Structure of the model

The implementation of the model follows the structure proposed by Lauster et al [75]. The data flow is illustrated in Fig. 3.2. First of all, ArcGIS -a commercial GIS software- is used to access the data about building geometry. The buildings are described through a shapefile, which is a popular geospatial vector data format developed by ESRI. Here, we call the result-ing model a quasi-3D model as information about the height are added as attributes to the

shapefile from external files. The age of construction is a further attribute that is linked to the shapefile. In Fig. 3.1, each building was given a color based on the age of construction: red buildings are the oldest, green ones are the newest. The reference envelopes are sometimes called archetypes. The archetype method, as it is sometimes called, is present in literature among the top-down engineering-based building stock models [76]. A building archetype describes the main building components that make up the envelope: pavement, external walls, ceiling, roof, windows and other glazed elements, and internal partitions. These components are described through their stratigraphy, i.e. an ordered list of layers with their corresponding thickness, thermal conductivity, density and specific heat capacity. The archetypes are chosen according to the prescriptions of national legislation in force at time of construction. Thus, each building of the district is given a reference envelope (archetype) according to its age. A MATLAB script reads the geometrical input file and the archetype file and parameterizes all the buildings according to the chosen simplified model. In other words, all the geometrical and physical information on the buildings is reduced to a set of thermal resistances and capac-itances. Indeed, lumped-capacitance models assume that the distributed thermal mass of the dwelling is lumped into a discrete number of thermal capacitances, as discussed above.

Figure 3.2 - Data flow of the model.

Therefore they are often referred to as xRyC models, where x is the number of thermal re-sistances and y the number of thermal capacitances of the equivalent electrical circuit. These models rely on the following assumptions: linearity of the heat transfer mechanism,

represen-tation of multilayer wall characteristics by lumped parameters and single zone approximation [77]. The parameterization was done according to the prescriptions of the ISO 13790, that reduces all the physical phenomena occurring within the building to a 5R1C model, as dis-cussed in Section 3.1.1. The other inputs are weather data (external air temperature and solar radiation) taken from the Test Reference Year of the chosen location and building occupancy profiles. Yet the model is not capable of reproducing patterns of presence of occupants in buildings, that would be very important to account for simultaneity of internal heat gains, do-mestic hot water demand and other variables that depend on users behaviour rather than on physical quantities. Finally, the one year simulation is performed for all the buildings, each of them being modelled as a single thermal zone. All the buildings are considered as singular entities and there are no interactions -such as shading effects- between them. This simplifica-tion allows to parallelize the calculasimplifica-tion, thus reducing the computasimplifica-tion time of the overall simulation. The equations are formulated in state-space form as suggested by Michalak [78].

Figure 3.3 shows the 5R1C lumped capacitance model proposed by ISO 13790 [56]. The lumped-capacitance model was solved by a linear system composed of n heat balance equa-tions, where n is the number of nodes of the corresponding thermal network.

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Figure 3.3 - 5R1C model of the building according to ISO 13790.

Figure 3.4 shows, by way of example, a temperature node of an equivalent circuit.

Figure 3.4 - Reference node. The following heat balance holds true for the reference node.

𝐻₀ (𝜃_0,𝑡− 𝜃_1,𝑡) + 𝐻₁(𝜃_2,𝑡− 𝜃_1,𝑡) + 𝜙_1,𝑡 = 𝐶₁ 𝜃_1,𝑡− 𝜃_{1,𝑡−𝛥𝑡} 𝛥𝑡

(3.1) As is usual in building simulations, the system has one degree of freedom unless one variable is fixed by the user. This leads to two possible model uses:

(a) Calculation of the heat load. The indoor air temperature 𝜃_𝑖 is set by the user and the output of the model is the heat load 𝜙_ℎ𝑐 , i.e. the energy needed by the building to maintain the specified set-point temperature 𝜃_𝑠𝑒𝑡;

(b) Calculation of the indoor air temperature. The heat load 𝜙_ℎ𝑐 is set by the user, and the output of the model is the indoor air temperature 𝜃_𝑖.

The models considered do not include the balance of water vapour in the indoor ambient which means that the calculation of the latent heat load (to be delivered to or extracted from conditioned spaces) is not included. The water vapour balance can, however, be included simply as a post-processing calculation after the thermal simulation. The parameters of the model are the thermal transmission coefficients of the glazed and opaque building compo-nents (𝐻_{𝑡𝑟,𝑤} and 𝐻_{𝑡𝑟,𝑜𝑝} respectively), the ventilation coefficient 𝐻_𝑣𝑒, the coupling conduct-ance 𝐻_{𝑡𝑟,𝑖𝑠} and the thermal capacitance 𝐶_𝑚. The final variable was calculated in accordance with the ISO 13786 International Standard [47] The thermal transmittance of opaque building components 𝐻_{𝑡𝑟,𝑜𝑝} is divided into 𝐻_{𝑡𝑟,𝑒𝑚} and 𝐻_{𝑡𝑟,𝑚𝑠} as shown in Figure 4. The nodes repre-sent the temperatures of the outdoor air 𝜃_𝑒, the supply air 𝜃_𝑠𝑢, the indoor air 𝜃_𝑎𝑖𝑟, the internal wall surface 𝜃_𝑠 and the thermal mass 𝜃_𝑚. Heat gains (𝜙_𝑖𝑛𝑡 and 𝜙_𝑠𝑜𝑙) are distributed to nodes 𝜃_𝑎𝑖𝑟, 𝜃_𝑠 and 𝜃_𝑚 in accordance with the coefficients indicated in the standard. The solar heat gain 𝜙_𝑠𝑜𝑙 includes here not only the solar radiation entering through external windows but also the short-wave radiation absorbed by the external walls and the long-wave radiation

emitted by the external surfaces to the outdoor environment. In the 1C model, all these terms sum up to give an additional heat gain, as shown in Equations (3.2) and (3.3). The first term is the short-wave radiation absorbed by opaque building components, where 𝐹_𝑠𝑜 is the shading reduction factor for the external obstacles, 𝐼_𝑠𝑜𝑙 the solar irradiance, 𝛼_𝑠 the absorption coeffi-cient, 𝑅_𝑠𝑒 the surface heat resistance, 𝐴_𝑜𝑝 and the projected area of the opaque building com-ponent. All these variables refer to the external surface of the exterior walls and must be con-sidered separately for each orientation (the subscript of the orientation is omitted for the sake of simplicity). The second term is the extra heat flow due to the thermal radiation to the sky, where 𝐹_𝑟 is the form factor between the building element and the sky (0.5 for vertical walls), 𝛼_{𝑐𝑜𝑛𝑣} is the heat transfer coefficient, and 𝛥𝜃_𝑒𝑟 is the difference between the external air tem-perature and the apparent sky temtem-perature.

𝜙_𝑠𝑜𝑙 = 𝐹_𝑠𝑜 (𝑎_𝑠 𝑅_𝑠𝑒 𝑈_𝑜𝑝 𝐴_𝑜𝑝) 𝐼_𝑠𝑜𝑙− 𝐹_𝑟𝜙_𝑟 (3.2) 𝜙_𝑟 = 𝑅_𝑠𝑒 𝑈_𝑜𝑝 𝐴_𝑜𝑝 𝛼_{𝑐𝑜𝑛𝑣} 𝛥𝜃_𝑒𝑟 (3.3) Using models from well-known Standards makes the results reproducible and allows to gen-eralize results pertaining to different climate conditions and building structures.