Assessment of the controlling envelope of a model-based multivariable controller for vapor compression refrigeration systems

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Applied Thermal Engineering 30 (2010) 1538e1546

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Assessment of the controlling envelope of a model-based multivariablecontroller for vapor compression refrigeration systems

Leonardo C. Schurt a, Christian J.L. Hermes a,*, Alexandre Trofino Neto b

a POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, BrazilbDepartment of Automation and Systems Engineering, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil

a r t i c l e i n f o

Article history:Received 17 February 2010Accepted 19 February 2010Available online 3 March 2010

Keywords:ControlRefrigeration systemModelingExperimentationControlling envelope

* Corresponding author. Present address: DepartmeFederal University of Paraná, P.O. Box 19011, 815Tel.: þ55 41 3361 3239; fax: þ55 41 3361 3129.

E-mail address: [email protected] (C.J.L. Hermes).

1359-4311/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.applthermaleng.2010.02.024

a b s t r a c t

This paper explores the controlling characteristics of a first-principles model-based controller speciallydeveloped for vapor compression refrigeration systems.Mathematical sub-modelswere put forward for eachof the system components: heat exchangers (condenser and evaporator), variable-speed compressor andvariable-orifice electric expansion device. The dynamic simulation model was then used to design a multi-variable controller based on the linear-quadratic-Gaussian technique using a Kalman filter for the estimatordesign. A purpose-built testing apparatus comprised of a variable-speed compressor and a pulse-widthmodulated expansion valve was used to collect data for the system identification, and model and controllervalidation exercises. It was found that the model reproduces the experimental trends of the working pres-sures andpower consumption in conditions far from thenominal pointof operation (�30%)with amaximumdeviation of �5%. Additional experiments were also performed to verify the ability of the controller oftracking reference changes and rejecting thermal load disturbances. Itwas found that the controller is able tokeep the refrigeration system running properly when the thermal load was changed from 340 to 580 W(460W nominal), and the evaporator superheating degreewas varied from 9.5 �C to 22 �C (16.6 �C nominal).

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Household and commercial refrigerators consume by 10% of theelectrical energy produced worldwide, a figure that has motivatedboth customers and governments to push the manufacturers forhigh efficiency products. However, in spite of the large effort put inrefrigeration systems advancements in the past years, just a modesteffect has been observed suggesting that the conventional vaporcompression refrigeration technology is reaching its limits. Vaporcompression refrigeration systems usually comprise a single-speedcompressor, a fixed-orifice expansion device, two heat exchangers(i.e., the condenser and the evaporator), and a volatile working fluid(named refrigerant) that undergoes a reversed Rankine thermo-dynamic cycle. Furthermore, the temperature of the refrigeratedcompartment is controlled by a thermostat that switches therefrigeration system on and off according to a cycling pattern.

It has been advocated in the open literature that the refrigerationsystems with electronic-controlled equipment (i.e., compressorspeed, and valve opening, among others) may improve significantlythe overall energy performance when compared to the conventional

nt of Mechanical Engineering,31-990 Curitiba, PR, Brazil.

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refrigeration appliances. For instance, variable-speed compressorshave been employed in the past decades to obtain a continuousmatching between the cooling capacity and the thermal load withremarkable gains in energy performance [1,2]. Moreover, it has beenalso reported [3,4] that the system performance improves further incases where a variable-orifice expansion valve is additionallyemployed, as the evaporator is kept fully activated (i.e., flooded)during most of the system runtime.

It is noteworthy that such refrigeration systems require propercontrol strategies to account for the simultaneous operation ofa variable-speed compressor and a variable-orifice expansion valve.Albeit there have been in the past years some publications withregard to the analysis and development of control strategies forvariable-speed compressors and variable-orifice expansion valvesoperating simultaneously [5e8], their focus was on large capacityrefrigeration applications (>6 kW).

Pottker and Melo [4] carried out an experimental study to eval-uate the effect of variable-opening electric expansion valves on theperformance of variable-speed (VSC) and single-speed (SSC) vaporcompression refrigeration systems. A purpose-built breadboardrefrigeration apparatus consisting of a variable-speed compressor, anelectric expansion valve (EEV), and two secondary flow rate andtemperature controlled aqueous solution loops (condensing andevaporating media) was designed and constructed to emulaterefrigeration systems with capacities ranging from 0.3 to 1.5 kW

Nomenclature

RomanA cross-sectional area, m2

A, B, C, E, W state-space matricesAv valve opening, %D coil inner diameter, mh specific enthalpy, J/kgJ cost functionK controller gain matrixL length, mL observer gain matrixm mass flow rate, kg/sN compressor speed, s�1

NTU number of transfer unitsp pressure, PaQ heat transfer rate, Wq heatflux, W m�2

Q, R weighting matricesT temperature, Kt time, su controllable inputs matrixUA overall conductance, W/KV secondary coolant flow rate, L/minv specific volume, m3/kgw disturbance input matrix

x dynamic state matrixy output matrix

Greeka heat transfer coefficient, W/m2 K3ll-sl internal heat exchanger effectivenessg void fractionl boundary position, mr specific mass, kg/m

Subscriptsc condensere evaporator, statici inlet, integralK Kalmanl saturated liquido outletr refrigerantref references secondary coolant, isentropic processsur surroundingsv saturated vapor

Other symbolshxi volume average of x_x ¼ dx=dt time derivative of x

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546 1539

(see Fig. 1). It was found that, independently of the secondary fluidtemperature entering the heat exchangers, the VSC/EEV systemalways operates with a maximum coefficient of performance (COP)once it keeps the evaporator flooded and also delivers a coolingcapacity that exactly matches the thermal load.

Later, Marcinichen et al. [9] put forward an empirical dual-SISO(single-input, single-output) strategy for the simultaneous control ofcompressor speed and expansion valve opening using the experi-mental rig constructed by Pottker and Melo [4]. The control strategywas devised to obtain a maximum COP within a range of coolingcapacity from 0.3 to 0.8 kW. The refrigeration systemwas identifiedusing the step-response method, which provided first order linearmodels for both the evaporator superheating and the brine outlettemperature. The empirical models were used to derive two single-input, single-output (SISO) proportional-integral (PI) controllers, onefor the evaporator superheating as a function of the EEVopening, andanother for brine outlet temperature as a function of the compressorspeed. Both controllers were implemented into a dual-SISO controlstrategy that operated satisfactorily in terms of reference trackingand disturbance rejection. Nonetheless, it was found that theempirical identification has constrained the controller to a regionclose to the point of operation.

In order to design controllers that can be applied to a broad rangeof operation, the use of first-principles simulation models have beensuggested in the literature [10] for thedynamic identificationof vaporcompression refrigeration systems. Albeit there are several publica-tions concerning the modeling for controlling vapor compressionrefrigeration systems [11e16], all of them conducted the systemidentification and model validation exercises for regions close to thepoint of operation (w5%).

Therefore, in a prior study, Schurt et al. [17] put forwarda multivariable model-driven controller for vapor compressionrefrigeration systems. Mathematical sub-models were developed foreach of the system components: heat exchangers (condenser andevaporator), variable-speed compressor and variable-orifice

expansion valve. The dynamic simulation model was then used todesign a MIMO (multi-input, multi-output) controller based on thelinear-quadratic-Gaussian (LQG) technique using a Kalman filter forthe estimator design. The purpose-built testing apparatus con-structed by Pottker andMelo [4] was then used to collect data for thesystem identification and model validation exercises. It was foundthat the model reproduces the experimental trends of the workingpressures and power consumption in conditions far from the oper-ation point (�30%) with a maximum deviation of �5%. However,Schurt et al. [17] have not explored the controlling envelope of theMIMO control system, which is, therefore, the main focus of thepresent study.

2. Dynamic simulation model

Modeling of the refrigeration system relies on the developmentof sub-models for each of the cycle components (see Fig. 1), whichhave followed both dynamic (refrigerant flow through the heatexchangers) and quasi-steady (compressor, expansion valve, internalheat exchanger, and condensing and evaporating media) modelingapproaches. Further details can be found in Schurt et al. [17].

2.1. Heat exchangers

The model for the refrigerant flow through the condenser andevaporator was based on the following key assumptions: (i) 1-Dflow; (ii) straight, horizontal and constant cross-sectional coil; (iii)negligible diffusion effects; (iv) negligible pressure drop; and (v)negligible thermal inertia of the walls. The governing equations,derived from the mass and energy conservation principles, arerepresented by

v

vtðrf Þ þ 1

Av

vzðmf Þ ¼ S (1)

Fig. 1. Schematic representation of the test rig (Pottker and Melo [4]).

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e15461540

where f ¼ 1 and S ¼ 0 retrieve the conservation of mass, whereasf ¼ h and S ¼ dp/dt þ 4q/D retrieve the conservation of energy,where q is the heat flux [W/K] and D the coil inner diameter [m]. Tobe solved Eq. (1) must be integrated in both time and spacedomains. The latter was performed following the so-called moving-boundaries approach [11], according to which the spatial domain isdivided into two zones namely single and two-phase flow regions,as illustrated in Fig. 2. Since the interface position varies with time,the Leibnitz's rule has to be invoked for the integration of Eq. (1),

Zl2ðtÞ

l1ðtÞ

v

vtðrf Þdz ¼ ðl2 � l1Þ

ddtðhrf iÞ

þ �hrf i � rl2 fl2�dl2dt

� �hrf i � rl1 fl1�dl1dt

ð2Þ

where l1¼0 and l2¼ l(t) for the two-phase region, whereas l1¼ l(t)and l2 ¼ L for the single-phase region (i.e., evaporator superheatingand condenser subcooling). It is worth noting that the condensergascooling region was modeled as if it was saturated since thevapor provides a lower heat transfer coefficient, but at higher log-mean temperature difference which compensate each other, thus

λ (t)

L

mi, hi

Qtp

Fig. 2. Schematic representation of the spati

Fðx;u;wÞ ¼ A�1�ðmi �moÞ miðhi � hlÞ þ Qtp m

supplying a heat flux with the same magnitude of that calculatedassuming two-phase refrigerant [17].

The averaged terms of Eq. (2) were approximated by a 2nd orderscheme (single-phase zone) and by the mean void fraction (two-phase zone), respectively:

hrf i ¼ 12ðrofo þ rlflÞ (3)

hrf i ¼ rvfvhgi þ rlflð1� hgiÞ (4)

The condenser and evaporator mean void fraction hgi werefitted to experimental data as a part of the system identificationexercise [17]. The spatial integration of the mass and energyconservation equations (1) in each flow region represented in Fig. 2provided three ordinary differential equations that were re-orga-nized in the form of a linear set of differential equations:

CðxÞ$ _x ¼ Fðx;u;wÞ (5)

where x ¼ [p l ho]T, u ¼ [mi mo]T, w ¼ [Qtp Qsp]T, and F(x, u, w) isobtained from

mλ,, hλ mo, ho

Qsp

al discretization of the heat exchangers.

oðhl � hoÞ þ Qsp�T (6)

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546 1541

The elements of matrix C(x) are shown in Table 1. It is noteworthythat, l ¼ v for the evaporator, l ¼ l for the condenser, and thesuperscripts ()0 and ()00 represent the pressure and enthalpy deriva-tives, respectively. Moreover, it is worth noting that mi ¼ meev andmo ¼ mvsc for the evaporator, whereas mi ¼ mvsc and mo ¼ meev forthe condenser. The heat transfer rates were calculated according tothe 3-NTU approach presented in [17].

2.2. Compressor

The compressor sub-model provides the refrigerant mass flowrate suctioned from the evaporator and discharged to thecondenser, and the power consumed by the compression process[18], being respectively calculated from

mvsc ¼�St � cSt

�ðpc=peÞcv;v=cp;v�1

��N=v1 (7)

Wvsc ¼ aþ b$mvscðh2s � h1Þ (8)

where cp,v and cv,v were calculated for the saturated vapor at theevaporating pressure, a, b, c and St were fitted to experimental dataas a part of the model identification exercise [17], and T2s wasobtained from

T2s ¼ Tc þ ðh2s � hvðpcÞÞ=cp;vðpcÞ (9)

where h2s ¼ h(pc,s1) and s1 ¼ s(pe,h1). The refrigerant enthalpy atthe compressor discharge was obtained from the following energybalance:

h2 ¼ h1 þ ðWvsc � UAvscðT2s � TsurÞÞ=mvsc (10)

where Tsur is the surrounding air temperature (¼25 �C), and UAvsc isthe conductance of the compressor obtained empirically from thesystem identification exercise [17].

2.3. Expansion valve

The refrigerant mass flow rate through the expansion valve,meev, was calculated from the orifice equation as follows

meev ¼ CeevAeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2$r4$ðpc � peÞ

p(11)

where meev is given in [kg/s], Ae ¼ AoAv/100 is the effective passageflow area, Av is the valve opening [%], Ao is the nominal orifice cross-sectional area (0.1238 cm2), r4 is the specific mass at the valve inlet,and Ceev is the discharge coefficient calculated as a power-law func-tion of the subcooling degree, Ceev¼ d(T3� Tc)e, where the constantsd and e were obtained from the system identification exercise [17].

2.4. Liquid line suction line heat exchanger

The liquid line/suction line heat exchanger (also named internalheat exchanger) aims to increase the amount of liquid refrigerant at

Table 1Elements of matrix C(x).

c11 lðhgir0v þ ð1� hgiÞr0lÞ þ ðð1=2ÞðL� lÞðr0lþ r0oÞÞ

c12 hgirv þ ð1� hgiÞrl � ð1=2Þðrl � roÞc13 ð1=2ÞðL� lÞr00oc21 lððhgirvh0v þ r0vðhv � hlÞÞ þ ð1� hgiÞðrlh0l þ r0lðhl � hlÞÞ � 1Þc22 hgirvðhv � hlÞ þ ð1� hgiÞrlðhl � hlÞc23 0c31 ð1=2ÞðL� lÞðrlh0l � hlr

0o � 2Þ

c32 ð1=2Þroðhl � hoÞc33 ð1=2ÞðL� lÞðro þ r00oðho � hlÞÞ

the evaporator inlet, improving the cooling capacity and avoidingcompressor slugging. The internal heat exchanger sub-modeldetermines the states of the refrigerant at the compressor (1) andvalve (4) inlet sections (see Fig. 1) based on the information avail-able at the condenser (3) and evaporator (6) exits (see Fig. 1),respectively. On one hand, the compressor inlet temperature T1 wasobtained from the temperature effectiveness:

T1 ¼ 3ll�slT3 þ ð1� 3ll�slÞT6 (12)

h1 ¼ hvðTeÞ þ cp;vðTeÞðT1 � TeÞ (13)

On the other hand, the refrigerant enthalpy at the evaporatorinlet h4 was obtained from the following energy balance:

h4 ¼ h3 þ ðh6 � h1Þ (14)

T4 ¼ Tc þ ðh4 � hlðpcÞÞ=cp;lðpcÞ (15)

2.5. Solution algorithm

The overall system simulation relies on the time integration ofan equation set comprised of six ordinary differential equations.The initial conditions are the evaporating and condensing pres-sures, the refrigerant enthalpies at the evaporator and condenseroutlet sections, and the positions of the evaporation and conden-sation boundaries, all obtained from the prior steady-state solutionof the equation set enforcing the time-derivatives to be nil. Theboundary conditions are the compressor speed, the valve opening,and the temperatures and flow rates of the secondary coolants, allof them obtained from experimental data. The numerical integra-tion of the differential equations was carried out through theODE23S adaptive method for a stiff set of equations [19]. Thethermodynamic properties were obtained from REFPROP7 [20].

3. Controller design

In order to simplify the controller design, a linear state-spacerepresentationwas adopted, which required the linearization of thedynamic simulationmodel using a Taylor expansion series, yielding

d _x ¼ A$dx þ B$duþW$dwdy ¼ C$dx þ E$dw

(16)

The vectors dx, dy, du and dw indicate respectively the differencebetween the static (x0, y0, u0,w0) and the dynamic values of the statevariables, x¼ [pe le ho,e pc lc ho,c]T, the controlled variables, y¼ [DTsupTs,e,o]T, the input variables, u¼ [N Av]T, and the perturbations,w¼ [VcVe]T. The matrices A, B, W, C and E were obtained from the modellinearization exercise and their values are summarized in [17].

The MIMO controller was designed based on a LQG scheme withan integrator, as illustrated in Fig. 3. The controller design wasfocused not only on matching the cooling capacity to the thermalloads but also on keeping the evaporator superheating at a pre-defined level by driving the compressor speed and the electric valveopening simultaneously. In Fig. 3, yref ¼ [DTsup,ref Ts,e,o,ref]T is thereference input signal, _x ¼ yref � y is the tracking error, x is theoutput of the integrator, xK is the estimated state, u ¼ �Ke$xK þ Ki$x

is the control signal, and Ke and Ki are the state-feedback and inte-gral gain matrices, respectively.

The LQG controller consists of a combination of a linear-quadratic-regulator (LQR) and an optimal state estimator, whichhave been developed independently according to the separationprinciple [21]. The former consists of finding the matrix K ¼ [KejKi]

xK

y

+

- -

yref + + u Ki

Ke

Cxy

BuAxx

=+=

x

KalmanFilter

Fig. 3. Schematic representation of the controller.

0 50 100 150 200 250 300 350 4001.0

1.5

2.0

2.5

3.0

Time [min]

Evap

orat

ing

pres

sure

[bar

]

ExperimentalSimulation

0 50 100 150 200 250 300 350 40011

12

13

14

15

16

Time [min]

Con

dens

ing

pres

sure

[bar

]

ExperimentalSimulation

a

b

Fig. 4. Model validation exercise: (a) evaporating pressure; and (b) condensingpressure.

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e15461542

of the control action, u(t) ¼ �K$xa(t), where xa ¼ [xK x]T, thatminimizes the following quadratic criterion:

J ¼ZN

0

hxTa ðtÞ$Q$xTaðtÞ þ uTðtÞ$R$uðtÞ

idt (17)

where Q and R are positive-definite Hermitian matrices deter-mined through the Bryson method [22]. It is noteworthy that Q andR lead to the elements of matrix K straightforwardly through thepositive-definite-solution P of the Riccati equation [17].

A discrete-time version of the LQG controller was designed andthe controller was then implemented on a digital computer witha sample time of 2 s. It is worth of note that the controller wasdesigned to reject perturbations due to the integrator, and also tokeep the time-response of the closed-loop system approximately thesame that of the open-loop system. The values ofmatricesQ, R andKof the discrete-time LQ regulator are summarized in [17].

Since some states cannot be measured directly (e.g., position ofthe evaporation and condensation boundaries), a state estimatorwas also required. The estimator was developed using a Kalmanfilter as indicated below,

_xKðtÞ ¼ ðA � L$CÞ$xKðtÞ þ B$uðtÞ þ L$yðtÞ (18)

where L and S are calculated according to the procedure describedin [17].

4. Experimental facility

Fig. 1 shows a schematic diagram of the HFC-134a refrigerationsystem used in this study. The compressor is of hermetic variable-speed reciprocating type, with minimum and maximum speeds of1800 and 4500 rpm, respectively. The condenser and evaporator areboth tube-in-tube heat exchangers, connected to two distincttemperature and flow controlled secondary heat transfer loops.Brine of 27% ethylene-glycol aqueous solution is used for the evap-orator, whereas pure water is used for the condenser. The expansiondevice is a pulse-width-modulation (PWM) electric expansiondevice with variable opening from 0 to 100%. The condenser andevaporator heat transfer rates are obtained from energy balances onthe refrigerant and secondary fluid sides. The refrigerant heattransfer rate was calculated by multiplying the refrigerant mass flowrate, measured by a Coriolis mass flow meter (measurementuncertainty of �0.1%) by the enthalpy difference between the heatexchanger inlet and outlet, which provided a maximum measure-ment uncertainty of 2% [4].

Immersion T-type thermocouples (measurement uncertainty of�0.2 �C) and pressure transducers (measurement uncertainty of�0.15%) were installed at selected points of the circuit to obtain therefrigerant enthalpies and the evaporator superheating. Thesecondary fluid heat transfer rate was calculated using the volu-metric flow rate, measured by a turbine flow meter (measurementuncertainty of �1%), the fluid density, the specific heat and thetemperature difference between the heat exchanger inlet and outlet.

Moreover, the compression power was measured with an uncer-tainty of�0.5% (full scale). For steady-state conditions, the differencebetween the refrigerant-side and the secondary fluid-side heattransfer rates was less than 2%. A control and data acquisition systemwas used for monitoring the experimental variables and for settingthe compressor speed and valve opening. Further details of theexperimental apparatus can be found in [4].

The facility was used to collect data for the system identificationand model validation exercises. It was found that the model repro-duces the experimental trends of the working pressures and powerconsumption in conditions far from the operation point (�30%) witha maximum deviation of �5%, as illustrated in Fig. 4.

5. Results and discussion

The experimental analyzes were carried out to assess thecontroller performance in terms of reference tracking, controllingenvelope, and disturbance rejection. The reference tracking test wasconducted imposing step changes to the controlled variables (i.e.,references), for instance the evaporator superheating and the brinetemperature at the evaporator outlet. The controlling envelope wasevaluated by changing the references until the maximum andminimum controlled states were reached. The disturbance rejectionwas investigated by varying the flow rate of the evaporating andcondensing media.

5.1. Reference tracking and disturbance analysis

The test was initiated when the steady-state condition was ach-ieved for the following operating conditions: compressor speed of3000 rpm, valve opening of 45%, evaporating medium flow rate andtemperature of 1.24 L/min and 10 �C, respectively, and condensingmedium flow rate and temperature of 1.17 L/min and 35 �C,

Table 2Changes imposed to the evaporator superheating and brine outlet temperature during the reference tracking exercise.

Time [min] t < 20 20 < t < 40 40 < t < 60 60 < t < 80 80 < t < 100 100 < t < 120 t > 120

DTsup [�C] 16.6 20.2 12.5 16.6 16.6 16.6 16.6Ts,e,o [�C] 4 4 4 4 5 3 4

a

b

c

d

Fig. 5. Reference tracking analysis: (a) controlled variables; (b) control actions;(c) working pressures; and (d) cooling capacity.

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546 1543

respectively. At this condition, an evaporator superheating of 16.6 �Cand a brine temperature at the evaporator outlet of 4 �C were ach-ieved and adopted as references. Independent changes in the refer-ences were imposed following the sequence shown in Table 2. Thedynamic behavior of the controlled variables is depicted in Fig. 5a,where it can be noted that the controller drove the refrigerationsystem towards the prescribed references. It is noteworthy thata reference change in one of the controlled variables produceda disturbance in its counterpart, which was identified and properlyrejected by the controller.

The highest disturbance level was observed during the secondchange in the evaporator superheatingdegree, at t¼ 40min, reachinga minimum value of 10.6 �C, which represents an undershooting of2 �C. Such an effect was induced by the magnitude of the imposedchange (7.7 �C), negligible in the first and third changes of references(at t¼ 20min and t¼ 60min, respectively). The disturbances causedby changing the reference of the brine outlet temperature were alsoproperly rejected by the controller, which affected the evaporatorsuperheating in 2 �C only (at t ¼ 120 min).

For the brine outlet temperature, the controller showed a betterperformance, with a maximum undershooting of 0.5 �C at theinstant when the highest change of reference was observed, i.e.,2 �C at t ¼ 100 min. Furthermore, the controller kept the brineoutlet temperature within a 0.5 �C band during the first 80 min.

Fig. 5b shows the variations experienced by the control actions(i.e., compressor speed and valve opening) in order to drive thecontrolled variables towards the desired references. It is worthynoting that, during the change of reference of the evaporatorsuperheating (t < 40 min), the control actions presented oppositebehaviors: the compressor speed increased when the valveopening decreased. This was so as the cooling capacity tends todecrease inasmuch as the superheating degree increases, forcingthe controller to increase the compressor speed in an attempt tokeep the cooling capacity (see Fig. 5d), thereby reducing theevaporating pressure (see Fig. 5c) and increasing the evaporatorlog-mean temperature difference. The opposite behavior wasobserved within the time interval 40 < t < 60 min.

During the change of reference of the brine outlet temperature(t > 60 min), it was noted that both compressor speed and valveopening showed the same behavior (see Fig. 5b). On one hand, toraise the brine outlet temperature (i.e., lower cooling capacity)without changing the evaporator superheating (t ¼ 80 min), thecontroller increased the evaporating pressure (see Fig. 5c) byreducing both the compressor speed and the expansion valveopening so as to diminish the refrigerantmass flow rate. On the otherhand, the cooling capacity increased at t¼ 100min as the compressorspeed and the valve opening were simultaneously brought up (seeFig. 5b).

The disturbance rejection test aims to check whether thecontroller is able to keep the controlled variables at the referencepreviously set, even if changes are imposed on the thermal load orthe surrounding conditions. Such changes have been respectivelyemulated in the experimental apparatus by modifying the evapo-rator and condenser secondary coolant flow rates, as shown inTable 3. Such variations have induced disturbances on thecontrolled variables that have been recognized by the controllerand then rejected, as can be seen in Fig. 6a. The largest deviationobserved in the evaporator superheating, in comparison to its

Table 3Changes imposed to the evaporator and condenser secondary coolant mass flow rates during the disturbance rejection exercise.

Time [min] t < 20 20 < t < 40 40 < t < 60 60 < t < 80 80 < t < 100 100 < t < 120 t > 120

Vs,c [L/min] 1.16 1.16 1.16 1.16 1.38 0.93 1.16Vs,e [L/min] 1.23 1.43 1.00 1.23 1.23 1.23 1.23

a

b

c

d

Fig. 6. Disturbance rejection analysis: (a) controlled variables; (b) control actions;(c) working pressures; and (d) cooling capacity.

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e15461544

reference, was 3 �C at t ¼ 40 min. Such deviation is related to thesudden reduction of the brine flow rate from 1.43 to 1.00 L/min,which represented a 30% reduction in the thermal load imposed(from 550 to 385 W, see Fig. 6d).

It can be also noted in Fig. 6 that the variations observed fort < 70 min are related to the thermal load disturbances, as thechanges in the condenser side had no practical influence on thecontrolled variables and control action, such as the compressorspeed and the valve opening (Fig. 6b), the mass flow rate (Fig. 6d),and evaporating temperature (Fig. 6c). The condensing pressure, onthe other hand, has experienced some disturbances, although thelargest variationwas due to the changes on the evaporator brine flowrate (Fig. 6c). Albeit the cooling capacity (see Fig. 6d) was practicallynot affected by the disturbance imposed in the condenser secondarycoolant flow rate (t > 70 min), it changed substantially with theevaporator brine flow rate (t < 70 min).

5.2. Assessment of the controlling envelope

The operational envelope beyond which it is not possible tomaintain the controlled variables at their references was evaluatedby artificially driven the controller through step changes promotedin the references. In addition, changes in the evaporator brine flowrate were also imposed to the system in order to identify themaximum and minimum controllable thermal loads.

First, increasing step changes (from 4.0 to 6.0 �C, step of 0.5 �C) onthe brine outlet temperature were imposed, as shown in Fig. 7a,

a

b

Fig. 7. Controlling envelope analysis with increasing brine temperature: (a) controlledvariables; (b) control actions.

a

b

Fig. 8. Controlling envelope analysis with decreasing brine temperature: (a) controlledvariables; (b) control actions.

a

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e1546 1545

which is equivalent to a thermal load reduction. In order to followthe references and also to adapt the system to lower coolingdemands, the controller decreased the valve opening while reducedthe compressor speed until 1800 rpm, the minimum allowed speed,

a

b

Fig. 9. Controlling envelope analysis with decreasing evaporator superheating:(a) controlled variables; (b) control actions.

as shown in Fig. 7b. When the minimum compressor speed wasreached, the evaporator superheating increased as the valve openingruled the system while the compressor speed became saturated.Similarly, decreasing step changes (from 4.0 to 2.5 �C, step of 0.5 �C)on brine outlet temperature were imposed, as shown in Fig. 8a,which is equivalent to a thermal load augmentation. At each newreference change, the controller increased the cooling capacity byopening the valve and raising the compressor speed up to themaximum allowed value (4500 rpm), as shown in Fig. 8b. As theevaporator superheating was fixed at 16.6 �C, the brine outlettemperature was limited within the range from 3 to 5.5 �C.

Alternatively, decreasing step changes (from 16.6 to 8.5 �C) on theevaporator superheatingwere imposed, as shown in Fig. 9a, until thecontroller is no longer maintaining the controlled variables at theirreferences. It is noteworthy that at superheating degrees lower than8.5 �C the system became unstable due to two factors. First, as theproposed controller is linear, it cannot identify the non-lineardynamics of the variables (see Fig. 9b). In addition, the evaporatordry-out location oscillates at lower superheating degrees due to thenatural two-phase flow instabilities [11], which affected the qualityof the control signal and therefore the control action.

b

c

Fig. 10. Controlling envelope analysis with increasing and decreasing thermal load:(a) controlled variables; (b) control actions; and (c) disturbances.

L.C. Schurt et al. / Applied Thermal Engineering 30 (2010) 1538e15461546

Additional tests were carried out varying the evaporator brineflow rate, but keeping the evaporator superheating degree and thebrine outlet temperature references fixed (see Fig. 10a). During thefirst 120 min, the flow rate was steeply increased to the maximumvalue that can be achievedwithout loosing the system controllability,as showed inFig.10b. Therefore, the refrigeration systemexperienceda thermal load variation from 460 to 580 W (see Fig. 10c). In otherwords, a thermal load 26% higher than the nominal was achieved butkeeping the preset references (see Fig. 10b). Between 120 and140 min, the system was then returned to its initial condition. After140min, the brine flow ratewas steeply decreased in order to reducethe thermal load to 340 W, i.e., a figure 26% lower than the nominalvalue (see Fig. 10c).

6. Summary and conclusions

A dynamic simulationmodel for the identification and control ofvapor compression refrigeration systems was developed and vali-dated for a broad range of operation. A breadboard refrigerationsystem comprised of a variable-speed compressor and an electricexpansion valve was used to gather the required experimental datafor both system identification and model validation exercises. Itwas found that the model reproduces well the experimental trendsof the working pressures even in conditions far from the operationpoint used in the system identification, with maximum discrep-ancies between calculated and measured counterparts within 5%error bands.

A model-driven multivariate linear strategy for controlling theevaporator feeding and for matching the cooling capacity to thethermal loads was devised. The dynamic simulation model waslinearized according to a Taylor expansion series and then used todesign a proportional plus integral type controller based on thelinear-quadratic-Gaussian (LQG) method, which is based on a stateestimator of the Kalman filter type. Additional experimental testswere also carried out using the breadboard facility to verify thecontroller ability for tracking reference changes and rejectingthermal load disturbances due to the integral action of the controller.

It was found that the controller was able to drive the facility forthermal loads spanning from 340 to 580 W, i.e., �26% in compar-ison to the nominal value. Similarly, the controller kept the refrig-eration system running properly for evaporator superheatingdegrees ranging from 9.5 �C to 22 �C, albeit it was designed for16.6 �C. However, it is worthy of note that the controller cannot beapplied for superheating degrees lower than 9.5 �C, when both thecontrolled system and the control signal became unstable.

Acknowledgements

The authors would like to thank the CNPq (Grant No. 573581/2008-8 e National Institute of Science and Technology in Refrig-eration and Thermophysics) for the financial support. Thanks arealso addressed to Embraco S.A. for sponsoring this research project.The authors duly acknowledge Prof. C. Melo and Prof. N. Roqueiro

(Federal University of Santa Catarina), and Dr. L. W. da Silva(Embraco S.A.) for their valuable technical contributions.

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