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NIFS7 (2001),l,10-21

Multi-objective Optimisation in Air-Conditioning Systems:Comfort/Discomfort Definition by IF SetsPlamen Angelov

Building ServicesEngineeringResearchGroupDepartmentof Civil and Building Engineering,Loughborough University, LoughboroughLEl l 3TU, Leicestershire,UK

tel:+44 (1509)223774 fax:+44 (1509)223981 e-mail:P.P.Angelov@Lboro.ac.UK

Abstract: The problem of multi-objective optimisationof air-conditioning(AC) systems s treated n thepaper in the framework of intuitionistic fvzzy (lF) set theory. The nature of the problem is multi-objective onewith requirements or minimal costs (generally, life cycle costs;more specifically, energy costs) and maximaloccupants' comfort (minimal discomforl). Moreover, its definition by conventional means is bounded to anumber of restrictions and assumptions,which are often far from the real-life situations.Affempts have beenmade to formulate and solve this problem by means of the fuzzy optimisation [4]. The present paper makesfurther step by exploring the innovative conceptof IF sets [6] into definition of the trickiest issue: comfort anddiscomforrdefinition. The new approach allows to formulate more precisely the problem which compromisesenergy saving and thermal comfort satisfactionunder given constraints.The resulting IF optimisation problemcould be solvednumerically or, under someassumptions, nalytically Il]-t21. An example llustrates he viabilityof the proposedapproach.A solution which significantly (with 3S%) improvescomfort is found which is moreenergeticallyexpensivewith only 0.6Yo.This illustrates he possibility to use he approach or trade-off analysisin multi-objective optimisationof AC systems.Keywords: IF sets, F optimisation, multi-objective optimisation, air-conditioning systems,comfort/discomfort

1. IntroductionIn the last few decadeshereare an increasing nterest n the optimum design and controlof AC systems I4l, t9l-t191. Besides of economicalefficiency, it has ecological aspectslowering the "green house" effect, fuel consumption and, hence, reducing emission ofpollutants while keeping occupants'comfort U 01, 22]. The aim is to designand control ACsystemsuch that to minimise the life cycle costs, ncluding capital, operationalcosts(mainlyenergyconsumption),maintenanceand commissioningcostsetc. while minimising the feelingof discomfort of occupantsand keeping as maximum as possible productive environmentalconditions. Solution of this problem is usually based on random search or gradient-basedtechniques 9]-[0]. In last few yearssuccessful pplicationof GA to this problemhas beenreported 5], [9].The specific of the AC systems,by differ from most othertechnicalsystems, s the factthat human and its comfort are in the centre of the problem and, as a result, control andoptimisation objectives as well as some constraints are highly subjective, ill defined and

' So far as the laws of mathematics refer to reality, theythey are certain, they do not refer to reality. are not certain.And so farAlbert Ehstein

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different factors ike clothing, metabolism,air movements,occupants'activity, solar radiationetc. have been seriouslystudied [15]-[20]. Human comfort is a complex non-linear, ime-varying function of temperature,humidity, airflow rate, clothing level, activity, lighting,shading,noise etc. It is important to mention that the comfort requirementsare specffic foreach ndividual or group of individuals and are not constant.The fact that natural temperaturecould be different in different locations and could vary with cultural and other differences,that occupantscould have different feeling of comfort has been eported122l-1231.Current practice is to design and operateAC systemsunder assumptionsof occupants'comfort requirements based on statistical design criteria contained within standards ikeISO7730 l2l), known as PPD and PMVr. In reality, the thermal sensationsand needs ofoccupantsvary from simple design standards 23]. Occupants'preferences are subiective,individual, their presencen the occupiedzone and evel of activity is time varying tl5]-[19].Another important issue in the AC systemscontrol is their energy efficiency. Thelatest data rom the InternationalEnergy Agency (http:i/www.iea.org) showsthat significantpart of the primary energy s used n buildings and AC systemaccount or the major part of it.This problem is increasingly important nowadays, when ecological and economicalconcurrent requirements (the ozone hole, gas emissions and surging petrol prices, costeffectivenessas a condition for higher competitivenessetc.) are pressing and have to bebalancedwith the comforl needs.Additional potential for higher energy efficiency exist infinding and keeping more economical set-points or regimes, based on more adequaterepresentation f humans' eelings of comfort anddiscomfort, of keeping a really productiveenvironment 15]. It could ead o a twofold improvement:./ more customised,specificuser-orientedAC system;,/ higher energy efficiency.

A general ormulation of the optimisationproblem of an AC system n the framework ofIF sets heory is proposed n the presentpaper. F setshave been ntroduced ecently [6] as anextensionand generalisationof the fuzzy set concept.They considernot only the degreeofmembership o a given set, but also the degreeof non-membership uch that the sum of bothvalues is less than one [7]. Due to the symmetrical treatmentof objectivesand constraintsthis multi-objective problem is transformed o a non-fuzzy (crisp, conventional) one [l-2],124l-1261hat in somecases ould be, solved analytically I25l.The building thermalmodeland plant models arenot consideredn this presentation n order o simplify the problem.An example of a multi-objective IF optimisation problem i llustrates the viability of theproposedapproach.A solution,which significantly improvescomfort andreducesdiscomfortfeeling, is found. This illustrates he possibility to use IF optimisation for trade-off analysis nmulti-objectiveoptimisationof AC systems.2. Somedefinitions over IF setsIF set basicdefinition [6]:Definition I

The IF setA is a mappingR + [0;l]x[0;l], which is definedby a pair t6lwhere pra(x)denotes he degreeof membershipof x to the setA;va(x) is the degreeof non-membership rejection)of x to A;pn(x) + vo(x) < 1.

' from PredictedPercentage f Dissatisfiedpeopleand PredictedMean Vote respectively

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It is interesting o note, hat A becomea conventional uzzy set,when pu(x)+vu(x) :1.2.1 Crispification: de-fuzzification over IF sets 3JCrisptfication operatorshave been ntroducedby the author n [3] as a mapping[0;1]x[0;l] + R in analogy o de-fuzzificationoperationover conventional uzzy sets.Theresult of this operation s a cr isp valuewhich is representativeor the given IF set as a whole.This operation is necessary n controllers to derive the final control action (fuzzy or IF setpoint can not be given to the servo n a control system)which will be realised. t is necessaryalso n decisionmaking andexpert systems or elicitation of information. A brief summaryofdefinitions of four crispification operatorsand their main propertiesarepresented ellow.Definition 2a l3lThe centre-of-gravity(CorC)crispification operationover IF sets s definedas follows

Z1t@,) v(x,))x,0 - t = l ; N : card(x); for p(x)> v(x) (1)lF co G NZUt@,) v(x,))j =lThis operator's definition is based on the difference between the degree ofmembershipand non-membership rejection),which have o bepositive.Definition 2b l3lThe mean-of-maxima MoM) crispification operator s defined n analogy o the MoM

de-fuzzification operatorover (p(xi)-v(x')).I.>'-t, N*,,,,_ru,t, ; xr:{* | p(*')-v(*r):Tg" (pr("i)-v(xJ)} Q)Definition c 3l

Mean-of Equilibria (MoE) is introduced as a new operator n [3], which differ fromde-fuzzification of conventional fuzzy sets). It averagesall points with maximal p(x1) (themost acceptable oints) and thesewith minimal v(xi) (the lessnon-acceptable):

M KZ*|' >.:, ?. . _ i=l + i=t (3)MoE 2M 2Kx*:{xl tt6*) - Mmax,':' (*)}; xo:{*l({:*min,': ' ,(*)}Definition2dFinally, an analogue f BADD operators introducedor crispification.t is over hedifference (x')-v(x')

NZUt@,) v(x,))" ,l= l for p(x) > v(x)IF _BADD NZUt@,) v(x,))"(s)Basic featuresof theseoperatorsare similar with theseof their analogies:The CoG crispification operator gives all possible solutions in which the degreeof

acceptances higher than the degreeof non-acceptancep(xi)>v(*i)). However, it averagesgood andpoor solutions (although it gives them different weights). The MoM and MoEcrispification operationsgive information about he bestsolution(s) MoE gives t in the senseof higher degreeof acceptance nd lower degreeof non-acceptance).However they ignore

x9- o,,-, .

t2

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information about the other possible solutions. The BADD crispification operator has theanalogousproperty as BADD de-fuzzification operator:- it implies CoG cispificationoperator or cr:l (i t impliesCoG, f v:0 also);- it approaches MoM crispification operator for o + c and approaches MoMcrispification when v:0 also;- it implies middle average MA) when cr:O:IF_BADD(a: I ; v>0) : IF_CoGIF_BADD(a: I ; v:0) : IF_CoGIF_BADD(a + a;v>0) : IF MoM

IF_BADD(q -+ *;v:0) : IF_MoMIF_BADD(a: 0) : IF_MA

a/v 0 Positive0 IF_CoG IF_CoGI IF_MoM IF MoM IF-MA Table ICrisptfication definitions t3] in combination with the conjunction and disjunctionoperationsover IF sets 7] allow to developnew engineeringapplicationsof IF setssuch as IFoptimisation, F decisionsupport system, F controllers, F expert systemsetc.

2.2. F optimisation l-[2]The conceptof IF optimisationproblem is introducedand investigatedn tll-t2l by theauthor. Ideasabout IF setsand optimisation and decisionmaking have been also consideredin [7]-[8]. The original definition of tll-t2l and the approach roposed herefor solving theIF optimisation problem will be briefly presentedhere, as it will be used in AC systemoptimisation problem later. The benefit of the IF optimisationproblems s twofold: they giverichest apparatus for formulation of optimisation problems and, on the other hand, thesolution of IF optimisation problem can satisfy the objective(s) with higher degree han theanalogous uzzy optimisationproblem and the crisp one.

Quite often, it is difficult to describe he constraintsof an optimisation problem by crisprelation's (equalitiesandlor non-equalities) 35]. Practically,a small violation of a givenconstraint s admissibleand it can lead to a more efficient solution of the real problem [35].Objective formulation represents, n fact, a subjective estimation of a possible effect of agiven value of the objective function. Fuzzy optimisation formulations f29,32,34f are moreflexible and allow finding solutions,which are more adequate o the real problem. One of thepoorly studied problems in this field is definition of degreesof membershipand rejection(non-membership)[30,36]. The principles of fuzzy optimisation problems are criticallystudied n pal-126]. However, mainly the transformationsand the solution procedures1231,1251,31], [33] havebeen nvestigated. pplying IF setsconcept t is possible o reformulateoptimisation problem using degreesof rejection of constraintsand values of the objective,which are non-admissible.In general, an optimisation problem includes objective(s) and constraints. n fuzzyoptimisation problems (fuzzy mathematicalprogramming [35], fuzzy optimal control [28],linear programming with fuzzy parameters[32] etc.) the objective(s) and/or constraints orparameters nd relationsare represented y fuzzy sets.These uzzy setsexplain the degreeofsatisfactionof the respectivecondition and are expressed y their membership unctions [36].Let us consideran optimisationproblem:i : I , . . . ,p,(x)---> min

l 3

(6)

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subjectto Sj6),_ v,(x) i -- I ,...,p+qF,6) + v,(x)

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G ,tC : { , xeRn} (10)where G denotesan IF objective (gain)C denotesan IF constraintThis operatorcan be easilygeneralised nd applied o IF optimisationproblem:

D: {, xeRn} (11)p+ql t o : ) l t ,i:lf.1v D :y , ,

where D denotes he IF set of the decisionMin-aggregatorcould be used or conjunction andmax-operator or disjunction:p*.sItD : \i! lt, x e R' (12)

ltn S ltiqv D : ^ ,? f ' ' x e R '

v o 2 v ,It canbe transformed o the following systemof equations:

e 1,_ l t i(x) i :I,...,p+q (13)B>,_ v,(x) i :I,...,p+qla > ,_p > , _ 0a ) , - 0a + p < , _ lwhere a denotes he minimal acceptable egreeof objective(s)and constraints

B denotes he maximal degreeof rejectionof objective(s)and constraintsFinally IF optimisation problem can be transformed tll-t2l to the following crisp(non-fuzzy) optimisation problem which can be easily solved numerically by simplex,gradient-based r geneticalgorithms [5] or, in somecases 25] analyically:(cr-p)-> max (14)

subject to 0 (,_ pri(x) i: 1 . ,p+qB >,_ vi(x) i : l ,...,p+qcr>r- BB > , _ 0c r + B ( , _ 1

In general, he solution of IF optimisationproblem is different than the solution of theanalogous uzzy problem and the degreesof satisfactionof a given objective or constraint nan IF optimisation problem can be higher or lower. It depends on the formulation of therespective unctions of acceptance nd rejection. This property of the IF optimisationproblemis analogous o the property of fuzzy optimisation problemswhen in some cases it dependson the membership function) the solution can satisfy the objective function better, but theprice for this is the worse satisfaction of some constraints.A practical example of multi-objective optimisationof AC systems s considered n the next section.

l 5

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3. optimisation of AC systemsas an IF optimisation problem3.1AC optimisation an multi-objectiveproblemIn general, the problem of AC systemoptimisation is formulated as a multi-objectiveoptimisation,which aims to minimise costs(life-cycle costs,maintenance, ommissioning,capital and running - mainly energy costs) and to maximise the comfort (minimisediscomfort) subject to number of constraints t4l. Building and thermal plant models areconsideredas constraintsas well as different physical and technical imitations over differentcomponentsof the AC system fans,coils, heaters,air-handlingunits etc.):Jt: Costs(AC systemvariqbles: T, RH, m, ...) -+ minJ2: Comfort (AC systemvariables: T, RH, m, ...) -> maxJ3: Discomfort (AC systemvariables: T, RH, m, ...) + minSubject o Constraints (AC systemvariables: T, RH, m, ...) : 0where RH denotes elativehumi dity, o/om denotesmass low rate,kg/s

Practically, the optimisation problem is solved as an uni-objective one where comfortrequirements re considered sa constraint 4]-[5], t9l-tl0l.J : Costs(I, RH, m, ...) + min

(1s)

(t6)

(r7)

Subject o I PPD (f , RH, m, ...) | < I0%Constraints (f, RH, m, ...):0This simplification makes he problem a rough approximationof the real intention of thedesigner. t is so, f conventional echnique s applied,because t simplifies the solution of theproblem. Fuzzy sets and IF sets, specifically, offer a tool for coming to grip with thisproblem.Discomfort, dissotisfaction as well as comfort, which is normally a more restrictivecategory, could be formulated with an appropriate, subjectively specified membershipfunction (Fig.2).The problem of AC systemoptimisationbecame uzzy multi-objective one:

Costs f, RH, m, ...) -+ minDiscomfort (f, RH,m, ...) + mrnComfort (I, RH, m, ...) -)max

Subject o Constraints (f, RH, m, ...) z 0where - denotes uzzyIt should be noted that constraintsand costscould be fuzzy, IF or crisp. We considerfuzzy description for them without loss of generality for the sake of simplification of theexpressions.Building model (which could be of the fuzzy rule-base ype) as well as modelsof componentsof the thermal systemare not considereddue to simplicity.3.2 Partial Load Ratio definition by IF setsFormulations of constraintswith IF setsallow a higher flexibility to describebetter thereally existing uncertainty. For example, fans and coils are used with a part of their designload range: he loads ower than given thresholdsare not so effective.ParameterPartial LoadRatio (PLR) represents he load usedas a ratio to the designed oad. Practically, loads lowerthan given thresholdare not practically used(dashed ine on the Fig.l). Very often, plBcriticalis 20ohof the design oad (Lo"'tn). Fuzzy constraintsover the fan's load (bold line p; onFig.l ) give additional flexibility in description.According to this fuzzy constraint, PLR in arangeof [0.15; 0.25] are acceptable, ut with degree ess than 1. Introduction of a non-

i pcostsI Vdiscomforti pcomfortI /lconstraints

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membership (rejection) function, however, (bold line v1 on Fig.l) enhanceseven further thisflexibility: combinationof prtandvsdefinesan IF set over PLR (Fig.1). ts meaning s that:

the rateof rejection (non-acceptance) ecreasing teeplyuntil 20% of the I.desisn'not ully acceptable acceptablewith some eservations)as pr1

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(l - conventionalcase;2 - comforl description; 3 - discomfort description)From the Fig. 2 one could also see hat for this particular exampleof thermal comfort,slightly higher temperatures,which are still in the standardcomfort band (25-26oC) are withhigher degree of rejection v1 (dissatisfoction, discomfort) than slightly lower temperatures(22-23 oC). That means that the non-membership (discomfort) function could be non-symmetrical. The practical meaning is that this person(s) tend to prefer slightly lowertemperature han normal to slightly higher. Such degreeof flexibility and freedom s difficult,if not possible, o haveevenusing descriptionwith conventional vzzy sets. n practice,whenstatisticalmeasuresare used insteadof fuzzy and IF sets, such nuancesare simply ignored.Our point is that using more complex and realistic description possesspotential for muchmore customisedand effective systems o be build in engineeringpractice ncreasingmachineintelligencequotient.4. Solving Multi-objective IF optimisation problem of AC systemcontrol

Solution of the optimisationproblem (17) could be found numerically Il-2].For somespecialcases t could be found analyically l25l.We considera simple exampleof an ACsystemoptimisation in order to illustrate he proposedapproach.Suppose hat cumulative costs (capital and running) dependson PPD as follows (Fig.3)[4]:

Costs :9 .9+ II P P D Iand suppose for simplicity that constraints are satisfied for all variables considered, hefollowing optimal solution is achieved:Costso I0 ktPPDO TI O%Practically, the dependence f Costson PPD is more complex and includesbuilding andplant models often it is dynamic problem [4]-[5], t9l-ll0l. Variablesare usually controlparametersof the plant (supply air temperature,mass low rate etc.). However, the natureofthe problem is the same.If apply conventional optimisation problem (15) comfort characteristicswill be in therequiredrange,but on one of its marginalvalues(+ or - 10% dissatisfied).Suppose hat comfort and discomfort are described n a similar way as shown on Fig.2(with the only difference that instead of T and marginal values of 22 and 26 oC thedependences in respect o PPDwith tl\% dissatisfactions boundaryconditions).

(18 )

(le)

(20)r ro. lPPD)2; PPD10%

C+lu;ooo

1 110.810.610.410.21 09. 89. 69.4

Costs unction

9 T q o N $ ( oFig.3Costsas a functionof PPD

1 8

3 PPD,o/o

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Suppose hat costs aregiven with the following membership unction [4]:I . . . . . . . . . . . . . . . . . . . . . . . ;osts 10 f0.ll(9.9 Costs); Costs 10 f

Costs 9.9+ IIPPDI

(2r)

1 . 2I# o e8,0'o

= 0 40.20

r ! t l - O c . ) ( O O )r c i d c t

Fig.4 Costs unction (conventional uzzy set)Then the following IF multi-objective optimisationproblem arise[4]:Fresutt> max (22)

Vdiscomfort minsubject to presutt : FcostsAND pcomfortAND pconstraintsIts solution could be found analytically or somecases 25] from:

trt"os,PPD) : min(trtromfortgPD,vdir"o.for) Q3)PPD"+I0IPPDI 100:0 (24)Note that objective function is a conventional fuzzy set, comfort/discomfort are IF setsandother constraints ouldbe.either uzzy,IF or crispsets.It is supposed or simplicity that constraintsare satisfied or all valuesof consideredvariables.The fuzzy optimal solution is:Costsoq' f, l0 060 (-0.6%) (2s)

PPDopt !4.2% (+3Bo% omparing o the standard10% dissatisfaction)It is seen hat significant improvementwith more than ll3 of thecomfort requirements ould beachieved with marginally higher (with only 0.6% or f60) costs. For the more complex (andrealistic) IF optimisation problems he solution could be found numerically using gradient-based rgeneticalgorithms 5].5. Conclusions

Optimisation of air-conditioning systems is treated in the paper in the framework of IFoptimisation. This new approach allows to formulate more precisely the problem, whichcompromisesenergy saving and thermal comfort/discomfortsatisfactionunder given constraints.This fuzzy optimisationproblem is solvedanalytically under someassumptions.IF optimisation of AC systemsallow us to formulate problem more flexibly, user-friendlyandto analysemore preciselyand deeply the whole rangeof possiblesolutions. t helps us to avoidnecessityof scaling and normalisation applied in other approaches when penalty functions areused) as well as infeasible solutions (infeasible solutionshas degreeof rejection I and are simplynot considered n the final result). The problem is formulated in more natural IF multi-objectivefashion and is solved efficiently. This is possibleanalytically in some casesand either numericallyin general.A practical engineering example illustrates the viability of the proposed approach. Morecustomised solution (significantly improving comfort - with more than 38%) is found for themarginally higher energy expenseof f60 or 0.6% more. In general, t is possibleto find less

Costs unction

t 9

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energetically expensivesolutions,which in the same ime are more customised.This, however,dependson the specific preferencesof occupantsand could not be guaranteedor all practicalcases.6. AcknowledgenentTheauthor would like to thank EPSRC or the financial support rough hegrantGNM27299.7. References[] AngelovP., Optimizationn an Intuitionisti Fvzzy Environment, wzy Sets ndSystems,86,299-306.1997[2] AngelovP., ntuitionisticFuzzyOptimization, oteson IntuitionisticFuzzySets,1,27-33, 995[3] AngelovP., CrispiJication: efuzzifrcationver IntuitionisticFuzzy Sets,BUSEFAL,64,5l-55,1995[4] Angelov P., A FuzzyApproach o Multi-Objective Optimization of Building ThermalSystems,Proc.ofthe 8'o FSA WorldCongress, aiwan,Taipei,17-19August 1999, ,429-434[5] Angelov P., Evolving Fuzzy Rule-basedModels,Journal of CIIE, special issueon IndustrialApplication of Soft Computing,17, 2459-468,2000, SSN l0I7 -0669[6] Atanassov ., IntuitionisticFuzzySets, uzzySets nd Systems,20,87-96,986[7] AtanassovK., Ideas or Intuitionistic Fuzzy Equations, nequalitiesand Optimisation,Notes onIntuitionisticFuzzySets, ,l7-24, 1995[8] BustinceH., Handling Multicriteria Fuzzy Decision Making ProblemsBasedon IntuitionisticFuzzySets,NotesonIntuitionisticFuzzySets (l) (1995)42-47l9l Wrieht J.A., HVAC Optimisation Studies: Sizing by GA, Building SystemsEngineeringReseqrch nd Technologt,17 1) (1996)7-14[ 0]Ren M.J., Optimal Predictive Control of Thermal Storage n Hollow Core Ventilated SlabSystems, h.D.Thesrs, oughborough niversity 1997)[11] Kajl S., P.Malinowski,E.Czogala,M.Blazinski,Flzzy logic and NN approach o thermaldescription of buildings, Proc. of the 3th European Congress on Fuzzy and IntelligentTechniques UFIT'9S, Aachen,Germanyv.l (1995)299-303[2] Skrjanc . et.al.,Flzzy Modelingof Thermal Behavior n Building, Proc. of the 6th EuropeanCongress n IntelligentTechniquesUFIT'98,Aachen,Germany, (1998)878-882[13]LovedayD.L., G.S.Virk, Artificial Intelligenceor Buildings,pplied Energt,4l, 1992,201-221[ 4] Culp C.et al.,The Impact of AI Technology within the FIVAC IndustrySHRAEJourna ,l2,l 990,I2 22[15]HagrasH., V. Callaghan,M. Colley,G. Clarke,A Hierarchical uzzy GeneticMulati-agentArchitecture for Intelligent Buildings Sensingand Control, Conferenceon Advances n Soft

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