Hybrid model predictive control in supermarket refrigeration systems

Hybrid model predictive control in supermarket refrigeration systems
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  HYBRID MODEL PREDICTIVE CONTROL INSUPERMARKET REFRIGERATION SYSTEMSLars 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 controlproblemofsupermarketrefrigerationsystems.Thelatterarehybridsystemswithswitchednonlinear 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. INTRODUCTIONA 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 of the 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.Themajordrawback,however,isthatthecontrolloopsare 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, 2004 a ), meaning that the open-ing and closing actions of the valves coincide. Con-sequently, the compressor periodically has to work hard 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 bythe fact that many of the control inputs are restricted todiscretevalues,suchastheopening/closingoftheinletvalves 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 of the 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 DESCRIPTIONIn 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 self service.Asimplifiedsupermarketrefrigerationcircuitisshownin 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-frigerantevaporateswhileabsorbingheatfromthesur-rounding air circulating through the evaporator. Theresulting air flow creates an air-curtain at the front of the display case. The fact that the air-curtain is colderthan the goods leads to a heat transfer from the goods˙ Q goods − air   and - as a side effect - from the surrounding˙ Q airload   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 Liquid Manifold  Suction Manifold Suction line Suction line Condenser Unit Compressor Rack Fig. 1. A simplified layout of a supermarket refrigera-tion system. Air-curtainGoodsEvaporatorRefrigerant outRefrigerant inSuction lineInlet valve goods T  air  T  air goods Q −  airload  Q wall T  e T  e Q wallair  Q − 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 CONTROLThe 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 temperature  T  air   reaches acertain upper temperature bound the valve is openedand  T  air   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 thermalcapacityoftheevaporatorwallactsasalow-passfilter.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 lowflowof evaporatedre-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 synchronizationeffectsleadtofrequentcompressorswitchingscausinglarge fluctuations in the suction pressure and a highwear on the compressors.4. MODELLINGWe start by summarizing the model of the supermar-ket refrigeration system in continuous-time derivedin (Larsen, 2004 b ). 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 rack and 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 Cases The dynamics in the display case can be described bythree states, namely the goods’ temperature  T  goods , thetemperature of the evaporator wall  T  wall  and the massof liquified refrigerant in the evaporator  M  ref  . 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. dT  goods dt  =  −  UA goods − air   M  goods · Cp goods · ( T  goods − T  air  )  (1) dT  wall dt  =  UA air  − wall  M  wall  · Cp wall · ( T  air   − T  wall ) −  UA wall − ref   (  M  ref   )  M  wall  · Cp wall · ( T  wall  − T  e ) , (2)where  UA  is the heat transfer coefficient with thesubscript denoting the media between which the heatis transferred,  M   denotes the mass and  Cp  the heatcapacity, where the subscript denotes the media theparameter refers to. Moreover  T  e  is the evaporationtemperature.As indicated in Eq. (2), the heat transfer coeffi-cient between the refrigerant and the evaporator wall UA wall − ref   is a function of the mass  M  ref   of the liqui-fied refrigerant in the evaporator. It is assumed herethat this relation can be described by the followinglinear function: UA wall − ref   (  M  ref   ) = UA wall − refrig , max ·  M  ref   M  ref  , max ,  (3)where  M  ref   =  M  ref  , max , when the evaporator is com-pletely filled with liquid refrigerant.The accumulation of refrigerant in the evaporator isdescribed by: dM  ref  dt  =  0 if   valve  =  1 , − ˙ Q e ∆ h lg if   valve  =  0 and  M  ref   >  0 , 0 if   valve  =  0 and  M  ref   =  0 . (4)where  ∆ h lg  is the specific latent heat of the remain-ing liquified refrigerant in the evaporator, which is anonlinear function of the suction pressure. ˙ Q e  is thecooling capacity that can be found by setting up theenergy balance for the evaporator:˙ Q e  = ( T  wall  − T  e ) · UA wall − ref   (  M  ref   )  (5)The amount of liquid refrigerant in the evaporator  M  ref   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 (  M  ref   = 0), then ˙  M  ref   =  0.Finally,theairtemperature T  air   canbefoundbysettingup the energy balance for the air-curtain. T  air   =  ˙ Q airload   + T  goods · UA goods − air  + T  wall · UA air  − wall , (6)where ˙ Q airload   is a given external heat load on the air-curtain. The Suction Manifold  The dynamic in the suction manifold is modelled bytwo states, namely the density of the refrigerant in thesuction manifold  ρ suc  and the suction pressure  P suc .Setting up the mass and energy balance, and assumingan ideal gas, the corresponding state equations areobtained: d  ρ suc dt  =  ˙ m in − suc −  ˙ V  comp · ρ suc V  suc ,  (7) dP suc dt  =  Rh in − suc  ˙ m in − suc − h out  − suc  ˙ V  comp ρ suc C  v · V  suc ,  (8)where ˙ V  comp  is the volume flow out of the suctionmanifold determined by the compressor(s), ˙ m in − suc  = ∑ ni = 1  ˙ m i  is the total mass flow from the display casesto the suction manifold,  h in − suc  is the inlet enthalpyto the suction manifold which is a refrigerant specificnonlinear function of   P suc  and the superheat,  n  is thenumber of display cases, and  m i  is the mass flowof the refrigerant out of the  i ’th display case that isdeterminedbythepositionofthevalveandtheamountof enclosed refrigerant when the valve is closed. The Compressor Inmostrefrigerationsystems,thecompressorcapacityis discrete-valued, as compressors can be switched  only either on or off. Let  q  denote the total numberof compressors. The compressor bank is modelledusing a constant volumetric efficiency  η vol  and themaximal displacement volume  V  sl . Thus, the volumeflow ˙ V  comp , i  out of the suction manifold created by the i ’th compressor can be determined as follows˙ V  comp , i  =  comp i ·  1100  · η vol  · V  sl  i  =  1 ,.., q  (9)where  comp i  is the  i ’th compressor capacity, and˙ V  comp  =  ∑  pi = 1 ˙ V  comp , i  is the total volume flow. For amore elaborate model derivation, the reader is referredto (Larsen, 2004 b ). 4.2 MLD Framework  The general MLD form of a hybrid system introducedin (Bemporad and Morari, 1999) is  x ( k  + 1 ) =  Ax ( k  )+  B 1 u ( k  )+  B 2 δ  ( k  )+  B 3  z ( k  )  (10a)  y ( k  ) = Cx ( k  )+  D 1 u ( k  )+  D 2 δ  ( k  )+  D 3  z ( k  )  (10b)  E  2 δ  ( k  )+  E  3  z ( k  )  ≤  E  4  x ( k  )+  E  1 u ( k  )+  E  5 ,  (10c)where  k   ∈  N  is the discrete time-instant, and  x  ∈ R n c ×{ 0 , 1 } n  denotes the states,  u  ∈ R m c ×{ 0 , 1 } m  the inputs and  y  ∈  R  p c × { 0 , 1 }  p  the outputs, withboth continuous and binary components. Furthermore, δ   ∈{ 0 , 1 } r   and  z ∈ R r  c 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 filtering  T  air  . Therefore,  T  air  is a state in the simplified model rather than  M  ref  .Moreover, the density of the refrigerant in the suctionmanifold described by Eq.(7) is assumed constant.Severalnonlinearitiesarepresentinthemodel,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,2004 b ). 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 interval  T  samp  is difficult, as the switched dy-namic in the display case exhibit greatly differing timeconstants. Specifically, the decrease of   T  air   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 for  less  timethan 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 time  t  open  as a continuous-valuedcontrol input that allows one to vary the opening  time of the valve within the sampling interval. This be-havior is well approximated by introducing  t  open  asa variable that varies the opening  degree  of the valvethus avoiding one additional nonlinearity.Summing up, each display case has the three states  x disp  = [ T  air  , T  goods , T  wall ] , and the input  u disp  = [ d  valve , t  open ] , where  d  valve  ∈ { 0 , 1 }  describes whether thevalveisopenorclosed,and t  open  ∈ [ t  min , T  samp ] denotesthe opening time of the valve. The opening time isconstrained to be less than the sampling interval andlarger than some minimum opening time  t  min . 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 by  comp  ∈ { 0 ,  100 q  ,  2 · 100 q  ,..., 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 capacities  comp  ∈{ 0 , 50 , 100 } . The input vector is defined as  u  =[ d  valve , 1 , t  open , 1 , d  valve , 2 , t  open , 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 function  comp ( k  ) = comp ( k   −  1 ) + ∆ comp ( k  )  to the MLD model. Thesampling interval is  T  samp  =  60sec. Assuming that ittakes20secto ensurethat theevaporatoris completelyfilled after opening the valve, we set  t  open  ∈  [ 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 and 52 inequality constraints. The derivationof the MLD system is performed by the compilerH YSDEL  (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-sure  P suc  close to its reference value of 4 . 2bar while  fulfilling the soft constraints on the air temperaturesin the display cases  T  air   ∈  [ 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 pressure  P suc  from its reference, (ii) the switch-ing of the manipulated variables (the compressor andvalve control actions) and (iii) the violation of thesoft constraints on  T  air  . 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 a  Mixed-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 (Bemporad  et al. , 2000). Details about MPC canbe found in (Maciejowski, 2002). 5.3 Objective Function According to (Bemporad  et al. , 2000) and Section 5.1,the following optimal control problem is considered:min u ( 0 ) ,.., u (  N  − 1 )  J   =  N  − 1 ∑ k  = 0   P suc ( k  ) − 4 . 2  ∞ Q 1  +  u ( k  )  ∞ Q 2  +  p ·  N  − 1 ∑ k  = 1  S  ( k  )+ S  ( k  )  subject to the evolution of the MLD model (10) overthe prediction horizon  N   and taking into account thediscrete-valued nature of some of the manipulatedvariables ( d  valve , i  and ∆ comp ).The deviation of   P suc  from its reference is weightedby  Q 1 . To keep the energy consumption low, the vari-ations on  P suc  should be kept at a minimum. There-fore, a large weight is chosen, i.e.  Q 1  =  500. Theweight matrix on themanipulated variablesisgiven by Q 2  =  diag ( q 1 , q 2 , q 3 , q 4 , q 5 ) . 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 set  q 5  to 2. Thus, switchingparts of the compressor on or off costs 2 · 50  =  100,andthedeviationin P suc  hastoamountto  100500  = 0 . 2barbeforeachangeinthecompressorcapacityisinitiated.Assuming that it is 100 times less expensive (in termsof wear) to open or close the valves than to change thecompressor capacity, we set the weights on  d  valve , 1  and d  valve , 2  to  q 1  =  q 3  =  100100  =  1. The weights on  t  open  areset to  q 2  =  q 4  =  0 . 1.The soft constraints on the temperature bounds aretaken into account by introducing slack variables forthe upper  S   and the lower  S   boundsand a large penalty,e.g.  p  =  10 4 .6. SIMULATION RESULTSThis 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  [ T  lower  , T  upper  ] = [ 0 , 4 ]  anda PI-type suction pressure controller with a dead bandof   ± 0 . 2bar around the reference of 4.2bar. MPC usesa horizon of   N   =  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 only 20min, 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 temperatures 1 . 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, C PLEX  9 . 0 run on a Pentium IV 2 . 0 GHzcomputer. For  T  samp  =  1min and a horizon of   N   =  10,the computation time is in average 3 . 9sec and always
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