Los recursos del mundo sensible

Various procedures from the very simple towards the much more complex are already developed [1, 7, 8]. Each procedure is intuitively best paired with a specific experimental data set. For example, daily averages of dynamic data sets are inadequate for applying linear regressions. Reversely, a dynamic analysis is not suited to static data sets since such sets don’t emphasize well the dynamic characteristics of the building.

96 Dynamic analysis

The preferred analysis methods are parameter identification methods applied on dynamic data sets. In this paper, a grey-box stochastic 1D-model is used to represent the entire building. It is rather simple and often appears suitable:

𝑑𝑇_𝑖 =(𝑇𝑒−𝑇𝑖) 𝑅𝑖𝑒𝐶𝑖 𝑑𝑡 + 𝑄ℎ 𝐶𝑖 𝑑𝑡 + 𝐴𝑤𝑞𝑠 𝐶𝑖 𝑑𝑡 − 𝑄𝑣 𝐶𝑖𝑑𝑡 + 𝜎𝑖𝑑𝜔𝑖 (1) 𝑑𝑇𝑒 = (𝑇_𝑅𝑖−𝑇𝑒) 𝑖𝑒𝐶𝑒 𝑑𝑡 + (𝑇𝑎−𝑇𝑒) 𝑅𝑒𝑎𝐶𝑒 𝑑𝑡 + 𝜎𝑒𝑑𝜔𝑒 (2) 𝐻𝐿𝐶 ≅_𝑅 1 𝑖𝑒+𝑅𝑒𝑎+ 𝑄𝑣 𝑇𝑖−𝑇𝑎 (3) where 𝑄𝑣 = 0.33𝑛𝑉(𝑇𝑖− 𝑇𝑎) (4)

where Ti, Te and Ta are respectively the indoor air, the building envelope and the ambient (outdoor

air) temperatures, Rie is the thermal resistance between the interior and the building envelope, Rea is

the thermal resistance between the building envelope and the interior thermal medium, Ci and Ce

are the heat capacities of the interior (internal walls) and of the building envelope (external walls),

Qh is the energy flux from the heating system, Awqs is the solar aperture multiplied by the energy flux

density from the solar radiation (i.e. solar gains) , Qv is the energy flux from the ex-filtrations, ωi and

ωe are standard Wiener processes, and σi and σe are their incremental variances. Finally, n is the air

change rate estimated by direct measurement, V is the volume of the building, and 0.33 is the volumetric heat capacity of the air expressed in Wh/m³K. Note that stochastic models make use of Wiener processes in order to cope with imperfections of the chosen model structure and with the noise included in the collected datasets that are both unavoidable and have to be accepted as the right balance with respect to useless or even problematic test and model complexity.

The corresponding equivalent RC-network is represented in Figure 1 :

Figure 1: Equivalent RC-network of the whole building envelope thermal model

The interior temperature is the output state of the model and is associated with a thermal capacity (air & furniture). The outdoor temperature is chosen as input. The (unobservable) building fabric envelope temperature is assumed to be aggregated in one single node and is obviously associated with a thermal capacity. The overall thermal resistance offered by the envelope against the heat losses is represented by two thermal resistances in series. Finally, the system is subjected to three other inputs: the electric heating power, the ventilation losses and the solar radiation, all predominantly acting on the inside air node temperature.

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Removing the envelope state and neglecting ventilation losses (Qv could also be hidden in a corrected

Qh term) would combine equations (1) and (2) into equation (5) only:

𝑑𝑇𝑖 =(𝑇_𝑅𝑎−𝑇𝑖) 𝑖𝑎𝐶𝑖 𝑑𝑡 + 𝑄ℎ 𝐶𝑖 𝑑𝑡 + 𝐴𝑤𝑞𝑠 𝐶𝑖 𝑑𝑡 + 𝜎𝑖𝑑𝜔𝑖 (5) Transient analysis

The transient method is typically applied on nightly data sets obtained from steps in the heating power (see Figure 8). This method has been shown to give reasonably accurate results under certain conditions. The measurement must be performed when there is no solar radiation (at night), and with good steadiness in Ta and Qh such that equation (5) becomes:

α ≜dTi dt ≅ dΔT dt = −(ΔT) RC + Q C (6)

This relation can be duplicated for two periods with significantly different values of Q (index h and c

here below stand for heating and cooling sequentially) and the resulting set of 2 equations with 2 unknowns yields the following expressions for HLC and C:

C =QhΔTc−QcΔTh αh∆Tc−αc∆Th and HLC = 1 R= αhQc−αcQh αh∆Tc−αc∆Th (7)

As explained in the introduction, the two measurements follow each other in one single night if the weather conditions are propitious and if the main time constant of the building is small enough (limited thermal resistance and mass) to produce clear and clean temperature ramps.

Static analysis

The most basic analysis is associated to the state-of-the-art co-heating experiments assuming quasi- steady state conditions and is obtained by further simplification of equation (5) for which Ti is

supposed invariable. When then can rewrite: 𝐶𝑖𝑑𝑇_𝑑𝑡𝑖 = 0 =(𝑇𝑎_𝑅−𝑇𝑖)

𝑖𝑎 + 𝑄ℎ+ 𝐴𝑤𝑞𝑠 (8) or HLC −A_ΔT𝑤q𝑠= Q_ΔTℎ (9) where HLC =_𝑅1

𝑖𝑎 and ΔT = 𝑇𝑖− 𝑇𝑎 (10) and (11)

In equation (9), both the HLC and Awqs terms are found implicitly from a bi-linear regression using

daily-averaged data points.

It is finally also possible to correct the used energy term by the ventilation losses if one requires estimating the building UA-value (HD according to ISO 13789) instead of the overall heat loss

coefficient HLC. Equation (9) can then be modified as: UA − A𝑤q𝑠

ΔT = 𝑄ℎ−𝑄𝑣

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This correction is used in order to better compare the experimental designs and analysis methods since the actual air change rate could strongly vary in time and hence impact the HLC. Further in this paper we will refer to building HLC and building UA-value, depending if the correction is applied or not.