Mehnil Engineering Letters Szent István University nnul ehnil-sientifi Journl of the Mehnil Engineering Fulty Szent István University Gödöllő Hungry Editor-in-Chief: Dr. István SZÓ Editor: Dr. Gábor KLÁCSK Exeutive Editoril ord: Dr. István RÓFI Dr. János EKE Dr. István FRKS Dr. László FENYVESI Dr. István HUSI Dr. László KÁI Dr. Sándor MOLNÁR Dr. Péter SZENDRŐ Dr. Zoltán VRG Cover design: Dr. László ZSIDI HU ISSN 060-3789 ll Rights Reserved. No prt of this publition my be reprodued stored in retrievl system or trnsmitted in ny form or by ny mens eletroni mehnil photoopying reording snning or otherwise with the written permission of Fulty. Páter K. u. 1. Gödöllő H-103 Hungry dekn@gek.szie.hu www.gek.szie.hu Volume 14 (016)
Institute for Mthemtis nd Informtis Multiple liner regression bsed models for solr olletors Rihárd KICSINY Deprtment of Mthemtis Institute for Mthemtis nd Informtis bstrt Mthemtil modelling is the theoretilly estblished tool to investigte nd develop solr therml olletors s environmentlly friendly tehnologil het produers. In the present survey reent multiple liner regression (MLR) bsed olletor models re presented nd ompred with one nother nd with physilly-bsed model used suessfully in mny pplitions by mens of mesured dt. he MLR-bsed modelslled MLR model SMLR model nd IMLR model prove to be rther preise with modelling error of 4.6% 8.0% nd 4.1% respetively whih mens tht ll MLR-bsed models re more or nerly the sme urte s the well-tried physilly-bsed model. he SMLR model is the most while the IMLR model is the lest esy-to-pply MLR-bsed model with the lowest nd the highest omputtionl demnd respetively. Nevertheless ll MLR-bsed models hve lower omputtionl demnd thn the physilly-bsed model. ordingly the MLR-bsed models re suggested for fst but urte olletor modelling. Nomenlture : olletor surfe re m ; : speifi het pity of the olletor fluid J/(kgK); I : globl solr irrdine on the olletor surfe W/m ; t: time s; : mbient temperture of the olletor C; : in inlet olletor (fluid) temperture C; : let olletor temperture (ssumed to be the sme s the homogeneous olletor temperture in se of the physilly-bsed model) C; U L : overll het loss oeffiient of the olletor W/(m K); v: (onstnt) flow rte inside the olletor m 3 /s; V : volume of the olletor m 3 ; 0 : optil effiieny of the olletor - ; : olletor fluid density kg/m 3 ; 35
Institute for Mthemtis nd Informtis : time dely before Cse or 3 s; : time dely before Cse s; 1: time of flowing inside the olletor from the inlet to the let when the pump is swithed on permnently s; : length of time between suessive mesurements on the olletor s. 1. Introdution Mthemtil modelling is the most widely used nd theoretilly estblished tool to investigte nd develop solr therml olletors s environmentlly friendly tehnologil het produers. he two min tegories of mthemtil models for olletors re physilly-bsed models whih represent ext physil lws (bsed on theory) nd blk-box models whih desribe empiril orreltions (bsed on experienes or mesurements). mong the most importnt physilly-bsed models the Hottel-Whillier-liss model (Duffie nd ekmn 006) my be the erliest whih is frequently used to dte. his model determines the olletor temperture s funtion of time nd spe. uzás et l. (1998) proposed simpler model ssuming tht the olletor temperture is homogeneous in spe. his model is liner ordinry differentil eqution (ODE) vlidted in (Kisiny 014) nd is likely the simplest physillybsed model used in the prtie (see e.g. (Kumr nd Rosen 010; uzás nd Kisiny 014)) but n still desribe the trnsient olletor proesses with n pproprite ury. his model will be lled physilly-bsed model in short below. he gretest dvntge of blk-box models is tht it is not needed to know the physil lws of olletor preisely in order to rete model. Nevertheless the model my be rther preise even if it is simple s in the se of (Kisiny 014). he most frequent blk-box model type is perhps the rtifiil neurl network (NN) in the field of olletors. Generlly NNs re urte but rther troublesome to pply beuse of the so-lled trining proess. he onvergene of the lgorithm inditing the end of trining session my be lso timeonsuming. ording to Fisher t l. (01) onveniently usble lgorithm ensuring relible nd fst determintion of n pproprite NN for olletor is still missing. euse of these problems simple nd generl but still urte blk-box model whih n be pplied esily nd fst for wide rnge of solr olletors hs been reently worked in (Kisiny 014). he model is bsed on the wellknown methods of mthemtil sttistis more preisely the multiple liner regression (MLR). sed on the literture MLR is rre blk-box modelling tehnique in the field of olletors despite of its simpliity. Considering the high preision (with modelling error of 4.6%) simple usbility nd low omputtionl demnd of the mentioned MLR-bsed model (MLR model in short) 36
Institute for Mthemtis nd Informtis it ws worth trying to simplify further the MLR model. Suh simplified model lled SMLR model (with modelling error of 8.0%) hs been worked in (Kisiny 015). On the other hnd it ws lso worth improving the MLR model to try to mximize the preision. Suh n improved modellled IMLR model (with n error of 4.1%) hs been worked in (Kisiny 016) where it hs been empirilly shown tht the ury nnot be signifintly improved ny more if the regression equtions re ll liner in terms of the input vribles. In the present survey s summry of former works the bove MLR-bsed models re presented nd ompred with one nother nd with the physillybsed model by mens of mesured dt.. Physilly-bsed nd MLR-bsed models For the Reder s onveniene the physilly-bsed model nd the MLR-bsed models re relled in this setion. he sheme of the studied solr olletor n be seen in Fig. 1. Figure 1. Sheme of the solr olletor.1. Physilly-bsed model he physilly-bsed model is the ODE of Eq. (1). d dt U L v 0 I t t t in t t (1) V V V.. MLR model he inputs of the MLR model re from ppropritely hosen vlues of in I nd. he put is from ppropritely hosen vlues of. he flow rte vlue v is fixed positive onstnt or 0. euse of the boundedness of the flow rte in 1 n ply role s n input in the MLR model if is the put where the positive onstnt 1 is time dely (more preisely the time of flowing inside the olletor from the inlet 37
Institute for Mthemtis nd Informtis to the let when the pump is swithed on permnently). Similr onsidertions hold for I nd s well beuse of the bounded propgtion speed of their effets so former I nd vlues n ply roles s inputs in forming the put. (he time delys of I nd re ssumed to be the sme ( for the ske of simpliity.) Nturlly pproprite former vlue of lso ffets the vlue of ) nd prtiiptes s the initil vlue of the MLR model t time t in essene. Considering the olletor s blk-box distint sub-models s prts of the MLR model hve been identified for signifintly different operting onditions. For exmple the olletor behves different if the pump is on (v>0) or off (v=0) permnently. Even the effet of in is negleted in permnently swithed off se sine there is no flow between the olletor inlet nd let. Considering typil dy when the temperture inrese of is signifint three different min operting ses re distinguished ording to Fig.. C 1 C1 C 3 3 IW/m 4 time h 4 pump on/off 100 10 1.5 1 4 1 C1 C 3 3 4 Figure. Outlet temperture solr irrdine nd pump opertion on typil dy time h Cse orresponds to permnently swithed off pump more preisely Cse ontins the term strted t the beginning of the dy nd finished when the pump is first swithed on. ll the terms whih begin t time when the pump is 38
Institute for Mthemtis nd Informtis permnently off for extly time nd finish t the next swith-on or t the end of the dy lso belong to this se. ( is the time whih is generlly enough for to beome not flututing but permnently monotone sine intentionlly frequent flututions re hrteristis of Cses C1 nd C.) Cse orresponds to permnently swithed on pump more preisely Cse ontins ll the terms whih begin t time when the pump is permnently on for extly time nd finish t the next swith-off. ( is the time whih is generlly enough for to beome not flututing but permnently monotone.) Cse C orresponds to frequent swith-ons nd -offs. It n be seen tht there re two further signifintly different operting ses within Cse C: bsilly inreses before the solr noon nd bsilly dereses fter the solr noon so Cse C is divided into Cses C1 nd C. he MLR model is omposed of the liner equtions ()-(d) whih desribe the orresponding sub-model of eh operting se. Cse : Cse : I I () mod mod in in 1 I I (b) Cse C1: Cse C: mod C mod C in C1in 1 I C1I C1 1 C1 in C in 1 I C I C C () (d) I in I inc1 I C1 C1 C1 C1 inc I C C C re onstnt prmeters. ording to the C definition of the mesurements tke ple t times t 0 3... he modelled vlue of (tht is ) is determined t times t 3... mod 39
Institute for Mthemtis nd Informtis from the mesured vlues of t t on Eqs. ()-(d). See (Kisiny 014) for more detils. I nd in 1 bsed.3. SMLR model he MLR model is simplified to the SMLR model in the wy of merging Cses C1 nd C. hus there is only one operting se with one mthemtil reltion (see Eq. (3)) here. I mod in in 1 I (3) in I nd re onstnt prmeters. See (Kisiny 015) for more detils..4. IMLR model he IMLR model is similr to the MLR model. he min differenes re the following (see lso Fig. ): 1. he (lrgest) operting se Cse is divided into four sub-ses Cses 1 3 nd 4 s follows: Cse 1 onsists of the time period from the beginning of the dy to the time when the solr irrdine is first greter thn 10 W/m. his se prtilly belongs to the term of no irrdine in the first hlf of the dy. Cse onsists of the time period from the end of Cse 1 to the time when the solr irrdine is first greter thn 100 W/m. Usully this time is followed by very intensive inrese in the irrdine so this is pprently the time of sunrise when the irrdine hnges from (mostly) diffuse to (mostly) diret. Cse 3 onsists of the time periods besides Cses 1 nd 4 (see below) within Cse. Cse 4 ontins the lst three hours of the dy. In essene the term fter Cse C orresponds to the free ooling of the olletor from reltively high temperture. sed on experiments this setion nnot be modelled well with single reltion so it should be divided into sub-setions. Empirilly the problem n be solved well with only two sub-prts if the lst three hours re seprted.. he oeffiients of the zeroth-order members (f. C1 nd C in Eqs. ()-(d) in the MLR model) re set zero in Eqs. (4)-(4g) below. his set is in line with the physil phenomenon tht the olletor (let) temperture must be zero if ll the inputs in I nd the previous olletor temperture re zero. sed on experiments this nturl onstrint results in bit lower preision in the identifition but higher preision in the vlidtion tht is the modelling error dereses. he liner equtions (4)-(4g) orrespond to the operting ses of the IMLR model. Cse 1: mod 1 1 (4) 40
Institute for Mthemtis nd Informtis Cse : I mod I (4b) Cse 3: I mod I 3 3 3 (4) Cse 4: mod 4 4 (4d) Cse : Cse C1: Cse C: mod in in 1 I I mod in C1in 1 I C1I C1 C1 mod in C in 1 I C I C C (4e) (4f) (4g) 1 1 I I 3 3 3 4 4 in I inc1 I C1 C1 C1 inc I C C C re onstnt prmeters. 1 nd re the sme s in the MLR model. See (Kisiny 016) for more detils. 3. Comprison he below results of the models used in this omprison re from (Kisiny 014; 015 nd 016). he identifition nd the vlidtion of the models re bsed on the sme dys. he used rel flt plte olletor field of 33.3 m (Frks et l. 000) t the Szent István University (SZIU) in Gödöllő Hungry (SZIU olletor in short) is lso the sme. in I nd v re mesured 41
Institute for Mthemtis nd Informtis one in every minute t the SZIU olletor. he mesured vlue of serves only for identifition nd omprison purposes the mesured vlue (0 ) is fed into the models s initil ondition. Identifition Four mesured dys ( nd July 01 4 th June 01 8 th June 01 nd 8 th June 01) hve been seleted for the identifition in suh wy tht they over wide rnge of possible operting onditions of seleted seson (summer). he onstnt U is identified in se of the physilly-bsed model in suh wy tht L the time verge of the bsolute differene between the modelled nd mesured let tempertures tht is the verge of bsolute error is miniml with respet to the whole identifition period. he onstnts I in I inc1 I C1 C1 C1 C1 inc I C C C re C identified in the MLR model the onstnts in I re identified in the SMLR model nd the onstnts 1 1 I I 3 3 3 4 4 in I inc1 I C1 C1 C1 inc I C C C re identified in the IMLR model. Independent stndrd MLR rines hve been pplied bsed on the mesured dt of eh seprte operting se for the identifition of the three MLR-bsed models. he stndrd MLR rine (bsed on lest squres method) is well-known nd vilble in most sttistil nd spredsheet progrms (SPSS Exel et.). ble 1 ontins the verge of error (time verge of the differene between the modelled nd mesured let tempertures) nd the verge of bsolute error (time verge of the bsolute differene between the modelled nd mesured let tempertures) vlues for two dys ( nd July 01 8 th June 01) of the identifition of ll models. he verge of bsolute error vlues re presented in proportion to the differene between the dily mximl nd miniml mesured let temperture vlues s well in %. he men of these vlues with respet to ll of the four dys of the identifition is lso presented in ble 1 (7.8 % for the physilly-bsed 4.7 % for the MLR 6.6 % for the SMLR nd 3. % for the IMLR model). Vlidtion In the vlidtion ll identified models re pplied with the orresponding mesured inputs of the remining two summer months. he let temperture is modelled in the vlidtion (not mesured s in the identifition). he modelled dys re from 3 rd July to 31 st ugust 01 whih re 56 dys ording to minor tehnil interruptions. 4
Institute for Mthemtis nd Informtis ble 1 ontins the verge of error nd the verge of bsolute error vlues for two dys (3 rd ugust 01 5 th ugust 01) of the vlidtion of ll models. he verge of bsolute error vlues re presented in proportion to the differene between the dily mximl nd miniml mesured let temperture vlues s well in %. he men of these vlues with respet to ll of the 56 dys of the vlidtion is lso presented in ble 1 (7.8 % for the physilly-bsed 4.6 % for the MLR 8.0 % for the SMLR nd 4.1 % for the IMLR model). ble 1. verge of error nd verge of bsolute error vlues with the models Physillybsed model MLR model SMLR model IMLR model Identifition Vlidtion nd July (smooth opertion) 8 th June (intermittent opertion) Men % vlue for the whole identifition (four dys) 3 rd ugust (smooth opertion) 5 th ugust (intermittent opertion) Men % vlue for the whole vlidtion (3 rd July 31 st ugust) verge of error verge of bsolute error verge of error verge of bsolute error verge of bsolute error verge of error verge of bsolute error verge of error verge of bsolute error verge of bsolute error -1.86 C -0.47 C 1.43 C -0.53 C 4.33 C;.79 C; 3.88 C; 1.64 C; 7.0% 4.6% 6.3%.7% -1.6 C -0.3 C -.87 C -0.80 C 4.35 C; 3.01 C; 3.39 C; 1.84 C; 7.5% 5.% 5.8% 3.% 7.8% 4.7% 6.6% 3.% -1.38 C -1.31 C -0.1 C -1.18 C 4.70 C;.85 C; 3.71 C; 1.9 C; 7.4% 4.5% 5.8% 3.0% -.57 C -1.58 C -1.00 C -1.90 C 4.66 C; 3.07 C; 3.95 C;.30 C; 8.0% 5.% 6.7% 3.9% 7.8% 4.6% 8.0% 4.1% s exmples of the omprison Figs. 3 4 show the modelled nd mesured let tempertures in se of the physilly-bsed nd MLR models nd in se of the MLR nd IMLR models respetively for the sme dy of the vlidtion. he operting stte (on/off) of the pump is lso shown in the figures. 43
Institute for Mthemtis nd Informtis Physilly-bsed model MLR model Outlet temperture C 80 70 60 50 40 30 Outlet temperture C 80 70 60 50 40 30 pump on/off 0 mod mod 0 mes mes 0 5 10 15 0 4 0 5 10 15 0 4 time h time h 1 1 0 0 5 10 15 0 4 0 0 5 10 15 0 4 time h time h Figure 3. Modelled nd mesured mod olletor tempertures mes on 3 rd ugust 01 in se of the physilly-bsed nd MLR models pump on/off MLR model IMLR model Outlet temperture C 80 70 60 50 40 30 Outlet temperture C 80 70 60 50 40 30 0 mod mes 0 5 10 15 0 4 0 5 10 15 0 4 time h time h 1 1 0 0 5 10 15 0 4 0 0 5 10 15 0 4 time h time h Figure 4. Modelled nd mesured mod olletor tempertures on 3 rd mes ugust 01 in se of the MLR nd IMLR models pump on/off pump on/off 0 mod mes 44
Institute for Mthemtis nd Informtis sed on Eqs. (1) ()-(d) (3) nd (4)-(4g) it is not diffiult to see tht the SMLR model is the most while the IMLR model is the lest esy-to-pply MLR model with the lowest nd the highest omputtionl demnd respetively. Nevertheless ll MLR-bsed models with simple liner lgebri equtions hve lower omputtionl demnd thn the physilly-bsed model relized with n ODE. Conlusion In the present survey three reent MLR-bsed modelslled MLR model SMLR model nd IMLR model hve been presented nd ompred with one nother nd with well-tried physilly-bsed model by mens of mesured dt. ording to the results ll the MLR SMLR nd IMLR models hve proved to be rther preise with modelling error of 4.6% 8.0% nd 4.1% respetively whih mens tht ll MLR-bsed models re more or nerly the sme urte s the physilly-bsed model with n error of 7.8%. he SMLR model is the most while the IMLR model is the lest esy-to-pply MLR model with the lowest nd the highest omputtionl demnd respetively. Nevertheless ll the MLR-bsed models pplying simple liner lgebri equtions hve lower omputtionl demnd thn the physilly-bsed model relized with n ODE. ordingly the MLR-bsed models n be suggested for fst but urte olletor modelling. knowledgement he uthor thnks his ollegues in the Deprtment of Mthemtis in the Fulty of Mehnil Engineering (Szent István University) for their support. his pper ws supported by the János olyi Reserh Sholrship of the Hungrin demy of Sienes. Referenes [1] uzás J. Frks I. iró. Németh R. (1998) Modelling nd simultion of solr therml system Mthemtis nd Computers in Simultion Vol. 48 pp. 33-46. [] uzás J. Kisiny R. (014) rnsfer funtions of solr olletors for dynmil nlysis nd ontrol design Renewble Energy Vol. 68 pp. 146-155. [3] Duffie J.. ekmn W.. (006) Solr engineering of therml proesses 3 rd ed. John Wiley nd Sons New York. [4] Frks I. uzás J. Lágymányosi. Klmár I. Kboldy E. Ngy L. (000) ombined solr hot wter system for the use of swimming pool nd 45
Institute for Mthemtis nd Informtis kindergrten opertion Energy nd the environment Vol. I. /ed. by. Frnkovi/ Crotin Solr Energy ssoition Optij 000. pp. 81-88. [5] Fisher S. Frey P. Drük H. (01) omprison between stte-of-thert nd neurl network modelling of solr olletors Solr Energy Vol. 86 pp. 368-377. [6] Kisiny R. (014) Multiple liner regression bsed model for solr olletors Solr Energy Vol. 110 pp. 496-506. [7] Kisiny R. (015) Simplified multiple liner regression bsed model for solr olletors Hungrin griulturl Engineering Vol. 8 pp. 11-14. [8] Kisiny R. (016) Improved multiple liner regression bsed models for solr olletors Renewble Energy Vol. 91 pp. 4-3. [9] Kumr R. Rosen M.. (010) herml performne of integrted olletor storge solr wter heter with orrugted bsorber surfe pplied herml Engineering Vol. 30 pp. 1764-1768. 46