* M.Sc.Eng.JolantaGintowt,InstituteofBuildingMaterialsandStructures,FacultyofCivilEngine-ering,CracowUniversityofTechnology.

**PaulinaŁysik,student,CracowUniversityofTechnology.

TECHNICAL TRANSACTIONSCIVIL ENGINEERING

5-B/2014

CZASOPISMO TECHNICZNEBUDOWNICTWO

JOLANTAGINTOWT,PAULINAŁYSIK*

COMPARATIVEANALYSISOFTHECRITERIAUSED TOSELECTTHEOPTIMALENERGYSAVINGVARIANTS

INBUILDINGS.SELECTEDISSUES

ANALIZAPORÓWNAWCZAKRYTERIÓWSTOSOWANYCHDOWYBORUOPTYMALNYCH

WARIANTÓWENERGOOSZCZĘDNYCH WBUDOWNICTWIE.WYBRANEZAGADNIENIA

A b s t r a c t

Comparative analysis of multi-criteria Pareto and SPBT optimization, for a single-family,detachedresidentialbuildingusedasanexample.Mainelementsofinterestincludeminimumpowerandminimumcostofinvestment.

Keywords: multi-criteria optimization, Pareto, Kuhn-Tucker, SPBT

S t r e s z c z e n i e

AnalizaporównawczadlakryteriumoptymalizacjiwielokryterialnejwsensieParetoiSPBTna przykładzie budynku mieszkalnego jednorodzinnego, wolnostojącego. Funkcjami kryte-rialnymisąminimumenergii iminimumkosztów inwestycyjnych,zmiennymidecyzyjnymi– izolacyjnośćtermicznaprzegródzewnętrznych,wielkośćprzeszklenia,orientacjabudynkuwzględemstronświata.

Słowa kluczowe: optymalizacja wielokryterialna, Pareto, Kuhn-Tucker, SPBT

98

Designations

f(x) – vectorofobjectivefunctionx – vectorofdecisionvariablesr – positivemultiplier,controlsthemagnitudeofthepenaltytermsh(x) – vectorofequalityconstraintsGj – Heavisideoperatorgj(x) – vectorofinequalityconstraints,ρi – randomnumber<0;1>P[f(x)] – preferencefunction(substitutefunction)F1 – criterionfunction,energy[kWh]F2 – criterionfunction,investmentcost[zł]g(i) – boundaryconditionsx(5) – thermalresistance[m2K/W]x(6) – simplexsidesofthebaseofthebuilding

1. Optimization method to determine reasonably low energy consumption

1.1.UnconstrainedMinimization

Oneofthemostdevelopedgroupsofnumericaloptimizationmethodsistheiterativetype [1].For thismethod,apoint isestablishedon thebasisof thepreviouslyobtainedresults,which indicateswhere theminimum is likely to be, or the general direction inwhichitislikelytolie.Thisapproachincludes,withoutlimitation,thefollowingmethods:Direct Search Method of Hooker and Jeeves, Simplex Method of Nelder and Mead,VariableMetricMethodofDavidon-Fletcher-Powell.Thesemethodshowever,shouldnotbeused toevaluateenergy-efficientbuildings,as theyprimarilydetermine the technicalrequirements,suchasmaximumheattransfercoefficient.Inthiscaseitispossiblefindingalocaloptima.

1.2.ConstrainedMinimization

Themostcommonapproachtosolvingconstrainedminimizationproblemsinvolvestheuseofpenaltyfunctionstoconverttheseproblemsintounconstrainedproblems.Themostpopularpenaltyfunctionistheone,whichassociatesapenalty–whichisproportionaltothesquareofaviolation–asinthefollowingformula(1).

min ( , ) ( ) [ ( )] [ ( )]φx Rn

jj

p

jj

m

x r f x r h x r Gj g x∈ = =

= + +∑ ∑2

1 1

2 (1)

where:Gj –HeavisideoperatorsuchthatGj=0forgj(x)≥0andGj=1forgj(x)<0.

Oneofthewidelyusedformulationsofthetransformedinteriorfunctionistheoneshowninformula(2):

99

φ( , ) ( )( )

x r f x rg xjj

n

= +=∑ 11

(2)

Ifanyofconstraintfunctionsgj(x)approaches0,thepenaltytermincreasesveryrapidly.Inthismethod,itisnecessarytostartthesearchfromaninterior,feasiblestartingpoint.

FlexibleToleranceMethod.TheflexibletolerancemethodwasdevelopedbyHimmelblau.Inthismethod,T(x)isdefinedasapositivesquarerootofthesumofsquaredvaluesofallviolatedinequalitiesor/andequalityconstraints.Formula(3)describesthedependency.

T x hj x Gj g xjj

n

j

p

( ) [ ( )] [ ( )]= +

==∑∑ 2 2

11

12

(3)

AsmallvalueoftheT(xt)impliesthatxtisrelativelyfarfromthefeasibleregion.ExploratoryMethods.Theaccuracyofthismethoddependsonhedensityofthegrid,that

iswhywesetupthegridwithpointsspacedtogethercloseenoughtodefineaminimum,asdeterminedbytheinspectionofeachpoint(CombinationalMethod).

RandomSearchMethod(MonteCarloMethod,ModalMethod).Thevalueofobjectivefunctionisevaluatedforeachpointandthebestresultistakenastheminimum.Thismethodofferstwoapproachesfordealingwithconstraints.Firstmethod:apenaltyisusedforviolatingthesolutionoutsidethefeasibleregion.Inthiscase,theobjectivefunctionisevaluatedforallgeneratedpoints.Secondmethod:eachgeneratedpointistestedforviolationanddiscarded,ifitisnotafeasiblesolution.Inthiscase,theobjectivefunctionisevaluatedonlyforafeasiblesolution.Weselectvaluesofxi–vectorofdecisionvariables.Usedformula(4):

x x x xi ik

i iu

ik= + −ρ ( ) (4)

Usually,thismethodwilllocatethesolutionintheneighborhoodoftheglobalminimum.MethodforDiscreteandIntegerVariables.Solvingoptimizationproblemswithdiscrete

variables directly is much more difficult than solving similar problems with continuousvariables.

1.3.Multi-criterionOptimizationMethods

Inordertosolvetheproblem,weusedthePreferenceFunctionMethod,describedbytheformula(5):

P f x P f xx X

[ ( )] min [ ( )]* =∈

(5)

asaWeightingObjectiveMethod,NormedWeightingObjectiveMethod,GlobalcriterionMethod,Min-MaxMethod,WeightingMin-MaxMethod,MethodofIdealVectorDisplace-mentorConstraintTransformationMethod.

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1.4.MethodofSelectingaSetofParetoOptimalSolution,Kuhn-Tuckermethod

TheParetoOptimalSolutionbasedonarandomsearchmethod.ForKuhna-TuckeramethodweformulatetheLagrangefunctionasa

L(x,l)=f(x)–Σligi(x) (6)

ThenecessaryconditionsforsaddlepointoftheLagrangefunctionL(x,l),whichhavetobefulfilledsimultaneously,areasfollows:

∂∂

≥{ < ≤ }

≤ < ≤= < ≤

Ll

i u

u i vv i m

i

0 0

00

∂∂

≤{ < ≤ }

≥ < ≤= < ≤

Lx

j s

s j tt j n

j

0 0

00

l Llii

∂∂

= 0 x Lxjj

∂∂

= 0 andforalli,j (7)

1.5.Taskdescription

Criterion functions were: minimum energy consumption for heating F1, minimuminvestmentcostsF2.Coolingwasnotanalyzed.Suchanalysistakesseveralhours[11].Theoptimization’sdecisivevariableswere:thethicknessofwallinsulation(thermalresistanceof the layer), the size of the glazing on all elevations, the ratio of the sides of the base.Fixedparameterswere:thermalresistanceonthegroundfloor,flatroof,buildingareaandventilationairstream.Theanalysis’subjectwasasingle-familybuilding;inparticular,thegroundfloor–fixedfloorareaof120m2.Theboundaryconditionslimitingtheinsulationwereequaltoabout10[m2K/W]<R<3.33[m2K/W].

1.6.Results

Table1describes the resultsof theSPBT[years]and table2describes the results fromKuhn-Tucker.Duetothelowthicknessoftheinsulation,theSPBTvalueliesbetween15and20cm.This is illustratedbyFig.1a.Thesolutionmethodofmulti-criteria thicknessof theinsulationisapprox.36cm.ThisisdepictedinFig.1b.Thedifferencebetweentheresultsofbothmethodsisalmost20cm(onemethodrecommendsinsulaiontwiceasthickastheother).

T a b l e 1

SPBT dependence onthetypeof energy carrier andthickness ofinsulation

Thicknessofinsulation[cm] SPBT[year]coal gas electricity

10 122 69 2620 119 67 2530 132 74 28

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Ta b l e 2

Dependence of thickness x (5) and F1 and F2 (method Kuhn–Tucker)

F1 314244 314634.6 315174.8 314950.6 314135.6

F2 5754.035 5739.386 5721.686 5728.87 5724.476

x(5) 10–3.33 max max max max max

x(6) 2.5–1 1.119 1.2457 1.23 1.25 1.3069

Fig.1a)Theoptimalthicknessofinsulation,SPBT,b)theresultsofKuhn–Tuckermethod

2. Conclusions

1.TheSPBTmethodofchoosingtheoptimumthicknessofinsulationseemstobeonlyanestimate.

2.ThedifferenceinresultsbetweentheSPBTandmulti-criteriaoptimizationmethodesisnotwithoutsignificance.

3.Itseemsthatoneshouldapplyadvancedmethodstoevaluatetheinsulationefficiency(asopposedtothesimplemethodwhichSPBTundoubtedlyis).

4.Theuseofsimpleandeasywaysisnotagoodchoiceamongtheavailablecomputationaltools.

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