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4.1 Introduction

4.1.5 Requirements

The building optimization method suggests design decisions by estimating numerical so-lutions to (3.8). This formulation enables the decision maker to specify optimality re-quirements to performance measures, and to specify inequality and equality rere-quirements to performance measures as well as decision variables.

However, there are requirements related to discrete decisions, such as selecting a window from a window database, which always must be satisfied, regardless of the requirements specified by the decision maker. These are described in the following. Furthermore, how to incorporate the requirements to the energy performance of buildings into the decision-making is also addressed.

Discrete decisions

As described in Section 4.1.2, window j is selected by using the weights αj = 1 and αi = 0 fori6=j. The weights are therefore always either 0 or 1, that is,αi ∈ {0,1} for i= 1, . . . , nwin.

However, the problem (2.4) requires that the variables are continuous, therefore the weights are allowed to assume any value between 0 and 1, that is:

0 ≤ α(1)≤1, (4.6)

0 ≤ α(2)≤1. (4.7)

There is no guarantee that the optimum weights found by solving (2.4) are integers;

however, practical experience with the method indicates that this is often the case.

The requirements (4.6) and (4.7) are used as permanent parts ofAIˆ andbIˆ, which are used for specifying inequality requirements to the decision variables.

Furthermore, in order to ensure that exactly one window is selected for each of the two thermal zones, the sums of the weights must be one:

nwin

X

i=1

α(1)i = 1, (4.8)

nwin

X

i=1

α(2)i = 1. (4.9)

These requirements are used as permanent parts ofAEˆandbEˆ, which are used for speci-fying equality requirements to the decision variables.

Building regulations

The Danish building regulations [13] specify upper limits on the annual amount of energy required by the building, as well as upper limits on the (linear) thermal transmittance for various parts of the building envelope.

These regulations implement the EU Directive [18] on the energy performance of buildings.

The approach described in the following, intended for incorporating energy requirements to buildings into the decision-making, may therefore be useful for other countries with similar regulations.

However, the reader should notice that the approach described in the following does not guarantee that the design decision suggested by the building optimization method comply with the Danish building regulations. This is because the method does not use the energy performance calculation method required by the building regulations. The required method is implemented in the program BE06 described by Aggerholm and Grau [1]. Including this method into the building optimization method is a possible topic for further research.

The Danish requirements to the energy performance of non-residential buildings are de-scribed in terms of the total annual energy Qtot delivered to the building for heating, cooling, ventilation, producing domestic hot water, and for artificial lighting. The follow-ing energy frame requirement (referred to asEF3) must be satisfied for new non-residential buildings:

Qtot

Atot ≤95kWh

m2 +2200 kWh

Atot , (4.10)

whereAtot is the total heated floor area, including internal and external walls.

The Danish building regulations also include energy labels, intended for motivating build-ing owners to reduce the energy required by buildbuild-ings even further. The followbuild-ing energy

frame (referred to asEF2) requirement must be satisfied in order to achieve the low energy class 2 label:

Qtot Atot

≤50kWh

m2 +1600 kWh Atot

, (4.11)

and the energy frame requirement (referred to asEF1) for achieving the low energy class 1 label is:

Qtot

Atot ≤35kWh

m2 +1100 kWh

Atot . (4.12)

The energy frame requirements can be formulated in the following way:

95kWh

m2 Atot+ 2200 kWh−Qtot ≥ 0 (4.13)

50kWh

m2 Atot+ 1600 kWh−Qtot ≥ 0 (4.14)

35kWh

m2 Atot+ 1100 kWh−Qtot ≥ 0, (4.15)

where the right hand sides of these expressions are used as performance measures:

EF3 = 95kWh

m2 Atot+ 2200 kWh−Qtot (4.16)

EF2 = 50kWh

m2 Atot+ 1600 kWh−Qtot (4.17)

EF1 = 35kWh

m2 Atot+ 1100 kWh−Qtot (4.18)

(4.19) The energy frame related to each of these performance measures is satisfied when a design decision is found where the performance measure is positive. The energy frames can thus be addressed by requiring that one or more of the performance measuresEF3, EF2 and EF1must be positive.

The Danish building regulations specify (among others) the following upper limits on the thermal transmittance for the components used in the building envelope:

1. The thermal transmittance of the ground slab must be less than 0.30 W/m2K.

2. The thermal transmittance of the external walls must be less than 0.40 W/m2K.

3. The thermal transmittance of the roof construction must be less than 0.25 W/m2K.

4. The thermal transmittance of windows and doors must be less than 2.30 W/m2K.

TheU-value requirements can be addressed by specifying upper limits on the performance measuresUg,Uwall,Ur,Uwin(1) andUwin(2).

The Danish building regulations furthermore establish an upper bound on the heat loss through the building envelope, not including windows and doors. The heat lossQe per m2 of the building envelope is required to be less than 6 W, that is:

Qe Ae ≤6W

m2 ⇔ 6 W

m2·Ae−Qe≥0, (4.20)

whereAeis the area of the building envelope, not including windows and doors. The left hand side of (4.20) is used as a performance measure:

BE= 6W

m2·Ae−Qe. (4.21)

The Danish building regulations also specify upper limits on the linear thermal trans-mittances for the interaction between various parts of the building envelope. However, calculating the linear thermal transmittance for the interaction between two building com-ponents requires numerical methods for solving partial differential equations. This could make the performance calculations more time-consuming, and thereby increase the time needed for solving the optimization problem (3.8).

All linear thermal transmittances are therefore assumed to be constant, and close to or equal to the upper limits specified by the building regulations, in order to provide a conservative estimate of the design decisions. Developing methods for performing fast and reliable calculations for the linear thermal transmittance is a possible topic for further research. Methods similar to the one described in the paper included in Appendix B might be useful for this purpose; however, this has not been investigated as part of this study.