This paper is the second part of the study initiated in [1] and is devoted to finite-time blow-up and global existence for a semilinear heat equation on infinite weighted graphs. We first establish basic results on mild and classical solutions (which, to the best of our knowledge, were not previously available in the setting of graphs) proving their equivalence under suitable assumptions and showing the existence of a solution between a given sub- and supersolution. We then analyze blow-up and global existence on ZN, providing proofs based on methods different from those used on ZN in the existing literature. Moreover, for graphs with positive spectral gap, we prove global existence for small initial data. In contrast with previous functional analytic approaches yielding mild solutions, our method relies on the construction of global-in-time supersolutions and leads to the existence of classical solutions.
Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of parameter-dependent high-dimensional dynamical systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs projects the high-dimensional full-order model (FOM) dynamics onto a low-dimensional manifold. These projection-based ROMs approaches often rely on classical model reduction techniques such as proper orthogonal decomposition (POD) or, more recently, on neural network architectures such as autoencoders (AEs). In the case that the ROM is constructed by the POD, one has approximation guaranteed based on the singular values of the problem at hand. However, POD-based techniques can suffer from slow decay of the singular values in transport- and advection-dominated problems. In contrast to that, AEs allow for better reduction capabilities than the POD, often with the first few modes, but at the price of theoretical considerations. In addition, it is often observed, that AEs exhibits a plateau of the projection error with the increment of the dimension of the trial manifold. In this work, we propose a deep invertible AE architecture, named inv-AE, that computationally improves upon the stagnation of the reconstruction error typical of traditional AE architectures, e.g., convolutional, and the reconstructions quality. Inv-AE is composed of several invertible neural network layers that allows for gradually recovering more information about the FOM solutions the more we increase the dimension of the reduced manifold. Through the application of inv-AE to a parametric 1-dimensional Burgers’ equation, a parametric 2-dimensional fluid flow around an obstacle with variable geometry, and a parametric 3-dimensional Korteweg–de Vries, we show that (i) inv-AE mitigates the issue of the characteristic plateau of (convolutional and fully connected) AEs and (ii) inv-AE can be combined with popular autoencoder-based ROM approaches, e.g., DL-ROM and POD-DL-ROM, to improve their accuracy.
Monitoring electromechanical oscillations and islanding is crucial for ensuring the stability of modern power systems, particularly with the increasing penetration of inverter-based resources. Existing model-based and data-driven methods have shown potential but face challenges in robustness and computational efficiency, limiting their real-time application in TSO’s control rooms. This study proposes a novel data-driven approach for detecting electromechanical oscillations and network islanding in large-scale power systems. The method leverages only streaming frequency measurements from Phasor Measurement Units (PMUs) and employs advanced techniques, including Model Order Reduction technique and clustering algorithm such as Hierarchical Agglomerative Clustering. This enables the identification of disconnected network areas or coherent oscillatory regions with high accuracy, robustness, and computational efficiency. The approach is validated through real-world case studies in the European grid, including normal operation, the 2021 Continental Europe Synchronous Area separation, and the December 2016 oscillatory event. Results confirm its suitability for real-time stability monitoring.
Nanosecond pulsed plasmas show strong potential towards H-2 production via CH4 reforming, owing to their highly non-equilibrium nature. In this work, we investigate how the pulse repetition pattern, i.e. comparing single pulse operation vs. high-frequency pulse packages, correlates to the plasma properties and reforming process performance in atmospheric pressure pure CH4 using a combination of plasma diagnostics and gas chromatography. We show that the electrical properties (voltage, current, pulse energy), as well as gas temperature, electron density and plasma volume are significantly altered after the first pulse cycle in the pulse package as a result of the pronounced "memory effect" active at very short inter-pulse periods, during which the gas properties in the discharge gap (temperature, pressure, and composition) do not have enough time to return to the original conditions. These changes lead to a product selectivity shift towards highly unsaturated hydrocarbons, specifically C2H2, and energy cost reduction (from 480 to 400 kJ.mol(- 1)). In contrast, the overall CH4 conversion remains unaffected (similar to 21 %) as complete dissociation in the plasma region is already achieved after the first pulse cycle. These results demonstrate that tailoring the pulse repetition pattern provides an effective strategy to manipulate plasma properties and, thus, optimise CH4 conversion, energy efficiency, and control product selectivity in nanosecond pulsed CH4 plasmas.
Accurate knowledge of the thermodynamic properties of CO2-rich mixtures is crucial for optimizing the design and operation of the Carbon Capture, Utilization, and Storage (CCUS) chain, including the transportation step. In this work, combining measurements and modelling activities, new experimental density data are presented for five gravimetrically prepared multicomponent CO2-rich mixtures with CO2 mole fractions ranging from 0.859 to 0.952 and different concentrations of impurities such as N2, O2, Ar, CH4, and H2. Density measurements were conducted using a Vibrating Tube Densimeter (VTD) at temperatures from (273 to 313) K and pressures up to 22 MPa. The VTD has been calibrated against pure CO2 and further validated with pure propane and known binary mixtures of CO2 + CH4 and CO2 + O2. A semi-empirical calibration model parameters has been fitted for the VTD. Measurements have been performed at vapor, liquid and supercritical phases. The combined expanded relative uncertainty in density (k = 2) is estimated to be 0.4 % <= U(rho) <= 2.5 % with values <= 1 % for most density data points. To evaluate the accuracy of existing thermodynamic models, the performance of several equations of state, including GERG-2008, EOS-CG-2021, PC-SAFT, and the cubic equations of state based on Peng-Robinson, is evaluated against the new experimental data. The Helmholtz-energy based GERG-2008 and EOS-CG-2021 formulations provided comparable results and outperformed other models, with Average Absolute Relative Deviations (AARD%) close to 1 % for most of the measured densities and across the different phases and are the most accurate equations of state for density computation for multi-component CO2-rich mixtures. PC-SAFT showed reasonable results in the liquid region with AARD% between (0.4 and 1.7) %, however becoming less accurate near saturation conditions or at supercritical temperatures. The Peng-Robinson cubic equation of state predicted densities within +/- 3 % of relative deviation for most data points, although larger deviations are reported when the critical region is approached.
