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* 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
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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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