Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 153 (2016) 862 – 865 XXV Polish – Russian – Slovak Seminar “Theoretical Foundation of Civil Engineering” A method of evaluation of polioptimal thermo-modernization schemes of buildings Arkadiusz WĊglarza* , Paweá Grzegorz Gilewskib b a Warsaw University of Technology, Faculty of Civil Engineering, al Armii Ludowej 16, Warsaw 00-637, Poland Warsaw University of Technology, Faculty of Buidling Services, Hydro and Environmental Engineering, ul Nowowiejska 20, Warsaw 00-653, Poland Abstract The authors proposed a polioptimal method for determining thermo-modernization schemes of buildings based on the theory of fuzzy sets As optimization criteria the minimization of the total cost of thermo-modernization and maximization of energy effect were taken into account The Excel worksheet with optimization algorithm was prepared and tested on a numerical example © Published by by Elsevier Ltd.Ltd This is an open access article under the CC BY-NC-ND license © 2016 2016The TheAuthors Authors Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Peer-review responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Foundation Foundationunder of Civil Engineering of Civil Engineering” Keywords: polioptimalization; thermo-modernization; fuzzy sets Introduction The standard approach towards selection of thermo-modernization scheme is to assess of investment costs and profits from its accomplishment Usually it is done using simple indicators of economic evaluation, such as minimization of payback period [3] Environmental criteria are rarely taken into account, but if so, it is usually reduction of CO2 emission as a result of thermo-modernization investment * Corresponding author Tel.: +48691956505 E-mail address: a.weglarz@il.pw.edu.pl 1877-7058 © 2016 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Foundation of Civil Engineering” doi:10.1016/j.proeng.2016.08.194 863 Arkadiusz Węglarz and Paweł Grzegorz Gilewski / Procedia Engineering 153 (2016) 862 – 865 However optimization done in such a manner frequently does not reach the technical potential of increasing the energy efficiency of thermo-modernized construction Thus the authors proposed polioptimal approach to effectively use the total energy efficiency potential of the building Description of the optimization model Decision maker wants to thermo-modernize the existing building [1,2] He want to: minimize the total cost of thermo-modernization [PLN], maximize the energy effect [kWh/year] In the deterministic model those two objectives are basically contradictory [1,2] It was therefore decided to apply the theory of fuzzy sets [5,6] to determine the polioptimal thermo-modernization schemes of buildings taking into account criteria of energy intensity The theory of fuzzy sets allows a situation when an element may partly belong to some set and this belonging may be expressed using a real number from the interval [0,1] Thus, the membership function μ(x) :[0,1] is defined as follows [4,5,6]: x x P ( x ) xU f ( x), x X , ® ¯ 0, x X (1) where f(x) is a function with values from the interval [0,1] The values of such a function are called degrees of belonging Suppose that n variants of thermo-modernization are possible and they have assigned investment costs and energy effects From that variants we create two fuzzy sets of the same power, called: x Low cost of thermo-modernization, x High cost of thermo-modernization To create a fuzzy set named “Low cost of thermo-modernization” we should designate the thermo-modernization variant vi with the lowest cost: cos t (vi ) min(cos t ( v1 ), cos t ( vi ), cos t (v n )) , (2) and the thermo-modernization variant vj with the highest cost: cos t (v j ) min(cost (v1 ), cos t (vi ), cos t (v n )) (3) Afterwards the variant vi should have assigned the membership function μK(vi) = and the variant vj the membership function μK(vj) = The next step is to assign the other variants of thermo-modernization values of membership function μK(vk) proportional to the difference between cost(vk) – cost(vi) In order to create a fuzzy set called “High energy effect of thermo-modernization” one should designate a thermo-modernization variant vi with the highest energy effect: effect(vi ) min(effect(v1 ), effect(vi ), effect(v n )) , (4) and a thermos-modernization variant vj with the lowest energy effect: effect(v j ) min(effect(v1 ), effect(v j ), effect(vn )) (5) Afterwards the variant vi should have assigned the membership function μE(vi) = and the variant vj the membership function μE(vj) = The next step is to assign to the other variants of thermos-modernization values of membership function μE(vk) proportional to the difference between effect(vi) – effect(vk) 864 Arkadiusz Węglarz and Paweł Grzegorz Gilewski / Procedia Engineering 153 (2016) 862 – 865 The fuzzy set O(v1…n) fulfil to some extent both criteria will be an intersection (common part) of fuzzy sets E(v1…n) and K(v1…n) From the point of view of fuzzy sets algebra it is determination of the common part of two fuzzy sets The commonly used approach was proposed by Zadeh [5] Degree of membership of the element x to the set A ŀ B is the minimum of degrees of membership to set A and to set B: PA B( x) min^PA( x), PB( x)` (6) The other approach to determine a common part of two fuzzy sets is so called algebraic or probabilistic product: PA B(x) PA(x) PB(x) (7) Another formula determining the product of fuzzy sets is called àukasiewicz t-norm or limited difference: PA B( x) max^0, PA( x) PB( x) 1` (8) Generally in order to determine the degree of an element membership to the common part, the operations called tnorm or triangular norm are applicable In the literature [4] one can find a few different suggestions on t-norms After determining a fuzzy set O(v1…n) it is possible to find the optimal solution The solution will be a variant with the highest value of membership function in the fuzzy set O(v1…n) The optimization algorithm described above with two optimization criteria can be easily developed with a great number of other optimization criteria, e.g the minimum CO2 emission in the life cycle of investment or minimum of particulate matters emissions Example Housing community wants to thermo-insulate two gable walls in the building managed by itself Currently the wall of the building is constituted with support layer made of aerated concrete 600 with a thickness of 24 cm and two layers of lime-cement with the thickness of cm each The total area of the gable walls is 120 m2 Two methods of wall thermo-insulation were analyzed: light-wet and dry-light The analyzed variants are shown in the Table Table Analyzed variants of thermo-modernization Variant V1 Thermo-insulation method Light-wet with mineral wool 14 cm thick Total cost [PLN] Energy effect [MJ/year] 18 700 17 200 V2 Light-wet with mineral wool 16 cm thick 20 000 17 700 V3 Dry-light with mineral wool 14 cm thick + PCV siding 16 500 17 100 V4 Dry-light with mineral wool 16 cm thick + PCV siding 17 800 17 900 V5 Dry-light with mineral wool 14 cm thick + wooden boards cm thick 16 740 17 200 V6 Dry-light with mineral wool 16 cm thick + wooden boards cm thick 18 040 18 000 V7 Dry-light with mineral wool 14 cm thick + trapezoidal metal sheet 17 460 17 000 V8 Dry-light with mineral wool 16 cm thick + trapezoidal metal sheet 18 760 17 700 MAXIMUM: 20 000 18 000 MINIMUM: 16 500 17 000 The results calculated according to the method described earlier on are presented in the Table Table Results of optimization algorithm Variant V1 Thermo-insulation method Light-wet with mineral wool 14 cm thick ȝK(vi) ȝE(vi) min{ȝK(vi), ȝE(vi)} ȝK(vi)• ȝE(vi) max{0, ȝK(vi) + ȝE(vi) í 1} 0,37 0,20 0,20 0,07 0,00 865 Arkadiusz Węglarz and Paweł Grzegorz Gilewski / Procedia Engineering 153 (2016) 862 – 865 V2 Light-wet with mineral wool 16 cm thick 0,00 0,70 0,00 0,00 0,00 V3 Dry-light with mineral wool 14 cm thick + PCV siding 1,00 0,10 0,10 0,10 0,10 V4 Dry-light with mineral wool 16 cm thick + PCV siding 0,63 0,90 0,63 0,57 0,53 V5 Dry-light with mineral wool 14 cm thick + wooden boards cm thick 0,93 0,20 0,20 0,19 0,13 V6 Dry-light with mineral wool 16 cm thick + wooden boards cm thick 0,56 1,00 0,56 0,56 0,56 V7 Dry-light with mineral wool 14 cm thick + trapezoidal metal sheet 0,73 0,00 0,00 0,00 0,00 V8 Dry-light with mineral wool 16 cm thick + trapezoidal metal sheet 0,35 0,70 0,35 0,25 0,05 MAXIMUM: 0,80 0,63 0,57 0,56 MINIMUM: 0,2 0,00 0,00 0,00 x x x according to t-norm-minimum, the optimal solution is variant V4, according to t-norm-algebraic product, the optimal solution is variant V4, according to àukasiewicz t-norm, the optimal solution is variant V6 The differences in the results according to different t-norms are related to the ambiguity of determining the product of fuzzy sets Therefore in the practical implementation of the proposed method it is necessary to choose one method and stick to it consistently while solving the others decision-making problems Summary and conclusions The proposed optimization method can find a practical application in the process of decision-making by investors or energy auditors, who want to use other methods of thermo-modernization selection than those proposed in the Regulation of the Minister responsible for building construction on the detailed scope and forms of energy audit [3] In the near future amendment of the Act on supporting the thermo-modernization and renovation, as well as Regulations associated with it, is expected This paper can be a point in the discussion on the optimization methods that could be applied in the new methodology of energy audit In particular, it will be important for selection of projects in the process of deep thermo-modernization (e.g to the level of low-energy or passive house) where the application of single criteria optimization method (e.g minimum of investments costs) will significantly limit the opportunity to get the most out of energy-saving potential in the existing buildings References [1] OlĊdzka D., WĊglarz A., The method of development of projects of realization the thermomodernization works with energy intensive criterion (in Polish), XVIII Polish-Ukrainian-Lithuanian Conference, „Theoretical Foundations of Civil Engineering”, Simferopol, 13-18 September 2010 [2] OlĊdzka D., WĊglarz A , The method of development of polioptimal projects of technology and organization of building with the energy intensive criterion (in Polish), XIX Polish-Russian-Slovak Seminar "Theoretical Foundation of Civil Engineering", Zylina 2010 [3] Regulation of the Minister of Infrastructure from 17 March 2009 on the detailed scope and form of the energy audit and the audit of renovation, audits cards, as well as the algorithm of assessing the profitability of the thermos-modernization investment (in Polish) [4] Rudnik K., Concept and implementation of the inference system for probability-fuzzy basis (in Polish), PhD thesis,, Opole Technical University, Faculty of Electronics, 2011 [5] Czogaáa E., Pedrycz W.: Elements and methods of fuzzy sets (in Polish) PWN, Warsaw 1985 [6] Dworniczak P., Fuzzy sets for beginers (in Polish), Paper during XXX School of visual singularity, January 2003 ... thermo- insulation were analyzed: light-wet and dry-light The analyzed variants are shown in the Table Table Analyzed variants of thermo- modernization Variant V1 Thermo- insulation method Light-wet... support layer made of aerated concrete 600 with a thickness of 24 cm and two layers of lime-cement with the thickness of cm each The total area of the gable walls is 120 m2 Two methods of wall thermo- insulation... Minister of Infrastructure from 17 March 2009 on the detailed scope and form of the energy audit and the audit of renovation, audits cards, as well as the algorithm of assessing the profitability of