HYBRID MODEL PREDICTIVE CONTROL INSUPERMARKET REFRIGERATION SYSTEMS

Lars F. S. Larsen∗ Tobias Geyer∗∗ Manfred Morari ∗∗

∗Central R&D - RA-DP, Danfoss A/S, Nordborg, Denmark∗∗Automatic Control Lab, CH-8092 Zürich, Switzerland

Abstract: This paper presents a Model Predictive Control scheme in connection with theMixed Logical Dynamical framework, which tackles the modelling and optimal controlproblem of supermarket refrigeration systems. The latter are hybrid systems with switchednonlinear dynamics and discrete-valued input variables such as valves. Simulation resultsindicate the potential increase in efficiency and reduced wear with respect to traditionalcontrol schemes used nowadays.Copyright©2005 IFAC

Keywords: Supermarket refrigeration systems, hybrid systems, MPC.

1. INTRODUCTION

A supermarket refrigeration system consists of a cen-tral compressor bank that maintains the required flowof refrigerant to the refrigerated display cases locatedin the supermarket sales area. Each display case has aninlet valve for refrigerant that needs to be opened andclosed such that the air temperature in the display caseis kept within tight bounds to ensure a high quality ofthe goods.For many years, the control of supermarket refrigera-tion systems has been based on distributed control sys-tems, which are flexible and simple. In particular, eachdisplay case used to be equipped with an independenthysteresis controller that regulates the air temperaturein the display case by manipulating the inlet valve.The major drawback, however, is that the control loopsare vulnerable to self-inflicted disturbances caused bythe interaction between the distributed control loops.In particular, practice and simulations show that thedistributed hysteresis controllers have the tendency tosynchronize (Larsen, 2004a), meaning that the open-ing and closing actions of the valves coincide. Con-sequently, the compressor periodically has to workhard to keep up the required flow of refrigerant, whichresults in low efficiency, inferior control performanceand a high wear on the compressor.The control problem is significantly complicated by

the fact that many of the control inputs are restricted todiscrete values, such as the opening/closing of the inletvalves and the stepwise control of the compressor.Furthermore, the system features switched dynamicsturning the supermarket refrigeration system into ahybrid system.Motivated by these difficulties, we present in thispaper a novel approach to the modelling and con-troller design problem for supermarket refrigerationsystems. The refrigeration system is modelled as ahybrid system using the Mixed Logical Dynamical(MLD) (Bemporad and Morari, 1999) framework al-lowing one to capture the switched dynamics and thediscrete-valued control inputs. Based on the MLDmodel, we propose a centralized constrained finite-time optimal controller, specifically Model PredictiveControl (MPC) (Maciejowski, 2002), which accom-modates the multivariate nature of the refrigerationsystem, handles the constraints explicitly, and mostimportant, allows for a systematic controller design.The paper is structured in the following way. Section 2describes the basic layout of a supermarket refrigera-tion system, and Section 3 provides an overview ofthe traditional control setup. Section 4 summarizes thenonlinear model of the refrigeration system, the MLDframework and the derivation of the MLD model. InSection 5, the control objectives are formulated and

an objective function is set up. The proposed hybridMPC scheme is compared with the traditional controlapproach in Section 6 through simulations. Conclu-sions are drawn in Section 7.

2. SYSTEM DESCRIPTION

In a supermarket many goods need to be refrigeratedto ensure preservation for consumption. These goodsare normally placed in open refrigerated display casesthat are located in the supermarket sales area for selfservice.A simplified supermarket refrigeration circuit is shownin Figure 1. The heart of the system are the compres-sors. In most supermarkets, the compressors are con-figured as compressor racks, which consist of a num-ber of compressors connected in parallel. The com-pressors supply the flow of refrigerant in the systemby compressing the low pressure refrigerant from thesuction manifold, which is returning from the displaycases. The compressors keep a certain constant pres-sure in the suction manifold, thus ensuring the desiredevaporation temperature. From the compressors, therefrigerant flows to the condenser and further on tothe liquid manifold. The evaporators inside the dis-play cases are fed in parallel from the liquid manifoldthrough an expansion valve. The outlets of the evap-orators lead to the suction manifold and back to thecompressors thus closing the circuit.In Figure 2, a cross section of an open display caseis depicted. The refrigerant is led into the evaporatorlocated at the bottom of the display case, where the re-frigerant evaporates while absorbing heat from the sur-rounding air circulating through the evaporator. Theresulting air flow creates an air-curtain at the front ofthe display case. The fact that the air-curtain is colderthan the goods leads to a heat transfer from the goodsQgoods−air and - as a side effect - from the surroundingQairload to the air-curtain. Inside the display cases,a temperature sensor is located, which measures theair temperature close to the goods. This temperaturemeasurement is used in the control loop as an indirectmeasure of the goods’ temperature. Furthermore, anon/off inlet valve is located at the refrigerant inlet of

Display Case

Display Case Liq

uid

Man

ifo

ld

Suction Manifold

Suction line

Suction line

Condenser Unit

Compressor Rack

Fig. 1. A simplified layout of a supermarket refrigera-tion system.

Air-curtain

Goods

Evaporator

Refrigerant outRefrigerant in

Suction line

Inlet valve

goodsT

airT

airgoodsQ − airloadQ

wallT

eTeQ

wallairQ −

Fig. 2. Cross section of a refrigerated display case.

the evaporator, which is used to control the tempera-ture in the display case.

3. TRADITIONAL CONTROL

The control systems used in today’s supermarket re-frigeration systems are decentralized as mentioned inSection 1. Each of the display cases is equipped with atemperature controller and a superheat controller thatcontrols the filling level of the evaporator. The com-pressor rack is equipped with a suction pressure con-troller. Furthermore, the condenser is equipped witha condenser pressure controller, and on top, variouskinds of supervisory controllers may be used that helpadjusting the set-points. In this paper, we will onlyconsider the display case temperature controller andthe compressor controller.The display case temperature is controlled by a hys-teresis controller that opens and closes the inlet valve.This means that when the temperatureTair reaches acertain upper temperature bound the valve is openedandTair decreases until the lower temperature boundis reached and the valve is closed again. Unfortu-nately, when the valve is closed, the air temperaturecontinues to decrease below the lower bound for tworeasons. Firstly, the remaining refrigerant contained inthe evaporator evaporates, and secondly, the thermalcapacity of the evaporator wall acts as a low-pass filter.In the supermarket many of the display cases are alikein design and they are working under uniform condi-tions. As a result, the inlet valves of the display casesare switched with very similar switching frequencies.Therefore, the valves have a tendency to synchronizeleading to periodic high and low flow of evaporated re-frigerant into the suction manifold thus creating largevariations in the suction pressure.Turning on and off the compressors in the compressorrack controls the suction pressure. To avoid exces-sive compressor switching, a dead band around thereference of the suction pressure is commonly used.

When the pressure exceeds the upper bound, one ormore additional compressors are turned on to reducethe pressure, and vice versa when the pressure fallsbelow the lower bound. In this way, moderate changesin the suction pressure do not initiate a compressorswitching. Nevertheless, pronounced synchronizationeffects lead to frequent compressor switchings causinglarge fluctuations in the suction pressure and a highwear on the compressors.

4. MODELLING

We start by summarizing the model of the supermar-ket refrigeration system in continuous-time derivedin (Larsen, 2004b). After recalling the MLD frame-work, we derive an MLD model of low complexityand sufficient accuracy to serve as a prediction modelfor MPC in the next section.

4.1 Nonlinear Continuous-time Model

The nonlinear model of the supermarket refrigerationsystem is composed of individual models of the dis-play cases, the suction manifold, the compressor rackand the condensing unit. The main emphasis in themodelling is laid on the suction manifold and the dis-play cases such that the dynamics relevant for control-ling the compressor(s) and display cases are captured.The condensing unit is not considered here since itonly has minor effects on the considered control.

The Display CasesThe dynamics in the display case can be described bythree states, namely the goods’ temperatureTgoods, thetemperature of the evaporator wallTwall and the massof liquified refrigerant in the evaporatorMre f . Themodel of the display case encompasses three parts,namely the goods, the evaporator and the air-curtain inbetween (see Figure 2). By setting up the energy bal-ance for the goods and the evaporator, the followingtwo state equations can be derived, assuming a lumpedtemperature model.

dTgoods

dt=− UAgoods−air

Mgoods·Cpgoods· (Tgoods−Tair) (1)

dTwall

dt=

UAair−wall

Mwall ·Cpwall· (Tair −Twall)

−UAwall−re f(Mre f )Mwall ·Cpwall

· (Twall−Te),(2)

where UA is the heat transfer coefficient with thesubscript denoting the media between which the heatis transferred,M denotes the mass andCp the heatcapacity, where the subscript denotes the media theparameter refers to. MoreoverTe is the evaporationtemperature.As indicated in Eq. (2), the heat transfer coeffi-cient between the refrigerant and the evaporator wallUAwall−re f is a function of the massMre f of the liqui-fied refrigerant in the evaporator. It is assumed here

that this relation can be described by the followinglinear function:

UAwall−re f (Mre f ) = UAwall−re f rig,max·Mre f

Mre f,max, (3)

whereMre f = Mre f,max, when the evaporator is com-pletely filled with liquid refrigerant.The accumulation of refrigerant in the evaporator isdescribed by:

dMre f

dt=

0 if valve= 1,

− Qe

∆hlgif valve= 0 andMre f > 0,

0 if valve= 0 andMre f = 0.

(4)

where∆hlg is the specific latent heat of the remain-ing liquified refrigerant in the evaporator, which is anonlinear function of the suction pressure.Qe is thecooling capacity that can be found by setting up theenergy balance for the evaporator:

Qe = (Twall−Te) ·UAwall−re f (Mre f ) (5)

The amount of liquid refrigerant in the evaporatorMre f follows a switched nonlinear dynamic governedby Eq. (4). It is assumed that the evaporator is filledinstantaneously when the valve is opened (valve= 1).Furthermore, when the valve is closed (valve= 0) andall of the enclosed refrigerant has evaporated (Mre f =0), thenMre f = 0.Finally, the air temperatureTair can be found by settingup the energy balance for the air-curtain.

Tair = Qairload+Tgoods·UAgoods−air +Twall ·UAair−wall ,(6)

whereQairload is a given external heat load on the air-curtain.

The Suction ManifoldThe dynamic in the suction manifold is modelled bytwo states, namely the density of the refrigerant in thesuction manifoldρsuc and the suction pressurePsuc.Setting up the mass and energy balance, and assumingan ideal gas, the corresponding state equations areobtained:

dρsuc

dt=

min−suc−Vcomp·ρsuc

Vsuc, (7)

dPsuc

dt= R

hin−sucmin−suc−hout−sucVcompρsuc

Cv ·Vsuc, (8)

where Vcomp is the volume flow out of the suctionmanifold determined by the compressor(s),min−suc=∑n

i=1mi is the total mass flow from the display casesto the suction manifold,hin−suc is the inlet enthalpyto the suction manifold which is a refrigerant specificnonlinear function ofPsuc and the superheat,n is thenumber of display cases, andmi is the mass flowof the refrigerant out of thei’th display case that isdetermined by the position of the valve and the amountof enclosed refrigerant when the valve is closed.

The CompressorIn most refrigeration systems, the compressor capacityis discrete-valued, as compressors can be switched

only either on or off. Letq denote the total numberof compressors. The compressor bank is modelledusing a constant volumetric efficiencyηvol and themaximal displacement volumeVsl. Thus, the volumeflow Vcomp,i out of the suction manifold created by thei’th compressor can be determined as follows

Vcomp,i = compi · 1100

·ηvol ·Vsl i = 1, ..,q (9)

where compi is the i’th compressor capacity, andVcomp = ∑p

i=1Vcomp,i is the total volume flow. For amore elaborate model derivation, the reader is referredto (Larsen, 2004b).

4.2 MLD Framework

The general MLD form of a hybrid system introducedin (Bemporad and Morari, 1999) is

x(k+1) = Ax(k)+B1u(k)+B2δ (k)+B3z(k) (10a)

y(k) = Cx(k)+D1u(k)+D2δ (k)+D3z(k) (10b)

E2δ (k)+E3z(k)≤ E4x(k)+E1u(k)+E5, (10c)

where k ∈ N is the discrete time-instant, andx ∈Rnc ×{0,1}n` denotes the states,u ∈ Rmc ×{0,1}m`

the inputs andy ∈ Rpc × {0,1}p` the outputs, withboth continuous and binary components. Furthermore,δ ∈ {0,1}r` andz∈Rrc represent binary and auxiliarycontinuous variables, respectively. These variables areintroduced when translating propositional logic orPWA functions into mixed integer linear inequalities.All constraints on states, inputs and auxiliary variablesare summarized in the inequality (10c). Note that theequations (10a) and (10b) are linear; the nonlinearityis hidden in the integrality constraints over the binaryvariables.

4.3 MLD Model

Before transforming the nonlinear model into MLDform, we simplify the model and disregard the empty-ing phenomena of the evaporator in Eq. (4), meaningthat the evaporator empties instantaneously when thevalve closes. Instead, the emptying dynamics are ap-proximated by low-pass filteringTair . Therefore,Tair

is a state in the simplified model rather thanMre f .Moreover, the density of the refrigerant in the suctionmanifold described by Eq.(7) is assumed constant.Several nonlinearities are present in the model, namelyin the second term in Eq. (2), and in the Eqs. (5), (7)and (8). One might approximate these nonlinearitiesby piecewise affine functions to obtain an arbitrarilyaccurate representation of the nonlinear model. How-ever, as the control experiments in the last sectionwill show, linearizing the nonlinearities around theoperating point yields for our control purposes a suf-ficiently accurate model. The resulting hybrid modelis a switched linear system as detailed in (Larsen,2004b). Specifically, depending on the position of theinlet valves (open or closed) and the number of com-pressor capacities running, different linear dynamicsare active.

By subsequently discretizing the model in time anMLD formulation is obtained. Choosing a suitablesampling intervalTsampis difficult, as the switched dy-namic in the display case exhibit greatly differing timeconstants. Specifically, the decrease ofTair when thevalve is open is significantly faster than the increasewhen the valve is closed. To avoid long predictionhorizons (in terms of steps) in MPC, a long samplinginterval needs to be chosen. In general, however, thisimplies that the valve should be opened forlesstimethan the sampling interval in order to bring the temper-ature down from the upper to the lower temperaturebound without excessive undershoot. The necessaryadditional flexibility is achieved by introducing the in-termediate opening timetopen as a continuous-valuedcontrol input that allows one to vary the openingtimeof the valve within the sampling interval. This be-havior is well approximated by introducingtopen asa variable that varies the openingdegreeof the valvethus avoiding one additional nonlinearity.Summing up, each display case has the three statesxdisp= [Tair ,Tgoods,Twall ], and the inputudisp= [dvalve,topen], where dvalve ∈ {0,1} describes whether thevalve is open or closed, andtopen∈ [tmin,Tsamp] denotesthe opening time of the valve. The opening time isconstrained to be less than the sampling interval andlarger than some minimum opening timetmin. Thelatter constraint is introduced to ensure a proper fill-ing of the evaporator whenever the valve is opened.The control input to the compressor, the compressorcapacity, is given bycomp∈ {0, 100

q , 2·100q , . . . ,100}.

4.4 Example Refrigeration System

In the following, we focus on a supermarket refrig-eration system consisting of two display cases andone compressor with the discrete capacitiescomp∈{0,50,100}. The input vector is defined asu =[dvalve,1, topen,1,dvalve,2, topen,2,∆comp]T , where∆comp(k) = comp(k)− comp(k− 1). Note that the∆comp formulation will be needed in the next sec-tion to penalize the switching of the compressor. Thuswe need to add the state-update functioncomp(k) =comp(k− 1) + ∆comp(k) to the MLD model. Thesampling interval isTsamp= 60sec. Assuming that ittakes 20 sec to ensure that the evaporator is completelyfilled after opening the valve, we settopen∈ [20,60].The procedure in Section 4.3 yields an MLD systemwith 8 states (7 states from the system and one ad-ditional state for∆comp(k− 1)), 2 z-variables,4 δ -variables and52inequality constraints. The derivationof the MLD system is performed by the compilerHYSDEL (Torrisi and Bemporad, 2004) generating thematrices of the MLD system starting from a high-leveltextual description of the system.

5. OPTIMAL CONTROL PROBLEM

5.1 Control Objectives

The control objectives are to bring the suction pres-surePsuc close to its reference value of4.2bar while

fulfilling the soft constraints on the air temperaturesin the display casesTair ∈ [0,4], and while switchingthe compressors as little as possible in order to mini-mize wear. Switch transitions in the inlet valves of thedisplay cases are by far less critical concerning wear.Furthermore in some cases, it is even desired that theair temperature has a zigzag behavior as experienceindicates that this gives a more compact icing on theevaporator improving the heat transfer between theevaporator and the surrounding air.

5.2 MPC

As introduced in (Tyler and Morari, 1999),ModelPredictive Control(MPC) is well suited for findingcontrol laws for hybrid systems described in the MLDframework. Here, an objective function is used thatpenalizes with the∞-norm over a finite horizon thefollowing three terms. (i) the deviation of the suc-tion pressurePsuc from its reference, (ii) the switch-ing of the manipulated variables (the compressor andvalve control actions) and (iii) the violation of thesoft constraints onTair . The control law is then ob-tained by minimizing the objective function subjectto the evolution of the MLD model (10) and thephysical constraints on the manipulated variables. Aswe are using the∞-norm, this minimization prob-lem amounts to solving aMixed-Integer Linear Pro-gram(MILP). For details concerning the set up of theMPC formulation in connection with MLD models,the reader is referred to (Bemporad and Morari, 1999)and (Bemporadet al., 2000). Details about MPC canbe found in (Maciejowski, 2002).

5.3 Objective Function

According to (Bemporadet al., 2000) and Section 5.1,the following optimal control problem is considered:

minu(0),..,u(N−1)

J =N−1

∑k=0

(‖Psuc(k)−4.2‖∞

Q1+‖u(k)‖∞

Q2

)

+ p·N−1

∑k=1

(S(k)+S(k)

)

subject to the evolution of the MLD model (10) overthe prediction horizonN and taking into account thediscrete-valued nature of some of the manipulatedvariables (dvalve,i and∆comp).The deviation ofPsuc from its reference is weightedby Q1. To keep the energy consumption low, the vari-ations onPsuc should be kept at a minimum. There-fore, a large weight is chosen, i.e.Q1 = 500. Theweight matrix on the manipulated variables is given byQ2 = diag(q1,q2,q3,q4,q5). Recall that step changesin the compressor capacity have a magnitude of 50(comp∈ {0,50,100}). To avoid compressor switchingwhen the suction pressure lies within a reasonabledead band of±0.2bar, we setq5 to 2. Thus, switchingparts of the compressor on or off costs2 · 50 = 100,and the deviation inPsuchas to amount to100

500 = 0.2barbefore a change in the compressor capacity is initiated.Assuming that it is 100 times less expensive (in terms

of wear) to open or close the valves than to change thecompressor capacity, we set the weights ondvalve,1 anddvalve,2 to q1 = q3 = 100

100 = 1. The weights ontopenareset toq2 = q4 = 0.1.The soft constraints on the temperature bounds aretaken into account by introducing slack variables forthe upperSand the lowerSbounds and a large penalty,e.g.p = 104.

6. SIMULATION RESULTS

This section presents control experiments simulatedon the nonlinear model described in Section 4.1. Toillustrate the performance improvements that can beachieved by using an MPC scheme, the control per-formance resulting from a traditional controller (asdescribed in Section 3) is compared with the MPCscheme. To illustrate the problems that often arisewith the traditional control scheme, we have chosena refrigeration system with two equally sized displaycases that has a pronounced tendency for synchroniza-tion.The traditional controller comprises a hysteresis basedtemperature controller with[Tlower,Tupper] = [0,4] anda PI-type suction pressure controller with a dead bandof ±0.2bar around the reference of 4.2 bar. MPC usesa horizon ofN = 10.The lower part of Figure 3 depicts the air temperaturesin the two display cases when using the traditionalcontrol scheme. Initially, the two temperatures exhibitan offset which vanishes within the first hour due tothe control actions. In other words, the valves of thedisplay cases get synchronized leading to large varia-tions in the suction pressure as shown in the lower partof Figure 4. As depicted in Figure 5, the compressorcontroller tries in vain to suppress these variationswhen they exceed the dead band of±0.2bar causingexcessive switching and wear. Unless something isdone to de-synchronize the valves, they will remainsynchronized.The upper part of Figure 3 shows the temperaturesin the display cases when employing MPC. The twotemperatures coincide in the beginning and the valvesare thus synchronized. However, after only20min, theswitching of the valves is de-synchronized, resultingin smaller variations in the suction pressure as canbe seen in the upper part of Figure 4. These reducedvariations not only result from the de-synchronization,but also from the significantly reduced undershoot inthe air temperatures1 . The traditional controller failsto respect the lower bound as it cannot predict theundershoot in contrast to MPC. Finally, as can beseen in Figure 5, the large penalty on the compressorswitching reduces the number of switch transitionsconsiderably with respect to the traditional controlscheme.To solve the optimal control problem online at eachtime step,CPLEX 9.0 run on a Pentium IV2.0 GHzcomputer. ForTsamp= 1min and a horizon ofN = 10,the computation time is in average3.9sec and always

0 20 40 60 80 100 120 140 160 180−5

0

5

time [min]

Air

tem

pera

ture

[o C]

MPC

Tair1

Tair2

0 20 40 60 80 100 120 140 160 180−5

0

5

time [min]

Air

tem

pera

ture

[o C]

Traditional Control

Tair1

Tair2

Fig. 3. Tair in the 2 display cases, controlled respec-tively with MPC and a traditional hyst. controller.

0 20 40 60 80 100 120 140 160 1803.8

4

4.2

4.4

4.6MPC

time [min]

Suc

tion

Pre

ssur

e [b

ar]

0 20 40 60 80 100 120 140 160 1803.8

4

4.2

4.4

4.6Traditional Control

time [min]

Suc

tion

Pre

ssur

e [b

ar]

Fig. 4.Psuc in the suction manifold, controlled respec-tively with MPC and a traditional PI controller.

less than9.7sec rendering an online implementationof the MPC scheme computationally feasible.

7. CONCLUSIONS

The nonlinear system described in Section 4.1 wascast into the MLD form by approximating the switchednonlinear dynamics by switched affine dynamics, in-troducing integer manipulated variables to describethe discrete-valued control actions, and discretizingthe model in time. The result is a hybrid model of lowcomplexity that is sufficiently accurate to serve as aprediction model for MPC and which is computation-ally tractable.The case study of a supermarket refrigeration systemillustrated the performance limitations of traditionalcontrol schemes. MPC in connection with the MLD

1 The small undershoot inTair for MPC, despite the soft con-straints, is mainly caused by the coarse sampling.

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100MPC

time [min]

Com

pres

sor

Cap

acity

[%]

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100Traditional Control

time [min]

Com

pres

sor

Cap

acity

[%]

Fig. 5. The compressor control input (comp) usingrespectively MPC and a traditional PI controller.

model proved to be better suited for handling the dis-crete control actions, taking into account the interac-tions between the display cases and the compressor,respecting the temperature constraints, minimizing thevariations in the suction pressure, and reducing theswitching in the compressor bank. This led to a lowerwear in the compressor and a higher energy efficiencyof the supermarket refrigeration system. Most im-portant, the design and tuning of the cost functionwas easy and intuitive, and the extension of the casestudy to larger refrigeration systems is expected tobe straightforward. Moreover, if needed, the explicitpiecewise affine state feedback controller might bederived by pre-solving the optimal control problemoffline for all states.

REFERENCES

Bemporad, A. and M. Morari (1999). Control of sys-tems integrating logic, dynamics and constraints.Automatica35(3), 407–427.

Bemporad, A., F. Borrelli and M. Morari (2000).Piecewise linear optimal controllers for hy-brid systems. In:Proc. American Control Conf..Chicago, IL. pp. 1190–1194.

Larsen, L. F. S. (2004a). Effects of synchronizationof hysteresis controllers in a supermarket refrig-eration system. Technical report. Danfoss A/S.www.control.auc.dk/~lfsl .

Larsen, L. F. S. (2004b). Modelling of supermarketrefrigeration system. Technical report. DanfossA/S. www.control.auc.dk/~lfsl .

Maciejowski, J. M. (2002).Predictive Control. Pren-tice Hall.

Torrisi, F. D. and A. Bemporad (2004). Hysdel — atool for generating computational hybrid modelsfor analysis and synthesis problems.IEEE Trans.Contr. Syst. Technol.12(2), 235–249.

Tyler, M. L. and M. Morari (1999). Propositional logicin control and monitoring problems.Automatica35(4), 565–582.