Rafael WISNIEWSKI,John LETH,Jakob G.RASMUSSEN

1.Department of Electronic Systems,Aalborg University,Fredrik Bajers Vej 7C,9220 Aalborg,Denmark;

2.Department of Mathematical Sciences,Aalborg University,Fredrik Bajers Vej 7G,9220 Aalborg,Denmark

Analysis of synchronization in a supermarket refrigeration system

Rafael WISNIEWSKI1?,John LETH1,Jakob G.RASMUSSEN2

1.Department of Electronic Systems,Aalborg University,Fredrik Bajers Vej 7C,9220 Aalborg,Denmark;

2.Department of Mathematical Sciences,Aalborg University,Fredrik Bajers Vej 7G,9220 Aalborg,Denmark

In a supermarket refrigeration,the temperature in a display case,surprisingly,influences the temperature in other display cases.This leads to a synchronous operation of all display cases,in which the expansion valves in the display cases turn on and off at exactly the same time.This behavior increases both the energy consumption and the wear of components.Besides this practical importance,from the theoretical point of view,synchronization,likewise stability,Zeno phenomenon,and chaos,is an interesting dynamical phenomenon.The study of synchronization in the supermarket refrigeration systems is the subject matter of this work.For this purpose,we model it as a hybrid system,for which synchronization corresponds to a periodic trajectory.To examine whether it is stable,we transform the hybrid system to a single dynamical system defined on a torus.Consequently,we apply a Poincar'e map to determine whether this periodic trajectory is asymptotically stable.To illustrate,this procedure is applied for a refrigeration system with two display-cases.

Synchronization;Hybrid systems;Stability;Limit cycles;Stochastic approximation

1 Introduction

The foodstuffs in supermarkets are typically stored in open display cases in the sales areas.By utilizing a refrigeration cycle,heat is transported from the display cases to the outdoor surroundings[1].The refrigeration cycle is coordinated by a number of dedicated controllers,which are distributed within the refrigeration hardware such as display cases,compressors,and cold storage rooms.This concept has many practical advantages;for instance,it is flexible and yet simple.However,it neglects the cross-coupling between the subsystems,and the interplay between continuous and discrete dynamics.These effects may cause the degradation of refrigeration quality.A case in point is the synchronization of display cases.Each display case is equipped with a hysteresis controller that opens and closes an expan-sion valve.It adjusts the flow of refrigerant such that the desired temperature is reached.Practical experience shows that the temperature in one display case influences the temperature in the neighboring ones.These interactions frequently lead to a synchronous operation of the display cases in which the expansion valves in the display cases turn on at the same time.As discussed in[1],this synchronization causes high wear of the compressors,inferior control performance,and increased energy consumption.

A number of inspiring publications have addressed the problem of synchronization in supermarket systems.In particular,references[2,3]suggest a centralized controller based on hybrid model predictive control that by design avoids synchronization.Recently,a patent has been issued that proposes to adjust the cut-in and cut-out temperatures for the refrigeration entities to desychronize them[4].

In this article,we leave the feedback design problem and focus entirely on the synchronization dynamics as a mathematically intricate phenomenon.We assume that the hysteresis control has been designed and ask whether synchronization takes place.Although,we do not consider any implementation issues at the current stage,we stress that the work is relevant for the reduction of energy consumption in supermarkets.Indeed,detecting a tendency to synchronize will eventually lessen the energy consumed.Our standpoint is that a deep insight into synchronization is important in general for understanding dynamical systems with discrete transitions.

The phenomenon of synchronization in dynamical systems has been studied before;the definitions of synchronizations have been formulated in[5],and numerous examples of synchronization have been analyzed in[6].As for studying any phenomena in dynamical systems,the very first challenge is to establish a convenient definition of a state space and the notion of a trajectory.What immediately follows is the examination whether the existing methods from the theory of dynamical systems can be adapted for the analysis of synchronization.This approach has been taken in this work.In the study of synchronization,we model the supermarket refrigeration system as a hybrid system.The hybrid system in this work consists of several linear subsystems and a rule that orchestrates the discrete transitions among them[7].The state space is a disjoint union of polyhedral sets,and the discrete transitions are realized by reset maps defined on the facets of the polyhedral sets.The reset maps are regarded as generators of an equivalence relation allowing 'gluing' the polyhedral sets together.The result of this construction is a quotient space.This idea has been used before in[8-10].The original contribution of this work is to show that by the process of gluing,a hybrid system is transformed to a dynamical system that is defined on a single state space,a smooth manifold with boundary.Ifnis the number of display cases,then this manifold is the product of a 2n-torus and the non-negative reals.Consequently,the trajectories are continuous,piecewise smooth paths.A novelty of the current approach is that this construction allows the application of standard method from analysis of differential equations for the study of a refrigeration system.In particular,(asymptotic)synchronization corresponds to an asymptotically stable periodic trajectory.

The stability analysis of periodic trajectories can be completed by applying a Poincar'e map.The Poincar'e map has previously been used for the stability analysis of switched systems[11-14]and control synthesis of these[15,16].It was assumed in these works that switchings took place on a family of hyperplanes that are disjoin subsets of the state space.Thereby,small perturbations of the state contribute to small changes in system behavior.Whereas in this work,we generalize the discrete transitions to take also place at intersection points of hyperplanes.This is an important generalization as at a corner point small perturbations yield conceptually different system behavior.As mentioned above,this phenomenon cannot be captured in the concepts presented in previous works.Allowing mode switching to take place on the boundaries of polyhedral sets complicates the Poincar'e map.In particular,we explicitly calculate the Poincar'e map for a refrigeration system consisting of two display cases and a compressor.

2 Refrigeration system

In majority of supermarket refrigeration systems,the display cases and the compressors,which maintain the flow of refrigerant,are connected in parallel.The compressors compress refrigerant drained from the suction manifold.Subsequently,the refrigerant passes through the condenser and flows into the liquid manifold.Each display case is equipped with an expansion valve,through which the refrigerant flows into the evaporator in the display case.In the evaporator,the refrig-erant absorbs heat from the foodstuffs.As a result,it changes its phase from liquid to gass.Finally,the vaporized refrigerant flows back into the suction manifold.The process described above is called a refrigeration cycle.

2.1 Motivation for the study

In a typical supermarket refrigeration system,the temperature in each display case is controlled by a hysteresis controller that opens the expansion valve when the air temperatureT(measured near to the foodstuffs)reaches a predefined upper temperature limitTu.The valve stays open untilTdecreases to the lower temperature limitTl.At this point,the controller closes the valve again.Practice reveals that if the display cases are similar,the hysteresis controllers have tendency to synchronize the display cases[1].It means that the air temperaturesTifori∈{1,...,N},whereNis the number of display cases,tend to match as time progresses.

2.2 Model for synchronization

For simplicity of this exposition,the model of a refrigeration system consists of two identical display cases and a compressor.

The dynamics of the air temperatureTifor display casei∈{1,2}and the suction pressurePfor the system of two display cases are governed by the following system of equations,which is further discussed in the appendix,

wherea,b,c,d,e,α and β are constants(their specific values are provided by equations(a3)in the appendix),and δi∈ {0,1}is the switching parameter for the display casei;it indicates whether the expansion valve is closed for δi=0 or open for δi=1.The switching law is given by the hysteresis control:

whereandare respectively the predefined upper and lower temperature limits for the display casei.By convention,δi=0 for any initial conditionof(1a)with[.Such an initial condition is assumed throughout this paper;hence,(2)is well defined.

We write equations(1)simply as

where

,andwith2={0,1}.Hence,for δ∈22,we study a Cauchy problem of the form

Proposition 1For anys∈ {l,u},the vector field ξδ,δ∈22,is transversal toand

ProofWith 〈.,.〉the standard inner product in R3andNi=(2-i,i-1,0),i=1,2,we find that

We note that the sign of(1b)determines whether the coordinate functionTiofxis increasing or decreasing.For two display cases,this information is provided in Fig.1.

Fig.1 The state space of the refrigeration system consisting of two display cases is illustrated.Here,the pressure axis is suppressed,andThe direction of the vector field ξδis indicated by the dark shaded triangles.

3 Supermarket systems as a hybrid system

In order to represent(2)in a mathematically satisfactory way,we will use the modeling formalism of hybrid systems on polyhedral sets with state-dependent switching.It is seen as a subclass of hybrid systems of[7];and yet,it is rich enough to model any system with multiple hysteresis control.

3.1 Hybrid systems

We writeF?PifFis a face of the polyhedral setP.A mapf:P→P′is polyhedral if

1)it is a continuous injection,and

2)for anyF?Pthere isF′?P′with dim(F)=dim(F′)such thatf(F)?F′.

Definition 1(Hybrid systems on polyhedral sets with state-dependent switching) For finite index setsJandD,a hybrid systems on polyhedral sets with state-dependent switching(of dimensionn)is a triple(P,S,R)=(PD,S,RJ),where

1)P={Pδ? Rn|Pδa polyhedral set,dim(Pδ)=n,δ∈D}is a family of polyhedral sets.

2)S={ξδ:Pδ→ Rn|Pδ∈P,δ∈D}is a family of smooth vector fields.

3)R={Rj:F→F′|F?P∈P,F′?P′∈P,dim(F)=dim(F′)=n-1,j∈J}is a family of polyhedral maps,called reset maps.

Next,we shall refer to hybrid systems with statedependent switching simply as hybrid systems.

Remark 1After identifyingDwith a finite subset of R,we can rewrite the hybrid system(PD,S,RJ)as

for(x,q)∈ˉC,and

for(x,q)∈ˉD,where

and

This is precisely the hybrid system in[7].

After this remark,we will show that a refrigeration system with two display cases furnished with a hysteresis control is a hybrid system.To begin with,we consider the following scenario.Letand δ=(0,0);thereby,both display cases are initially switched off.Suppose that at timet,the air temperatureTiof theith display case reaches the upper temperature limitTui,then theith display case is switched on,and δi=1.This scenario indicates that the refrigeration system comprises four dynamical systemsdefined on the polyhedral set

A discrete transition between these four systems takes place whenever a trajectory reaches the boundary ofQ.The polyhedral setQhas five facets;four of them will be instrumental in the sequel,

where α ∈2and δ0[a,b]={a},δ1[a,b]=.

To sum up,the set P consists of four copies of a polyhedral setQin(5)

Formally,in(6),we have separated(made disjoint)each of the copies ofQ.

The set S consists of four dynamical systems given by(3)for δ∈22.To characterize the set R,we define

where the results of the summation are computed modulo 2.Intuitively,the mapltakes a polyhedral set enumerated by δ to the future polyhedral set.The variableiindicates that the discrete transition takes place when the temperatureTireaches its upper or lower boundary.The set R consists of eight reset maps R={Ri(δ)|(i,δ) ∈ {1,2}X22},where the mapsare defined byObserve that the reset maps are identities in the first argument.By abuse of notation,we frequently identify(x,δ)withx.

Remark 2For a supermarket refrigeration system withNdisplay cases,the definition of the facets on the polyhedral setis generalized tofori∈ {1,...,N}and α ∈2.The facet operators commute in the following sense

As a consequence,the set R of the reset maps is

For the maplgiven by

the mapsRi(δ):are defined by

The hybrid refrigeration system with two display cases is illustrated in Fig.2.Here,each element ofPhas been(orthogonally)projected onto the(T1,T2)-space.Hence,the polyhedral setsPδare represented by cubes.The three cubesP(0,1),P(1,0),P(1,1)have been vertically and/or horizontally reflected(compare with Fig.1).The stippled lines in the drawing indicate the reset maps in R.

Fig.2 The T1T2-state space of the refrigeration system consists of two display cases.The reset maps are indicated by the stippled lines(see Fig.1 and its caption for further explanation).The pressure axis has been suppressed;thus,each Pδ =QX{δ}is illustrated by a square.By abuse of notation,the facets of Pδare denoted by(instead of

3.2 Trajectories of a refrigeration system

We bring in a concept of a(hybrid)time domain[7].Letk∈N∪{∞};a subset Tk?R+XZ+will be called a time domain if there exists an increasing sequencein R+∪ {∞}such that

where,and

Note thatTi=[ti-1,ti]for alliifk=∞.We say that the time domain is infinite ifk=∞ortk=∞.The sequencecorresponding to a time domain will be called a switching sequence.

Definition 2(Trajectory) A trajectory of the hybrid system(PD,S,RJ)is a pair(Tk,γ)wherek∈ N∪{∞}is fixed,and

?Tk? R+XZ+is a time domain with corresponding switching sequence{ti}i∈{0,...,k},

1)For eachi∈ {1,...,k-1},there exist δ ≠ δ′∈Dsuch that γ(ti;i) ∈ bd(Pδ),and γ(ti;i+1) ∈ bd(Pδ′),where bd(P)is the boundary ofP.

3)For eachi∈{1,...,k-1},there existsj∈Jsuch that

A trajectory atxis a trajectory(Tk,γ)with γ(t0;1)=x.

The next definition formalizes the notion of a periodic trajectory,which will be used in defining synchronization of the refrigeration system.

Definition 3((T,l)-periodic trajectory) Let(T,l)∈R+XZ+.A trajectory(Tk,γ)is(T,l)-periodic(or just periodic)if 1)Tkis an infinite time domain,and 2)for anyi∈{1,...,k}andt∈p(Tk),wherep:Tk→ [t0,∞[is the projectionp(t,i)=t,we have γ(t+T;i+l)= γ(t;i).

In particular,if(Tk,γ)is a(T,l)-periodic trajectory,andTis nonzero thenp:Tk→ [t0,∞[is surjective.

3.3 State space of the refrigeration system

To study any dynamical system,the starting point is a convenient definition of the state space.It was suggested in[9,10]to glue the state spaces of respective subsystems of a hybrid system together along the subsets identified by the reset maps.

Let R-1={R-1|R∈R}.Then,the equivalence class ofx∈Xis denoted by

In particular,for a refrigeration system with two display cases,letx=(T1,T2,P)∈X.Ifxis in the interior ofPδfor some δ ∈22,the equivalence class[x]={(T1,T2,P;δ)}.Ifxis in the interior ofthen[x]={(T1,T2,P;0,0),(T1,T2,P;1,0)},and if{(0,0)},we have[x]={(T1,T2,P;0,0),(T1,T2,P;1,0),(T1,T2,P;0,1),(T1,T2,P;1,1)}.Furthermore,X?is the product of a 2-torus with the non-negative reals,X?=T2XR+,i.e.,a smooth manifold with boundary.We note that the system consisting ofNdisplay cases will give rise toX?=TNXR+.In[18],we have explicitly constructed a differentiable structure on the state spaceX?of a system withNhystereses,thereby of a refrigeration system withNdisplay cases.Whereas,in[19],we have studied local stability of such a system.

4 Periodic trajectory and stability

We direct our attention to the subject matter-the synchronization of refrigeration systems.A refrigeration system is said to exhibit asymptotic synchronization if there exists a(T,l)-periodic trajectory which is asymptotically stable inX?[20,Definition 13.3].We remark that asymptotic synchronization is described in a more general setting in[5],and that the above definition is local as only trajectories sufficiently close to the periodic trajectory are considered.Using the Poincar'e map[20,Theorem 13.1],we prove the following theorem.

Theorem 1The refrigeration system with two display cases exhibits asymptotic synchronization.

Theorem 1 is a direct consequence of Lemmas1and 2 below.

Lemma 1There exist initial temperatures on the diagonal,and initial pressure which give rise to a(Tp,2)-periodic trajectory.

ProofFori∈N,the analytic expression for a trajectory

atof the system(3)is

forj=1,2,where

with all constants given in the appendix.We observe from equation(8)that the projection of a trajectory on theT1T2-plane is(Tp,2)-periodic if its initial conditionx0is of the formwithand initial δ is(0,0).As a consequence,the trajectory is on the diagonalT1=T2of the polyhedral setsP(0,0)andP(1,1).Using the aboveT1T2-plan analysis as a guideline,we find an initial conditionof a(Tp,2)-periodic trajectory in the state spaceX.Indeed,by choosing,the initial pressureˉP0is determined by solving,forandt2,the system of equations

With the notation as in the proof above,letand(T∞,γˉx0)denote the(Tp,2)-periodic trajectory atˉx0;thus,

Lemma 2The(Tp,2)-periodic trajectory atis asymptotically stable inX?.

ProofWe remark that stability of a periodic trajectory can be determined by a Poincar'e map.Indeed,asymptotic stability of a periodic trajectory(inX?)is equivalent to asymptotic stability of the fixed point of a corresponding Poincar'e map[20,Theorem 13.1].

Next,we describe the Poincar'e map.For this purpose,the evolution of a trajectory starting at a point nearby the(Tp,2)-periodic trajectory γˉx0is outlined.

Letx0denote any of the two pointsfor some ?>?′>0.Thus,whereis the max-norm.We show that for sufficiently small ?,wherex1∈P(0,0)is the point at which the trajectory γx0atx0meets the hyperplaneT1=0(inP(0,0))for the first time.From this,we will conclude that this Poincar'e map is a contraction.

Forx=(x′;δ)andy=(y′;δ′),both inX,we writex?y(resp.x?y)wheneverFurthermore,letBelow,we describe one period of the trajectory γx0in the following four steps.

1)From the choice of initial conditionx0,and from(8),it follows thafor someandforNote thatfor

3)Since γx0(t1;3)∈P(1,1),we conclude that there exists asuch thatandforNote thatfo

Having described one period of the trajectory γx0,we are now ready to prove thatSince δ=(1,1)andwe conclude by(8)thatFrom symmetry of the systems(3)for δ =(0,1)and δ =(1,0),itfollowsthatMoreover,since γx0(.;1)and γˉx0(.;1)are solutions to the stable affine linear system(3),with δ=(0,0),starting(t=t0)at a distancewe conclude thatTogether with item 1)and 2),this implies thathence,Therefore,we only need to show thatwhich follows by straightforward computations involving the explicit expression(7)ofP.The case withis conceptually the same as above and is,therefore,left to the reader.

As shown in Section 3.3,the dynamics of the refrigeration system carries over to the manifoldX?;thus,the above procedure defines a standard Poincar'e map.Using the fact that the periodic trajectory can be covered by charts,the results in[20]applies.The Poincar'e map takes a pointx0with|x0-ˉx0|m=? to the pointx1with|x1-ˉx0|m

We numerically investigated a subset B of the basin of attraction of the stable limit cycle corresponding to synchronization that contains the pointˉx0.We determined that

5 Conclusions

In this paper,we have studied the synchronization phenomenon in supermarket refrigeration systems.We have associated synchronization with a periodic trajectory in a hybrid system.To determine whether asymptotic synchronization occurs,i.e.,the periodic trajectory is asymptotically stable,we have used a Poincar'e map.This approach has been carried out for a refrigeration system with two display cases.

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23 May 2013;revised 29 November 2013;accepted 29 November 2013

DOI10.1007/s11768-014-0077-2

?Corresponding author.

E-mail:raf@es.aau.dk.Tel.:+45 9940 8762.

This work was supported by the Danish Council for Technology and Innovation.

?2014 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Appendix

Model of refrigeration systemThe mathematical model presented here is a summary of the model developed in[21].For theith display case,dynamics of the air temperatureTair,ican be formulated as

where the process parameters are specified in Table a1,and δi∈ {0,1}is the switch parameter for theith display case.When δi=0,theith expansion valve is switched off,whereas when δi=1 it is switched on.The suction manifold dynamics is governed by the differential equation

whereNis the number of display cases,andfor

We denoteTi=Tair,iandP=Psucand write the dynamics of the air temperature and suction pressure in the concise form(with the process constants in(a1a)collected ina,b,c,d,e,α,β and then replaced by their numerical values)

Table a1 Parameters for a simplified supermarket refrigeration system.

Rafael WISNIEWSKIis a professor in the Section of Automation&Control,Department of Electronic Systems,Aalborg University.He receives his Ph.D.in Electrical Engineering in 1997,and Ph.D.in Mathematics in 2005.In 2007-2008,he was a control specialist at Danfoss A/S.His research interestis in system theory,particularly in hybrid systems.E-mail:raf@es.aau.dk.

John LETHreceived his M.S(2003)and Ph.D.(2007)degrees from the Department of Mathematical Sciences,Aalborg University,Denmark.Currently,he is employed as Assistant Professor at the Department of Electronic Systems,Aalborg University.His research interests include mathematical control theory and(stochastic)hybrid systems.E-mail:jjl@es.aau.dk.

Jakob Gulddahl RASMUSSENis an associate professor at the Department of Mathematical Sciences,Aalborg University,Denmark.He received his Ph.D.in Statistics in 2006 also at the Department of Mathematical Sciences.His research interests include spatial statistics and stochastic processes(including stochastic hybrid systems).E-mail:jgr@math.aau.dk.