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Performance study of a multi-objective mathematical programming modelling approach for energy decision-making in buildings Christina Diakaki a, *, Evangelos Grigoroudis a, Dionyssia Kolokotsa b a b

Department of Production Engineering and Management, Technical University of Crete, University Campus, Kounoupidiana, 73100 Chania, Greece Department of Environmental Engineering, Technical University of Crete, University Campus, Kounoupidiana, 73100 Chania, Greece

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 February 2013 Received in revised form 13 July 2013 Accepted 17 July 2013 Available online 12 August 2013

The improvement of energy efﬁciency in buildings is among the ﬁrst priorities worldwide. To this end, several measures are available, and the decision maker faces a decision problem with multiple objectives having to compensate several energy, ﬁnancial, and other factors in order to make a satisfactory selection. To solve this problem, a decision modelling approach is proposed herein, based upon the principles of multi-objective mathematical programming, thus capturing only these elements, which affect the decisions to be taken. To evaluate its performance under realistic operational conditions in a building, the proposed approach is applied to an existing building for retroﬁt purposes, and several simulation investigations are performed in order to study and evaluate the quality of the retroﬁt alternatives proposed by the decision model. The results of these simulation investigations conﬁrm, that despite its reduced precision compared to the corresponding simulation model of the building, the decision model allows for the realistic comparative evaluation of the considered alternatives. The example case study reported herein, demonstrates also the functionality of the proposed approach, exploits its qualities, and highlights its strengths, weaknesses and limitations. 2013 Elsevier Ltd. All rights reserved.

Keywords: Building Energy efﬁciency Energy improvement Mathematical programming modelling Multi-objective optimisation

1. Introduction The energy sector nowadays faces signiﬁcant challenges that are expected to become even more acute in the upcoming period. The current energy trends as well as the related carbon emissions raise great concerns about the “three Es”, i.e. the environment, the energy security and the economic prosperity as deﬁned by the IEA (International Energy Agency) [1]. Improving the energy performance of buildings is a key measure to achieve the ambitions of Europe, particularly EU Climate & Energy targets to reduce Greenhouse gases emissions by 20% and achieve energy savings of 20%, both by 2020. Being responsible for the 40% of the energy consumption and 36% of the carbon emissions worldwide, buildings are targeted as the sector with the most signiﬁcant energy efﬁciency margin. Energy efﬁcient economy should be the main focus in the buildings and construction sector as mentioned in the 2012/27/EU directive announced in October 2012 [2]. In order to shift to a more sustainable future, the spread of innovative technological solutions should be accelerated.

* Corresponding author. Tel.: þ30 28210 37346; fax: þ30 28210 69410. E-mail address: [email protected] (C. Diakaki). 0360-5442/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.07.034

From the energy perspective, the adaptation of buildings in the climate change involves solutions considering the building envelope and its insulation, the space heating and cooling systems, the water heating systems, the lighting appliances and other equipment. In contrast, however, to other systems, most buildings have a long life span. This means that more than half of the current global building stock will still be standing in 2050, while, at the same time, even new buildings under the present economic environment can be energy inefﬁcient [3,4]. As a consequence, most of the energy and CO2 savings’ potential lies in the retroﬁtting and procurement of new technologies for the existing building stock, as well as in the efﬁcient design and establishment of improved standards for new buildings. To this end, various measures may be considered in the design but also in the operational stage of a building, when renovation or retroﬁt actions are performed [5]. According to Wulﬁnghoff [6], there are over 400 alternative measures which may be considered separately or combined in groups. With such a variety of available measures, the main challenge is to identify those that will be the most effective and reliable in the long term considering environmental, energy, ﬁnancial or other factors. In other words, the DM (decision maker), who may be the architect, the engineer or the building expert, faces a decision problem with multiple objectives, where the search of an optimal

C. Diakaki et al. / Energy 59 (2013) 534e542

solution is meaningless, since the criteria, which have to be satisﬁed are generally competitive (e.g. energy efﬁcient solutions are more expensive than less efﬁcient ones). The dominant approach to the aforementioned problem involves, mainly, simulation-based approaches, whereby alternative scenarios, which are expected to improve the energy performance of the building under study, are initially prescribed by the DM [7]. These speciﬁc scenarios, which may vary according to buildings’ characteristics, type, use, climatic conditions, etc., are then evaluated through simulation using more or less advanced/detailed calculations [8e18]. Sometimes, the DM employs complementary to simulation, advanced decision support techniques. This procedure may involve for example multicriteria-based decision making methods (see e.g. [19e26]) that are introduced to assist him/her in examining the trade-offs among the pre-deﬁned and pre-evaluated alternative actions and reach a ﬁnal decision (see Ref. [27] for a complete state-of-the art review). The approach described above allows the DM to obtain a quite precise quantiﬁcation of the alternative scenarios and solutions’ energy performance. Moreover, this approach requires advanced knowledge and expertise from the DM in order to successfully select the predeﬁned energy efﬁciency scenarios, solutions and technologies. Therefore, in the simulation based approach the decision problem is discretised, and some of the resulting discrete solutions and scenarios are examined. As a consequence the optimality of the ﬁnal solution depends in a great extent on the DM’s ability and expertise as well as on the number of the examined discrete solutions. For example, if the DM decides to evaluate only a small number of solutions, the optimality of the ﬁnal solution is not at all guaranteed. In fact, there is no guarantee that the ﬁnal solution is among the set of good solutions, in the sense that there might be other energy efﬁcient ones that perform better in all the considered criteria. Furthermore, the selection of a representative set of alternatives is usually a difﬁcult problem, while the ﬁnal solution is heavily affected by these predeﬁned alternatives. On the opposite case, i.e. if the DM deﬁnes a large number of solutions, the required evaluation and selection process may become extremely time-consuming and difﬁcult to handle. As a consequence, the DM’s work is limited to a potentially large but certainly ﬁnite number of alternative scenarios, while the real opportunities are enormous. In order to avoid the predeﬁnition of the alternative scenarios and/or solutions that will be evaluated, various alternative approaches have been proposed. In one of these approaches the decision problem is deﬁned considering multiple objectives, while energy simulation models are combined with GAs (genetic algorithms). The recent literature reports on several such approaches (see Ref. [28] for a complete relevant review), where GAs are combined with known tools like the DOE 2.1 [29] thermal analysis software [30e32], the TRNSYS [33] simulation software [34,35], or other less known energy calculation models [36e43]. In all these approaches, GAs are employed to search the decision space, while the simulation or other energy analysis tools are used to evaluate the solutions proposed by the GAs. This approach however, is still computationally expensive, since the time associated with optimisation can become prohibitively high due to the usually large number of simulations that need to be performed. Nevertheless, the DM has the ability to examine a potentially inﬁnite solution space, while obtaining at the same time a quite precise quantitative evaluation of the examined solutions. By nature, the GAs may provide also several satisfactory solutions, as they result in a population of good solutions rather than a single solution, while they allow the utilisation of quite complex mathematical models. The necessity of applying GAs is justiﬁed by the size and the complexity of the deﬁned optimisation problems. The usage,

535

however, of the simulation or the other employed tools may considerably increase the computational effort, which is already overloaded due to the requirements of developing and running the GA itself. In addition, the GAs do not guarantee the ﬁnding of the optimal solution, while in the majority of the cases reported in the literature, the DM’s preferences are not taken into account during the decision making process [28]. Instead, the GA is limited in ﬁnding the Pareto frontier, i.e. the set of solutions that are not dominated by any other solution. This means, however, that other more or less sophisticated techniques have then to be applied upon it, in order to identify this single solution that will satisfy the DM’s preferences. On the other hand, a multi-objective methodology can be developed without any need of coupling with other tools, based on a mathematical programming rather than a building simulation type of modelling. According to Williams [44], a mathematical programming model involves a set of mathematical relationships such as equations and inequalities, which correspond to some down-to-earth relationships in the real world. In many occasions, the data utilised to build such models are not precisely quantiﬁed and the developed mathematical relationships do not reﬂect all the realworld problem details. Nevertheless, with mathematical programming, it is still possible to result in little inaccuracies in the solutions, if the elements of reality that are important in decision making are captured. After all, within the context of decision making, the purpose of mathematical programming models is not to precisely represent reality but to assist the whole process through the creation of a realistic basis for the comparative evaluation of the available alternative solutions at the less possible computational effort. For this reason, such models should be used as one of a number of tools for decision making, and the answers that they produce should be always subjected to close scrutiny [44]. Following the mathematical programming type of modelling, Diakaki et al. [5] investigated and developed [45] a multi-objective decision modelling approach for the problem of energy efﬁciency in buildings, which has then been adopted by other researchers too (see e.g. [46e49]). It is the aim of the present study to further investigate and highlight the strengths, weaknesses and limitations of the speciﬁc approach, as well as its potential synergies with other methods, within the context of decision analysis for the energy efﬁciency in buildings. To this end, a decision problem concerning the retroﬁt of an existing building is deﬁned. The problem is modelled as a multi-objective mathematical programming problem and solved via available relevant solution techniques. The suggestions of the decision model are then evaluated under real operational conditions using a validated simulation model of the building, while useful results and recommendations are extracted. It should be noted here that the focus of this particular study is not on the speciﬁc decision problem addressed herein, but on the methodological approaches, which may be adopted for its solution. The rest of the paper is structured in ﬁve more sections. Section 2 introduces the considered retroﬁt decision problem, while Section 3 presents its solution via the proposed multi-objective mathematical programming approach. The simulation evaluation of the decision model’s recommendations is presented in Section 4, while a discussion of the ﬁndings of the study, as well as of the strengths, weaknesses and limitations of the proposed approach follows in Section 5. Section 6, ﬁnally, summarises the conclusions of the study. 2. The decision problem The decision problem considered herein focuses on the energy efﬁciency of the building depicted in Fig. 1. The speciﬁc building is located in the suburbs of Iraklion, Greece within the campus of TEIC

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daytime from the building’s windows and doors, while during night, artiﬁcial lighting is provided using energy-efﬁciency lamps. The energy-efﬁciency lamps are also used during winter, depending on the prevailing natural lighting conditions. The building is permanently occupied by three persons performing, seated, light work such as typing, while quite often, the occupation increases by visiting students. Finally close to the building a weather station is installed. Based on the indoor environmental conditions’ analysis of the considered building [50], the energy requirements are mostly due to increased cooling load during the summer season that is attributed mainly to the climatic conditions in Crete, Greece. Due to the same reason, the energy consumption for heating is only a small portion of the total energy demand [50]. To reduce therefore the energy load of the building, the following alternatives are considered as being the most appropriate:

Fig. 1. The building of the case study.

(Technological Educational Institute of Crete), and covers a total area of approximately 50 m2. It is actually a uniform area constructed in 1997 as an administrative ofﬁce for research programs [50]. The walls and roofs of the building have increased insulation. The structure of the walls consists of three layers that are, from inside to outside: 0.025 m of insulation, 0.200 m brick and 0.050 m plasterboard. The ceiling’s structure consists mainly of two layers that are, from inside to outside: 0.200 m concrete and 0.030 m insulation. Moreover, approximately half (i.e. 25 m2) of the building’s ceiling (at the north side) is covered by a pitched roof that is, in turn, covered by tiles. The structure of the building’s ﬂoor consists basically of one layer that is 0.150 m massive concrete, while in its foundations, an underground exists. In addition, the building has 10 windows and 2 doors of the same construction covering a total surface of 10.89 m2. At the south of the building, there is a shading mat of dimensions 2.60 m 2.50 m 3.00 m (height width length). Heating and cooling is provided by an A/C Inverter of 3.5 kW and COP (coefﬁcient of performance) equal to 2. Table 1 summarises the characteristics of the considered building. As mentioned earlier, the building is used as an administrative ofﬁce. It is therefore, equipped with personal computers, printers and other related ofﬁce equipment. Lighting is provided during

Table 1 Summary of building’s characteristics (adopted by Ref. [50]). General information Location Iraklion, Crete, Greece Surface area 50 m2 Operation hours 08:00e14:30 MondayeFriday and 17:00e21:00 Tuesday, Thursday and Friday excluding national holidays Orientation NortheSouth Building envelope Walls Three-layer (insulation, brick, plasterboard) outer walls with thermal conductivities of insulation, brick and plasterboard layers equal to 0.04, 0.89, and 0.14 W/mK, respectively Roof Two-layer (concrete, insulation) ceiling with thermal conductivities of concrete and insulation layers equal to 2.1 and 0.04 W/mK, respectively Windows Double glazing windows with thermal transmittance U ¼ 1.4 W/m2K and effective total solar energy transmittance g ¼ 58.9% Floor One layer (concrete) ﬂoor with thermal conductivity equal to 2.1 W/mK Building services Cooling system A/C Inverter of 3.5 kW (Electricity) Heating system Same with cooling system

increase of roof insulation; replacement of doors and windows with others that will prevent the high levels of solar radiation; replacement of the heating/cooling system with a more efﬁcient one. Tables 2 and 3 summarise the characteristics of the considered alternative solutions, including their costs. A cost of 150 Euros/m3 is also considered for the ceiling’s insulation material, which due to space limitations may not exceed the 0.10 m totally. The values of the thermal and solar transmittance in Table 2 have been taken from the ASHRAE database [51], while all the cost values, as well as the heating/cooling systems data of Table 3 have been obtained through a short market survey. For this particular building, a detailed TRNSYS simulation model has been developed by Kolokotsa et al. [50], which allows for the evaluation of alternative retroﬁt solutions. The simulation model has been validated against real data, thus ensuring an acceptable level of representation of reality (see Ref. [50] for details on model development and validation). 3. Multi-objective mathematical programming modelling and solution of the decision problem The development of a multi-objective mathematical programming modelling approach for the improvement of energy efﬁciency in buildings requires: The deﬁnition of decision variables, discrete and/or continuous, to reﬂect the total set of alternative measures to be considered; The identiﬁcation and formulation into appropriate linear and/or non-linear mathematical expressions of the objectives to be achieved; The delimitation of the set of feasible solutions through the identiﬁcation of linear and/or non-linear constraints concerning either the decision variables and their intermediary relations or the objectives of the problem (natural and logical constraints may also be considered as necessary); and The development of a solution technique capable of handling the deﬁned variables, objectives and constraints. Table 2 Characteristics of alternative window/door types. Type

U value (W/m2K)

Effective total solar energy transmittance (%)

Cost (Euros/m2)

1 2 3

0.40 0.70 1.26

0.408 0.407 0.212

85 90 100

Insulating xenon Insulating krypton Low SHGC argon, gold

C. Diakaki et al. / Energy 59 (2013) 534e542 Table 3 Characteristics of alternative electrical heating/cooling systems. Type 1 2 3

A/C inverter of 3.8 kW A/C inverter of 5.3 kW A/C inverter of 7.0 kW

COP (coefﬁcient of performance)

Cost (Euros)

2.10 2.30 2.50

700 900 1300

The mathematical model should be as precise as possible to preserve validity of suggested solutions and as simple as possible to reduce the required computation effort. Based on the aforementioned principles, Diakaki et al. [45] developed a multi-objective decision model, which is adopted herein for the decision problem deﬁned in Section 2. This model, which may easily be solved via compromise programming, allows for decision making regarding: Building envelope components (for walls, ceilings and ﬂoors) and materials; Doors and windows types; and Heating, cooling and DHW (domestic hot water) systems; based on the minimisation of the following three criteria: Total annual primary energy consumption; Total annual CO2 emissions’ release; and Initial investment cost. The model can be used as a screening tool for the various solutions in the sense that it captures those elements only, which affect the criteria of concern to the DM. The application of the aforementioned multi-objective decision modelling approach to the particular decision problem examined herein (see Section 2), leads to the mathematical model, which is summarised in the Appendix, and has the following two pursued goals:

537

consequently the overall energy requirements of the building. At the same time, the maximum permissible roof insulation is added to reduce the negative effects of its cement-based structure, and the most energy efﬁcient heating/cooling system is selected to ensure maximum utilisation of the primary energy. These decisions, which in the case of the building are the most energy efﬁcient ones, result in the maximisation of the corresponding cost, which reaches its minimum value, when the cheapest, but least energy efﬁcient, window/door type and heating/cooling system are selected with no additional roof insulation (see again Table 4). When both criteria are optimised simultaneously, the decisions suggested by the model are also affected by the preferences of the DM, which are expressed via the weight coefﬁcients p1 and p2, with p1þp2 ¼ 1, for the energy and cost criterion, respectively. These weight coefﬁcients imply the trade-off among values of the criteria, which are normalised with regard to their optimum feasible values, and may be deﬁned by the DM through questions appropriate to elicit this trade-off. As the weight coefﬁcient of a criterion increases from 0 to 1, its corresponding values decrease and ﬁnally reach its optimal value, when only this criterion is optimised independently from the other. For intermediary values of the weight coefﬁcients, several solutions may be obtained that favour each criterion at a higher or lower level, depending upon the speciﬁc values that have been selected. This behaviour is demonstrated through Fig. 2, which displays graphically the changes of the criteria values in relation to the weight coefﬁcient values, indicating that the more a criterion is considered important by the DM, the more the ﬁnal decision is in favour of this criterion. In this ﬁgure, the DM may also deﬁne maximum acceptable levels regarding the values of the criteria (aspiration levels). In addition, Table 5 summarises the decisions suggested by the model for different values of the weight coefﬁcients. 4. Simulation investigations

Decrease of the total annual primary energy consumption Q due to space heating and cooling, which reﬂects the operational costs of the building; and Decrease of the cost C for the acquisition of windows/doors and roof insulation, which corresponds to the initial investment cost for the retroﬁt of the building. The emissions criterion is omitted in this particular study because all considered systems for space heating and cooling are electrical (see Table 3), which means that their related emissions will be analogous to the consumed primary energy. Table 4 displays the, so-called, payoff matrix, which provides the criteria values when each of them is optimised independently from the other, and demonstrates their competitiveness. When the energy consumption criterion is optimised independently from cost, the window/door type with the minimum effective total solar energy transmittance is selected to reduce the level of solar radiation entering the building via the openings during the summer season, thus reducing the corresponding cooling load and

As mentioned earlier, the decision model described in Section 3 focuses on these elements of the building, which affect the criteria of concern to the DM. It is therefore necessary to evaluate the suggested decisions under realistic operational conditions in the building. To this end, several simulation runs are performed using the TRNSYS building model aiming at quantifying and evaluating the building performance under the retroﬁt proposals suggested by the decision model. The scenarios examined and compared via simulation include the following: Scenario 1: The building as it is; this is the reference case against which the decision model’s suggestions will be compared and evaluated. Scenario 2: Application of the decision model’s retroﬁt suggestions for the case where the DM is only interested in minimising the initial investment cost, i.e. when the weight coefﬁcient of energy and cost criteria are p1 ¼ 0 and p2 ¼ 1,

Table 4 Payoff matrix. Type of solution

Selected window/ door type

Thickness of added insulation (m)

Selected heating/cooling system

Total annual primary energy consumption (KWh)

Initial investment cost (Euros)

Minimisation of total primary energy consumption Minimisation of initial investment cost

Low SHGC argon, gold Insulating xenon

0.10

A/C inverter of 7.0 kW

4329.29

2916.62

0.00

A/C inverter of 3.8 kW

5974.72

1625.91

538

C. Diakaki et al. / Energy 59 (2013) 534e542 Table 5 Decision model suggestions for different values of weight coefﬁcients. Type of solution p1

p2

0.00 0.25 0.50 0.75

1.00 0.75 0.50 0.25

1.00

0.00

Selected window/ door type

Thickness of added insulation (m)

Selected heating/ cooling system

Insulating xenon Insulating xenon Insulating xenon Low SHGC argon, gold Low SHGC argon, gold

0.000 0.022 0.016 0.025

A/C A/C A/C A/C

0.070

A/C inverter of 7.0 kW

inverter inverter inverter inverter

of of of of

3.8 3.8 5.3 5.3

kW kW kW kW

Fig. 2. Criteria values vs coefﬁcient values (results of decision model).

respectively (see Table 5 for the details of the corresponding retroﬁt suggestions). Scenario 3: Application of the decision model’s retroﬁt suggestions for the case where the DM is interested in minimising the initial investment cost, but not at the total expense of energy improvements, i.e. when the weight coefﬁcient of energy and cost criteria are p1 ¼ 0.25 and p2 ¼ 0.75, respectively (see Table 5 for the details of the corresponding retroﬁt suggestions). Scenario 4: Application of the decision model’s retroﬁt suggestions for the case where the DM is indifferent between the cost and the energy consumption criteria, i.e. when their respective weight coefﬁcients are both equal to 0.5 (see Table 5 for the details of the corresponding retroﬁt suggestions). Scenario 5: Application of the decision model’s retroﬁt suggestions for the case where the DM is interested in minimising the primary energy consumption, but not at the total expense of the corresponding cost, i.e. when the weight coefﬁcient of energy and cost criteria are p1 ¼ 0.75 and p2 ¼ 0.25, respectively (see Table 5 for the details of the corresponding retroﬁt suggestions). Scenario 6: Application of the decision model’s retroﬁt suggestions for the case where the DM is only interested in minimising the primary energy consumption, i.e. when the weight coefﬁcient of energy and cost criteria are p1 ¼ 1 and p2 ¼ 0, respectively (see Table 5 for the details of the corresponding retroﬁt suggestions).

standards. Moreover, the initial investment cost is calculated via the following equations:

ðInitial investment costÞ ¼ ðCost for doors=windowsÞ þ ðRoof insulation costÞ þ ðSystem costÞ

(2)

ðCost for doors=windowsÞ ¼ ðTotal area of doors=windowsÞ Cost of selected type per m2 (3) ðRoof insulation costÞ ¼ ðInsulation areaÞ ðInsulation thicknessÞ Cost of insulation per m3

(4)

considering the choices that each scenario assumes (see Table 5), and the cost data of Tables 2 and 3. The simulation results conﬁrm the correctness of the choices made by the decision model. The more the DM spends, the more energy efﬁciency is achieved. This is also demonstrated via Fig. 3, which displays the changes of total primary energy consumption and initial investment cost for the different examined scenarios.

5. Discussion Tables 6 and 7 summarise the results for each of the aforementioned, examined scenarios. The criteria against which these scenarios are evaluated include the energy consumption for heating and cooling purposes as well as the primary energy consumption and the initial investment cost. The heating and cooling energy consumption are calculated by TRNSYS, while primary energy consumption is calculated according to the following equation

ðPrimary energy consumptionÞ ¼

The application of the proposed decision modelling approach presented in Section 3, as well as the simulation investigations of Section 4, reveal some interesting points. The values of the considered criteria (i.e. the primary energy consumption and the initial investment cost) show the same trend either calculated via simulation, as Fig. 3 displays, or calculated via the decision model, as Fig. 2 displays. This behaviour suggests that the decision model

ðEnergy consumptionÞ=ðGeneration efficiency of systemÞ ðReturn rate of the electricity power plantÞ

where the values for the generation efﬁciency are taken from Table 3 considering the system that each scenario assumes (see Table 5), while for the return rate of the electricity plant, the value 0.35 has been considered according to the Greek technical

(1)

successfully managed to capture the signiﬁcant elements of the building operation and provided solutions that can indeed allow the DM to make a satisfactory decision without the need to run several and more complex simulation runs.

C. Diakaki et al. / Energy 59 (2013) 534e542

539

Table 6 Simulation results. Scenario ID

1 2 3 4 5 6

Annual energy consumption (kWh)

% change compared to scenario 1

Heating

Cooling

Total

Heating

Cooling

Total

453.7 399.6 383.2 386.2 499.9 446.4

3900.0 3915.0 2971.0 3154.0 2807.0 2160.0

4353.7 4314.6 3354.2 3540.2 3306.9 2606.4

e 11.92 15.54 14.88 þ10.18 1.61

e þ0.38 23.82 19.13 28.03 44.62

e 0.90 22.96 18.69 24.04 40.13

If the DM had chosen the common approach to make a decision, he/she would have to run several simulations. Taking into account that the alternative choices within the presented study included 3 alternative types of doors/windows, 3 alternative types of heating/cooling systems and inﬁnite alternatives for additional ceiling insulation (from 0 to 0.70 m), a DM would have to run 3 3 n simulation runs, where n is the number of the alternative choices that he/she would have selected to examine for additional roof insulation. For example, if he/she would have selected to examine the addition of 0, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60 and 0.70 m of ceiling insulation, i.e. 8 alternative choices, he/she should have to run 3 3 8 ¼ 72 simulation runs and then examine empirically their results or employ a multicriteria decision analysis approach to assist him/her in this examination. The simulation runs are quite high although the example case study has been kept, for demonstration purposes, rather simple, while in reality, the DM may have to examine several more alternative solutions. The decision model, which was developed based on the principles of mathematical programming modelling, allowed the DM to consider in an easy, straightforward and quick way all the available options, taking into account at the same time his/her preferences. Despite its reduced precision compared to the corresponding building simulation model, the decision model allowed for the realistic comparative evaluation of the alternatives. In addition, having developed the decision model following the principles of mathematical programming, the DM may easily expand it, if necessary, in order to consider additional retroﬁt alternatives and/ or use it in case of other buildings. The adoption of mathematical programming allows the development of generic and ﬂexible tools, which may be applied with little modiﬁcations (of mainly the input data) to other buildings, independent of size and type. It also allows the introduction of the DM’s preferences, directly in the models, thus simplifying the overall decision process. Finally, mathematical programming offers a variety of alternative, more or less traditional, solution techniques, which may be employed as appropriate depending on the size and complexity of the deﬁned decision problems [5,28].

Table 7 Total annual primary energy consumption vs initial investment cost. Scenario ID

Total annual primary energy consumption (kWh)

% change compared to scenario 1

Initial investment cost (Euros)

1 2 3 4 5 6

6219.57 5870.20 4563.54 4397.76 4107.95 2978.74

e 5.62 26.63 29.29 33.95 52.11

e 1625.91 1788.72 1942.48 2177.31 2916.62

Fig. 3. Total primary energy consumption vs initial investment cost for the examined scenarios (simulation-based results).

Of course, as with all models, the validity of the results of the proposed approach depends upon the quality of the structure and the data utilised in the development process; poor models may lead to inaccurate and misleading results. The decision model should be precise enough to preserve the validity of the results and, at the same time, simple enough to reduce the required computational effort. Finally, it should be noted that despite its undoubted strengths, as discussed above, the proposed approach does not provide a precise quantiﬁcation of the performance of the suggested decisions. To get such a precise quantiﬁcation, an analytic model is necessary. The adoption of an analytic model, however, leads to the simulation-based approaches to decision making, described earlier in Section 1, thus losing all the aforementioned advantages of the mathematical programming approach. In real-life studies though, prior to any retroﬁt or new-construction project, decisions should be reached in a rather straightforward manner supplemented also by results of detailed analyses. Within the frame of such studies, the proposed approach could be really useful, since it could be used as a screening tool to easily and effectively search the entire decision space for solutions that satisfy the needs and preferences of the DM. Then, a detailed simulation model could be adopted to perform the required detailed analyses focussing, however, only on the alternatives identiﬁed by the decision model as being signiﬁcant to the DM. This way, the whole decision-making process can be simpliﬁed and accelerated. 6. Conclusions The improvement of energy efﬁciency in buildings is among the ﬁrst priorities worldwide. To this end, several measures are available, and the DM has to compensate several energy, ﬁnancial, and other factors in order to make a satisfactory selection. The problem of the DM is characterised by the existence of multiple and in several cases competitive objectives, each of which should be optimised against a set of feasible solutions that is prescribed by a set of parameters and constraints that should be taken into account. In simple words, the DM is facing a decision problem with multiple objectives, which is usually approached through simulation-based methodologies that focus on particular aspects of the problem rather than dealing with it in to its global dimension. In contrast to the aforementioned approaches, the multiobjective mathematical programming approach discussed herein allows for the consideration of a potentially unlimited number of available options without the need to be combined and/or complemented by any other method. The approach has been applied to an existing building considering different retroﬁt options.

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The decision model results have been evaluated through simulation, which conﬁrmed their validity under realistic operational conditions in the considered building. This has been the result of a careful model development process, which is a prerequisite in order to reach quality results. The proposed methodological approach allowed for the development of a generic and ﬂexible decision model, which takes into account the DM’s preferences and may lead to useful recommendations. However, it should not be considered as a panacea for all cases and all types of decision problems. The pursued goals and the particular characteristics of each considered problem should always be the compass towards the selection of the appropriate methodological approach and corresponding tools, keeping in mind that there will always be a trade-off between the complexity and the corresponding computational effort.

Acknowledgements This work was partially supported by the Special Account of Research Funds of the Technical University of Crete under the “Development of a Decision Support System for the Improvement of Energy Efﬁciency in Buildings” Research Project within which the decision model adopted herein was developed, and the IEE (Intelligent Energy Europe) program SAVE 2007 under the CoolRoofs (Contract N : EIE/07/475/SI2.499428) Research Project within which the simulation model adopted herein was developed and validated. The contents of the paper reﬂect the views of the authors, who are responsible for the accuracy of the data presented herein. The contents do not necessarily reﬂect the ofﬁcial policy of either the Technical University of Crete or the European Commission.

Appendix A. The multi-objective decision model A1. Parameters, variables and criteria

(continued ) Notation Description FS,dw FCM,dw ISL,dw,n

gv Uv dWAL,wl dFL dRF,rl kWAL,wl kFL kRF,rl k nel COPs dmin dmax BLC QHD,n QCD,n QVEN,n QVNHG,n CD/W,v Cd Cs xv

xd xs

Q C x

l G1(x) G2(x)

Notation Description V S DW W WL RL v s dw w wl rl n HSn CSn Tn

qIH qIC qE,n

AD/W,dw AWAL,w AFL ARF bD/W,dw bWAL,w bFL bRF FF,dw

Number of available doors/windows types Number of available electrical heating/cooling systems Total number of building’s doors and windows Total number of building’s walls Total number of layers of the wall structure Total number of layers of the roof structure Index to V; v ¼ 1, ., V Index to S; s ¼ 1, ., S Index to DW; dw ¼ 1, ., DW Index to W; w ¼ 1, ., W Index to WL; wl ¼ 1, ., WL Index to RL; rl ¼ 1, ., rL Month index; n ¼ 1,.,12 Parameter indicating whether heating is required in month n (when HSn ¼ 1), or not (when HSn ¼ 0); n ¼ 1, ., 12 Parameter indicating whether cooling is required in month n, (when CSn ¼ 1), or not (when CSn ¼ 0); n ¼ 1, ., 12 Duration (s) of month n; n ¼ 1, ., 12 Internal design temperature during the heating season ( C) Internal design temperature during the cooling season ( C) Average external temperature ( C) during month n; n ¼ 1, ., 12 Area (m2) of door/window dw; dw ¼ 1, ., DW Area (m2) of wall w; w ¼ 1, ., W Area (m2) of ﬂoor Area (m2) of roof Temperature correction factor (%) for door/window dw; dw ¼ 1, ..., DW Temperature correction factor (%) for wall w; w ¼ 1, ., W Temperature correction factor (%) for ﬂoor Temperature correction factor (%) for roof Frame factor (%) of window dw; dw ¼ 1, ., DW

G1min

G2min

p1 p1

Correction factor for shading (%) of window dw; dw ¼ 1, ., DW Correction factor for movable devices (%) of window dw; dw ¼ 1, ., DW Solar radiation on window dw, having a certain orientation and tilt angle during month n (MJ/m2month); dw ¼ 1, ., DW, n ¼ 1, ., 12 Effective total solar energy transmittance (%) of window type v; v ¼ 1, ., V Thermal transmittance (W/m2 C) of window type v; v ¼ 1, ., V Thickness (m2) of wall layer wl; wl ¼ 1, ., WL Thickness (m2) of ﬂoor Thickness (m2) of roof layer rl; rl ¼ 1, ., RL Thermal conductivity (W/mK) of wall layer wl; wl ¼ 1, ., WL Thermal conductivity (W/mK) of ﬂoor Thermal conductivity (W/mK) of roof layer rl; rl ¼ 1, ., RL Thermal conductivity (W/mK) of added roof insulation material Return rate (%) of the electricity power plant Coefﬁcient of performance of heating/cooling system s; s ¼ 1, ., S Minimum permissible thickness of added roof insulation (m) Maximum permissible thickness of added roof insulation (m) Building load coefﬁcient (W/K) Energy demand for space heating in month n (MJ) Energy demand for space cooling in month n (MJ) Ventilation heat losses in month n (MJ); n ¼ 1, ., 12 Internal heat gains in month n (MJ); n ¼ 1, ., 12 Cost of door and window v (Euros/m2); v ¼ 1, ., V Cost of added roof insulation material (Euros/m) Cost of heating/cooling system (Euros) s; s ¼ 1,., S Alternative choices for doors/windows; equals to 1, if door/window type v is selected, else equals to 0; only one type may be selected for all doors and windows of the building; v ¼ 1, ., V Thickness of additional roof insulation (m) Alternative choices for electrical heating/cooling systems; equals to 1, if type s is selected, else equals to 0; only one type may be selected for the building; s ¼ 1, ., S Total annual primary energy consumption (MJ) Initial investment cost (Euros) The vector of decision variables; includes xv, v ¼ 1, ., V, xd and xs, s ¼ 1, ., S Tchebyshev distance The total annual primary energy consumption criterion as function of the decision variables The initial investment cost criterion as function of the decision variables The optimum (minimum) value of Q when total annual primary energy consumption is optimised independently from the investment cost (MJ) The optimum (minimum) value of C when initial investment cost is optimised independently from the annual primary energy consumption (Euros) Weight coefﬁcient of the total annual primary energy consumption criterion; p1þp2 ¼ 1 Weight coefﬁcient of the initial investment cost criterion; p1þp2 ¼ 1

A2. Multi-objective decision model

P12 ½minG1 ðxÞ ¼ Q ¼

½minG2 ðxÞ ¼ C ¼

n¼1

QHD;n þ nel

DW X

P12

AD=W;dw

dw ¼ 1

þ ARF xd Cd þ

n¼1

S QHC;n X xs COPs s¼1

V X

xv CD=W;v

v¼1 S X s¼1

ðxs Cs Þ

C. Diakaki et al. / Energy 59 (2013) 534e542

541

Subject to the constraints:

QHD;n ¼

QCD;n ¼

BLC ¼

8 > > < > > :

1 BLC qIH qE;n Tn þ QVEN;n Tn QINHG;n Tn ! C B DW V P P HSn @ A; ðxv gv Þ AD=W;dw FF;dw FS;dw FCM;dw ISL;dw;n 0

0; 0

8 > > <

else

DW P

V P

!1

F F F I ðx g Þ C A B CSn @ dw ¼ 1 D=W;dw F;dw S;dw CM;dw SL;dw;n v ¼ 1 v v A; > > þQAINHG Tn BLC qIC qE;n Tn QVEN;n Tn : 0; DW X

AD=W;dw bD=W;dw

dw ¼ 1

V X

PW ðxv Uv Þ þ

v¼1

w ¼ 1 AWAL;w bWAL;w PWL dWAL;wl wl ¼ 1 kWAL;wl

Subject to the following constraints V X

xv ¼ 1

v¼1 S X

if positive

v¼1

dw ¼ 1

xs ¼ 1

s¼1

xv ˛f0; 1g cv ¼ 1; :::; V xs ˛f0; 1g cs ¼ 1; :::; S x˛½dmin ; dmax

A3. Single-objective compromise programming formulation

½minz ¼ l Subject to the constraints: All constraints of the decision model presented in Appendix section A.2

l ðG1 ðxÞ G1 min Þðp1 =G1 min Þ l ðG2 ðxÞ G2 min Þðp2 =G2 min Þ l0

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if positive else

AFL;f bFL;f dFL kFL

ARF bRF dRF;rl RL rl ¼ 1 kRF;rl

þP

þ kx

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