PREDICTING THE TEMPERATURE PROFILE OF INDOOR BUILDINGS BY USING ORTHONORMAL BASIS FUNCTIONS

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1 Eleventh Internatonal IBPSA Conference Glasgow, Scotland July 27-30, 2009 PREDICTING THE TEMPERATURE PROFILE OF INDOOR BUILDINGS BY USING ORTHONORMAL BASIS FUNCTIONS Bruno C. Regnato¹, Roberto Z. Frere², Gustavo H. C. Olvera¹, Nathan Mendes² and Marc O. Abade³ ¹, ² - Pontfíca Unversdade Católca do Paraná / ¹ - PPGEPS / ² - PPGEM Curtba/Brasl - Zp Code ³ - Unversty of La Rochelle La Rochelle/France bruno.regnato@yahoo.com.br; {roberto.frere;gustavo.olvera;nathan.mendes}@pucpr.br; mabade@unv-lr.fr ABSTRACT Orthonormal Bass Functons (OBF) s a structure of dynamc models that have been appled n dfferent classes of dynamc systems. Several works descrbng the theory and applcablty of OBF (Orthonormal Bass Functons) n dentfcaton and control can be found n the lterature. Ths work s focused on the problem of fndng a Multple- Input/Sngle-Output (MISO) OBF model for predctng ndoor ar temperature and energy consumpton. The am s to analyse an alternatve way to do so n relaton to well establshed buldng energy smulaton tools. The model s bult n terms of the followng varables, that s, the nput data, are heatng power, outdoor temperature, relatve humdty and total solar radaton, and the output data nvolved s the ndoor temperature. The methodology has been tested for the low thermal mass case of the BESTEST model and the output data has been generated by usng a buldng hygrothermal smulaton tool. Valdaton procedures have shown very good agreement n terms of temperature predcton errors between the model and the smulaton tool data for both wnter and summer perods. INTRODUCTION Snce gas and ol resources are dmnshng ths century, the world faces an energy crss. The current energy problem has a strong connecton to global warmng and both problems are rooted n the unsustanable use of fossl fuels as our prmary energy source. Moreover, when the buldng desgn processes are consdered n the energy savngs concept, HVAC systems are almost crtczed as one of the man responsble for the energy consumpton n resdental and commercal buldngs. Most refrgerants used n ar condtonng and refrgeraton contrbute to global warmng n addton to ozone depleton. Even the new non-ozone depletng alternatve refrgerants add to the global warmng problem. Tryng to reduce the HVAC carbon emssons, researches are tryng to avod the energy waste of HVAC systems based on development of buldng models. In general, the man obectves of obtanng models for thermal analyss n offces, resdental buldngs and shoppng malls are: ) estpulate better ndoor clmate condtons for the occupants; ) avod the energy waste to decrease the HVAC equpment operatng cost and carbon emssons and ) smulate buldngs nteractng wth HVAC equpment. A classfcaton usually found n the lterature for buldng models s: analytcal, sem-emprcal and emprcal models. The analytcal models depend on many parameters and are based on parametrc equatons. In a smlar way, the sem-emprcal models depend on test data and less parameters, combnng analytcal and emprcal calculatons. Smulaton programs such as PowerDomus (Mendes et al., 2003a) and ASTECCA (Mendes et al., 2003b) have analytcal and sem-emprcal models for HVAC systems. Fnally, emprcal models (or black box models) are usually appled to a system when t s vewed n terms of ts nput, output and transfer characterstcs wthout any knowledge requred of ts nternal workngs. Black box models can be extremely useful for a varety of purpose, namely predcton, control, relablty aspects and system management. In hygrothermal systems modelng procedures, when the combnaton of the HVAC and buldng envelope study are based on the black box concepts, several detaled analyses can be done, for nstance, temperature, thermal comfort and energy consumpton predctons (Athents et al., 1990; Chen and Chen, 2000). An nterestng applcaton s the synthess of advanced control technques for HVAC equpments, where a model s usually requred for the controller synthess. An example of an advanced controller n the HVAC context s the Model Predctve Control Technque (Frere et al., 2008b; Donasky et al., 2007). Over the last decades, a large number of models have been developed n order to understand buldng behavors submtted to dfferent clmate condtons. As presented by (Frere et al., 2008a) black box models can be used to verfy a buldng hygrothermal system behavor based on data provded by a

2 buldng energy smulaton software. The models have been used for thermal comfort control based, affectng the ndoor temperature relatve humdty. Another example of such knd of modelng has been proposed by Gvon and Krüger. Based on Gvon (1999) method, they presented results of the applcaton of formulae to predct daly ndoor temperatures n three montored low-cost houses n Curtba, Brazl. In (Papst and Lamberts, 2001), a thermal performance analyss based on Gvon's regresson model has been presented. Thermal performance comparsons between three resdental buldngs have been made. Gvon and Krüger (2003) have presented predctve regresson equatons to evaluate maxmum, average and mnmum temperatures of specfc houses occuped by dfferent famles. Vrk and Loveday (1994) have presented a model n whch the multvarable stochastc dentfcaton technque s appled to a full-scale test room wth dedcated heatng, ventlaton and ar condtonng plant. Ths model could predct ndoor temperature of a test-zone n cold clmates. Consderng the wde applcablty of black box models found n the lterature, the present paper s focused on a methodology, known as system dentfcaton, and on the proposal of the use of orthonormal bass functons structure for a HVAC ntegrated to a buldng model. The am s to obtan models capable to descrbe the HVAC system and buldng behavor n a structure that could be useful for advanced control law synthess. Moreover, ths paper can be seen as an evoluton of the work presented n (Frere, et al., 2005a) and (Frere, et al., 2008a) where a new black box modelng strategy has been evaluated. The next secton of ths paper descrbes the smulaton tool used to generate data for the dentfcaton process and the buldng structure. Then, the buldng dentfcaton method has been presented followed by the model estmaton and valdaton procedures. Fnally, an applcaton example of such methodology s dscussed and conclusons about ths research are addressed. SIMULATION TOOL AND ENVIRONMENT The buldng smulaton tool used n the dentfcaton procedure s the Brazlan PowerDomus software (Mendes et al., 2003), whch s based on a lumped formulaton. Its energy balance consders: sensble and latent conductve heat transfer loads; convectve heat transfer; long- and short-wave radaton; nfltraton; ventlaton; loads related to the HVAC system beyond other varables. Fgure 1 shows the user nterface of PowerDomus software. It can be seen the BESTEST proect geometry used as a buldng structure n the dentfcaton procedures. PowerDomus s capable to provde all the output reports necessary for the buldng dentfcaton procedures proposed n ths paper, becomng data acquston easer. Fgure 1. PowerDomus BESTEST geometry. The BESTEST buldng structure has been choosed because of the avalable data to compare and valdate buldng smulaton software (IEA, 1995). PowerDomus has provded results n agreement to dfferent buldng smulaton software (Frere, et al., 2008a). The buldng structure s defned as follows. The BESTEST buldng s based on the sngle-zone model of the BESTEST methodology for the low thermal mass case (600). Table 1 presents the buldng envelope materal propertes. The room dmensons are 8.0m X 6.0m X 2.7m of length, wdth and heght, respectvely. There are also two Southorented 6.0m² wndows. The nput sgnals appled on the buldng smulaton tool are the Denver s weather varables, that s, outdoor temperature, outdoor relatve humdty and dffuse and drect solar radatons. The nput sgnals used on the dentfcaton procedure are the clmate varables descrbed before and the control sgnal appled to a heatng system of 5 kw power. The output sgnal s the ndoor temperature. IDENTIFICATION METHOD The system dentfcaton s a theory where models are constructed from observed data. In the black box system dentfcaton approach, a par of nput/output data s collected from the system and, by means of an optmzaton procedure, the best model that fts the collected data are computed. When a sngle par of nput/output data s used, one obtans a SISO

3 (Sngle-Input-Sngle-Output) dentfcaton procedure and f more than one nput and output data are nvolved, a MIMO (Multple-Input-Multple- Output) procedure s performed. Table 1 Materals for lghtweght BESTEST 600 case. MATERIAL k ρ c p (J/kg.K) d (m) (W/m²) (kg/m³) Exteror wall (nsde to outsde) Plasterboard Fberglass qult Wood sdng Floor (nsde to outsde) Tmber floorng Insulaton Roof (nsde to outsde) Plasterboard Fberglass qult Wood sdng A thermal system, as an ndoor envronment, could be defned as a MIMO (Multple-Input-Multple- Output) system. It could be dvded n subsystems that relate all the nput varables to each output. In the present paper, the consdered output s the ndoor temperature. The nput sgnals used n the dentfcaton process are heatng power, outdoor temperature, outdoor relatve humdty and total solar radaton. The MIMO system dentfcaton procedure follows the flow depcted n Fgure 2 (Lung, 1999) (Johansson, 1993). Ths procedure, n the context of the present paper, s dscussed as follows. Fgure 2. The system dentfcaton loop. Experment Desgn and Data A two-year smulaton perod has been performed usng the PowerDomus software. Results for the last year have been used n the dentfcaton procedures n order to reduce the ntal condtons effects. A tme step of 60s and the weather fle from Denver cty USA have been used as nput data n the software as well as a random control sgnal appled to a heatng system (a heatng system of 5 kw maxmum power). The reported data are: the control sgnal, outdoor temperature, outdoor relatve humdty and total solar radaton and the reported output s the ndoor temperature. The expermental data (recorded nputs and outputs over a tme nterval 1 k N) for the system dentfcaton procedure can be put n the followng structure: N = Z { u(1), TEXT (1), H EXT (1), S EXT (1), y(1),..., u( N), T ( N), H ( N), S ( N), y( N)} EXT EXT EXT (1) y( s the ndoor temperature (n C). T EXT (, H EXT ( and S EXT ( are the outdoor temperature (n C), outdoor relatve humdty (n %) and total solar radaton (n W/m²) respectvely. u( s the nput sgnal appled on the HVAC system that vares from 0 to 1, representng the turn off and turn on states, respectvely. SYSTEM MODELING USING ORTHONORMAL BASIS FUNCTIONS As mentoned before, the choce of the model structure s an mportant step on the dentfcaton procedures. In (Frere, et al., 2005a) and (Frere, et al., 2008a), the ARMAX structure has been adopted for developng a buldng coupled to the HVAC model. In present paper, the use GOBF structure n buldng smulaton context s analyzed, so ts concepts are revewed n followng. A SISO lnear causal dynamc system, wth fnte memory, can be descrbed by ts transfer functon H(z) or ts mpulse response h(, that s: Y = H U (1) or y ( = h( k ) u( ). (2) = 0 In ths equaton, u( and y( are the nput and output dscrete-tme sgnals, respectvely. Moreover, f the mpulse response has fnte memory, h( can be represented by a seres of orthonormal functons, as follows:

4 h( = c φ (, (3) = 1 where φ (, =1,,, s a base of orthonormal functons and c s the -th seres coeffcent. By substtutng (2) nto (1): y( = c φ ( k ) u(, (4) = 1 = 0 and by applyng the Z-Transform, one obtans: Y = c Φ U. (5) = 1 The seres coeffcents converge to zero as ncreases. So, the seres approxmaton error, defned as: n ε n ( = y( c φ ( k ) u(, = 1 = 0 (6) can be made as small as wshed ust by augmentng the seres approxmaton truncaton value n. Therefore, the model output y ˆ( s gven by: Y ˆ = n c Φ U ( Z). = 1 (7) The key dea of such approach s to descrbe the system mpulse response, and consequently, the system transfer functon, by a seres of orthonormal functons. So, an orthonormal bass s defned. Dfferent orthonormal functons can be used n such context and a class of these functons s the one constructed by usng the domnant system dynamc(s), gven n terms of poles, n a ratonal transfer functon format (Heuberger et al., 2005). An ssue related to the problem of defnng a ratonal orthonormal bass s the choce of the value and number of dfferent modes n the bass functons. For nstance, ratonal orthonormal functons characterzed by one sngle mode, whch s gven by a real pole or a par of complex ones, are known as Laguerre and Kautz functons, respectvely. However, by defnng bass wth more than one dynamc response, one can mprove the seres convergence, leadng to a more parsmonous representaton n relaton to bass havng only one mode. A way of defnng bass functons wth several modes s to use nner functons, as t wll be shown n the followng. Such knd of bass s known as Generalzed Orthonormal Bass Functons (GOBF) (Van Den Hof et al., 1995) or Orthonormal Bass Generated by Inner Functons (OBGIF). Laguerre and Kautz bass can be vewed as a specal case of GOBF/OBGIF ones. Lets G (z) a transfer functon wth all-pass behavor, defned by: 1 p * z G ( ) = z, (8) p z 1 where p s a complex pole and p * s ts complex conugate. Snce G (z)g (z -1 )*=1, G (z) can be called an nner functon. Lets G b (z) a transfer functon gven by a cascade realzaton of n b functons G (z), each one havng a dfferent pole (real or complex), knowng that the complex ones come n conugate pars. These n b poles contan the a pror nformaton about the system domnant dynamcs. G b (z) s also an nner functon. Lets the quadruple (A,B,C,D) the space state realzaton of G b (z), and: V z zi A 1 1 ( ) = ( ) B. (9) An orthonormal bass can be defned by a cascade realzaton of n g functons G b (z) as follows. Lets ϕ ( the -th element of the sgnal vector v ( and V (z),.e., the Z nverse transform of v (, gven by: 1 1 V = ( zi A) BG, b = 1, Kn g (10) So ϕ (, =1,, n b and =1,, n g s a set of functons that forms a complete bass n the Lebesgue space (Van Den Hof et al., 1995). Equaton (7) can thus be rewrtten as follows: n g Y ˆ = C V U ( Z), (11) = 1 where C vector contans the n b coeffcents of the system orthonormal bass realzaton. By defnng L (z) equal to the product V (z)u(z), one obtans: n g Y ˆ = C L = C L( z), (12) = 1 where C and L(z) are the concatenaton of C and L (z), respectvely, for =1,, n g. The model structure s llustrated by Fgure 1 and the space state realzaton s gven by: L( k + 1) = AL( + Bu(, (13) yˆ( = CL( where the state vector and matrces A e B are obtaned by usng the number of nner functons and ts prevously selected poles locatons. The procedure for system dentfcaton by usng OBGIF model structure can be vewed as presented n Fgure 1. ) Frst, t s necessary to select the bass structure, bascally, the functon poles (Equaton (8)) and the number of functons (n g ). These parameters wll

5 defne matrces A and B order and elements. ) Second, the model parameters, represented by matrx C, are estmated. Such procedure can be performed teratvely and ntalzed by usng some pror knowledge about the system dynamcs. 600FF Wnter The nput data are presented n Fgure 4. It represents the data collected for a 15-day wnter perod n northern hemsphere (from January 1 st to January 15 th ). As presented before, these nput sgnals have been obtaned from smulatons performed wth the PowerDomus software. Fgure 3. Orthonormal bass model structure. IDENTIFICATION PROCEDURES The steps reported by Fgure 2 were descbed n the prevous secton, where the Model Set s represented by the OBF structure. By usng the Least Square Crteron to ft the model, n ths secton, the parameter estmaton procedure s presented. It s dvded nto two dentfcaton procedures n order to obtan two buldng models: ) 600FF Wnter; and ) 600FF Summer. For both wnter and summer models, the control sgnal has been turned on for determnng the buldng dynamc n relaton to such nput. In the dentfcaton procedure presented here, the collected data has a 15-day sample perod. The system has 4 nputs and 1 output,.e., the ndoor temperature. The legend for the nput sgnals s shown on Table 1. The MSE (Mean Square Error) crteron s used here to evaluate the ftness of the computed model. Fgure 4. Wnter nput data for model estmaton. The frst step s to defne the bass structure, whch s gven by the set of bass functons poles and by the number of functons n the bass. The bass poles selecton procedure s dscussed n (Regnato et al. 2007) and the selected poles are gven n Table 2. Table 2 presents the estmaton parameters acheved by the algorthm, the system output s shown wth the model output n Fgure 5, and the MSE between the system output and the model output s It can be notce that, even after 15 days, the model, feeded wth the same nput data, s able to predct the output wth a qut small error. Table 1 Input Data Descrpton. INPUT LEGEND INPUT DESCRIPTION #1 (ºC) Outdoor Temperature #2 (-) Outdoor Relatve Humdty #3 (kw/m²) Total Solar Radaton #4 (kw) Control Sgnal (HVAC) Fgure 5. GOBF model estmaton for wnter season

6 Table 2 Wnter Estmaton Parameters. INPUT NUMBER POLES LOCATION LEGEND OF BASIS #1 (ºC) [ ] 2 #2 (-) [ ] 2 #3 (kw/m²) [0.9929] 1 #4 (kw) [ ] 2 600FF Summer The nput data are presented n Fgure 6. As seen on that fgure, the fourth nput sgnal s zero almost for all the 15 days of the summer perod (from July 1 st to July 15 th ), except for the ntal samples. Ths behavor occurs because the envronment temperature s hgh enough, so that the Control Sgnal actuaton (generated by a heatng unt) s not necessary. Fgure 6. Summer nput data for model estmaton. Fgure 7. GOBF model estmaton for summer season. MODEL VALIDATION The valdaton procedures use the same data prevously shown n Fgure 4 and 6, except that t does not compute the nput sgnal denoted Control Sgnal for both seasons. Ths controlled stuaton has been performed n order to change the nput behavor and consequently ts output n order to test the computed models. 600FF Wnter The system and model outputs are shown n Fgure 8 and ts MSE s Smlar to the prevous analyss, when the same outdoor model s appled n the model, t s able to predct the ndoor temperature several days n advance showng that the dynamc of the buldng could be captured n representaton. Table 3 presents the estmaton parameters acheved by the algorthm, the system output s shown wth the model output n Fgure 7 and the MSE between the system output and the model output s The mean error and the open loop predcton behavor almost the same for the wnter case. Table 3 Summer Estmaton Parameters. INPUT NUMBER POLES LOCATION LEGEND OF BASIS #1 (ºC) [ ] 2 #2 (-) [ ] 2 #3 (kw/m²) [ ] 2 #4 (kw) [0.9796] 1 Fgure 8. Wnter valdaton curves

7 600FF Summer Fgure 9 shows the system and model outputs for the summer season and ts MSE of An equvalente bevavor of the prevous secton can be appled here. the BESTEST 600 buldng case showed very hgh correlaton coeffcents. The heavy mass BESTEST 900 case wll be analysed n future works. For valdaton purposes, the same weather data has been used to predct the ndoor ar temperature for the wnter and summer perods. The results were compared wth those obtaned wth the same buldng smulaton that created the data for the estmaton. The equatons obtaned n the dentfcaton procedure have shown that t can be used n a very relable way, provdng results as accurate as the orgnal data, whch can be gathered from numercal or expermental means. In addton, the equatons can be mplemented and results n terms of buldng hygrothermal and energy response can be rapdly obtaned for any weather condton, any HVAC capacty and any nteror heat gan. ACKNOWLEDGMENTS Fgure 9. Summer valdaton curves. Table 4 summarzes the MSE of all smulatons, that s, the estmaton and valdaton cases. It can be see that mean error between the actual data and the model predcton over a future wndow of ffteen days ahead s close to 1 C, whch s acceptable for control and comfort applcatons. In addton to such small mean square errors, t can be depcted by the fgures that the bggest ampltude of errors plots can be estmated as smaller than 3 or 4 C. Table 4 Models summary ndexes ESTIMATION VALIDATION MEAN WINTER ,7868 SUMMER ,3448 CONCLUSION Ths paper has descrbed the development of buldng thermal Orthonormal Bass Functons model based on an dentfcaton process. The collected data was obtaned from smulatons performed wth a buldng smulaton tool - PowerDomus. GOBF models can be adequate to a great varety of purposes, such as thermal comfort and energy consumpton predcton, HVAC equpment control and system management, n a rapd and easy way. In order to test the presented procedure, a Denver meteorologcal weather fle has been used to generate the collected data from the frst 15 days of January (wnter) and July (summer) usng a samplng tme of 1 mn, provdng 21,600 pars of nput/output for the wnter perod and also the same number for the summer perod. In ths way, a model developed for The authors would lke to acknowledge CAPES/Brazl (Coordenação de Aperfeçoamento de Pessoal de Nível Superor), FINEP/Brazl (Fnancadora de Estudos e Proetos) and CNPq/Brazl (Conselho Naconal de Desenvolvmento Centífco e Tecnológco) for supportng ths work. REFERENCES Athents, A.K., Stylanou, M., Shou, J A Methodology for Buldng Thermal Dynamcs Studes and Control Aplcatons. ASHRAE Transactons, 96 (2), Bllngs, S.A., Voon, W.S.F Correlaton Based Model Valdty Tests for Non-Lnear Models. Internatonal Journal of Control. Vol. 44. N 1. pp Chen, Y., Chen, Z A Neural-Network-Based Expermental Technque for Determnng z- Transfer Functon Coeffcents of a Buldng Envelope. Buldng and Envronment, 35, Clarke, D.W Advances n Model Based Predctve Control. Oxford Unversty Press. Donasky, E., Olvera, G.H.C. Mendes, N PMV-Based Predctve Algorthms for Controllng Thermal Comfort n Buldng Plants, 16 th IEEE Conference on Control Applcatons (CCA), Suntec Cty, Sngapore. Fanger, P.O Thermal Comfort, McGraw-Hll Inc.. New York, USA. Frere, R.Z., Olvera, G.H.C., Mendes, N. 2005a. Development of Sngle-Zone Predctve Equatons Usng Lnear Regresson for Advanced Controllers Synthess. Proc. Of the 9 th

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