GEOMETRIC OPTIMIZATION OF FENESTRATION. Jonathan Wright and Monjur Mourshed

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1 Eleventh Internatonal IBPSA Conference Glasgow, Scotland July 27-30, 2009 GEOMETRIC OPTIMIZATIO OF FEESTRATIO Jonathan Wrght and Monjur Mourshed Department of Cvl and Buldng Engneerng, Loughborough Unversty, Loughborough, Lecestershre, LE11 3TU, UK ABSTRACT Fenestraton and ts desgn have a sgnfcant mpact on the energy use assocated wth the artfcal lghtng, heatng and coolng of a buldng. To date, research nto wndow optmzaton has been for wndows that are constraned to have a regular geometrc shape and poston. Ths paper descrbes an approach n whch a buldng façade s dvded nto a number of cells, each cell havng one of two possble states, a sold wall constructon, or a wndow. A Genetc Algorthm search method has been used to optmze the state of each cell, wth the number or aspect rato of the wndows beng constraned were desrable. It can be concluded that the approach results n desgn solutons that have nterestng and nnovatve archtectural forms and whch mnmze buldng energy use. ITRODUCTIO Fenestraton and ts desgn have a sgnfcant mpact on the energy performance of buldngs; t mpacts on daylght penetraton, artfcal lghtng energy use, and heatng and coolng energy use. To date, optmzaton methods have been used to mnmze buldng energy use by optmzng the dmensons of the wndows and shadng elements, but wth the wndow shape always beng rectangular and only mnor movement n the poston of the wndows on the façade beng allowed (among others; Caldas and orford, 2002). Although ths approach can result n reduced buldng energy use (and assocated carbon emssons), the fact that the shape and postons of the wndows on the façade are fxed lmts the extent to whch energy use can be reduced. Restrctng the shape of the wndows to be rectangular also nhbts the archtectural form of the fenestraton. The objectve of the research presented n ths paper s to nvestgate an approach to fenestraton optmzaton n whch the shape, number, and poston of wndows can be optmzed for mnmum buldng energy use. The optmzaton s mplemented by dvdng the buldng façade nto a number of small rectangular cells, each cell beng defned by the optmzaton to be ether a sold constructon or glazed. Snce the desgner may wsh to mantan some control over the general shape and number of wndows, metrcs have been developed for the cellular wndow structure that allow the aspect rato and number of wndows to be constraned durng the optmzaton. The cellular optmzaton approach can result n desgn solutons that resonate wth the form of fenestraton developed by Le Corbuser n the desgn of the Chapel of otre Dame du Haut (a wdely known example of modern archtecture). The South façade of ths buldng, known as the wall of lght, has wndows of varyng sze that are randomly placed on the wall (Fgure 1). It s consdered by some (Buser, 2005), that ths results n a fenestraton desgn that goes beyond the general purpose of llumnatng the nteror space and enables an aesthetc nterplay of lght and form that embodes a feelng of ntmacy and an aura of mystery. Fgure 1, Façade Detal, Le Corbuser s Chapel of otre Dame du Haut OPTIMIZATIO APPROACH The optmzaton of buldng fenestraton nvestgated n ths paper s based on the buldng façade beng dvded nto a number of rectangular cells. Each cell has one of two possble states, a wndow element, or an opaque constructon element. The number and locaton of the cells havng a wndow constructon s set by an optmzaton algorthm, the objectve of the optmzaton beng to mnmze buldng energy use. In order to allow the desgner to have some control over the form of wndows generated, several metrcs

2 have also been developed that can be used both as desgn constrants durng the optmzaton, and n the analyss of the desgn solutons. Wndow Metrcs Fve metrcs have been defned for use as desgn constrants and/or n the analyss of solutons: the number of wndows; wndow area; wndow aspect rato; wndow densty; the centre of gravty of the wndows. umber of wndows: Fgure 2 llustrates a façade of 15 cells wde by 8 cells hgh (a total of 120 cells). The ndvdual wndow cells are shaded (blue), and have a bold outlne. A wndow s defned by a set of adjonng wndow cells, there beng 3 such wndows n Fgure 2. ote also that as ndcated for Wndow 1, cells that are adjacent at ther corner pont are consdered to be part of the same wndow. Fgure 2, Cellular Wndow Façade llustrated as a dashed lne. The aspect rato of a wndow, s gven by: AR = (2) cellshgh cellswde where; AR s the aspect rato of wndow, cellshgh s the number of cells formng the heght of the boundng-box, and s the number of cellswde cells formng the wdth of the boundng-box. As such, Wndow 1 n Fgure 2 has an aspect rato of 1.0, Wndow 2, 1.0, and Wndow 3, Wndow densty: the majorty of conventonal wndows are completely rectangular. The extent to whch the optmzed wndows are rectangular s defned by the wndow densty, ths beng a measure of the extent to whch the wndow cells fll the rectangular boundng-box: wndowcells = totalcells D (3) where; wndow, 100 D s the percentage densty, of wndowcells cells assocated wth wndow, and s the number of wndow totalcells the total number of cells assocated wth wndow. Wndow 1 n Fgure 2 has a densty of 50%, Wndow 2 100%, and Wndow 3 40%. Wndow poston: the bas n poston of the wndows s measured through the mean centre of gravty of all wndows. Ths s gven as a vertcal and horzontal coordnate poston: Wndow area: the total wndow area s expressed as a percentage of the total façade area and s defned by: A = wndowcells total wndowtota l 100 (1) where; area, nt AwndowTotal wndowcells total s the total percentage wndow s total number of wndow cells, and nt s the total number of cells on the façade. The total percentage area of all wndows n Fgure 2 s 9.2%. Wndow aspect rato: aspect rato has been defned as heght dvded by wdth. The heght and wdth of a wndow s descrbed by the rectangular box that forms a boundary on all cells of the wndow. The boundng-box of the wndows n Fgure 2 s CoG CoG v nr j= wndowcells j wndowcells total j = 1 (4) nc wndowcells k k k= h = 1 (5) wndowcells total where; CoGv and CoG h are the vertcal and horzontal coordnate postons respectvely, j and k are the ndex of the cellular row and column respectvely (Fgure 2), and wndowcells j wndowcells k are the number of wndow cells n row j wndowcells total and column k respectvely, and total number of wndow cells on the façade. s the

3 The centre of gravty of the wndows n Fgure 2 s marked wth a +; the assocated arrow ndcates the drecton and dstance of the centre of gravty from the centre of the façade. Problem Formulaton The optmzaton problem s defned by: nl heat cool elect ( Q + Q + Q ) l l l mn f ( X ) = (6) subject to: l= 1 { 1, ng} g ( X ) 0.0;, where; (7) Q, heat l Q, and elect are the heatng, cool l coolng and electrcal energy use (kwh) at buldng load condton l; nl s the number of load condtons; the electrcal energy use ncludng the energy used n artfcal lghtng. g (X ) s constrant functon, there beng a total of ng constrant functons. The constrant functons are formulated from the wndow metrcs; n the experments descrbed n ths paper, two metrcs have been used to constran the desgn solutons; the total number of wndows; and the wndow aspect rato. X s a vector of dscrete problem varables whch s mapped to a nc by nr matrx of bnary decson varables x k, j, one varable for each cell on the façade: x1,1, x x1,2, x X x1, nr, x x {0,1}; k, j 2,1 2,2 k {1,, nc}; j {1,, nr},, x,, x 2, nr nc,1 nc,2,, x Q l nc, nr ; (8) where, nc s the number of columns of cells, and nr the number of cell rows on the façade (Fgure 2). A decson varable value of 0 results n cell (k, j) havng a sold opaque constructon element, whereas a value of 1 results n a wndow element. Optmzaton Algorthm A bnary encoded Genetc Algorthm (GA) has been used as the optmzaton method n ths research. ot only have GA been shown to be effectve n solvng buldng optmzaton problems (Wetter and Wrght, 2004), but the GA bnary encodng of the varables naturally lends tself to the characterstcs of the cellular wndow optmzaton problem varables, these also beng bnary. The form of GA mplemented here s based on: random ntalzaton of the problem varables; a Gray bnary encodng of the varables (De Jong, 2006); bnary tournament selecton (De Jong, 2006), wth the soluton ftness beng assgned by the stochastc rankng algorthm (Runarsson and Yao, 2000); 100% probablty of chromosome crossover and a 50% probablty of bnary gene crossover (known as unform crossover; De Jong, 2006); a probablty of 1 gene mutaton per chromosome; eltsm n the form of the sngle best soluton from the prevous generaton beng ncluded n the new generaton; completon of the search after a fxed number of unque solutons have been evaluated; ths beng set to a 1000 for the unconstraned optmzaton, and 3000 for the more demandng constraned optmzaton; a populaton sze of 30 ndvduals wth automatc re-ntalzaton of the populaton f t collapses before completon of the search. EXAMPLE OPTIMIZATIO The example optmzaton s based on an atrum of a three-storey commercal buldng located n Chcago, USA. The atrum s 15m wde by 15m long by 8.2m hgh wth only the southern façade beng exposed to the external envronment. The other three sdes of the atrum are connected to nteror spaces that are controlled to have the same thermal condtons as the atrum. The roof, nternal partton walls and the external façade have a lght-weght constructon; the floor s constructed from unnsulated concrete; and the wndow cells have a double-glazed constructon. Performance Smulaton The EnergyPlus whole buldng performance smulaton (verson ; Crawley et al. 2001), has been used to evaluate the atrum energy use of each desgn soluton. The atrum has been modelled as an ndependent zone wth the nternal partton walls beng treated as adabatc heat transfer surfaces. The performance of the atrum HVAC system has been modelled usng a pseudo-system havng an dealzed 100% effcency (the EnergyPlus

4 purchased ar model). Ths model s sutable for estmatng the heatng and coolng energy use durng the early desgn stages (where there s less focus of the detaled desgn of the HVAC system; Mourshed et al. 2003). The system s operated 24 hours/day wth coolng and heatng setponts for the occuped perods beng set to 25.6 C and 20 C respectvely; nght setback setpont temperatures are 30.0 o C for coolng operaton and 15.0 o C for heatng operaton. The artfcal lghtng (and ts energy use), s used to supplement daylght llumnance levels by an amount that mantans the llumnance setpont (500lux), calculated at two reference ponts (both reference ponts havng equal weght n the control of the artfcal lghtng). The two reference ponts are located along the md-pont of the façade wdth, and at a dstance of 25% and 75% of the depth of the atrum. The buldng s fully occuped from 09:00 to 17:00; the occupancy s reduced durng the perods from 07:00 to 09:00, and 17:00 to 22:00. Optmzaton Problem Varables The southern façade of the atrum has been dvded nto 15 cells wde by 8 cells hgh (nc=15; nr=8; a total of 120 cells and problem varables). Each cell can have one of two possble states (opaque or glazed), and therefore, there are a total of possble solutons to ths optmzaton problem (1.3x10 36 solutons). ote that cell (k=1, j=1), corresponds to the top-east corner cell on the façade and cell (k=nc, j=nr) to the bottom-west corner cell (expresson 8; Fgure 2). Experments Four dfferent sets of results have been generated, an ncremental traverse of the soluton space, and three dfferent optmzaton runs. The ncremental traverse s not a formal optmzaton run but has been used to examne the general characterstcs of the optmzaton problem. In ths case, the traverse ncrementally adds wndow cells to the façade (Fgure 3) startng from the top-east corner (k=1, j=1), and endng n the bottom-west corner (k=nc, j=nr). Fgure 3, Incremental Traverse of the Soluton Space The three optmzaton runs presented n ths paper are: 1. an unconstraned optmzaton; 2. mnmzaton of energy use wth the aspect rato constraned to be n the range 1.5 to 1.75, ths range beng set to allow several alternatve wndow szes to occur wthn an 8 cell hgh façade and havng an aspect rato close to the Golden Rato (Tables 1); the Golden Rato (Heght/Wdth=1.62), s often used n defnng the geometrc proportons of a buldng, and n ths case s set to produce tall-thn wndows that emphasze the heght of the atrum; Heght (cells) Table 1, Possble Range of Solutons wth Aspect Rato Constraned Wdth for an approxmate Golden Rato (cells) Aspect Rato (-) mnmzaton of energy use wth the total number of wndows constraned to be less than or equal to 3. RESULTS AD AALYSIS Fgure 4 llustrates that the optmum wndow area s n the order of 22%, ths correspondng to 27 wndow cells (whch n the case of the ncremental traverse are located n the top two rows of the façade). ote that, although t s expected that the wndow cells would be dstrbuted towards the top of the façade (as ths maxmses daylght penetraton and reduces lghtng energy use), t s lkely that the optmzed number and dstrbuton of wndow cells wll be dfferent to that found from the smple ncremental traverse of the soluton space. Fgure 5, llustrates that the energy use s domnated by the heatng of the atrum. However, the optmum wndow area s dctated by the trade-off between the reducton n lghtng (electrcal) energy and the ncrease n coolng energy wth the number of wndow cells. The greatest rate of reducton n lghtng (electrcal) energy use occurs between 1 and 27 wndow cells, wth no sgnfcant reducton for a further ncrease n number of cells. Although ncreasng the number of cells further results n a reducton n heatng energy use, there s a much greater rate of ncrease n coolng energy use, ths

5 resultng n an ncrease n total energy use for more than 27 wndow cells. Fgure 4, Total Energy Use wth umber of Cells Fgure 6, Unconstraned Soluton (W=west; E=east; vewed from nsde the buldng) Table 2, Unconstraned Soluton Metrcs Fgure 5, Components of Energy Use (electrcal energy ncludes artfcal lghtng) Energy Use (kwh) Area Densty Aspect Rato (-) umber of Wndows Fgure 7, Soluton Scatter around the Unconstraned Soluton Unconstraned Optmzaton Fgure 6 llustrates the soluton wth the mnmum energy use resultng from the unconstraned optmzaton. The total energy use s 36,858 kwh wth a glazed area of 21.7% (Table 2), these beng smlar to the energy use and percentage area found from the ncremental traverse (Fgure 4). Fgure 6 also llustrates that the wndow cells are based towards he top-west quadrant of the façade (as ndcated by the cross and arrow, the base of the arrow beng located n the centre of the façade). However, there s consderable scatter across the façade, whch suggests that the optmum energy use s nsenstve to the locaton of ndvdual wndows and the percentage wndow area. Ths s reflected by the number of solutons n close proxmty to the optmum (Fgure 7). Although a number of alternatve solutons are avalable for nspecton and possble use by the desgner, they all have the same characterstc that the poston of the wndows cells are based towards the top-west quadrant of the façade. The reason for ths has been examned by smulatng the atrum energy use for a 7 cell wde, by 4 cell hgh rectangular wndow located n one of four postons; the top-east corner, the top-west corner, the bottomeast corner, and the bottom-west corner. The wndow has 28 cells, and occupes 23.3% of the façade area (whch s n the regon to the perceved optmum area). The mpact of wndow locaton on energy use s llustrated n Fgure 8 (note that the range of values for the vertcal energy use axs s the same for all forms of energy use; 500kWh)

6 Fgure 8, Impact of Wndow Poston on Energy Use Fgure 9, Fractonal Lghtng Load, Control Reference Pont Closest to the Façade and 10 llustrate the fracton of 500lux requred from artfcal lghtng for the two lghtng control (reference) ponts. In the case of the reference pont closest to the façade (Fgure 9), postonng the wndows at the top of the façade results n an ncrease n wnter lghtng load, but a decrease n summer load. The lghtng load assocated wth the control pont located furthest from the façade (Fgure 10), s reduced n all months, the hghest reducton beng when the wndow s postoned n the top-west corner of the façade. As well as havng an mpact on lghtng energy use and the resultng nternal heat loads, the wndow poston has a drect mpact on the dstrbuton of beam solar radaton on the nternal surfaces of the atrum. In partcular, the fracton of total beam radaton fallng on the floor s reduced n all months, the greatest reducton occurrng durng the wnter months (Fgure 11). A reducton n solar radaton fallng on the floor results n a reducton n heatng energy use, but an ncreases n coolng energy use (the pattern of heatng and coolng energy use wth wndow poston evaluated wth no lghtng load s smlar to that n Fgure 8, although there s a smaller dfference n energy use between east and west locatons). Snce, the nternal partton walls are modelled as adabatc surfaces and are thermally lght-weght, an ncrease n the dstrbuton of radaton onto these surfaces s lkely to result n an ncrease n the more nstantaneous heat gan to the zone, resultng n a reducton n heatng load, but ncrease n coolng load. Fgure 11, Fractonal Dstrbuton of Beam Solar Energy to the Floor Fgure 10, Fractonal Lghtng Load, Control Reference Pont Furthest from the Façade It s clear that locaton of the wndow has the greatest mpact on lghtng (electrcal) energy use. Fgures 9 Constraned Aspect Rato Fgure 12, llustrates the soluton wth the wndow aspect rato constraned to be n the range 1.5 to Agan, the energy use and percentage wndows area are of the same order as the soluton found from the ncremental traverse and the unconstraned optmzaton. In ths respect, t can be concluded that the optmum energy use has not been restrcted by

7 the aspect rato constrants. However, even though the total energy use s unaffected, the dstrbuton of the wndow cells across the façade dffers sgnfcantly from the unconstraned soluton. In comparson to the unconstraned soluton, the range of aspect rato s reduced from 0.71 to 2.0 n the unconstraned case, to 1.5 to 1.67 n the constraned case), Tables 2 and 3); the number of wndows s also reduced from 8 to 3 (Tables 2 and 3). The reducton n number of wndows s a result of; there beng only 5 alternatve wndow geometres that satsfy the aspect rato constrants (Table 1); and that the total number of wndow cells beng avalable to form a wndow of vable aspect rato s restrcted by the optmum wndow area beng n the regon of only 22% of the façade. In the case of ths soluton, two of those geometres appeared n the soluton (5 cell hgh by 3 cells wde, and 6 cells hgh by 4 cells wde). Gven that there are fewer wndows, and the percentage areas are smlar, t mght be expected that the densty (compactness) of the wndows would be hgher for the constraned solutons. However, ths s not the case as the unconstraned soluton has many sngle cell wndows havng a densty of 1.0; as a result, the mean densty of the wndows n the unconstraned soluton s 71% (Table 2), compared to only 49% for the constraned soluton (Table 3). Fgure 12, Constraned Aspect Rato Soluton (W=west; E=east; vewed from nsde the buldng) dfference n the energy between ths soluton and that of the other solutons s nsgnfcant); the percentage wndow area n ths case s also slghtly lower than for the solutons from other experments (Tables 2, 3, and 4). The fact that the geometry of the wndows n ths experment s unconstraned has resulted n a wndow havng a short-wde aspect rato, whereas the aspect rato constrant experments forced a soluton havng a tall-thn wndows (Tables 3 and 4). Even though the soluton found for the aspect rato constrant experment would have satsfed the number of wndows constrant, allowng the wndows to have a free-form n ths experment, naturally resulted n denser wndows (49% when the aspect rato was constraned, and 64% when the number of wndows was constraned; Tables 3 and 4). ote also that, ntutvely, the soluton found for the number of wndows constraned (Fgure 13), appears to be more compact (denser), than that found for the unconstraned optmzaton (Fgure 6). However, ths s not reflected n the wndow densty metrc, as the unconstraned soluton has a hgher average densty than the constraned number of wndows soluton (Tables 2 and 4); the reason for ths s that there are several sngle cell wndows n the unconstraned, these havng the hghest densty of 1.0. Fgure 13, Soluton wth umber of Wndows Constraned (W=west; E=east; vewed from nsde the buldng) Energy Use (kwh) Table 3, Constraned Aspect Rato Soluton Metrcs Area Densty Aspect Rato (-) umber of Wndows Constraned umber of Wndows Constranng the number of wndows to be less than or equal to 3 resulted n a soluton havng only two wndows the locaton of whch s heavly based towards the top-west quadrant of the façade (Fgure 13). Ths resulted n the soluton wth the lowest energy use of all experments (although the Table 4, Constraned umber of Wndows Soluton Metrcs Energy Use (kwh) Area Densty Aspect Rato (-) umber of Wndows DISCUSSIO AD COCLUSIOS Ths paper nvestgates the optmzed desgn of fenestraton that s based on the façade of the buldng beng dvded nto a number of small regularly spaced cells. Each cell on the façade s represented by a dscrete problem varable havng two possble states (sold constructon, or glazed constructon). A wndow on the façade s defned

8 to be a set of adjonng cells. Metrcs have been developed that descrbe the number, poston, total percentage area, cellular densty, and aspect rato of such wndows. The metrcs have been used n the analyss of the desgn solutons, and n the case of aspect rato and number of wndows, to constran the form of wndows found by the optmzaton. Three optmzaton experments are descrbed n ths paper; an unconstraned mnmzaton of buldng energy use, and two constraned mnmzatons of energy use, the frst wth the wndow aspect rato constraned, and the second wth the number of wndows constraned. In all three cases, the optmzaton was for the desgn of a three-storey atrum located n Chcago, USA. Gven the bnary nature of the problem varables, all three optmzaton problems have been solved usng a bnary-encoded Genetc Algorthm (GA). The results ndcate that the GA was able to fnd near optmum solutons to all three optmzaton problems. An ncremental traverse of the soluton space confrmed that both the energy use and percentage wndow area found by the optmzaton was nearoptmal. The ncremental traverse also ndcated that although, for ths partcular example, the magntude of the buldng energy use was domnated by heatng energy use, t was the rate of change n lghtng and coolng energy, and the trade-off between the two, that dctated the optmum percentage wndow area. Gven that, each optmzaton experment resulted n a dfferent dstrbuton of wndow cells, but that the optmzed energy and wndow area was of the same order of magntude n each case, t s concluded that, for the example buldng studed here, the poston of the wndow cells has only a second-order effect on energy use. However, n the results from all experments, the optmzed poston of the wndows cells was based towards the top-west corner of the façade. Locatng the wndows towards the top of the façade results n the penetraton of daylght to a greater depth n the atrum; correspondngly, ths reduces the energy use from artfcal lghtng, partcularly when the wndows are postoned towards the top-west quadrant of the façade. The poston of the wndows also has an mpact on the dstrbuton of the beam solar radaton on the nternal surfaces of the atrum, whch n turn affects heat loads through the dfferent heat loss and storage effects of the varous constructon elements. The optmzaton results also ndcate that the desgn constrants for wndow aspect rato and the number of wndows are effectve n producng dfferent desgn solutons. The unconstraned optmzaton resulted n the hghest number of wndows and scatter across the façade. Constranng the aspect rato of the solutons resulted n fewer wndows, but wndows havng a more regular geometrc shape and whch were postoned more evenly across the façade. Ths effect s a functon of the possble number of cells n the façade and the extent to whch ths lmts the number of alternatve wndows geometres that meet the specfed aspect rato. Constranng the total number of wndows (predctably), resulted n a more compact wndow desgn. Snce the form of wndows was unconstraned n ths case, the poston of the wndows was heavly based towards the top-west corner of the façade. It can be concluded, that a cellular façade, wth the state of each cell (glazed or sold) beng optmzed by a GA, s not only effectve n the mnmzaton of buldng energy use, but can also result n desgn soluton havng nnovatve and nterestng archtectural forms. The wndow metrcs developed here, also allow the desgner to control the general form of the solutons. Further research s requred to evaluate the performance of the GA, the effectveness of the metrcs n controllng the solutons, the characterstcs of the solutons, and the applcablty of the approach to real buldng desgn. REFERECES Buser, T Experencng Art Around Us, Thomson Wadsworth, Florence, KY, USA. Caldas, L. and orford L A desgn optmzaton tool based on a genetc algorthm, Automaton n Constructon, 11, Crawley, D.B., Lawre, L.K., Wnkelmann, F.C., Buhl, W.F., Huang, Y.J., Pedersen, C.O., Strand, R.K., Lesen, R.J., Fsherm, D.E., Wtte, M.J., Glazer, J EnergyPlus: creatng a new generaton buldng energy smulaton program, Energy and Buldngs, Vol. 33, o. 4, pp De Jong K.A Evolutonary Computaton: A Unfed Approach. The MIT Press. ISB Mourshed, M., Kellher, D., & Keane, M ArDOT: A tool to optmze envronmental desgn of buldngs, Proceedngs of IBPSA Runarsson T.P. and Yao X Stochastc Rankng for Constraned Evolutonary Optmsaton. IEEE Transactons on Evolutonary Computaton, 4(3), Wetter M, and Wrght J A A Comparson of Determnstc and Probablstc Optmzaton Algorthms for onsmooth Smulaton-based Optmzaton, Buldng and Envronment, 39(8), ISS: