Using fractional experimental designs to establish multi-parametric expressions for energy consumption of commercial buildings

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1 Using fractional experimental designs to establish multi-parametric expressions for energy consumption of commercial buildings Sila Filfli, Dominique Marchio Center for Energy and Processes-CEP, Ecole des Mines de Paris, France Corresponding SUMMARY The large advancement in the market of building materials and of HVAC systems leads to a wide variety of technical solutions that can be employed to build an efficient low energy commercial building. The comparison between different combinations of parameters representing the main elements of the envelope, lighting management and solar blinds, office automation, plug consumptions and efficiency of the HVAC system is always of special interest to building owners and HVAC design engineers; however, such a study is rarely totally accomplished because of the time required. The paper describes a fast method for organizing simulations through the use of fractional factorial designs. The method permits reduction of the number of simulations needed to cover all combinations of influent parameters. The reduction result shows that it is feasible and quick to determine mutli-parametric models for a large range of building types. As a result, consumption can be calculated bypassing steps of projects description, sizing of components and running the simulations. The precision of such multi-parametric models is established by comparison with full simulation. INTRODUCTION The purpose of the project is to build a large library of analytical multi-parametric models giving the consumption as a function of key parameters of typical building/system. The project is addressed mainly to engineers and commercial buildings design departments who aim to obtain optimized solutions quickly and to identify the impact of main choices on total yearly energy consumption. In a first time, the description of French commercial sector, mainly for offices and hospitals, has been established. It permits to draw up a representative typology of these buildings [1] [2]. On the basis of this typology, geometrical description and different values or scenarios necessary for accomplishing the simulation are defined [3][4]. Simulations are then carried out with ConsoClim [5], a building energy simulation software package. Its particularity is that systems models [6] require simple and available parametric inputs and are coupled to the building model. The variables predominantly affecting energy consumption are then listed. The high number of possible simulations needed to cover all the variation of variables in addition to the

2 organization of a huge quantity of results was initially the main reasons for introducing experimental design notion. METHODS The aim of the method is to obtain maximum information with a minimum number of simulations. For a defined problem, various solutions can be tested, evaluated and compared. A systematic organization of simulations helps to analyze the results and decreases the risk of errors at the same time. The experimental designs are essentially based on multi factorial experiments and on the use of multiple regressions besides variance analysis while treating the results. Their application supposes a good knowledge of the studied phenomenon before realizing the tests. This has to be accompanied by the ability to detect the parameters most likely to influence the system operation [7]. It is necessary to represent the diversity of powerful energetic technical solutions already present on the market, then to be able to establish a correspondence between the chosen solutions and a level of the cluster parameter representing the variety of these solutions. If the number of considered parameters and the number of their levels are very high, the solution consists in reducing the number of levels and/or factors followed by control of the precision of obtained results. When defining the design of experiments, a preliminary step is to study the linearity (or not) of variables in order to set the number of levels. y y x A 1 A 2 x A 1 A 3 Figure 1. Choice of number of levels: 2 levels on the left (linear evolution), 3 levels at least on the right (non linear evolution) COMPLETE FACTORIAL DESIGN Factorial design or complete factorial design is to experiment all the possible combinations between the levels of the variables. The number of combinations is a product of the numbers of levels of the factors. [8] In the study we considered 13 variables for office building application, they are: Table 1. Low and high values of the considered variables building/system Variables Level Level (-1) Level (+1) U op Insulation of ceiling and walls (opaque) 0.2, 0.4 W/m 2 /K 0.3, 0.6 W/m 2 /K U bay Characteristics of glazed surfaces 2 W/m 2 /K 3 W/m 2 /K Ori Orientation Glazed facades N/S Glazed facades E/W Management of solar protection as a function of natural lighting Occupant not so reactive Occupant very reactive Off_Aut Mode of management and effeciency of office automation equipments 15 W/m² 7.5 W/m² A 2

3 Management and efficiency lighting Interrupter 18 W/m² Interrupter + graduator 10 W/m² Iner Inertia level light heavy eability 1.2 m 3 /(h.m²) under 4 Pa 2.4 m 3 /(h.m²) under 4 Pa ilation Reduction in flow rate during occupancy -30% Normal flow rate during occupancy Aux_Eff Fans and pumps efficiency Boil_Eff Boiler efficiency 0.98 (70 C), 1.08 (40 C) 0.89 (70 C), 0.88 (40 C) Dist_Ins Insulation of distribution network 0.14 W/m.K 0.28 W/m.K Chillers - Split,VRV (/COP) / /1.95 This paper presents an application of the method to a small area office building (1000 m 2 ) existing in the industrial suburban zones and simulated in the climatic region of Paris; for other cases see [11]. The selected system for this article is made of a chiller for cooling, gas boiler for heating, air-handling unit for fresh air, and 4 tubes fan coil units (domestic hot water is not taken into consideration). The HVAC system is described from market real devices after sizing all the needed elements in cooperation with engineer design departments. Simulations are realized to check the consumption tendency between the levels of each variable (the linear response justifies keeping only two levels). The variables used in the experimental design are centered and reduced. To pass from a variable, A, to the reduced centered variable, x, the formula is: x = (A-A 0 )/ A (1) Where A 0 is the central value of the interval [- 1, +1], expressed in current unit, A being the variation between the average position of the variable and the domain extremity. Figure 2a (left) gives consumption while passing from the basic case (all variables at level 1) to another case each time one variable passes to level 0 (central value of the interval [- 1, +1]) then on the level (+1). The value obtained is compared with the calculated value at level (0) obtained from a linear regression. Annual total consumption at level (0) - value obtained by simulation (kwh/m 2 ) Accuracy of central values Annual total consumption at level (0) - calculated value in (kwh/m 2 ) Bisector Uop Ubay Orientation Solar Protections Office automation ing Inertia eability ilation Auxiliaries Efficiency Boiler Efficiency Distribution Insulation Annual total consumption at level (0) - value obtained by simulation (kwh/m 2 ) Accuracy of central values Annual total consumption at level (0) - calculated value in (kwh/m 2 ) Figure 2. Precision of the assumption of the answer (consumption) linearity Bisector Uop Ubay Orientation Solar Protections Office automation ing Inertia eability ilation Auxiliaries Efficiency Boiler Efficiency Distribution Insulation Figure 2b (right) is obtained by the same principle, this time starting from a basic case where all the variables are at the high level (+1). For each simulation, a variable passes to the level (- 1) then to the level (0). The variation obtained with the calculated values is negligible. To carry out a 2 13 complete factorial design, 2 13 simulations have to be done. Each simulation gives an answer y i. The answer is represented by a matrix-column [Y] including 2 13 lines. The

4 mathematical model associated to the 2 13 design contains 2 13 unknowns: one constant, 13 principal effects and interactions. The effects are represented by a matrix-column [A] constituted of 2 13 lines. The matrix of calculation of coefficients X contains a number of lines equal to the number of answers y i that is 2 13, and as many columns as there are unknown factors i.e Matrix X is thus a square matrix (2 13, 2 13 ). It is reversible, we can write: Y (2 13, 1) = X(2 13, 2 13 ) A (2 13,1) (2) A = X -1 Y (3) The researched model is constituted from 13 parameters of Table 1. The matrix representation usually employed in experimental designs is replaced by YATE notation [13]. The variable is multiplied by the level it takes (-1 or +1); the interaction between two variables or more is represented by the product of their levels. So the simplified writing of the model becomes (Refer to table 1 for nomenclature): C tot (kwh/m 2 ) = M tot + U op + U bay + Ori + + Off_Aut + + Iner Aux_Eff + Boil_Eff + Dist_Ins + + all interactions of the 1 st order (= 78 interactions) (4) M tot is the average and C tot is the total yearly consumption. This model considers that the interactions of 2 nd order and above are negligible, which is validated by comparing the results of the model to the real answer: that of the simulation. In this study, the energy ratios refer to total energy consumption (lighting, heating, cooling, plug loads and auxiliaries) without considering specific areas of the building such as data processing centers. Conversion into primary energy is given by applying the French rules [10] ratios (2.58 for electrical energy and 1 for gas energy). The number of simulations needed to accomplish such a model is: 2 13 (13 variables with 2 levels each) or 8192 simulations. This number is huge in terms of results and organization. As mentioned in the introduction, the methodology is to be applied to two main commercial sectors: offices and hospitals represented by five types of office buildings and two types of health buildings. The building/system matrix is representative of the real park in France and each system is sized per climatic region. With this model of the park, just for the small area building analyzed in this paper we have: 5 HVAC systems, 2 climatic regions i.e. 10 times 8192 simulations. The time for a series of 8192 simulations is around 820 hours with a relatively powerful computer. These figures show that complete factorial design is not a realistic solution. For us, it is only a reference set of results to compare with the fractional factorial design or fractional design presented below. FRACTIONAL DESIGN The fractional design is a "fraction" of the factorial design; it is an orthogonal subset of combinations of the complete factorial design. Its main advantage is revealed in the significant reduction of the number of simulations and therefore rapidity in giving reduced models. Nevertheless, the reduced number of experiments makes it impossible to determine all coefficients of the multi-polynomial model related to the complete factorial design. The

5 reason is that the number of coefficients is higher than the number of experiments. In fact, the fractional design makes it possible to determine the sum of certain coefficients identified by the complete factorial design; these coefficients are called aliases. A half-design is less expensive in terms of simulation number than a complete one, but this profit is offset by an ambiguity in the estimation of effects [13]. However, when building fractional designs, it is necessary to alias as much as possible the principal effects with interactions of a higher order, these latter being in general negligible. To simplify the application of fractional designs, we use the Taguchi method or Taguchi tables [15]. The construction of an experimental design according to the method of Taguchi is based on two principles [9]: 1- the use of tables, facilitated thanks to linear graphs, provides the mode for filling the matrix. 2- considering the actions of 2 nd order negligible except in particular cases. Some interactions are immediately neglected when physical reasons deem it possible; for example, the interaction between the boiler efficiency and the is null. On the other hand, there are some interactions that cannot be neglected: - interactions between the thermal transmittance of the glazing and the internal contributions or lighting for a largely glazed building (thus strong internal contributions and lighting), - interaction between the boiler efficiency and the insulation of the distribution for a significant length of the network; etc. The reduction of the number of interactions considered leads to a reduction of the degree of freedom of the model and consequently the number of experiments to be realized. The first order interactions, which are considered significant, are kept. The influence of this choice is then evaluated. Cooling Heating U op U bay Ori Off_Aut U op U bay Ori GPS Off_ Aut Aux_ Eff Iner Dist_ Ins Aux_ Eff Iner Boil_ Eff Dist_ Ins ing AHU auxiliaries FCU auxiliaries Ori Aux_Eff Distribution auxiliaries Office automation U op U bay Ori Off_ Aut Aux_ Dist_ Eff Ins Aux_Eff Off_Aut Figure 3. Variables that have an effect on the consumption of each sub model

6 Selection of the most significant interactions requires advance experience in the building/system simulation. To avoid this expertise requirement, we suggest the establishment of different models representing categories of annual consumption (figure 3). The addition of these elementary models of subparts constitutes the total consumption. The variables that influence each category of consumption are the following: The global model is then composed of sub-models (Refer to table 1 for nomenclature): C cool (kwh/m 2 ) = M cool + U op + U bay + Ori + + Off_Aut + + Iner Aux_Eff + Dist_Ins +. +.U bay +.U bay +.Off_Aut +.Off_Aut + U bay.off_aut +.Aux_Eff U bay Off_Aut. +.Ori +.Uop +.Iner (5) C heat (kwh/m 2 ) = M heat + U op + U bay + Ori + + Off_Aut + + Iner Aux_Eff + Dist_Ins + Boil_Eff.U op + Off_Aut.U bay + Ubay. + U bay.+ Boil_Eff.U bay + Off_Aut. + Off_Aut. + Boil_Eff.Off_Aut +. + Boil_Eff. + Boil_Eff.Iner + Boil_Eff. + Boil_Eff. + Boil_Eff.Aux_Eff + Boil_Eff.Dist_Ins (6) C (kwh/m 2 ) = M Ori +. +.Ori +.Ori (7) C Aux_AHU (kwh/m 2 ) = M Aux_AHU + + Aux_Eff +.Aux_Eff (8) C Aux_FCU (kwh/m 2 ) = M Aux_FCU + U op + U bay + Ori + + Off_Aut + + Iner Aux_Eff + Dist_Ins + U op.aux_eff + Iner.U bay +.U bay + U bay.aux_eff + Iner.Off_Aut + OFF_AUT.Aux_Eff +.Aux_Eff + Iner.Aux_Eff +.Aux_Eff +.Aux_Eff + Aux_Eff.Dist_Ins (9) C Aux_Dist (kwh/m 2 ) = MA ux_dist + Aux_Eff (10) C Off_Aut (kwh/m 2 ) = M Off_Aut + Off_Aut (11) To find the table of Taguchi adapted to the model, two criteria must be validated [11]: criterion of the degrees of freedom and criterion of orthogonality. We give an example for the cooling sub-model. The number of degrees of freedom is 28 (12 variables, 15 interactions, 1 for average). The number of experiments should be equal to or higher than 28. The PPCM between disjoined orthogonal actions is (4,8,16) = 16. The Taguchi table L 32 (2 31 ) is then chosen - 32 simulations, 31 columns that can be assigned to variables or interactions [14] - with a resolution plan IV, i.e. variables of order 1 are aliases of interactions of order 3, and interactions of order 2 are aliases with other interactions of order 2. The associated linear graph given by Taguchi has been adapted to our model [11]. The most significant variable in the model of cooling is obviously the. Iner U op U bay Ori Aux_eff 14 Dist_Ins Off_Aut

7 RESULTS The total consumption C tot is deduced from the sub models distinguishing gas and electricity. C tot (kwh/m 2 ) = C cool (kwh/m 2 ) + C heat (kwh/m 2 ) + C (kwh/m 2 ) + C Aux_AHU (kwh/m 2 ) + C Aux_FCU (kwh/m 2 ) + C Aux_Dist (kwh/m 2 ) + C Off_Aut (kwh/m 2 ) (12) The results of the reduced model have been compared with the reference simulations (32 in the case of cooling). The precision is in the interval of + 0.5%. Neglecting some interactions may influence the results precision; this latter improves systematically when more interactions are taken into consideration. In the selected example, we also compared the results of the additive model (12) with the results of a complete factorial design (4). In other words, we compare the model obtained by 112 simulations (32 for cooling, 32 for heating, 8 for lighting, 38 auxiliaries and office automation (without counting identical simulation) with the model obtained by 8192 simulations. The precision is shown in figure 4. Table 1. Average deviation (with respect to the complete model) is 0.66 (kwh/m 2 ) -1 % Deviation 1 % -3 % Dev.< -1 % -5 % Dev. < -3 % 1 % < Dev. 3 % 3 % < Dev. 5 % N of simulations (total = 8192) % % % 0.26 % Yearly total consumption (kwh/m 2 )- resultats of reduced additive model Comparison between the consumption of reduced additive model and the consumption obtained by a complete factorial design Yearly total consumption (kwh/m 2 ) - results of complete design Figure 4. Precision of reduced model with respect to complete model Typical results like effects of the elements are deduced for each model, we give that of cooling as an example. 2.5 Effects of variables on the yearly consumption of cooling in kwh/m Uop Ubay Ori Off_Aut Iner Aux_Eff Dist_Ins -1

8 Uop Values of interactions (and their aliases) on the yearly energy consumption of cooling in kwh/m 2 Ubay.Off_Aut Ubay. Ubay..Ubay.Ori. Off_Aut..Off_Aut.Off_Aut Figure 5. Example of effects and interactions values for cooling model, average = 7.8 kwh/m 2...Iner..Aux_eff DISCUSSION It has been shown that the reduction of the number of simulations does not have an important influence on the quality of the answer, granted precaution with significant variables and interactions. The time requirements for running and, more importantly, preparation of simulations are greatly reduced when compared up to the factorial design. Evidently, the preparation of the fractional plan must be carried out with firm knowledge of the method. The user must be attentive while launching simulations of validation. This method is applicable to all types of buildings and systems in all sectors. The multi-parametric model is useful for an economical optimization when disposing of cost libraries of different technical solutions. ACKNOWLEDGEMENT The authors thank the ADEME for its financial support. REFERENCES 1. Marchio, D, Filfli, S, Alessendrani, J.M, 2004 Solutions to design air-conditioned office buildings consuming less than 100 kwh/m²/year. IEECB04 proceedings. 2. Filfli, S, Marchio, D, 2005 : Study of air-conditioned office buildings consuming less than 100 kwh/m²/year in France, Clima 2005 proceedings. 3. Filfli, S, Marchio, D, Fleury, E, and al What solutions for air-conditioned office buildings consuming less than 100 kwh/m²? Final Report. 4. Filfli, S, Marchio, D, Fleury, E, and al What solutions for air-conditioned health buildings consuming less than 100 kwh/m²? Final Report. 5. Marchio, D, Fleury, E, Millet J.R and al, "Méthode de calcul des consommations d énergie des bâtiments climatisés Consoclim". 6. Algorithms of ConsoClim, version Souvay, P, Les plans d expériences, Méthode Taguchi. A Afnor collection A savoir 8. Benoist, D, Tourbier Y., et Germain-Tourbier S Plan d expériences : Construction et analyse. Lavoisier TEC & DOC. 9. Vigier M Pratiques des plans d expériences Les Editions d Organisation. 10. French rules, RT Filfli, S, Thèse de doctorat. Ecole des Mines de Paris. Optimisation bâtiment/système pour minimiser les consommations dues à la climatisation 12. «Chiffres clés du bâtiment», ADEME, Pillet, M, 1999, Les plans d expériences par la méthode Taguchi. Les Editions d Organisation. 14. Bendell T. 1989, Taguchi Methods, Proceedings of the 1988 European Conference. 15. Tables published by ASI: American Supplier Institute