Aalborg Universitet. Non-linear and adaptive control of a refrigeration system Rasmussen, Henrik; Larsen, Lars F. S.

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1 Aalborg Univrsitt Non-linar and adaptiv control of a rfrigration systm Rasmussn, Hnrik; Larsn, Lars F. S. Publishd in: IET Control Thory & Applications DOI (link to publication from Publishr):.9/it-cta.9.56 Publication dat: Documnt Vrsion Early vrsion, also known as pr-print Link to publication from Aalborg Univrsity Citation for publishd vrsion (APA): Rasmussn, H., & Larsn, L. F. S. (). Non-linar and adaptiv control of a rfrigration systm. IET Control Thory & Applications, 5(), DOI:.9/it-cta.9.56 Gnral rights Copyright and moral rights for th publications mad accssibl in th public portal ar rtaind by th authors and/or othr copyright ownrs and it is a condition of accssing publications that usrs rcognis and abid by th lgal rquirmnts associatd with ths rights.? Usrs may download and print on copy of any publication from th public portal for th purpos of privat study or rsarch.? You may not furthr distribut th matrial or us it for any profit-making activity or commrcial gain? You may frly distribut th URL idntifying th publication in th public portal? Tak down policy If you bliv that this documnt brachs copyright plas contact us at vbn@aub.aau.dk providing dtails, and w will rmov accss to th work immdiatly and invstigat your claim. Downloadd from vbn.aau.dk on: april, 8

2 Nonlinar and adaptiv control of a rfrigration systm Hnrik Rasmussn a, Lars F. S. Larsn b a Dpartmnt of Elctronic Systms, Aalborg Univrsity, Dnmark (hr@s.aau.dk) b Danfoss A/S, Nordborg, Dnmark (Lars.Larsn@Danfoss.com) Abstract In a rfrigration procss hat is absorbd in an vaporator by vaporating a flow of liquid rfrigrant at low prssur and tmpratur. Controlling th vaporator inlt valv and th comprssor in such a way that a high dgr of liquid filling in th vaporator is obtaind at all comprssor capacitis nsurs a high nrgy fficincy. Th lvl of liquid filling is indirctly masurd by th suprhat. Introduction of variabl spd comprssors and lctronic xpansion valvs nabls th us of mor sophisticatd control algorithms, giving a highr dgr of prformanc and just as important ar capabl of adapting to varity of systms. This papr proposs a novl mthod for suprhat and capacity control of rfrigration systms; namly by controlling th suprhat by th comprssor spd and capacity by th rfrigrant flow. A nw low ordr nonlinar modl of th vaporator is dvlopd and usd in a backstpping dsign of a nonlinar adaptiv controllr. Th stability of th proposd mthod is validatd thortically by Lyapunov analysis and xprimntal rsults show th prformanc of th systm for a wid rang of oprating points. Th mthod is compard to a convntional mthod basd on a thrmostatic suprhat controllr. NOMENCLATURE p L l ṁ h i h g h o h lg T f comp f min T SH T w ṁ w c w c p, α B H γ. Introduction tim drivativ oprator d/dt lngth of th vaporator lngth of th vaporator two phas sction rfrigrant mass flow rat spcific nthalpy, inlt vaporator spcific nthalpy, nd of two phas sction vaporator spcific nthalpy, outlt vaporator spcific vaporation nrgy, rfrigrant rfrigrant boiling tmpratur rfrigrant prssur, vaporator comprssor spd minimum comprssor spd f min = 5Hz suprhat, vaporator tmpratur of watr into th vaporator mass flow of watr spcific hat capacity of watr constant prssur spcific hat of rfrigrant hat transfr cofficint rfrigrant-watr width of vaporator hight of vaporator void fraction Rfrigration systms ar widly usd as wll in applications for privat consumrs as for th industry. Dspit diffrncs in siz and numbr of componnts, th main construction with an xpansion valv, an vaporator, a comprssor and a condnsr, rmains to a considrabl xtnt th sam. Larg parts of th sam tchnological challngs ar thrfor ncountrd in both markts. In this papr w focus on a small watr chillr, howvr th gnrality of th rsults applis to a largr family of so-calld : systms, i.. systm with vaporator and comprssor. Rfrigration and air conditioning, accounts for a hug part of th total global nrgy consumption, hnc improving nrgy fficincy in ths systm can potntially lad to a trmndous rductions in th nrgy consumption. Optimizing th st-points of ths systms has bn provd to nabl a substantial rduction in th powr consumption as shown in []. In [] a mthod for on-lin optimization of th st-points to minimiz powr consumption is prsntd. In a rfrigration systm on of th ky variabls to control, which gratly affcts th fficincy of th systm, is th suprhat. Th suprhat is usd as an indirct masur of th liquid fraction of rfrigrant in th vaporator. To utiliz th potntial of th vaporator to its maximum, th suprhat should b kpt as low as possibl, i.. th liquid fraction should b as high as possibl. Th suprhat is traditionally controlld by adjusting th opning dgr of th xpansion valv. This is a common control stratgy and xampls can b found in.g. [] and []. Mchanical thrmostatic xpansions valvs (TXV) is currntly th prfrrd choic as xpansion dvic in numrous applications. TXV s ar rlativly inxpnsiv and dlivr a good control prformanc if dsignd and sizd corrctly. Dsigning and sizing TXV s is not always straight forward and onc installd, th possibility of adjusting it to fit th spcific application, is rathr limitd. Furthrmor rgarding production it rquirs many diffrntiatd vrsions to fit th various applications. Ths shortcomings hav opnd for th introduction of lctronic valvs, which nabl th us of mor sophisticatd control algorithms that potntially can ovrcom ths difficultis. Controlling th suprhat using standard SISO PID control, howvr oftn lads to poor prformanc, causd by mainly two major challngs. Firstly; th suprhat is strongly coupld with th opration of th comprssor. Nglcting this oftn lads to instability or th so-calld hunting phnomna, s [5]. Scondly; th fact that th suprhat acts highly nonlinar, dpnding on th point of opration and th vaporator dsign, limits th obtainabl prformanc with standard PID controllrs. Prprint submittd to IET Control Thory & Applications Octobr, 9

3 Prvious works by [6] and [7] hav provd that gain schduling is a way to handl gain variations. In [8] a nw promising modl basd control dsigns that tak th cross couplings btwn th (uncontrollabl) comprssor and th valv into account, has startd to mrg. By th introduction of variabl spd comprssors, an additional control variabl that can b activly usd has bn introducd. [9] prsnts a nw non-linar control stratgy whr th comprssor is controlling th suprhat and th valv is controlling cooling capacity. Rcntly this rsult has bn improvd in [], whr a nw non-linar control stratgy using a backstpping mthod basd on Lyapunov thory is applid for improving stability. Ths tchniqus dfinitly show an improvd prformanc Howvr thy rly on a dtail knowldg of spcific systm paramtrs, which ar typically not availabl for a larg part of th applications. Furthrmor th focus on limiting th us of rfrigrants (grnhous gass) and incrasing prizs on raw matrial hav drivn th introduction of nw vaporator dsigns on a markt, that is charactrizd by a low intrnal volum. Exampls of such vaporators ar micro channl and plat hat xchangrs. Du to th low intrnal volum and thrby fastr dynamics, ths vaporators add to th abov mntiond control challngs. To accommodat th control challngs introducd by ths vaporator typs and th rquirmnts for adaptation this papr furthr dvlops th rsult prsntd in [] with an adaptation routin to stimat unknown systm spcific paramtrs. With th nw controllr it is possibl to mak continuous control down to zro cooling powr. Bcaus th backstpping dsign is basd on Lyapunov stability, th stability of th control and th adaptation can b guarantd. By using this approach a narly prfct dcoupling btwn capacity and suprhat tmpratur, for rasonabl choic of gains in th controllr, can b obtaind. Exprimnts on a tst systm show an xcllnt prformanc during startup as wll as for variation of cooling capacity by stp chang of th comprssor spd btwn minimum and maximum. Th nw controllr is also compard to a convntional controllr basd on a thrmostatic xpansion valv (TXV) for controlling of th suprhat.. Systm dscription Rfrigration systms typically us a vapor-comprssion cycl procss to transfr hat from a cold rsrvoir (.g. a cold storag room) to a hot rsrvoir, normally th surroundings. Th main ida is to lt a rfrigrant circulat btwn two hat xchangrs, i.. an vaporator and a condnsr. In th vaporator th rfrigrant absorbs hat from th cold rsrvoir by vaporation and rjcts it in th condnsr to th hot rsrvoir by condnsation. In ordr to stablish th rquird hat transfr, th vaporation tmpratur (T ) has to b lowr than th tmpratur in th cold rsrvoir (T cr ) and th condnsation tmpratur (T c ) has to b highr than th tmpratur in th hot rsrvoir (normally th surroundings T a ), i.. T < T cr and T c > T a. Th rfrigrant has th proprty (along with othr pur fluids and gass) that th saturation tmpratur (T sat ) uniquly dpnds on th prssur. At low prssur th corrsponding saturation tmpratur is low and vic vrsa at high prssur. This proprty is xploitd in th rfrigration cycl to obtain a low tmpratur in th vaporator and a high tmpratur in th condnsr simply by controlling rspctivly th vaporating prssur ( ) and th condnsing prssur (P c ). Btwn th vaporator and th condnsr is a comprssor. Th comprssor comprsss th low prssur rfrigrant ( ) from th outlt of th vaporator to a high prssur (P c ) at th inlt of th condnsr, hrby circulating th rfrigrant btwn th vaporator and th condnsr. To uphold th prssur diffrnc (P c > ) an xpansion valv is installd at th outlt of th condnsr. Th xpansion valv is basically an adjustabl nozzl that hlps upholding a prssur diffrnc. Th tst systm fig. is a simpl rfrigration systm with watr circulating through th vaporator. Th vaporator is a plat hat xchangr, i.. an vaporator typ with a low intrnal volum. Th hat load on th systm is maintaind by an lctrical watr hatr with an adjustabl powr supply for th hating lmnt. Th comprssor, th vaporator fan and th condnsr pump ar quippd with variabl spd drivs so that th rotational spd can b adjustd continuously. Th systm is furthrmor quippd with an lctronic xpansion valv that nabls a continuous variabl opning dgr. Th systm has tmpratur and prssur snsors on ach sid of th componnts in th rfrigration cycl. Mass flow mtrs masur th mass flow rats of rfrigrant in th rfrigration cycl and watr on th scondary sid of th vaporator. Tmpratur snsors masur th inlt and outlt tmpratur of th scondary mdia on rspctivly th vaporator and th condnsr. Th applid powr to th condnsr fan and th comprssor is masurd. Finally th ntir tst systm is locatd in a climat controlld room, such that th ambint tmpratur can b rgulatd. For data acquisition and control th XPC toolbox for SIMULINK is usd.. Modling and vrification.. Modl ovrviw A dtaild modl for an vaporator is basd on th consrvation quations of mass, momntum and nrgy on th rfrigrant, air and tub wall. This lads to a numrical solution of a st of diffrntial quations discrtizd into a finit diffrnc form, s []. This modl givs vry dtaild information to th control dsignr comparabl to th ral systm. This mans that it is usful for tsting controllrs, but du to th high complxity not for dsign of nw control principls. A simplr modl may b obtaind by using a so calld moving boundary modl for th tim dpndnt two phas flows and by assuming that spatial variations in prssur ar ngligibl, which mans that th momntum quation is no longr ncssary. Th numrical solution may dscrib th systm quit wll and rsults ar shown in [] and []. Th moving boundary modl is vry gnral and may b fittd to most vaporator typs. By simplifying th moving boundary modl furthr a vry simpl nonlinar modl dscribing th dominating tim constant and th nonlinar bhavior btwn input and output is

4 obtaind. Th gain and tim constant variations as a function of th inputs and disturbancs ar xprssd analytically. Following approximations mad ar fluid flow is on-dimnsional spatial variations in prssur ar ngligibl axial conduction is ngligibl cross sctional ara of flow stram is constant th hat transfr cofficint from watr to wall is small compard to th hat transfr cofficint from wall to boiling rfrigrant th nrgy for supr hating th gas is ngligibl compard to th nrgy for vaporating th rfrigrant th hat capacity of th wall btwn watr and rfrigrant is considrd to b ngligibl... Enrgy and mass balanc two phas sction Th mass and nrgy of th two phas sction ar givn by M (t) U (t) = (ρ l ( γ ) + ρ g γ )BHl (t) = (ρ l ( γ )h l + ρ g γ h g )BHl (t) whr it is assumd that th work associatd with th rat of chang of prssur with rspct to tim is ngligibl. From () th following rlation is obtaind () U h g M = ρ l ( γ )(h g h l )BHl () Insrtion of () in () thn givs d(ρ l ( γ ) + ρ g γ )BHl d d dt = ṁ ṁ comp (7) Assuming th liquid to b incomprssibl (7) bcoms with κ = dρ g d... Suprhat sction BHl κ d dt = ṁ ṁ comp (8) If th axial conduction is ngligibl and th hat capacity of th watr c w ṁ watr >> c p, ṁ th suprhat T SH bcoms [ { }] T SH = (T w T ) xp α B(L l )) c p, ṁ (9).. Comprssor Th piston comprssor modl is dvlopd from factory givn data as ṁ comp = α c f comp () whr α c is a function of and P c. Assuming P c = P c,r f du to control of th condnsr fan th variation of α c is only causd by variation of. In th working ara for th systm this variation is lss than 5% and α c is considrd as a constant. Equ. () in (8) thn givs If it is furthr assumd that void fraction γ is constant indpndnt of l, and variation of h g and h l du to prssur variation is nglctd, th following rlation is obtaind U h g Ṁ = ρ l ( γ )(h g h l )BH dl dt Th mass and nrgy balanc is givn by Ṁ = ṁ ṁ comp U = h i ṁ h g ṁ comp + α Bl (T w T ) Combining () and () thn givs ρ l ( γ )(h g h l )BH dl = (h g h i )ṁ α Bl (T w T ) (5) dt Th first trm on th right sid corrsponds to th nrgy diffrnc btwn th rfrigrant laving and ntring th two phas sction of th vaporator. Th scond trm is th rat of th hat transfr from watr to rfrigrant. Th lft sid dscribs th chang of nrgy of th two phas sction. From rfrigrant data [] w hav h g = HDwP( ) h i = HBubP(P c ) h l = HBubP( ) T = T DwP( ) ρ g = VDwP( ) ρ l = VBubP( ) () () (6).5. Combind modl BHl κ d ṁ = + () α c f comp dt α c f comp T = T DwP( ) c ẋ = (h g h i )ṁ c (T w T )x f min ṁ c Ṗ = + f comp α c f comp T SH = (T w T ) [ xp { }] x x δ with: a) c = ρ l ( γ )(h g h l )BH b) c = BHl κ/(α c f min ) b) c = α BL c) x δ = c p, ṁ /(α BL ) d) x = l /L.6. Control input and masurmnt () Th control inputs ar f comp and ṁ and th masurd valus ar T SH, and T w. From ths masurmnts th rlativ lngth x of th two phas sction is obtaind by x,mass = x δ log T w T T w T T SH ()

5 .7. Modl vrification Th modl paramtrs to b stimatd ar (c, c ) and θ = (α c, x δ, c ). A sris of xprimnts giving larg signal xcitation of th systm for diffrnt working conditions ar prformd. Simulation using th modl () with th sam input (ṁ ν, f comp,r f ) as usd in th xprimnt thn givs th output (, T SH ). Th constants c and c ar first found by visual fitting of simulatd and masurd valus of th output. Using ths valus for all xprimnts θ may now b dtrmind by minimizing th prformanc function J(θ) = t t t {K t (,mass ) + (T SH T SH,mass ) () }dt Th paramtr K dtrmins th wight btwn squard valus of th variation of (,mass ) and (T SH T SH,mass ). A valu K >> givs a valu of α c rsulting in th bst fit to th prssur quation, but bcaus th modl only is an approximation th influnc of th othr paramtrs ar hiddn in nois. If K << th opposit is th cas. A rasonabl wight btwn variation of and T SH with rspct to modl rrors sms from many xprimnts with diffrnt K to b th valu K = 5. Th rsult is shown in tabl Simulatd and masurd valus for xprimnt and ar shown in fig. () and (). It is sn that th modl givs a good dscription of th dominating dynamics of th systm whn optimizd valus ar usd. Fig. (5) shows th simulatd output using th stimatd man valus. Th dynamics ar again wll dscribd but DC valus ar badly modld. This mans that th DC valu problm nds a spcial tratmnt.. Control objctivs and challngs Th main focus of this papr is to driv a nw control schm that improvs th control prformanc and th nrgy fficincy compard to xisting control schms. This is mainly obtaind by utilizing a gnric modl basd control. A furthr advantag, by utilizing simpl and gnric modls is that th scalability of th controllrs ar maintaind. This is an important aspct for mass producd applications lik rsidntial air conditioning systms bcaus it nabls th control schm to work on a divrsity of systm compositions and sizs, with only minor changs. Bing mor spcific following objctivs should b fulfilld by th control: Maintain a constant low suprhat Maintain stabl opration undr varying oprational conditions Ability to rspond fast to changs in th rqustd cooling powr Fast sttling tim aftr startup Ability to supprss disturbancs and follow load variations Th main control challngs in th systms ar as prviously dscribd th cross couplings and th nonlinaritis. Furthrmor saturations of th actuators i.. comprssor and valvs also complicat th control dsign. If a variabl spd driv for th comprssor is usd th comprssor capacity can continuously b opratd btwn a maximal and a minimal spd. At low spd th lubrication of th comprssor stops working, hnc limiting th minimally allowabl spd th comprssor can b opratd at. This causs a discontinuity in th lowr nd of th oprating rang of th comprssor capacity complicating th control dsign, for xmplification s [5]. 5. Nw control mthods Th stady stat valu of th prssur givn by th modl f min ṁ c Ṗ = + (5) f comp α c f comp is proportional to ṁ /α c. In th modl vrification sction th uncrtainty of α c was shown. Th rfrigrant flow ṁ was masurd, but in a practical control schm an stimat of ṁ has to b usd. This mans that th gain ṁ /α c may hav an rror up to % of th bst guss. Bcaus th masurd prssur is of good quality a way to ovrcom this problm is to control th prssur by an PI-controllr. Th controllr u = f comp = α c + c τ ṁ p f min f comp p u for u min < u < u max (,r f ) with u min > givs th closd loop for th prssur (6) τ Ṗ = +,r f (7) Th gain of th PI-controllr (6) is givn by c f min α c f comp τ ṁ whr τ is th spcifid closd loop tim constant. Th gain is sn to b schduld with th comprssor frquncy and rfrigrant mass flow. If th mass flow is not masurd an stimat basd on () may b usd. Th rsulting cascadd structur shown in fig. (6) thn givs th following modl for th rlativ filling x and th suprhat tmpratur T = T DwP( ) (a) c ẋ = (h g h i )ṁ c (T w T )x (b) τ Ṗ = +,r f (c) T SH = (T w T ) [ xp { }] x x δ (d) (8) Th variation in th gain ṁ /α c in (5) thn only influnc th tim constant τ in (8.c). Th output nonlinarity (8.d) is compnsatd for by using th invrs nonlinarity in th masurmnt of x () Th multiplication of th two stats x and

6 Tabl : Exprimnts for modl vrification Exprimnt c c α c x δ c J. f comp = and. < ṁ < f comp = 5 and.6 < ṁ < f comp = 6 and.9 < ṁ < ṁ =. and 5 < f comp < ṁ =.8 and 5 < f comp < random random random man valus T call for a nonlinar dsign mthod if th controllr has to b b valid in th hol working ara. Th rfrnc valu for th prssur may b calculatd basd on th rfrnc of th vaporating tmpratur T r f by,r f = PDwT(T r f ) (9) Bcaus th rlation btwn T and is narly linar in a larg oprating intrval (7) is quivalnt to τ Ṫ = T + T r f () Th following bilinar modl for th rlativ filling x and th vaporation tmpratur T may thn b usd for th controllr dsign c ẋ = (h g h i )ṁ c x (T w T ) () τ Ṫ = T + T r f In () x has to b controlld to a valu x by T r f.ift was th control input thn for constant x c p(x x ) = k (x x ) () is obtaind by th valu T calculatd by () c x (T w T ) = (h g h i )ṁ + k (x x ) () Th valu for th tuning paramtr k is found by th Lyapunov analysis. Insrtion of () in () givs c p(x x ) = (k + c (T w T ))(x x ) + c x (T T ) τ p(t T ) = (T T ) + T r f T τ pt For k > th Lyapunov function candidat () P = c (x x ) + τ k (T T ) (5) is positiv dfinit and has th tim drivativ Ṗ = (k + c (T w T ))(x x ) k (T T ) +(T T )k (T r f ( + τ p)t + c x (6) (x x k )) 5 For a control input T r f givn by T r f = ( + τ p)t c x (x x k ) (7) th tim drivativ of th Lyapunov function bcoms Ṗ = (k + c (T w T ))(x x ) k (T T ) (8) This function is ngativ dfinit for positiv k +c (T w T ) > and k >, lading to a stabl closd loop systm. Th nw backstpping controllr T = T w ( (hg h c x i )ṁ + k (x x )) (a) T ff = ( + τ p)t (a) T r f = T ff + c x k (x x ) (b),r f = PDwT(T r f ) (b) (9) f +c min fcomp p τ ṁ u = α c p (,r f ) (c) u = sat(u, u min, u max ) (c) f comp = u (c) Th structur of th dvlopd backstpping controllr (9) is shown in fig. 7 and is tstd on a simulation modl basd on stimatd man valu modl paramtrs. Th rsult is shown in fig. 8 for th following controllr paramtrs τ = k = k = x =.9 () It is sn that th variation in x causd by th variation in m is small du to th small tim constant τ for th prssur controllr. In th controllr c is assumd known lading to a stady stat m qual to th rfrnc. Fig. 9 shows th simulatd output if c is changd during th simulation. Th figur shows th nd for an adaptation of th c valu. 6. Adaptiv backstpping control Adaptation of c is basd on th following modification of th Lyapunov function (5) P = c (x x ) + τ k (T T ) + γ (c ĉ ) ()

7 whr ĉ is th stimat of c. P is positiv dfinit for k > and γ>. Dfining T by ĉ x (T w T ) = (h g h i )ṁ + k (x x ) () lads for constant x to th quation c p(x x ) = (k + ĉ (T w T ))(x x ) + ĉ x (T T ) (c ĉ )x (T w T ) τ p(t T ) = (T T ) + T r f T τ pt Th tim drivativ of P thn bcoms Ṗ = (k + ĉ (T w T ))(x x ) k (T T ) +(T T )k (T r f ( + τ p)t + ĉx (x x k )) (c ĉ ) ( (x x )x (T w T ) + γ (pĉ ) ) For a control input T r f givn by () () T r f = ( + τ p)t ĉx (x x k ) (5) and th adaptation law (pĉ ) = γ(x x )x (T w T ) (6) th tim drivativ of th Lyapunov function bcoms Ṗ = (k + ĉ (T w T ))(x x ) k (T T ) (7) This function is ngativ dfinit for positiv k + ĉ (T w T ) > and k > (8) lading to a stabl closd loop systm. Th adaptiv backstpping controllr T = T w ( (hg h ĉ x i )ṁ + k (x x )) T ff = ( + τ p)t T r f T ff ĉx (x x k ),r f = PDwT(T r f ) u = α f + c min c f comp p (,r f ) τ ṁ p u = sat(u, u min, u max ) f comp = u dĉ = γ(x x dt )x (T w T ) (9) Th adaptiv controllr (9) is idntical to (9) with th xtnsion of th updat law for ĉ and th constant c in (9) has bn rplacd with th stimat ĉ in (9). Th simulation shown in Fig. 9 whr c is changd during th simulation is rpatd with th adaptiv backstpping controllr givn in qu. (9). Th rsult is shown in fig. whr stimation of ĉ givs corrct stady stat valu of x. Stability Th stability is provn by th Lyapunov analysis, xcpt for th situation whr u saturats. Bcaus th systm is opn loop stabl thn saturation of u will not caus instability and implmntation of th PI-controllr with anti intgrator windup [6] givs propr opration as sn in th startup xprimnt in fig Exprimnts Figur shows th controllr for constant x,r f and variation of th cooling capacity by a stp up of Q = (h g h i )ṁ. Th controllr kps T SH at a narly constant valu indpndnt of th cooling. Figur shows a stp down of Q = (h g h i )ṁ and again tracking tim for ĉ givs th dviation from th rfrnc of T SH. Figur shows th stimatd paramtr c for a stp in Q at a narly constant tmpratur T w,in of th watr inlt. Th tracking tim constant shown is sn to xplain th tim for th dviation of T SH from th rfrnc in fig. and. Figur shows th stimatd paramtr c for a constant Q and variation of th watr inlt tmpratur T w,in. Figur 5 shows a startup of th systm. Th sttling tim is mainly du to th rspons of th condnsator prssur controllr which for th vaporator controllr is sn as a disturbanc. Fig. 6 shows th rsult from th convntional control systm shown in Fig.. Th suprhat is controlld by th opning of th thrmostatic xpansion valv (TXV) and th cooling capacity by th comprssor spd. This figur should b compard to Fig. and Fig.. Th coupling btwn suprhat and cooling capacity is sn to b considrably largr than for th nw controllr. This mans that tuning of th controllr for cooling of th systm influncs th total prformanc considrably [7]. A common way to obtain a propr prformanc is to tun th cooling controllr to b slow compard to th suprhat controllr. 7.. Enrgy fficincy Du to th limits of th comprssor spd continuous suprhat control is only possibl for f min f comp f max () If th rfrigrant flow ṁ = ṁ,max givs a comprssor spd f comp = f max thn ṁ > ṁ,max givs a rducd suprhat tmpratur and may lad to liquid rfrigrant into th comprssor. This mans that th uppr limit for th cooling capacity should b st to Q,max = (h g h i )ṁ,max. If th rfrigrant flow ṁ = ṁ,min givs a comprssor spd f comp = f min thn ṁ < ṁ,min givs an incrasd suprhat. This is accptabl sn from a control point of viw, but th nrgy fficincy is dcrasd. This mans that continuous control of th cooling Q = (h g h i )ṁ is possibl for all Q < Q,max. 6

8 For Q < Q,min = (h g h i )ṁ,min th suprhat controllr is saturatd lading to an incrasd suprhat tmpratur. Efficincy κ is basd on a stady stat xprimnt by calculating Q, (k) = kt T Q (k )T (τ)dτ Q comp, (k) = kt T Q (k )T comp (τ)dτ () κ (k) = Q, (k)/ Q comp, (k) A way to ovrcom th problm with dcrasd fficincy for Q < Q,min is to priodically (T ) start and stop th rfrigrant flow/comprssor and thn control th man valu of th rfrigrant flow by th duty cycl. If th systm to b coold down has a dominating tim constant T systm T, thn th variation in tmpratur du to th start/stop is small and th cooling controllr may act as a discrt tim controllr with sampling tim T, and th duty cycl as control input. Th two situations ar rfrrd to as continuous control and PWM control in th following. A comparison btwn continuous control and PWM control for th nw controllr is shown in Fig. 7. It is sn that PWM control is mor nrgy fficint for small cooling capacitis. Th rason for that is that if th continuous controllr givs a rfrigrant flow ṁ < ṁ,min th comprssor spd saturats at f comp = f min rsulting in an incrasd suprhat. This is accptabl sn from a control point of viw, but th nrgy fficincy is dcrasd du to th rducd filling of th vaporator Fig. 8 shows th sam comparison btwn continuous control and PWM control for a convntional systm controlld by a thrmostatic xpansion valv. In this systm it is not possibl to mak continuous control for Q < Q,min. Othrwis th prformanc is vry lik th prformanc of th backstpping controllr shown in Fig Conclusion A nw control stratgy whr th suprhat tmpratur is controlld by th comprssor and th cooling capacity by th rfrigrant mass flow is compard to a convntional control stratgy basd on a thrmostatic xpansion valv for control of th suprhat. A low ordr modl for th highly nonlinar systm with comprssor spd as input to th suprhat output is drivd. This modl is usd in a nonlinar backstpping dsign mthod. Th dvlopd mthod givs a suprhat control which is narly indpndnt of th cooling capacity. Th stability of th proposd mthod is validatd thortically by th Lyapunov analysis and xprimntal rsults show th stabl prformanc of th systm for a wid rang of oprating points. Compard to othr mthods no gain schduling of th suprhat controllr is ncssary to covr a larg rgion of opration. Th xprimnts and simulations show that th backstpping controllr maintain continuous control at all rqustd cooling capacitis, thus nabling prcis tmpratur control. Th pric of having continuous control at low capacitis is howvr a rducd fficincy compard to PWM control. Comparison of th backstpping control with a TXV showd that th backstpping control can provid similar prformanc to th TXV. Th advantag of th backstpping control is howvr that it offrs a 7 highr flxibility in th systm control than th TXV and it provids th possibility to switch to PWM control at low capacitis thus optimizing th ovrall fficincy at all capacitis. Rfrncs [] Larsn,L.F.S., and Thybo,C.: Potntial nrgy savings in rfrigration systms using optimal st-points. Confrnc on Control Applications, Taiwan, [] Larsn,L.F.S., and Thybo,C., and Stoustrup,J., and Rasmussn,H.: A mthod for onlin stady stat nrgy minimization, with application to rfrigration systms. Confrnc on Dcision and Control, Bahamas, [] Parkum,J., and Wagnr,C.: Idntification and control of a dry-xpansion vaporator. th IFAC symposium on systm idntification, 99 [] Larsn,L.F.S.: Modl Basd Control of Rfrigration Systms. Ph.D. Thsis ISBN , Aalborg Univrsity / Danfoss A/S, 5 [5] Changquin T., and Chunpng D., and Xinjiang Y., and Xianting L.: Instability of automotiv air conditioning systm with a variabl displacmnt comprssor. Part. Exprimntal invstigation. Intrnational Journal of Rfrigration, no8, 5. [6] Xiang-Dong,H., and Liu,S., and Assada,H.H., and Itoh,H.: Multivariabl control of vapor comprssion systm. VAC&R Rsarch, 998 [7] Zhao,L., and Zahruddin,M.: Dynamic simulation and analysis of watr chillr rfrigration systm. Applid Thrmal Enginring, 5, 5, [8] Xiang-Dong,H., and Asada,H.H.: A nw fdback linarization approach to advancd control of multi-unit HVAC systms. Amrican Control Confrnc, [9] Rasmussn, H.: Nonlinar Suprhat Control of a Rfrigration Plant. Confrnc on Control Applications, San Antonio, USA, 8 [] Rasmussn, H.: Nonlinar Suprhat Control of a Rfrigration Plant using Backstpping. IECON 8, Orlando, Florida, USA, Nov.8 [] Jia,X., and Tso,C.P., and Jolly,P., and Wong,Y.W.: Distributd stady and dynamic modling of dry-xpansion vaporators. Int. Journal of Rfrigration, 999,, 6 6. [] Grald,E.W., and MacArthur,J.W.: Moving boundary formulation for modling tim-dpndnt two-phas flows. Int. J. hat and Fluid Flow, 99,, [] H, X.D., and Asada,H.H., and Liu,S., and Itoh,H.: Multivariabl control of vapor comprssion systms. HVAC&R Rsarch, 998,, 5. [] Skovrup,M.J.: Thrmodynamic and Thrmophysical Proprtis of Rfrigrants. Vr.., Tchnical Univrsity of Dnmark, [5] Thybo,H., and Larsn,L.F.S., and Stoustrup,J., and Rasmussn,H.: Control of systms with costs rlatd to switching applications to aircondition systms, CCA, Saint Ptrsburg, Russia, 9 [6] Aaström, K.J., and Wittnmark,B.: Adaptiv Control - Scond Edition, Addison-Wsly, 995. [7] Brorson,P.M.T., and Jagt,M.T.G. van dr: Hunting of Evaporators Controlld by a Thrmostatic Expansion Valv. ASME J. Dynamic Systms, Masurmnts, and Control, 98,, 5.

9 f comp,rf Hat load Q load Tcr Pump T o Evaporator Suction prssur control N C Comprssor Suprhat control OD W C T aoc Surroundings Condnsr T ic P c Condnsr prssur control N CF Fan T a W CF [Hz] : modlld and masurd Expansion Valv m rf T oc.5 Figur : Layout of th tst rfrigration systm including convntional control loops T : modlld and masurd sh [C] Two-phas sction l Suprhat sction Figur : Modld and masurd and T sh for variation of input f comp m m [kg/s].5..5 m hi h g ho. 6 8 P : modlld and masurd.5 T L [C] 6 8 T sh : modlld and masurd Figur : Schmatic drawing of th vaporator 6 8 Figur 5: Modld and masurd and T sh for variation of input ṁ using stimatd man valus m.5 [kg/s] : modlld and masurd P, rf m - PI Equ.(6) f c Modl Equ.(8) P T SH.5 Figur 6: Modl with PI control of input f c. 6 8 T sh : modlld and masurd Eq.(9.a) [C] 6 8 x - x cx k T ff T rf PDwT, rf - Eq.(9.c) f c m Modl Equ.(8) P T SH Eq.() Figur : Modld and masurd and T sh for variation of input ṁ Figur 7: Backstpping controllr structur 8

10 .5 m 5 Cooling: Q r [kg/s] x Suprhat tmpratur: T SH [Hz] Comprssor spd: f comp Figur 8: simulatd x and for variation of input ṁ using th backstpping controllr for known paramtrs Figur : Control of suprhat du to disturbanc causd by a stp up of th cooling. c 5 Cooling: Q r [watt/c] x Suprhat tmpratur: T SH [Hz] Comprssor spd: f comp Figur 9: simulatd x and for variation of c using th backstpping controllr for constant c qual to th valu bfor th chang. Figur : Control of suprhat du to disturbanc causd by a stp down of th cooling. Simulatd and stimatd c 5 Cooling: Q r [watt/c] x T watr,in Estimatd paramtr: c Figur : simulatd x and for variation of c using th adaptiv backstpping controllr. Figur : Estimatd ĉ for variation in cooling capacity Q and constant tmpratur of th watr inlt T w,in 9

11 5 Cooling: Q r T watr,in Man Efficincy vs. Q,man Estimatd paramtr: c.5 PWM continuous Figur : Estimatd ĉ for constant cooling capacity Q and variation of th tmpratur of th watr inlt T w,in.5 6 Cooling: Q r Suprhat tmpratur: T SH Figur 7: Nw controllr: Man valu th fficincy as a function of man valu of th cooling. [Hz] Comprssor spd: f comp Man Efficincy vs. Q,man Figur 5: Startup of th systm..5 continuous 5 Cooling capacity: Q r PWM Suprhat tmpratur: T SH Figur 8: TXV controllr: Man valu th fficincy as a function of man valu of th cooling for th convntional controllr Figur 6: Convntional control by a thrmostatic valv.