Power systems are being reshaped by decarbonization, digitalization, and high shares of renewables. At the same time, increasingly severe extreme conditions expose the limits of traditional reliability frameworks, calling for risk-aware, resilience-oriented approaches to address high-impact, low-probability (HILP) events. In this context, this paper presents a comprehensive overview of the foundations of power system resilience. It revisits the transition from reliability to resilience, formalizes key concepts and metrics, and introduces advanced approaches for resilience assessment, including fragility-based modeling, cascading failure analysis, and tail-risk indicators. The paper further examines resilience-oriented investment planning, operational strategies across all event phases, and the role of distributed energy resources, microgrids, and cybersecurity. The analysis highlights that resilience extends reliability by focusing on extreme conditions, fundamentally reshaping decision-making and requiring coordinated strategies across infrastructure, operation, and governance.
In this paper, we propose a new stabilised finite element method for the Darcy-Forchheimer model based on the jump stabilisation technique. We prove the well-posedness of the discrete scheme and develop an a priori error analysis. We provide two academic examples that support the theory and study the effect of the Forchheimer number on convergence. Finally, we adapt the method to solve the coupling of the Darcy-Forchheimer model with a heat equation and provide several numerical experiments, including an application to the simulation of wood drying and a three-dimensional example.
Direct wind-to-hydrogen systems, in which a wind farm supplies power to an electrolyser without a firm grid connection, must accommodate the full variability of the wind resource, yet many existing techno-economic and design studies rely on input data with temporal resolutions of several minutes to one hour. This coarse averaging can mask short-term power fluctuations that drive electrolyser performance, cycling and degradation. This paper presents a novel wind-to-electrolyser model that synthesises effective wind speed suitable for a power curve wind model at 1 Hz, combining low-frequency trend variation, mesoscale wind speed changes and Kaimal turbulence, validated against aeroelastic simulations. The resulting power output is supplied to a PEM electrolyser model incorporating bank-level thermal dynamics and steady-state, cyclic and on/off degradation, enabling the effect of wind-power variability on long-term electrolyser performance to be quantified. Using 10-year simulations of a wind farm at a site in Orkney, Scotland, two scenarios are investigated: the effect of wind farm size (1–10 turbines) and input temporal resolution (1 s, 1 min, 10 min) on power variability, degradation and hydrogen production; and the effect of turbine configuration for a fixed installed capacity (three 5 MW turbines versus one 15 MW turbine). Results show that coarser temporal resolution substantially overestimates hydrogen production, particularly for small wind farms, where 10-minute-averaged data increase predicted output by up to 428% relative to 1 s data, primarily due to underestimated on/off and cyclic degradation. Aggregating turbines and, to a lesser extent, using larger rotors both smooth the power input and reduce this sensitivity, though the effect remains significant even for 10-turbine farms. These findings demonstrate that high-frequency wind power variability must be explicitly considered in the design and performance assessment of wind-to-hydrogen systems.
We study the distribution of mesh patterns of length 2 on layered permutations, extending recent work on the distribution of mesh patterns on arbitrary permutations. We classify all mesh patterns of length 2 into exactly 35 equidistribution classes and derive a formula for the distribution of each class. With one exception, the corresponding avoidance sequences appear in the Online Encyclopedia of Integer Sequences, revealing connections with the Fibonacci and Tribonacci numbers, certain integer compositions, and restricted words. We also refine several of our enumerative results by jointly tracking five classical permutation statistics: descents, left-to-right maxima and minima, and right-to-left maxima and minima.
Power systems face increasing uncertainties that create nonlinear regime-dependent dynamics. Critical Clearing Time (CCT) remains a key transient stability metric, yet analytical relations with operating conditions are rarely tractable. Advanced machine learning techniques offer accurate CCT prediction and partial interpretability, but fail to uncover the governing functional dependencies. This paper introduces a Piecewise Symbolic Regression (Pc-SR) framework that automatically discovers regime-conditioned equations linking system variables to CCT. Pc-SR combines cost-complexity-pruned decision trees for task-aware partitioning with symbolic models in each region. Validation on synthetic data confirms recovery of correct partitions and equations under noise, while tests on single-machine infinite-bus variants rediscover analytical CCT equations. Applied to a modified 39-bus system with inverter-based resources, Pc-SR produces interpretable, regime-specific CCT surrogates matching black-box accuracy while exposing nonlinearities and interactions. This framework advances beyond descriptive explainability, providing transparent models to accelerate stability screening and support operator insight into complex dynamic behaviors.
