• mean water temperature inside the FCU, Tw, is approx-

imately the average of water inlet temperature T inw and

water outlet temperature T outw , i.e. Tw = 0.5(T in

w + T outw ),

• heat transfer from water to air is driven by the temperaturedifference (Tw − T in

a ),• air intake temperature T in

a is assumed to be equal to zonetemperature,

• properties of air and water are assumed to be constant.With set assumptions, the following dynamics equations arederived for each FCU:

mwcwToutw = qwcw(T in

w − T outw )− Uo

(Tw − T in

a

), (11)

macaTouta = qaca(T

ina − T out

a ) + Uo(Tw − T in

a

), (12)

where T outa is the outgoing air temperature, qw is medium mass

flow through the FCU, ca and cw are the specific heat capacityof dry air and specific heat capacity of water, respectively.Parameter ma is the mass of air and mw is the mass of waterinside the FCU, available from manufacturer’s catalogue. Heattransfer coefficient Uo = f(qa, qw) is a nonlinear function ofmedium mass flow qw and airflow qa defined as [34], [38],[43]:

Uo(qa, qw) =a · qβa

1 + b ·(qa

qw

)β , (13)

where a, b and β are parameters determined based on physicalsystem properties or through identification. For FCUs withfixed set of fan speeds, the air mass flow qa for a certainfan speed x does not deviate over time significantly (if thereare no external impacts blocking the air path). Thus, it isreasonable to estimate separate functional dependencies forall available fan speeds avoiding thus the need for knowingthe exact, hardly measurable, information on the airflow. Bylinking the airflow information to a fan speed, (13) obtains theform:

Uo(x, qw) =

aoff

1 + boff · q−βw, for x = off,

aL

1 + bL · q−βw, for x = L,

aM

1 + bM · q−βw, for x = M,

aH

1 + bH · q−βw, for x = H,

(14)

with individual parameters a := {aoff, aL, aM, aH} and b :={boff, bL, bM, bH} defined for every fan speed. Parameter β doesnot depend on the airflow so one common parameter for allfan states is defined. For switched-off fan a FCU behaves as anormal radiator unit with a constant heat transfer coefficient,thus for fan switched off boff = 0.

The thermodynamic performance of the floor mounted unitsis downgraded during the cooling season. While during theheating season, incoming air temperature is considered equalto the zone temperature, during the cooling season cooledoutgoing air tends to settle at the floor without mixing withthe zone air. As a result, incoming air temperatures are lowerthan the zone temperature. The described seasonal effect is

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anticipated through introduction of correction coefficients εx

for every fan speed:

Uo(x, qw) = εx · ax

1 + ax · q−βw, x ∈ {off, L, M, H}. (15)

The ma/qa ratio is typically less than 1 s and thereforenegligible compared to water time constant. The air sidethermal process of a FCU is therefore observed as a stationaryprocess (T out

a = 0):

qaca(Touta − T in

a )︸ ︷︷ ︸Pa

= Uo(Tw − T in

a

)︸ ︷︷ ︸Pt

. (16)

This further means that the thermal power affecting the zone,Pa, is equal to the overall transmitted thermal power Pt.The important feature of this approach is that the hardlymeasurable and unreliable T out

a measurement is omitted. For afixed medium mass flow qw, the final thermodynamic modelof a FCU is in a form of a switched-linear model:

T outw =

[− qw

mw− Uo(x, qw)

2mwcw

]T out

w

+

[qw

mw− Uo(x, qw)

2mwcw

Uo(x, qw)

mwcw

] [T in

wT in

a

],

(17)

Pa =

[Uo(x, qw)

2

]T out

w

+

[Uo(x, qw)

2−Uo(x, qw)

] [T in

wT in

a

],

(18)

where Uo(x, qw) is defined in (14) and the fan speed x is usedfor switching.

V. THE IDENTIFICATION METHODOLOGY

The methodology for identification of an energy-model ofa system of FCUs consists of three major parts: i) sensorscalibration, ii) identification of a hydraulic system model,and iii) identification of thermodynamic FCUs models. Theconsidered system configuration consists of: i) multiple FCUsconnected in parallel, ii) temperature sensors installed on theFCU return pipes measuring the return medium temperature,iii) zone units for measuring the zone air temperature and fanspeed, and iv) central calorimeter for measuring the overallmedium mass flow through the system, supply and returnmedium temperatures and thermal consumption of the entiresystem. Supply and return pipes are assumed to be isolated.All measurements are collected with a time resolution of Ts.

A. Sensors calibration

Indirect measurement of the return medium temperature,typically performed with temperature sensor mounted on theFCU return pipe is subject to various effects (e.g. leadwires acting as a thermal sink, sensor insulation, effects ofambient temperature, etc.) that cause its deviation from thereal temperature. The so-called two-point calibration method,comparison with trusted sensor at lower and upper boundof the operating range, essentially re-scales the output andis capable of correcting both slope and offset errors. Forsystems with three-way valves (Fig. 3) and well-insulated

DS18B20

Fig. 3: FCU installation with three-way valve and 1-wire returnmedium temperature sensor (DS18B20) mounted on the return pipe.

supply pipelines, the sensor characteristics can be determinedby using historical measurements. In intervals with switchedoff fan and closed FCU valve (total flow goes through thebypass branch), calibrated sensor measurements should beequal to measurements of supply temperature Tcal. The sensorcalibration curve, defined with slope p1,i and offset p2,i, isthen found by solving the following optimization problem:

minp1,i,p2,i

Mi∑k=1

(T out,c

w,i (k)− Tmcal(k)

)2, (19)

where k denotes measurement samples, Mi is overall numberof samples used for calibration, T out,c

w,i = p1,iTout,raww,i + p2,i is

the calibrated sensor measurement and T out,raww,i is a raw sensor

measurement of the ith FCU. If temperature sensor is mountedclose to the bypass branch, measurements may be additionallydistorted due to the high thermal conductivity of the pipes.Since supply and return pipes are thermally coupled throughthe bypass, large thermal gradient between them influences thesensor measurements proportionally to temperature differencebetween the pipes. True temperature measurement T out,m

w,i isthus defined as:

T out,mw,i = T out,c

w,i − φ(T in,mw,i − T out,m

w,i ), (20)

where φ is the unknown heat transfer coefficient equal for allFCUs with the same bypass pipe configuration and T in,m

w,i isthe ith FCU supply temperature considered equal to Tm

cal. Forideal mixing of the return medium from different FCUs andonly the ith FCU operating at the time (valves on all otherFCUs closed) once stationary state is reached, the followingholds:

qw,i(Tinw,i − T out

w,i) = qw,o(∆Tcal,i −∆Tcal,0). (21)

where ∆Tcal,i is temperature difference between the systemsupply and return measured on the central calorimeter and∆Tcal,0 is the temperature difference in the system with valvesof all FCUs closed. Since

∑ni=1 qw,i = qw,o, by combining (20)

and (21) heat transfer coefficient φ is defined as:

φ = 1− 1

/ n∑i=1

(∆Tcal,i −∆Tcal,0

T inw,i − T out,c

w,i

), (22)

where n is the total number of FCUs in the system.

B. Identification of the hydraulic system model

The prerequisites for development of an analogous electricalmodel of the heating/cooling system are i) the availability

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of system documentation with known diameters and lengthsof individual pipe segments and ii) FCUs manufacturers’catalogues with specified pressure drop characteristics. Basedon the developed analogous electrical model, the flow sharesthrough different FCUs are defined by the solution of opti-mization problem (10). If the pressure drop at the systementrance is changed, new flow distribution is found by re-solving (10) based on new measurement of overall flowthrough the system qm

w,o.If the piping data or FCUs pressure drop characteristics are

not available, the approach based on running an individualexperiment on every FCU is proposed. It is important tonote that so obtained flow distribution is valid only foroperating points for which the experiments are performed. Ifthe overall pressure drop of the system is changed, experimentshave to be performed again under new conditions. Thus, theapproach is not advisable for systems with variable flow. Sincesuch approach is time consuming for large systems, hereinit is used for validation of the approach based on electric-hydraulic analogy. The individual experiments are performedby switching off all the units in the system (or assuringtheir constant operation) and running a test sequence on oneparticular unit. Valves remained fully opened for all units.In such a set-up, the central calorimeter measures the heatconsumption of the particular unit with a constant offset equalto the thermal power of the remaining part of the system.To assure constant losses, supply medium mass flow andtemperature are required to be constant during the test. Thetest sequence consists of switching on the highest fan speedon the ith unit and keeping it on until the stationary stateis reached. The hydronic systems are inevitably subject totransport delays. To account for the effect, calorimeter andreturn medium temperature measurements are considered asideal with variable transport delay τ estimated based on theknown pipe length and diameter as well as the medium massflow. After performing experiments on every FCU, flow sharethrough the ith unit ηi is found by solving the followingoptimization problem:

minηi,Pd

Mi∑k=1

(ηi · P a

w,i(k)− (Pmcal(k + τcal)− Pd)

)2s.t. P a

w,i(k) = qmw,o(k)cw,cal

(T in,m

w,i (k)− T out,mw,i (k + τfc)

),

0 ≤ ηi ≤ 1,(23)

where k denotes measurement samples, Mi is the overallnumber of the samples in the experiment with the ith FCU,cw,cal is a nominal water heat capacity used by the calorime-ter (usually set to heat capacity of distilled water cdw =4180 J/(kgK)), Pm

cal is thermal power measurement from thecalorimeter, P a

w,i is a-priori water-side thermal power of the ith

FCU and Pd is constant thermal power consumed by the rest ofthe system. For clarity, the transport delay τfc is assumed to bealready accounted when using the return medium temperaturemeasurements in the rest of the paper.

For available measurement of the system return mediumtemperature T out,m

w,cal and medium mass flow and supply tem-perature constant during the experiment, simple algebraic

equations for calculation of flow share through the ith FCUare derived. Flow share of the ith FCU is defined as theratio of temperature differences of the system and FCU returntemperature with all FCUs’ fans switched off (denoted withsuperscript 'off') and measured once stationary state of returnmedium temperature of the excited unit is reached (denotedwith superscript 'on'):

ηi =

∣∣∣∣∣ (Tout,m,offw,cal − T out,m,on

w,cal )

(T out,m,offw,i − T out,m,on

w,i )

∣∣∣∣∣ . (24)

C. Identification of a thermodynamic FCU model

In the majority of water-based heating and cooling systems,medium mass flow is controlled to a constant value whilesupply temperature is altered to meet the building thermaldemand. Thus, from the operation data of one FCU onlythe number of points on the Uo = f(x, qw) characteristicsequal to the number of available distinct fan speeds canbe obtained. To obtain multiple points for different mediummass flows the non-uniform distribution of the flow throughthe system is exploited, i.e. several FCUs of the same typewith different estimated flow shares are examined. For afixed medium mass flow and fan speed, overall heat transfercoefficient is a scalar value (14). The identification of theUo characteristic is thus divided into two parts. First, a setof scalars Uo = {U off

o , ULo , U

Mo , U

Ho } is found by solving the

following optimization problem for every dataset related to theconsidered FCUs of the same type:

minUo,cw

Mi∑k=1

|T out,mw,i (k)− T out

w,i(k)|

s.t. qw,i(k) = ηi · qmw,o(k),

(14).

(25)

Index k denotes measurement samples, cw is unknown waterheat capacity, T out,m

w,i is measured return medium temperatureof the ith FCU and Mi is the length of the considered dataset. The output of the thermodynamic FCU model T out

w,i(k) isdefined as:

T outw,i(k+1) = h

(xmi (k), qw,i(k), T

in,mw,i (k), T

in,ma,i (k), T out

w,i(k)), (26)

with function h(·) representing numerical integration of (17)over the interval [k, k+ 1]Ts and model inputs assumed to beconstant within that interval. For systems with known heatcapacity of the medium cw operating in a stationary state,values of the Uo set for a fixed medium mass flow qw,i andfan speed xi are defined as:

Uxo =qw,icw(T in,m

w,i − T out,m,xw,i )

(0.5 · (T in,mw,i + T out,m,x

w,i )− T in,ma,i )

. (27)

The result of the first part of the thermodynamic modelidentification is a set of value triplets {(q∗w,i, x∗i , Ux∗o )j : j =1, ...,K}, where K is their overall number and q∗w,i is meanvalue of the medium mass flow of the considered data set if thetriplets are obtained by solving (25) or it is equal to mediummass flow used in (27).

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The second part of the identification is related to findingthe unknown coefficients of the Uo characteristics (15). Theunknown coefficients are found by minimizing the squarederror between the model (15) and value triplets obtainedexperimentally:

mina,b,εεε,β

K∑j=1

(Uo(x∗i , q∗w,i)− Ux∗o )2, (28)

where εεε is a set off FCU efficiencies in every fan speedεεε := {εoff, εL, εM, εH} and Uo(x∗i , q

∗w,i) is defined as in (15).

To improve the performance of the model outside the currentoperating range of the system additional value pairs arecalculated form the catalogue data based on the stationaryequation (16).

Following from (14), it is evident that heat transfer coeffi-cient Uo increases with the medium mass flow qw. However,due to distinctively higher heat capacity of the medium com-pared to air, for fixed fan speed the Uo value starts to stagnateafter some amount of medium mass flow:

limqw→∞

Uo(x, qw) = ax, x ∈ {off, L, M, H}. (29)

This part of the Uo characteristics is typically covered inmanufacturer’s catalogue. The heat capacity tables from thecatalogue consist of stationary values of the sensible thermalpower data P cd

a , supply T in,cdw and return T out,cd

w temperature ofthe heating/cooling medium and entering air temperature dataT in,cd

a for different fan speeds. Thus, if such data are availableit is possible to calculate values in the a coefficient set fromthe stationary condition (16) as:

ax =

∣∣∣∣∣∣ P cd,xa(

T in,cd,xa − 0.5

(T in,cd,x

w + T out,cd,xw

))∣∣∣∣∣∣ . (30)

It is important to note that, even without the use of themanufacturer’s catalogue data, it is possible to estimate themodel covering the operating range of the system. Since thecatalogue data cover the constant part of the Uo characteristics,it represents a sort of performance bound for the consid-ered FCUs.

General-purpose compact identification methods are unableto cope with the problem of identifying the presented energymodel due to the multiple local minima occurring as a result ofa large number of unknown parameters and high nonlinearityof the model (14). The problem is avoided by breaking theprocedure into several parts, the workflow of which is givenin Fig. 4. The optimization problems (19) and (23) are in aform of a Quadratic Program (QP), efficiently solved by usingQP or any generic NLP optimization solver in MATLAB. Theoptimization problems (25) and (28) belong to a class of NLPsand are efficiently solved by using e.g. Levenberg-Marquardtalgorithm in MATLAB [42].

VI. TEST-SITE CONFIGURATION

The considered living-lab at University of Zagreb Facultyof Electrical Engineering and Computing spans over 12 floorsof the university skyscraper. The building has east-west ori-entation with overall 248 controllable zones. Each floor in

Pre

para

tion

Hyd

rau

lic

mod

elid

enti

fica

iton

Th

erm

odyn

am

icm

od

elid

enti

fica

tion

YES NO

Return medium temperature sensors calibration

Is heat capacity of the medium known?

YES NO

{cw,j : j = 1, ...,K}

cw

Repeat if operating point

is changed

{(q∗w,i, x∗i , U

x∗o )j : j = 1, ...,K}

{ηi : i = 1, ..., n}

Are the piping length and size dataand FCUs pressure drop characteristics available?

Build analogue electrical

model (Section III)

Perform experimental tests

on every FCU

Find flow distribution bysolving (10)

Solve problem

(23) or (24) for every FCU

Repeat if operating point

is changed

Use historical data or perform experimental testson selected FCUs

Determine optimal valuetriplets and heat capacity ofthe medium by solving (25)for all K available data sets

Find value tripletsby solving (27)

Estimate thermodynamic model of the FCU byfinding the unknown coefficient of Uo characteristics (28)

a := {aoff, aL, aM, aH}b := {boff, bL, bM, bH}εεε := {εoff, εL, εM, εH}

β

Fig. 4: The workflow of the methodology for identification of anenergy model for a system of FCUs.

the building has separate north-side and south-side pipingand thus a separate north-side and south-side FCU system asconsidered in this paper. The central part of the living lab isa database with data acquisition operating on a minute time-scale. The two-pipe system is used for seasonal heating andcooling. The FCUs, produced by manufacturer Trane (modelsFCC06 and FCC04) [44], are equipped with a centrifugal fanwith four different fan speeds (off, Low, Medium and High).Additionally, FCUs on the 9th and 10th floor are equippedwith a three-way valve (on-off type). Both fan speed andvalve position are controlled by Siemens RXC21.1/RXC21.5zone temperature controllers operating on LonWorks network.Each controllable zone includes a separate user interface fortemperature reference selection (QAX34.1 device). The exist-ing communication network is enhanced such that the RXCcontrollers are reconfigured to be able to pass the informationto a central database (current zone temperature, fan speed andvalve actuation) and to be able to receive the commands fromthe database (fan speed, valve actuation). All the FCUs in the

8

same zone are actuated simultaneously. The system is furtherupgraded with low-cost 1-wire digital temperature sensorsDS18B20 installed on the FCUs return pipes (Fig. 3). Ontwo selected FCUs, additional 1-wire sensors are mounted toinvestigate the effect of downgraded FCU performance duringthe cooling season and monitor incoming and outgoing airtemperatures (Fig. 5).

(position 1) (position 2)

(position 1) (position 2)

Fig. 5: Placement of additional 1-wire sensors for monitoring incom-ing and outgoing air temperatures.

Siemens calorimeters UH50-A50-00 operating on M-Bus pro-tocol are installed on every floor supply duct. Calorimetersmeasure supply and return medium temperature, temperaturedifference, medium flow, thermal power and consumed thermalenergy with one minute time resolution. All the systems areintegrated together with a network controller unit employedto enable two-way communication between devices operatingon different protocols. Logical organization of the describedsystem is shown in Fig. 6.

RXC RXC RXC

DS18B20 DS18B20 DS18B20

UH50-A50-00

PM3200

NETWORK

CONTROLLER

Central database with data acquisition and control unit

LonWorksOne wireM-BusModbusEthernet

One wire/Modbus

Fig. 6: Logical organization of the living-lab sensor-actuator network.

The article is focused on the FCU system of the south-sidepiping on the 9th floor consisted of 13 zones with 17 verticalFCUs mounted on the floor, 12 units of type FCC06 and 5units of type FCC04 (Fig. 7). The arrangement of units withincluded geometry of supply pipes (length and diameter) isgiven in columns 2 through 4 in Table I. The length of thepipe is defined as the length of the horizontal segment betweentwo consecutive FCUs or the length between the FCU andthe calorimeter (see Fig. 7). The equivalent length of verticalsupply and return pipes (including fittings) is identical for allunits and amounts l +

∑leq = 6.26 m. In the following

subsections, transmission heat losses are neglected due tothe good thermal insulation of the pipeline. This means thatthe FCU water inlet temperature is considered to be equalto the supply temperature measured by the calorimeter Tcal(T in

w,i = Tcal). If the temperature drop along the network issignificant, it should be modelled or additional temperaturesensors have to be mounted at the FCU water inlet.

Supply line

Return line

Zone13

Zone10

Zone8

Zone9

Zone7

Zone1

UH50-A50-00

FCC06

FCC06

FCC06

FCC04

FCC06

FCC06

FCC06

Fig. 7: Layout of the southern supply duct on the 9th floor.

VII. EXPERIMENTAL VALIDATION ON THE TEST-SITE

Since validation of both thermodynamic and hydraulicmodel requires experiments, one common set of experimentsper FCU is used. The experiments are performed by shuttingdown all the units and running a test sequence on oneparticular unit. Valves remained fully opened for all units.The test sequence consists of sequential fan speed changesfrom off to other possible fan speeds. The duration of everyspeed engagement is chosen to be 8 min, which proved to beenough to cover both the transient and steady-state behaviour.Alternatively, the identification of the thermodynamic modelis also applicable on FCU historical data with the requirementof recorded stationary operation for each fan speed. Mea-surements obtained after running the experiment test on theselected Zone 7 during the heating season 2016/17 are shownin Fig. 8.

Fansp

eed

Supply temperature T in,mw,10 Return temperature T out,m

w,10

Zone temperature

0

987654

Tem

perature

[oC]

Tem

perature

[oC]

Thermalpow

erP

m w,10[kW

]

H

M

L

0

70

60

50

40

28

26

24

22

Time [min]0 5 10 15 20 25 30 35 40 45

Fig. 8: Measurements obtained after running the experiment test inthe exemplary Zone 7 (FCU i=10).

A. 1-wire return medium temperature sensors calibration

Due to the well-insulated supply pipelines, large thermalconductivity of the copper pipes and 1-wire sensor mountednear to the bypass branch, offset characteristics is determined

9

using historical measurements of supply medium temperature.To avoid the transient impact of the medium stalled insidethe heat exchanger, only stationary values are used. Figure 9shows the calibration curve obtained by calibrating the 1-wiresensor mounted on the FCU return pipe in Zone 7.

Raw sensor measurement T out,raww,10 [oC]

0 10 20 30 40 50 60 70 80 90 100

CalibratedsensormeasurementT

out,c

w,10

[oC]100

90

80

70

60

50

40

30

20

10

0

Heating season data

Cooling season data

Sensor characteristics

Fig. 9: 1-wire return medium temperature sensor characteristics(FCU i = 10).

To determine the coefficient φ, the identification procedureaccording to (22) was performed. The resulting φ value forthe test-site and described scenario amounts φ = 0.1534.

B. Identification of the hydraulic test-side model

With known topology and geometry of the pipes (seethe first four columns in Table I), an analogous electricalmodel of the test-site hydraulic installations is developed. Thecorrelation between pressure drop and mass flow for both FCUtypes is found by identifying the unknown coefficients Rfc andαfc based on the data from the manufacturer’s catalogue (seee.g. Fig. 10 for FCC06 FCU type).

Pre

sure

dro

p[k

Pa]

Mass flow [kg/s]0 0.05 0.1 0.15 0.2 0.25 0.3

50

40

30

20

10

0

Catalogue dataFitted function

Fig. 10: Identified pressure drop function for Trane model FCC06.

To set up the optimization problem (10), a single measurementof the overall medium mass flow qm

w,o from the calorimeter isused. The flow distribution through the entire network is foundby solving the optimization problem (10) in MATLAB with thegenetic algorithm from [42]. The resulting flow distribution,defined as ηi = qw,i/qw,o and rounded to two decimals, islisted in the sixth column of Table I.

TABLE I: Estimated flow share for the south supply lineon the 9th floor for qm

w,o ≈ 0.36 kg/s.

ZoneNo.

FCUNo. i

d[mm]

l[m]

Unittype

ηi[%]

ηi[%]

Error|ηi − ηi|

ηi[%]

1 1 18 1.7 FCC06 4.62 4.73 0.042 22 3.5 FCC06 4.85

2 3 28 1.7 FCC06 5.40 5.49 0.134 35 3.5 FCC06 5.60

3 5 35 3.5 FCC06 5.81 5.88 1.224 6 35 3.5 FCC06 6.09 5.97 1.875 7 42 3.5 FCC06 6.49 6.50 0.24

6 8 42 1.7 FCC04 5.31 x x9 42 3.5 FCC04 5.51

7 10 42 3.5 FCC06 7.20 7.40 2.848 11 42 1.7 FCC06 7.58 7.66 1.069 12 42 2.1 FCC04 6.01 6.15 2.2710 13 28 5 FCC06 7.08 x x11 14 28 1.7 FCC06 6.77 6.71 0.79

12 15 28 3.5 FCC04 5.00 4.94 1.1316 22 1.7 FCC04 4.77

13 17 18 3.5 FCC06 5.62 x x

Flow shares, identified based on (23) and individual exper-iments, for 8 tests performed in the exemplary Zone 7 duringwinter 2015 and 2016, are shown in Table II.

TABLE II: Estimated flow share for Trane FCC06 based on (23) inZone 7 (FCU i = 10).

Test No. 1 2 3 4 5 6 7 8qw,10 [kg/s] 0.027 0.028 0.027 0.028 0.029 0.023 0.022 0.030100 · η10 [%] 7.26 7.35 7.35 7.57 7.34 7.26 7.78 7.31

The mean flow share is η10 = 7.40%, which deviates fromthe calculated value based on the electric-hydraulic analogyby only 2.84% (see Table I). The identified flow distributionthrough the test-site is listed in the seventh column of Table I.Since all FCUs in a single zone are actuated simultaneously,for zones with more than one FCU, the mean flow share ofall units is calculated instead of individual shares. Averagerelative error, mainly due to sensor accuracy, is 1.16%, whichproves the adequate accuracy of electric-hydraulic analogybased calculation of flow distribution through the system. Inzones marked with ’x’, measurements were unavailable.

C. Identification of the thermodynamic model of test-siteFCUs

Value triplets for identification of the thermodynamic modelof the selected FCU type (FCC06) are identified by solving theoptimization problem (25) for the data collected during 32 testsequence runs on the units of type FCC06 in different zonesduring heating season 2015/2016, 2016/2017 and coolingseason 2017. The optimization problem is solved by usingLevenberg-Marquardt algorithm in MATLAB [42]. The heatcapacity of water cw identified according to (25) for all theexperiments is shown in Fig. 11. The model (26) used in opti-mization (25) is initialized by using the known measurementsof the return medium temperature at the beginning of eachof the experiments. During the heating season incoming airtemperature can be considered equal to the zone temperature,

10

Experiment Number

Heatcapacity

[J/(kgoC)]

4504

4800

4180

4000

5000

0 5 10 20 25 30 3215

Heat capacity of the medium (experimental)Mean heat capacity of the mediumNominal heat capacity of clear water

Fig. 11: Identified heat capacity of the heating/cooling medium.

thus εx coefficients are set to one for all speeds. During thecooling season cooled outgoing air tends to settle at the floorwithout mixing with the zone air (Fig. 12).

0 10 20 30 40 50 60 70 80 90 100

35

30

25

20

H

L

M

off

Time [min]

Fan

speed

0 10 20 30 40 50 60 70 80 90 100

30

20

15

10

H

L

M

off

Time [min]

Fan

speed

25

Hea

ting

seaso

nT

emp

eratu

re[o

C]

Cooling

seaso

nT

emp

eratu

re[o

C]

Zone temperature Ta,10

Outgoing air temperature T outa,10

Incoming air temperature T ina,10

Fan speed x10

Fig. 12: Measurements of incoming and outgoing air temperature forFCU i = 10.

To anticipate the effect, separate efficiency coefficients εx

are identified for every fan speed, see (15). The identifiedparameter sets a, b and εεε and parameter β for Trane FCC06thermodynamic model obtained through (28), based on theabove obtained value triplets, are shown in Table III.

TABLE III: Estimated Uo(x, qw) function parameters for TraneFCC06.

Fan speed x/Model parameters

off L M H

ax 5.30 96.45 152.90 201.80bx 0 1.73·10−3 3.58·10−3 5.40·10−3

εx (cooling) 0 0.35 0.47 0.52εx (heating) 1

β 1.86

Time responses of the identified thermodynamic model forone exemplary zone, tested on the verification data set, areshown in Fig. 13. The model is simulated by using knownmodel input data to calculate the prediction of the return

Thermalpow

er[kW

]Thermalpow

er[kW

]Tem

perature

[oC] Simulated return temperature T out

w,10Measured supply temperature Tcal

Measured return temperature T out,mw,10

Calorimeter measurement P ∗cal

Simulated water side thermal power Pw,10

Calorimeter measurement P ∗cal

Simulated transmitted thermal power Pt,10

Time [min]0 5 10 15 20 25 30 35 40 45 50

60

50

40

30

4

3

2

1

0

5

4

3

2

1

0

5

Fig. 13: Identified FCU model (FCU i=10) response over the verifi-cation data set.

medium temperature without considering available measure-ments during that period (so-called open-loop prediction).Estimated heat capacity of the medium is considered (seeFig. 11) such that calorimeter power measurements are scaledand de-offsetted for the remaining piping consumption P ∗cal =Pm

cal·cw/cw,cal−Pd. As it can be seen from the figure, the modelsuccessfully captures the FCU dynamics. The comparisonbetween calculated water side thermal power Pw,i and air sidethermal power Pa,i for one selected FCU operating in a coolingseason is given in Fig. 14.

0 20 40 60 80 100 120 140 160 180

Time [min]

0.0

-0.5

-1.0

-1.5

-2.0

-2.5

Ther

mal

pow

er[k

W] off

H

Fan

speed

Calculated water side thermal power Pw,10

Calculated air side thermal power Pt,10

Fan speed x10

Fig. 14: The comparison between calculated air and water sidethermal power for FCU i=10.

The estimated functional dependence Uo(x, qw) for three non-zero fan speeds of Trane FCC06 is shown in Fig. 15. In Fig.16, normalized root mean squared error (NRMSE) of modelreturn temperature response compared with minutely sampled

11

Over

all

hea

ttr

ansf

erco

effici

entU

M o[W

/oC

]

Over

all

hea

ttr

ansf

erco

effici

entU

H o[W

/oC

]

Over

all

hea

ttr

ansf

erco

effici

entU

L o[W

/oC

]

Water mass flow qw [kg/s] Water mass flow qw [kg/s] Water mass flow qw [kg/s]0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3

Low speed Medium speed High speed

Experimental data Manufacturer’s catalogue data

Heating seasonCooling seasonHeating season Cooling seasonFitted functions

220

200

180

160

140

120

100

80

60

40

20

0

220

200

180

160

140

120

100

80

60

40

20

0

220

200

180

160

140

120

100

80

60

40

20

0

Fig. 15: Identified heat transfer coefficient function for low, medium and high fan speed of Trane FCC06.

measurements is given. NRMSE is calculated as:

NRMSE =1

Tout,mw,i

√∑Mi

k=1(T out,mw,i (k)− T out

w,i(k))2

Mi, (31)

where Tout,mw,i =

∑Mi

k=1 Tout,mw,i (k)/Mi is the mean value of

return medium temperature taken over considered data samplesfor a particular ith FCU, Mi is overall number of samplescollected during the experiment on the particular FCU andk denotes the measurement sample.The prediction error isbelow 6% for all data sets, even for prediction of the systembehaviour 1 h in advance (Mi ≥ 60 samples).

1 2 3 4 5 6 7 8Validation data set No.

NRMSE

[%]

10

8

6

4

2

09 10 11 12 13 14 15 16

i=2

i=5

i=7

i=10

i=11

i=14

i=2

i=5

i=5

i=10

i=10

i=10

i=10

i=10

i=10

i=14

Mi=

32

Mi=

31

Mi=

19

Mi=

35

Mi=

36

Mi=

35

Mi=

51

Mi=

35

Mi=

36

Mi=

36

Mi=

62

Mi=

45

Mi=

53

Mi=

36

Mi=

44

Mi=

31

Fig. 16: NRMSE of the identified model return medium tem-perature response compared to the measurements for FCUi = {2, 5, 7, 10, 11, 14}.

VIII. CONCLUSION

A majority of modern commercial buildings are equippedwith Building Energy Management Systems (BEMSs) to mon-itor and control different components of heating, ventilation,and air conditioning systems. This paper encloses a method-ology for the development of a control-oriented energy model

for a system of fan coil units, suitable for advanced model-based control algorithms. The developed models are validatedon the experimental data collected from a 248-office living-lab.The proposed methodology stands out in its simplicity, cost-effectiveness (minimal sensor configuration), non-invasivenessand amount of time required for model identification. Theincorporation of the developed energy model into the BEMSoffers several advantages such as i) acquiring thermal powersper zone required for identification of building thermodynamicmodel, ii) estimation of unmeasured thermal loads affectingthe zone for a very broad use in building monitoring andcontrol, iii) usage of an advanced FCU control algorithm fordirect control of thermal energy inputs per zone via fan speedswhich makes it possible to realize optimized thermal energyinputs computed via predictive energy management schemesfor maintaining zones thermal comfort and iv) developmentof fault detection and diagnosis algorithms for FCUs. Themodel error of both hydraulic and thermodynamic parts ofthe model is largely below 6% which is one of the mostimportant conclusions of the paper about the applicability ofthe proposed identification methodology and presented model.

ACKNOWLEDGEMENT

This work has been supported by the Croatian ScienceFoundation under the project No. 6731 - Control-based Hi-erarchical Consolidation of Large Consumers for Integra-tion in Smart Grids (3CON, http://www.fer.unizg.hr/3con)and by the Danube Transnational Programme through theproject Smart Building - Smart Grid - Smart City (3Smart,http://www.interreg-danube.eu/3smart), grant DTP1-502-3.2-3Smart. This research has been also supported by theEuropean Regional Development Fund under the grantKK.01.1.1.01.0009 (DATACROSS). We also gratefully ac-knowledge the contribution of Elma Kurtalj company who pro-vided us with Brightcore environment (https://brightcore.biz)for bidirectional building data communication.

12

APPENDIX

The minor losses in a pipe section include the losses infittings, valves, diameter changes and other components thatdisturb the flow and it is represented in terms of leq/d ratio,where leq is the equivalent length of the straight pipe to give thesame pressure drop as the fittings, and d is the inner diameterof fittings. The ratio for most common types of fittings usedin heating/cooling installations is listed in Table A.1.

TABLE A.1: Equivalent length of fittings [40], [41].

Type of fitting leq/d

Tee - along the straight 20Tee - to the branch 60Elbow 90 (smooth radius) 30Three way valve (fully opened - through flow) 30Sudden pipe diameter expansion 4*Sudden pipe diameter contraction 20** used with inlet velocity.

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