ANALIZA PORÓWNAWCZA KRYTERIÓW STOSOWANYCH DO WYBORU OPTYMALNYCH WARIANTÓW ENERGOOSZCZĘDNYCH W BUDOWNICTWIE. WYBRANE ZAGADNIENIA

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1 TECHNICAL TRANSACTIONS CIVIL ENGINEERING CZASOPISMO TECHNICZNE BUDOWNICTWO 5-B/204 JOLANTA GINTOWT, PAULINA ŁYSIK * COMPARATIVE ANALYSIS OF THE CRITERIA USED TO SELECT THE OPTIMAL ENERGY SAVING VARIANTS IN BUILDINGS. SELECTED ISSUES ANALIZA PORÓWNAWCZA KRYTERIÓW STOSOWANYCH DO WYBORU OPTYMALNYCH WARIANTÓW ENERGOOSZCZĘDNYCH W BUDOWNICTWIE. WYBRANE ZAGADNIENIA Abstract Comparatve analyss of mult-crtera Pareto and SPBT optmzaton, for a sngle-famly, detached resdental buldng used as an example. Man elements of nterest nclude mnmum power and mnmum cost of nvestment. Keywords: mult-crtera optmzaton, Pareto, Kuhn-Tucker, SPBT Streszczene Analza porównawcza dla kryterum optymalzac welokryteralne w sense Pareto SPBT na przykładze budynku meszkalnego ednorodznnego, wolnostoącego. Funkcam kryteralnym są mnmum energ mnmum kosztów nwestycynych, zmennym decyzynym zolacyność termczna przegród zewnętrznych, welkość przeszklena, orentaca budynku względem stron śwata. Słowa kluczowe: optymalzaca welokryteralna, Pareto, Kuhn-Tucker, SPBT * M.Sc. Eng. Jolanta Gntowt, Insttute of Buldng Materals and Structures, Faculty of Cvl Engneerng, Cracow Unversty of Technology. ** Paulna Łysk, student, Cracow Unversty of Technology.

2 98 Desgnatons f(x) vector of obectve functon x vector of decson varables r postve multpler, controls the magntude of the penalty terms h(x) vector of equalty constrants G Heavsde operator g (x) vector of nequalty constrants, ρ random number <0;> P[f(x)] preference functon (substtute functon) F crteron functon, energy [kwh] F2 crteron functon, nvestment cost [zł] g() boundary condtons x(5) thermal resstance [m 2 K/W] x(6) smplex sdes of the base of the buldng. Optmzaton method to determne reasonably low energy consumpton.. Unconstraned Mnmzaton One of the most developed groups of numercal optmzaton methods s the teratve type []. For ths method, a pont s establshed on the bass of the prevously obtaned results, whch ndcates where the mnmum s lkely to be, or the general drecton n whch t s lkely to le. Ths approach ncludes, wthout lmtaton, the followng methods: Drect Search Method of Hooker and Jeeves, Smplex Method of Nelder and Mead, Varable Metrc Method of Davdon-Fletcher-Powell. These methods however, should not be used to evaluate energy-effcent buldngs, as they prmarly determne the techncal requrements, such as maxmum heat transfer coeffcent. In ths case t s possble fndng a local optma..2. Constraned Mnmzaton The most common approach to solvng constraned mnmzaton problems nvolves the use of penalty functons to convert these problems nto unconstraned problems. The most popular penalty functon s the one, whch assocates a penalty whch s proportonal to the square of a volaton as n the followng formula (). 2 mn φ( xr, ) = f( x) + r [ h ( x)] + r G[ g ( x)] x R n = = p where: G Heavsde operator such that G = 0 for g (x) 0 and G = for g (x) < 0. One of the wdely used formulatons of the transformed nteror functon s the one shown n formula (2): m 2 ()

3 99 n φ( xr, ) = f( x) + r g ( x) = (2) If any of constrant functons g (x) approaches 0, the penalty term ncreases very rapdly. In ths method, t s necessary to start the search from an nteror, feasble startng pont. Flexble Tolerance Method. The flexble tolerance method was developed by Hmmelblau. In ths method, T(x) s defned as a postve square root of the sum of squared values of all volated nequaltes or/and equalty constrants. Formula (3) descrbes the dependency. p n 2 2 T( x) = [ h( x)] + Gg [ ( x)] = = 2 (3) A small value of the T(x t ) mples that x t s relatvely far from the feasble regon. Exploratory Methods. The accuracy of ths method depends on he densty of the grd, that s why we set up the grd wth ponts spaced together close enough to defne a mnmum, as determned by the nspecton of each pont (Combnatonal Method). Random Search Method (Monte Carlo Method, Modal Method). The value of obectve functon s evaluated for each pont and the best result s taken as the mnmum. Ths method offers two approaches for dealng wth constrants. Frst method: a penalty s used for volatng the soluton outsde the feasble regon. In ths case, the obectve functon s evaluated for all generated ponts. Second method: each generated pont s tested for volaton and dscarded, f t s not a feasble soluton. In ths case, the obectve functon s evaluated only for a feasble soluton. We select values of x vector of decson varables. Used formula (4): k u k x = x + ρ ( x x ) (4) Usually, ths method wll locate the soluton n the neghborhood of the global mnmum. Method for Dscrete and Integer Varables. Solvng optmzaton problems wth dscrete varables drectly s much more dffcult than solvng smlar problems wth contnuous varables..3. Mult-crteron Optmzaton Methods In order to solve the problem, we used the Preference Functon Method, descrbed by the formula (5): * P[ f( x )] = mn P[ f( x)] x X (5) as a Weghtng Obectve Method, Normed Weghtng Obectve Method, Global crteron Method, Mn-Max Method, Weghtng Mn-Max Method, Method of Ideal Vector Dsplacement or Constrant Transformaton Method.

4 00.4. Method of Selectng a Set of Pareto Optmal Soluton, Kuhn-Tucker method The Pareto Optmal Soluton based on a random search method. For Kuhna-Tuckera method we formulate the Lagrange functon as a L (x, l) = f (x) Σ l g (x) (6) The necessary condtons for saddle pont of the Lagrange functon L (x, l), whch have to be fulflled smultaneously, are as follows: { 0 0< u} l 0 u < v = 0 v < m x { 0 0< s} 0 s < t = 0 t < n l l = 0 x x = 0 and for all, (7).5. Task descrpton Crteron functons were: mnmum energy consumpton for heatng F, mnmum nvestment costs F 2. Coolng was not analyzed. Such analyss takes several hours []. The optmzaton s decsve varables were: the thckness of wall nsulaton (thermal resstance of the layer), the sze of the glazng on all elevatons, the rato of the sdes of the base. Fxed parameters were: thermal resstance on the ground floor, flat roof, buldng area and ventlaton ar stream. The analyss subect was a sngle-famly buldng; n partcular, the ground floor fxed floor area of 20 m 2. The boundary condtons lmtng the nsulaton were equal to about 0 [m 2 K/W] < R < 3.33 [m 2 K/W]..6. Results Table descrbes the results of the SPBT [years] and table 2 descrbes the results from Kuhn-Tucker. Due to the low thckness of the nsulaton, the SPBT value les between 5 and 20 cm. Ths s llustrated by Fg. a. The soluton method of mult-crtera thckness of the nsulaton s approx. 36 cm. Ths s depcted n Fg. b. The dfference between the results of both methods s almost 20 cm (one method recommends nsulaon twce as thck as the other). SPBT dependence on the type of energy carrer and thckness of nsulaton Thckness of nsulaton [cm] SPBT [year] coal gas electrcty Table

5 0 Table 2 Dependence of thckness x (5) and F and F2 (method Kuhn Tucker) F F x(5) max max max max max x(6) a) b) Fg. a) The optmal thckness of nsulaton, SPBT, b) the results of Kuhn Tucker method 2. Conclusons. The SPBT method of choosng the optmum thckness of nsulaton seems to be only an estmate. 2. The dfference n results between the SPBT and mult-crtera optmzaton methodes s not wthout sgnfcance. 3. It seems that one should apply advanced methods to evaluate the nsulaton effcency (as opposed to the smple method whch SPBT undoubtedly s). 4. The use of smple and easy ways s not a good choce among the avalable computatonal tools. References [] Osyczka A., Computer aded mulcrteron optmzaton system (CAMOS), Software Package n Fortran, Internatonal Software Publsher, 992, [2] Hobler T., Ruch cepła wymennk, wyd. 3, WNT, Warszawa 968. [3] Bogosłowsk W.N., Procesy ceplne wlgotnoścowe w budynkach, Arkady, 985,

6 02 [4] Recknagel, Sprnger, Hofmann, Schramek, Poradnk. Ogrzewane klmatyzaca, wyd., EWFE, Gdańsk 994, [5] Gordon B. Davs, Thomas r. Hoffmann, Fortran a structured, dscplned style, Unversty of Mnnesota, McGraw-Hll Book Company, 970, ISBN , 978. [6] Morrs H. de Groot, Optma statstcal decsons, McGraw-Hll, Inc., 22-33, 98. [7] Mrkowska G., Elementy matematyk dyskretne, Wyd. PJWSTK, Warszawa [8] Fndensen W., Szymanowsk J., Werzbck A., Metody oblczenowe optymalzac, Wyd. Poltechnk Warszawske, Warszawa 974, [9] Kttler R., Mkler J., Zaklady vyużvana slnećneho żarena, VEDA, 986, [0] Kossecka E., Bzowska D., Oblczane dobowych sum promenowana słonecznego na płaszczyzny nachylone przy wykorzystanu zależnośc dla współczynnków przezroczystośc atmosfery, Materały konferencyne KNT Łódź `93, Fzyka budowl w teor praktyce, na prawach rękopsu, [] Dz. U. 203 poz. 45, Rozporządzene Mnstra Transportu, Budownctwa Gospodark Morske z dna 3 styczna 203 r. zmenaące rozporządzene w sprawe metodolog oblczana charakterystyk energetyczne budynku lokalu meszkalnego lub częśc budynku stanowące samodzelną całość technczno-użytkową oraz sposobu sporządzana wzorów śwadectw ch charakterystyk energetyczne. [2] Jędrzeuk H., Marks W., O korzyścach z optymalzac budynków meszkalnych, Fzyka budowl w teor praktyce, Tom II, 2007, [3] Jędrzeuk H., Marks W., Evolutonal optmzaton of energy-savng buldngs, Archves of Cvl Engneerng, LI, 3, 2005, [4] Mkulsk L., Teora sterowana w problemach optymalzac konstrukc systemów, Wyd. Poltechnk Krakowske, Kraków 2007, ISBN: [5] Jędrzeuk H., Klemm K., Marks W., Multcrtera optmzaton of buldngs arrangement based on wnd crtera, Archves of Cvl Engneerng, LIV, 4, 2008,

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