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The district heating and cooling sector, which accounts for more than half of total energy consumption in many industrialised countries, is experiencing new challenges with the current transformation of the energy systems. The required decarbonisation will result in a more complex energy system, as a few large plants which utilise fossil fuels will be replaced by many small production plants distributed around the system, that use renewable or waste energy sources. Many of these sources are volatile; their output is affected by weather and other factors. Furthermore, district heating and cooling systems must be operated more efficiently and flexibly to ensure a consistent and cost-effective thermal energy supply, as well as effective participation power system balancing market. Because of the necessary changes in the energy system, district heating becomes increasingly beneficial to both end users as well as other energy sectors, and the number of connections is increasing in many countries in conjunction with the phase-out of fossil fuels such as oil and gas for space heating and hot water supplies. In this changed environment, increased adoption of digital technology in the district heating and cooling sector provides a chance to make systems smarter, more flexible, efficient and reliable, hence accelerating the necessary integration of additional renewable and waste energy sources into supply systems. This change affects the entire district heating energy chain, from generation to end user usage, and supports the transformation of the overall energy system. The digitalisation of district heating systems represents a paradigm shift in the production, distribution, and consumption of heat. Sensor technology, IoT connectivity, and AI algorithms advancements provide prospects for higher energy efficiency, system stability, and increased integration of renewable energy sources. However, for successful deployment, issues such as data privacy and system

In this paper we give an introduction to the Boltzmann equation for neutrino transport used in core collapse supernova models as well as a detailed mathematical description of the \emph{Isotropic Diffusion Source Approximation} (IDSA). Furthermore, we present a numerical treatment of a reduced Boltzmann model problem based on time splitting and finite volumes and revise the discretization of the IDSA for this problem. Discretization error studies carried out on the reduced Boltzmann model problem and on the IDSA show that the errors are of order one in both cases. By a numerical example, a detailed comparison of the reduced model and the IDSA is carried out and interpreted. For this example the IDSA modeling error with respect to the reduced Boltzmann model is numerically determined and localized.

We examine magnetorotationally driven supernovae as sources of $r$-process elements in the early Galaxy. On the basis of thermodynamic histories of tracer particles from a three-dimensional magnetohydrodynamical core-collapse supernova model with approximated neutrino transport, we perform nucleosynthesis calculations with and without considering the effects of neutrino absorption reactions on the electron fraction ($Y_{e}$) during post-processing. We find that the peak distribution of $Y_{e}$ in the ejecta is shifted from $\sim0.15$ to $\sim0.17$ and broadened toward higher $Y_{e}$ due to neutrino absorption. Nevertheless, in both cases the second and third peaks of the solar $r$-process element distribution can be well reproduced. The rare progenitor configuration that was used here, characterized by a high rotation rate and a large magnetic field necessary for the formation of bipolar jets, could naturally provide a site for the strong $r$-process in agreement with observations of the early galactic chemical evolution., Comment: 5 pages, 4 figures

We examine magnetorotationally driven supernovae as sources of $r$-process elements in the early Galaxy. On the basis of thermodynamic histories of tracer particles from a three-dimensional magnetohydrodynamical core-collapse supernova model with approximated neutrino transport, we perform nucleosynthesis calculations with and without considering the effects of neutrino absorption reactions on the electron fraction ($Y_{e}$) during post-processing. We find that the peak distribution of $Y_{e}$ in the ejecta is shifted from $\sim0.15$ to $\sim0.17$ and broadened toward higher $Y_{e}$ due to neutrino absorption. Nevertheless, in both cases the second and third peaks of the solar $r$-process element distribution can be well reproduced. The rare progenitor configuration that was used here, characterized by a high rotation rate and a large magnetic field necessary for the formation of bipolar jets, could naturally provide a site for the strong $r$-process in agreement with observations of the early galactic chemical evolution., Comment: 5 pages, 4 figures

In this paper we give an introduction to the Boltzmann equation for neutrino transport used in core collapse supernova models as well as a detailed mathematical description of the \emph{Isotropic Diffusion Source Approximation} (IDSA). Furthermore, we present a numerical treatment of a reduced Boltzmann model problem based on time splitting and finite volumes and revise the discretization of the IDSA for this problem. Discretization error studies carried out on the reduced Boltzmann model problem and on the IDSA show that the errors are of order one in both cases. By a numerical example, a detailed comparison of the reduced model and the IDSA is carried out and interpreted. For this example the IDSA modeling error with respect to the reduced Boltzmann model is numerically determined and localized.

We present a numerical method for handling the resolution of a general transport equation for radiative particles, aimed at physical problems with a general spherical geometry. Having in mind the computational time difficulties encountered in problems such as neutrino transport in astrophysical supernovae, we present a scheme based on full spectral methods in 6d spherical coordinates. This approach, known to be suited when the characteristic length of the dynamics is much smaller than the domain size, has the potential advantage of a global speedup with respect to usual finite difference schemes. An analysis of the properties of the Liouville operator expressed in our coordinates is necessary in order to handle correctly the numerical behaviour of the solution. This reflects on a specific (spherical) geometry of the computational domain. The numerical tests, performed under several different regimes for the equation, prove the robustness of the scheme: their performances also point out to the suitability of such an approach to large scale computations involving transport physics for mass less radiative particles., Comment: 18 pages

We present a numerical method for handling the resolution of a general transport equation for radiative particles, aimed at physical problems with a general spherical geometry. Having in mind the computational time difficulties encountered in problems such as neutrino transport in astrophysical supernovae, we present a scheme based on full spectral methods in 6d spherical coordinates. This approach, known to be suited when the characteristic length of the dynamics is much smaller than the domain size, has the potential advantage of a global speedup with respect to usual finite difference schemes. An analysis of the properties of the Liouville operator expressed in our coordinates is necessary in order to handle correctly the numerical behaviour of the solution. This reflects on a specific (spherical) geometry of the computational domain. The numerical tests, performed under several different regimes for the equation, prove the robustness of the scheme: their performances also point out to the suitability of such an approach to large scale computations involving transport physics for mass less radiative particles., Comment: 18 pages