Dynamic Operational Optimization of Air Source Heat Pump Heating System with the Consideration of Energy Saving

Size: px

Start display at page:

Download "Dynamic Operational Optimization of Air Source Heat Pump Heating System with the Consideration of Energy Saving"

Melissa Lindsey
6 years ago
Views:

Pepins of he 9h Inenaional Symposium on Advanced Conol of Chemical Pocesses The Inenaional Fedeaion of Auomaic Conol June 7-1, 15, Whisle, Biish Columbia, Canada TuA.

1 Pepins of he 9h Inenaional Symposium on Advanced Conol of Chemical Pocesses The Inenaional Fedeaion of Auomaic Conol June 7-1, 15, Whisle, Biish Columbia, Canada TuA.6 Dynamic Opeaional Opimiaion of Ai Souce Hea Pump Heaing Sysem wih he Consideaion of Enegy Saving Changxin Xing, Qiang ding, Aipeng jiang, Wen Cheng, Dahan Zhou Enegy uiliaion sysem and auomaion insiue, Hanghou Diani Univesiy, Zhejiang, Hanghou 3118, P.R. China (Tel: +86 Absac: I is uie impoan o impove he enegy saving of ai souce hea pump heaing sysem hough opimal opeaion and sysem engineeing mehod. In his wok, he nonlinea dynamic pocess of he hea pump heaing pocess was esablished and he inegal opimal objecive funcion was hen fomulaed o balance he conol accuacy and enegy saving. To solve he diffeenial and algebaic opimiaion poblems (DAOPs efficienly, simulaneous mehod was used o disceie he poblem by collocaion of finie elemens. Compuing esuls show ha poenial enegy saving of moe han 16% can be achieved unde pemied accuacy eo of one empeaue conol, and he ambien ai empeaue and eleciciy pice have significan effec on he opimal opeaion and pefomance of he hea pump heaing sysem. Base on he compuing esuls and analysis of opeaional paamees, a saegy o dynamically adjus he weigh of objecive funcion o balance he conol accuacy and enegy cos was poposed fo he popose of fuhe enegy saving, compuing esuls show ha: if we dynamically adjused he weigh of objecive funcion wih his mehod, abou 4.7% of fuhe enegy cos saving can be obained. Ou eseach is of significan meaning fo he opimal opeaion of hea pump heaing sysem unde dynamical condiions. Keywods: dynamic; opeaional opimiaion; hea pump; enegy saving; 1. INTRODUCTION Wih he emeging of enegy sueee and envionmen polluion, hea pump echniues ha ae chaaceied as muliple advanages of high efficiency enegy-saving and non-polluion povide he way ou fo building heaing. Fo he eason of he ousanding pefomance of i and he coninuous impovemen of he conol echniues, ai souce hea pump is apidly acceped by moe and moe uses. The applicaion of ai souce hea pump is absoluely adapive in a lage aea of China [Ju e al. (1996]. The climae of he Souh of Yange Rive in China sho o be ho in summe and cold in wine, whee he numbe of days ha daily aveage empeaue is blow 5 is abou 45 o 6 [Wang and Tan (4]. How o complee he opimal opeaion of hea pump and save enegy and minimie cos uns o be an impoan poblem. To solve his poblem, eseaches analyed and impoved hea pump sysem on seveal fons. Fo he applicaion and economy, Feng e al. (5 analyed he low-empeaue adapabiliy and invesigaing economy of hea pump sysem in diffeen aea of China. Fo he model level, Shi and Xue (7 eview he developmen of hea pump model. Gilles e al. ( simplified he models wihou disoion which sill capue he conol elevan dynamics. Fo he opimiaion, Claa e al. (1 analyed diffeen cases of hea pump model and pu ou opimiaionbased conol saegies. These wok lays a solid foundaion fo he eseach of his pape. As he coefficien of pefomance of ai souce hea pump is significanly affeced by ambien empeaue and he empeaue of inle and oule of wae cycle, he acual enegy consumpion of hea pump is diffeen a vaious ambien empeaues and one empeaue euiemens. Fo he eason of swing of he eleciciy pice in a day, opeaion cos of hea pump is no same in diffeen ime. Consideing ambien empeaue, one empeaue euiemen, eleciciy pice and he dynamic pocess of opeaion of hea pump, o opimie he conol of heaing sysem in muliple objecives has vey impoan significance.. SYSTEM ANALYSIS The fundamenal pinciple of hea pump heaing sysem shown Fig.1 is ha he hea pump exacs enegy fom ai by he way of compessing o make he empeaue of igean ise. Then he igean ansfes he hea o he wae in pipeline, which makes he wae empeaue un fom he inle T o he oule T. Fig.1. Schemaic diagam of hea pump heaing sysem. The ho wae wih he empeaue T is anspoed o esidenial building by pipeline o ise he one empeaue, compleing he goal of heaing. Fo he sysem shown in Fig.1, he dynamic pocess of heaing can be defined as: dt m c m c Q T T d C C C w w w w (1 Copyigh 15 IFAC 741

2 IFAC ADCHEM 15 June 7-1, 15, Whisle, Biish Columbia, Canada dt ( w ( f w, m T T wcw T T d C C dtf C f wf ( Tw, Tf f ( T Tf (3 d dt Cb f ( Tf T b ( Tamb T (4 d whee, T, T, T f, T and T amb epesen especively he supply wae empeaue, he eun wae empeaue, he floo empeaue, he one empeaue and he ambien empeaue. mw is he mass flow ae of he wae cicui and cw[j/kg.k] he specific hea capaciy of wae. Cw,s and Cw, epesen he hemal capaciy of he wae a empeaue T and he wae a empeaue T, K w, K f and K w epesen oveall hea exchange coefficien coespondingly. Q [W] is he hemal powe deliveed by he hea pump, elaed o he opeaion feuency of compesso. dt /d, dt /d, dtf /d and dt /d ae he deivaive of especively T, T, T f and T. The specific paamee values of his model ae lised in Table 1. Table 1 Paamee values of he model Iem Value Iem Value C w,s(j/k m w (kg/s.6 C w, (J/K K w,f (W/K 115 C f (J/K K f,(w/k 61 C b(j/k K b(w/k 8 Fo enegy saving of hea pump, he heaing specific enegy consumpion elaes o he eleciciy consumpion and pefomance of hea pump [Sakellai e al. (6], while he enegy efficiency of hea pump is expessed by he coefficien of pefomance (COP: ( Q ( P ( COP( (5 Whee P epesens he inpu powe (compesso powe. In dynamic pocess, he COP is elevan o he empeaues, which can be expessed as follow [Radu e al. (11]: COP( F( T, T Cop amb T T T T T T 1 amb 3 amb 4 5 amb The hea pump heaing sysem conols he compesso opeaion feuency o deemine he supply wae empeaue T. When he ambien empeaue flucuaes, he building hea losses change. So in elaively seady sae, he hea deliveed by hea pump maches he building hea loss expessed as follow: T b, ss b amb (6 Q ( T T (7 * T Q b, ss wf Whee, Q b,ss is he building hea loss, T * epesens he eence supply wae empeaue and β (β>1 he aio of he acual hea loss and he hea eleased o he one. Fo he f (8 hea pump, he elaion of he deliveed hea, opeaion feuency and empeaues can be obained by daa-fiing [Liang e al. (14]:. Q h b1t amb bt b3 fe b4t b5t (9 b f b T T b T f b T f 6 e 7 amb 8 amb e 9 e whee, fe is he opeaion feuency of he compesso and b, b1 b9 ae he coefficiens of he euaion. 3. OPTIMIZATION OBJECTIVES AND SOLUTION ANALYSIS As he main funcion of he floo heaing sysem wih ai souce hea pump is o mee he hemal comfo euiemen in buildings, he eence one empeaue nomally anges fom The eence one empeaue is assumed o be T and he acual one empeaue is T (. As a geneal objecive of conol is o make he acual one empeaue appoach o he eence one empeaue and o minimie he conol eo, which can be fomulaed by: Min J T T T ( ( (1 On he ohe hand, he opeaion cos of he sysem is in consideaion. So sueeing he enegy cos in he pomise of a ceain conol accuacy and achieving he opimal conol of he opeaion is also desiable. The eleciciy cos is mainly embodied in he eleciciy consumpion of he sysem and diecly elaive o he eleciciy pice in diffeen peiod. So he cos objecive is: MinJ e ( P ( e p (11 Whee ep ( is he eleciciy pice a diffeen momens. The opeaion powe and COP of hea pump and he building hea loss ae closely elaed o ambien condiion(such as empeaue, while he ambien condiion changes wih daily weahe, which makes he sysem balance he cos objecive and he hemal discomfo objecive deping on he ambien condiion. So in a ceain peiod [ ], an objecive funcion is given as follow including boh eleciciy cos and conol accuacy. p( ( (1 ( ( (1 J K e P K T T Hee K epesens he weighing faco of he wo objecives. Fo K=, he opimal conol pofile only sueees he enegy cos unde he given consains. Fo K=1, he hea pump opeaion minimies he hemal discomfo cos, egadless of he coesponding enegy cos. In ode o ge exceped esul, he value of K can be chosen deping on he oleance o empeaue acking eo se by uses o saisfy boh he hemal comfo euiemen of use and decease he enegy cos. So he hemal comfo euiemen should be inoduced ino sysem model euaions, which can be: T T (13 Fo he whole sysem, despie he clea ambien empeaue change in a day, he change beween adjacen days indicaes o be uasi-peiodiciy o some exen. So fo he convenience of sudy, opimiaion objecive funcion (1 akes 4 hous Copyigh 15 IFAC 74

3 IFAC ADCHEM 15 June 7-1, 15, Whisle, Biish Columbia, Canada as he cycle and makes he eminal and iniial saes he same. The empeaue TT=[T, T, Tf, T] T is defined o lec he main empeaue saes of heaing. So he eminal and iniial empeaue saes of he sysem should be: TT( TT(4 (14 And hese saes saisfy he consain below: TT TT( TT (15 lb The poblem ha akes (1 as objecive funcion and (1-9 and (13-15 as consains is named as Op1. Since he opimal poblem includes a se of song nonlinea DAEs (Diffeenial and Algebaic Euaions and many consains, i is faily ough o solve his kind of poblem by diec ways. Cuenly, DAEs opimiaion poblems ae solved by vaious saegies ha apply nonlinea pogamming (NLP solves o he DAE model [Biegle e al. (, (7]. To solve he Op1, simulaneous mehod was used o conve he DAEs ino an NLP by appoximaing sae and conol pofiles by a family of polynomials on finie elemens. Hee, we use he following monomial basis epesenaion fo he diffeenial pofile, which is popula fo Runge-Kua disceiaions [Biegle (1]: i 1 d ( i 1 hi ( (16 h d ub 1 i, Hee i-1 is he value of he diffeenial vaiable a he beginning of elemen i, hi is he lengh of elemen i, d/di, denoes he value of is fis deivaive in elemen i a he collocaion poin, and Ω is a polynomial of ode, saisfying ( 1..., (17 ' (, 1..., (18 Whee ρ is he locaion of he h collocaion poin wihin each elemen. Coninuiy of he diffeenial pofiles is enfoced by: d ( i 1 hi (1 d (19 1, Based on a numbe of sudies of Lay goup, adau collocaion poins wee seleced in his sudy because hey allow consains o be se a he of each elemen and o sabilie he sysem moe efficienly if high index DAEs ae pesen. In addiion, he conol and algebaic pofile ae appoximaed using a Lagange basis epesenaion which akes he fom: i1 ( ( 1 hi y y i, i1 ( ( ui, 1 hi u ( (1 Hee yi, and ui, epesen he values of he algebaic and conol vaiables especively in elemen i a collocaion poin, is he value saisfy i-1 i, ψ is he Lagange polynomial of degee saisfying: ( ( / h ( / h i i ( 1 ( 1 ( / hi 1 ( ( And a collocaion poins: 1, (, (, 1,...,, (3 The diffeenial vaiables ae euied o be coninuous houghou he ime (o space hoion, while he conol and algebaic vaiables ae allowed o have disconinuiies a he boundaies of he elemens. By he mehod of collocaion he opimal poblem wih DAEs is ansfeed ino NLP. Lage scale NLP solves such as IPOPT [Biegle and Zavala (9] ae vey suiable fo he soluion. 4. CASE STUDY To analye he pefomance of cos saving and empeaue conol wih he mehod above, his pape akes a siuaion of ai souce hea pump heaing in a ceain aea of Zhejiang as he objec of sudy. Fig. sho he ypical daily ambien empeaue of his aea in wine. Fig.3 depics he fied cuves of ai souce hea pump COP. The ime-of-use (TOU eleciciy pice in he aea can be seen in Fig.4 (a. The eence one empeaue is se o be. Accoding o he ealiy of he opeaion of hea pump sysem, he uppe and lowe bound of he sae empeaues ae idenified o be TTlb=[,5,1,11], TTub=[65,5,35,5]. The opimiaion poblem is solved wih he simulaneous opimiaion echnology, which spli 4 hous ino 48 conol unis o make he sysem egulae he inpu powe. Then 1 finie elemens ae allocaed in evey uni and evey finie elemen has 3 collocaion poins. And hen he IPOPT solve is used unde he plafom of GAMS o solve he disceied NLP. In his way, he opimal opeaion and inenal saes of sysem meeing coesponding euiemen is obained. 4.1 Opimiaion esuls wih diffeence weighing facos K The opimiaion objecive funcion includes wo conflicing pas. The fis pa is opeaion eleciciy cos, namely he economy objecive ha aims o save money. The second pa is consideaion of he hemal discomfo, which is he hemal comfo objecive o minimie he gap beween he acual one empeaue and he eence one empeaue. The bigge he value of K is, he moe aenion he opimiaion pays o value he hemal comfo. On he conay, he opimiaion will sueee he acual eleciciy cos. Suae oo of he second pa (T ( T idenified o be he hemal discomfo. While he value of K changes fom o 1, he pupose of opimiaion vaies. So he choice of he value of weighing faco will significanly influence hemal discomfo and acual eleciciy cos. To make sue he balance of values of he wo objecives, he opimiaion objecive funcion is nomalied. The nomalied funcion is as follow: J (1 K [ e ( Q ( / COP( K ( T ( T p is (4 Hee α is a consan obained by compuing, and when is value is.3, he objecives un o be balanced well. Copyigh 15 IFAC 743

IFAC ADCHEM 15 June 7-1, 15, Whisle, Biish Columbia, Canada Fig.. The ypical daily ambien empeaue of a ceain aea of Zhejiang in wine. (a Fig.5. Acually eleciciy cos (a and hemal discomfo wih diffeen weighing facos.

Dispaae values of he weighing faco is adoped and he oleance of one empeaue acking eo is undesigned. In his siuaion, solving he opimal conol poblem, he esuls ae shown in Fig.

4 IFAC ADCHEM 15 June 7-1, 15, Whisle, Biish Columbia, Canada Fig.. The ypical daily ambien empeaue of a ceain aea of Zhejiang in wine. (a Fig.5. Acually eleciciy cos (a and hemal discomfo wih diffeen weighing facos. Fig.3.The fied cuves of ai souce hea pump COP. (a (a Fig.4. Time-of-use eleciciy pice EP-1 (a and EP-. Dispaae values of he weighing faco is adoped and he oleance of one empeaue acking eo is undesigned. In his siuaion, solving he opimal conol poblem, he esuls ae shown in Fig.5-Fig.6. Fig.5 (a sho ha he acual eleciciy cos of daily heaing changes subsanially as weigh faco vaies. When K=1, he cos is moe han 3CNY pe day. While K=, i deceases o less han 1 CNY pe day. Of couse, his is he esul of he sacifice of hemal comfo, which can be seen fom Fig.5. Table povides he maximum diffeence beween he acual one empeaue and eence one empeaue, he hemal discomfo,he eleciciy cos and he cos-saving aio. Even hough he cos-saving aio eaches an amaing level of 68.66%, meanime he maximum empeaue diffeence also comes o an unaccepable value of 11. and he hemal discomfo is Bu when he weighing faco anges fom.3 o 1, his compomise, despie he sacifice of hemal comfo, demonsaes desied cos-saving pefomance, he empeaue diffeence displayed in Table is accepably limied and he hemal discomfo is oleable. Fo example, when K=.7, he hemal discomfo is only 7 and he maximum empeaue diffeence is meely.85. The cos, by conas, deceases fom 3.34 o 5.47 CNY. The lage he value of K is, he lage he hemal powe becomes, as depiced in Fig.6. A midnigh, he ambien empeaue is low and he value of COP is small. In ode o ensue a peable one empeaue, highe supply wae empeaue is needed which means moe hemal powe euiemen. Wih he climbing of ambien empeaue, he hemal powe declines he same as he supply wae empeaue. When he ambien empeaue ouches he peak, he hemal powe and supply wae empeaue come o especive valley poins. (a Fig.6. Opimal hemal powe (a, supply wae empeaue, eun wae empeaue (c and one empeaue pofiles (d wih diffeen weighing facos. Table Maximum empeaue diffeence, eleciciy cos, saving ae and hemal discomfo wih diffeen K K maximum empeaue diffeence ( eleciciy cos (CNY saving ae (% hemal discomfo Opimal opeaion wih diffeen TOU eleciciy pice Eleciciy pice is a diec faco ha deemines he acual eleciciy cos. Fo inspecing he effec of diffeen eleciciy pice on opimal opeaion, his pape inoduces anohe indusial eleciciy pice policy and consan eleciciy pice. Anohe ime-of-use (TOU eleciciy pice (EP- can be seen in Fig.4. Fig.7 depics he hemal powe and one empeaue based on he eleciciy pice EP- wih he value of K especively.7,.8 and.9. The esuling pofiles show ha he hemal powe swings wih Copyigh 15 IFAC 744

5 IFAC ADCHEM 15 June 7-1, 15, Whisle, Biish Columbia, Canada he eleciciy pice. A he condiion of same K and peiod, high eleciciy pice coesponds o low hemal powe and vice vesa. The elaion beween one empeaue and eleciciy pice is same as he hemal powe. Wha is moe, he lage he weighing faco is, he smalle he effecs of eleciciy pice on hemal powe and one empeaue ae, which is because lage K means lile aenion o economy objecive. (a (a Fig.7. Themal powe cuves (a and one empeaue pofiles wih he eleciciy pice EP Analysis of opimal opeaion wih ambien empeaue flucuaion Ambien empeaue is an impoan condiion fo he unning of ai souce hea pump, which diecly impacs coefficien of pefomance (COP. To obain he oveall opimal pefomance, he coesponding enegy consumpion a diffeen ambien empeaue vaies obviously. Fo inspecing he effec of ambien empeaue on opimal objecives, he ambien empeaue in Fig. is changed. Then he change of enegy consumpion is obained by opimal solving. Table 3 liss enegy coss when he ambien empeaue oveall ises and dops 1 and he K is 1. The esuls eveal ha ambien empeaue has an impoan effec on enegy cos wih opimal objecive. One degee ises of he empeaue cases 1.78% cos saving and one degee dop leads o 11.39% cos inceasing. The opimal conol ajecoy and elaed saes go by opimal saving show ousanding pefomance when he empeaue disibuion changes in a whole day. Fig.8 lays ou he hea pump inpu and coesponding oupu powe of wo diffeen ambien empeaue pofiles and he supply wae empeaue. When he ambien empeaue is high, he euiemen of he inpu powe is compaaively low. Ohewise, i would be high and lead o moe enegy cos. As he opimal inpu powe changes disincly wih diffeen ambien empeaue pofiles, o complee he oally opimal objecive of he sysem needs o egulae he inpu powe accoding o he flucuaion of he ambien empeaue. Table 3 The effec of ambien empeaue on opimal eleciciy cos K=1 Eleciciy Cos (CNY/day Rae of change (% One degee ise Tamb-1 One degee dop (c (d Fig.8. Two diffeen ambien empeaue cuves in he aea (a, he pofiles of especively he hemal powe, he inpu powe (c of hea pump and he supply wae empeaue (d wih he ambien empeaue in Fig.8 (a. 4.4 Analysis of opimiaion based on he dynamic adjusmen of he weigh of objecives The analyses discussed in he pevious paagaph indicae ha he ambien empeaue diecly impac he opimiaion objecive. Fo he enegy cos, i means ha he highe he ambien empeaue is, he lage he COP is, namely moe economic. On he ohe hand, he flucuaion of eleciciy pice in diffeen peiods leads o he fac ha o saisfy he opimal objecive, he opeaion cos vaies in diffeen peiods. In ode o boh minimie he one empeaue conol eo and ge bee cos-saving pefomance, a moe advisable choice is ha when he eleciciy pice is compaaively low and he COP is elaively high, he opimiaion pays moe aenion o he conol pecision, namely do moe o saisfy he euiemen of one hemal comfo. On he conay, he opimiaion sueees he enegy cos of sysem fuhe. Accoding o he pinciple above, his pape adjuss he opimiaion objecive funcion depiced in (1 o dynamically egulae he weighing faco beween he enegy cos objecive and he hemal discomfo objecive. The egulaion execues once in evey half hou fo he expecaion of bee opimiaion pefomance. The adjused objecive funcion is: J (1 K ( e ( / e [ e ( Q ( / COP( ] p pmin p COP( K ( T ( T (5 Whee e p( epesens he dynamic egulaion faco posiively coelaing o eleciciy pice and e pmin he minimum value of eleciciy pice in he whole peiod. Based on he esul analyses of Table3, he one empeaue acking eo is lile and he pefomance of cos saving is desiable when he weighing faco anges fom.5 o.7. So o e-solve he opimiaion conol poblem, he weighing faco is se a.7 and he maximum diffeence beween one empeaue and he eence one empeaue is euied o no moe han. The e-solving bases on he Copyigh 15 IFAC 745

6 IFAC ADCHEM 15 June 7-1, 15, Whisle, Biish Columbia, Canada eleciciy pice EP-1and he ambien empeaue Tamb-1. Conasing he new opimal esuls wih he ones based on he fome opimiaion objecive funcion eveals ha he adjused opimiaion objecive funcion leads o a bee pefomance of cos saving. By he adjusing, he eleciciy cos changes fom o CNY pe day and he hemal discomfo fom 4.6 o Despie a lile ise of he hemal discomfo, he one empeaue sill mees he euiemens and he sysem obains a moe cos saving close o 4.7%. The calculaing esul sho ha he adjusmen of he opimiaion objecive funcion has visible pefomance. Fig.9 depics he changes of inpu powe and ohe elaed saes. Case A epesens he esuls based on opimiaion objecive funcion (3 and Case B he esuls based on (5. As can be seen in Fig.9, he inpu powe ansfoms appaenly afenoon (afe1 o clock when he opimiaion objecive funcion ales. A he same ime he one empeaue pofile lecing he hemal comfo is in he oleance ange despie he flucuaion. (c (d Fig.9 Compaison of he opimal esuls befoe and afe he objecive funcion adjused By he way, he oal numbe of vaiables is 15554, he oal numbe of eualiy consains is 1537 and ineualiy consains 5. Then, he compuaion ime of he poblem is abou 59s. 5. CONCLUSION As he flucuaion of ambien empeaue and eleciciy pice significanly influence he opeaion cos and he conol pefomance of ai souce hea pump heaing sysem, opimiing conol of he hea pump heaing sysem is meaningful hough sysem opimiaion mehod. This pape, based on he basic dynamic pocess euaions and he oveall opimiaion objecive, analyes and sudies he opimal opeaion conol of he ai souce hea pump heaing sysem fo cos saving. A fis, he pape analyes he opimal opeaion conol of he sysem wih diffeen weighing facos of wo opimiaion objecives and he opimal esuls meeing a ceain oleance of hemal discomfo eveal ha he opimiaion can achieve moe han 16% cos saving by a lile sacifice of one empeaue conol accuacy. Then, he pape sudies he effec of eleciciy pice and ambien empeaue on he opimal opeaion conol of he sysem and poposes a saegy ha dynamically adjuss he objecive funcion o sueee cos. The calculaing esuls sho ha his saegy can lead o a fuhe cos saving aio which can be abou 4.7%. The sudy of his pape has ceain significance on diecing he opeaion and cos saving of ai souce hea pump heaing sysem. Based on his pape, he sudy abou he opeaion of hea pump sysem wih load flucuaion and inenal hea soage will be in consideaion as well as he elevan conol echniues. REFERENCES Biegle, L.T, Auo, M. C. and Andeas, W. ( Advances in simulaneous saegies fo dynamic pocess opimiaion. Chem. Eng. Sci., 57 ( Biegle, L.T. (7 An oveview of simulaneous saegies fo dynamic opimiaion. Chemical Engineeing and Pocessing, 46, Biegle, L.T. and Zavala, V.M. (9 Lage-scale nonlinea pogamming using IPOPT: An inegaing famewok fo enepise-wide dynamic opimiaion. Compues & Chemical Engineeing, 33 ( Biegle, L.T. (1 Nonlinea Pogamming: Conceps, Algoihms and Applicaions o Chemical Pocesses, Sociey fo Indusial and Applied Mahemaics (Cambidge Univesiy Pess: Pisbugh, PA Claa, V., Filip L., Jan, V.I., Lieve, H. (1 Sudy of he opimal conol poblem fomulaion fo modulaing aio-wae hea pumps conneced o a esidenial floo heaing sysem. Enegy and buildings, Feng, J.P., Xu, T. and Yang, F. (5 Applicaion and developmen of ai souce hea pump o HVAC in China. Building Enegy &Envionmen, 4 (5-3. Gilles, F., Chiselle, V., Olivie, L. and Gilbe, A. ( Developmen of a simplified and accuae building model based on elecical analogy. Enegy and Buildings, 34( Ju, Y.P, Zhang, Y.Q and Lv, C. R (1996. Enegy saving effecs and economic feasibiliy of ai-souce hea pumps, Jounal of Tian jin univesiy, 9 ( Liang, K., Nan, X. H. and Yue, Y.F. (14 Reseach on he effecs of fan feuency convesion on pefomance of an ai-souce hea pump wae heae sysem. Building Science, 3 ( Radu, B., Joshua, C., Kuo, M. and Chao, Segiu, T. (11 Paamee idenificaion and model based pedicive conol of empeaue inside a house. Enegy and Buildings, 43 ( Sakellai, D., Fosén, M. and Lundvis, P. (6 Invesigaing conol saegies fo a domesic lowempeaue hea pump heaing sysem. Inenaional Jounal of Refigeaion, 9 ( Shi, L. and Xue, Z.F. (7 Simulaion and conol eseaches fo hea pump/ai condiioning sysems: a eview. Hv&Ac, 37 ( Wang, E.C. and Tan, H.W. (4 Applicaion sudy on floo heaing sysem wih ai souce hea pump in shanghai aea, Building Enegy & Envionmen, 3 ( Wang, X.D., Zhang, C.H. and Zhang, Z. (14 Expeimenal eseach on pefomance of ai-souce hea pumps. HV&AC, 44 ( Copyigh 15 IFAC 746

On Control Problem Described by Infinite System of First-Order Differential Equations

On Control Problem Described by Infinite System of First-Order Differential Equations Ausalian Jounal of Basic and Applied Sciences 5(): 736-74 ISS 99-878 On Conol Poblem Descibed by Infinie Sysem of Fis-Ode Diffeenial Equaions Gafujan Ibagimov and Abbas Badaaya J'afau Insiue fo Mahemaical