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Flexible district heating grids form an important part of future, low-carbon energy systems. We examine probabilistic state estimation in such grids, i.e., we aim to estimate the posterior probability distribution over all grid state variables such as pressures, temperatures, and mass flows conditional on measurements of a subset of these states. Since the posterior state distribution does not belong to a standard class of probability distributions, we use Markov Chain Monte Carlo (MCMC) sampling in the space of network heat exchanges and evaluate the samples in the grid state space to estimate the posterior. Converting the heat exchange samples into grid states by solving the non-linear grid equations makes this approach computationally burdensome. However, we propose to speed it up by employing a deep neural network that is trained to approximate the solution of the exact but slow non-linear solver. This novel approach is shown to deliver highly accurate posterior distributions both for classic tree-shaped as well as meshed heating grids, at significantly reduced computational costs that are acceptable for online control. Our state estimation approach thus enables tightening the safety margins for temperature and pressure control and thereby a more efficient grid operation. • Novel approach combining classic statistical analysis and AI. • The estimated state distributions are not standard distributions. • MCMC and Bayesian calculus can be used to approximate state distributions. • MCMC requires mapping demands to grid states in every step. • Deep Neural Networks can speed this up by orders of magnitude. [ABSTRACT FROM AUTHOR]

• Compares results of spectral microphysical with parametrized fog model. • Shows influence of clouds on evolution of radiation fog. • Uses two-dimensional size distribution for aerosol particles and cloud droplets. • Shows influence of radiation term in droplet growth equation. The spectral microphysics cloud scheme MiFog and the two-moment cloud scheme PaFog are implemented in a single-column model of the atmospheric boundary layer for the numerical simulation of a radiation fog event. Results of numerical sensitivity studies are presented where the influence of cloudiness on the evolution of the fog has been investigated. It is shown that the appearance of cloudiness in the morning hours triggers an efficient dissolution of the fog. During situations with weak turbulence, quasi-periodic oscillations of the liquid water content occur in the MiFog simulations. It turns out that these fluctuations are caused by the direct interaction of diffusional droplet growth and gravitational settling of the droplets. Comparison of the MiFog and the PaFog model runs reveals that most of the time PaFog is capable of simulating the bulk fog properties sufficiently well. [ABSTRACT FROM AUTHOR]