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There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Čech Compactification, W. de Gruyter, Berlin, 2012) as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017). [ABSTRACT FROM AUTHOR]

• Use of solar energy source in ground source heat pump applications. • Comparison of solar thermal collectors and photovoltaic hybrid panels. • Effects of climate on thermal and energy behaviour of the SAGSHP system. • The use of PVT panels increases the global COP of the system. • The multi-source energy system increases RES exploitation. The use of multi-source energy systems is a promising technical solution for the reduction of greenhouse gases emissions and, at the meantime, for enhancing the exploitation of renewable energy sources. The present paper analyses a Solar Assisted Ground Source Heat Pump system and tries to fill the gap in the literature, where a direct comparison between the consolidated solar thermal technology and the emerging and widely discussed photovoltaic thermal technology is missing. Indeed, the two different devices for the exploitation of solar energy have been analysed through dynamic simulations carried out in TRNSYS environment. Through this approach, it is possible to investigate and predict in detail the thermal and electrical behaviour of the energy system, composed by a water-to-water heat pump, which withdraws from two thermal reservoirs that are connected to a solar loop, composed either of solar thermal collectors or of photovoltaic thermal panels and to a ground loop. The heat pump is used to provide space cooling and space heating to a residential building, through fan coil terminal units or radiant floor system. Furthermore, the heat pump system assists the solar field in the production of domestic hot water. This study looks at the development of a flexible system able to work in different operating conditions. The model's boundary conditions can be changed for simulating a variety of case studies as the weather condition, the building load and the domestic hot water profile can be easily changed, together with specific components' parameters. The control strategy makes it possible to vary the setpoint temperatures for simulating different operating conditions and management choices. In this way, the user can easily adapt the model functions to a specific plant system. The operating conditions of the two systems layouts have been compared and the energy performances evaluated for three case studies characterised by the weather conditions of the European cities of Berlin (mild-cold climate), Venice (mild climate) and Barcelona (mild-warm climate), and the corresponding thermal load profile of a two-storey building with a useful floor area of 170 m2. The study shows the possibility to directly exploit the solar energy for the domestic hot water production and as a heat source for the heat pump. The results from the comparison between the two systems' configurations, with photovoltaic thermal panels and with solar thermal collectors, prove that there is the possibility to obtain a relevant improvement in the efficiency of the heat pump using photovoltaic thermal panels. The investigated multi-source energy system, moreover, allows an optimal employment of the available renewable heat sources and could easily be applied to case studies characterized by different boundary conditions. [ABSTRACT FROM AUTHOR]