In this paper, we consider the problem of a Principal aiming at designing a reward function for a population of heterogeneous agents. We construct an incentive based on the ranking of the agents, so that a competition among the latter is initiated. We place ourselves in the limit setting of mean-field type interactions and prove the existence and uniqueness of the equilibrium distribution for a given reward, for which we can find an explicit representation. Focusing first on the homogeneous setting, we characterize the optimal reward function using a convex reformulation of the problem and provide an interpretation of its behaviour. We then show that this characterization still holds for a sub-class of heterogeneous populations. For the general case, we propose a convergent numerical method which fully exploits the characterization of the mean-field equilibrium. We develop a case study related to the French market of Energy Saving Certificates based on the use of realistic data, which shows that the ranking system allows to achieve the sobriety target imposed by the European commission.
Due to the integration of more and more intermittent renewable energy sources, electrical Transmission System Operator (TSOs) embrace a more dynamic grid management, and zonal controllers have emerged as critical components. These controllers, powered by optimization algorithms, oversee specific zones within the power grid, intervening on the power grid to mitigate line overflows. However, their choice of actions and effectiveness depends on predefined target plans provided by human operators, necessitating decision-support tools to streamline this process. Because of reinforcement learning (RL) performance in sequential problems, RL holds promise in devising such plans, yet its reliance on extensive training iterations presents a time-consuming challenge. In this paper, we propose an emulator for zonal controllers, facilitating sufficient training iterations within reasonable computation times. Our methodology combines RL with heuristic techniques tailored to alleviate common challenges encountered in RL applications, such as reducing the action space and incorporating expert knowledge where beneficial. We further introduce a framework for modeling this hybrid approach within a specific Markov Decision Process tailored to the training phase. To systematically tackle the complexities inherent in this problem, we adopt a phased approach, progressively identifying and resolving key challenges. We obtained an emulator that can handle dangerous real-time situations while following the target plan when possible. By integrating RL with domain-specific heuristics and leveraging a structured problem decomposition strategy, our methodology offers a promising avenue for efficiently training decision-support systems for zonal controllers in power grid management.
Reliable ground-motion measurements are essential for seismic hazard assessment and require seismological stations to be installed in free-field conditions, away from structural interferences. However, global network analyses show diverse installation configurations that can affect measurements. Although topographic effects are natural, their influence can extend over several meters, making them closely related to operators' installation choices. Although summit effects are well studied and known to cause strong amplifications, the impact of cliffs or escarpments remains less explored. In this study, 16 SmartSolo IGU-16HR 3C nodes were placed on either side of a 30-m cliff located in Cephalonia (western Greece), for 4-10 months, recording 307 low-to-moderate-magnitude seismic events. The main results reveal that de-amplifications recorded at the cliff base are greater than amplifications determined at the cliff top, particularly between 5 and 15 Hz. This study shows that amplification and de-amplification are primarily driven by topography, whereas their directional dependence is likely shaped by both topographic and lithologic factors. Furthermore, the anisotropy related to cliff fracturing interacts with the effects of source back azimuth. Amplification and de-amplification can be correctly predicted using the frequency-scaled curvature (FSC) and illuminated FSC proxies, provided that an exponential functional form is used for determining de-amplifications at the base of near-vertical cliffs. Furthermore, the azimuthal dependence observed, inducing strong de-amplifications depending on the direction observed, seems related more to the anisotropy of geologic formations than to the cliff topography itself. These effects should be considered when using data from stations in similar topographic settings. Neglecting them can bias ground-motion models and underestimate earthquake magnitudes, particularly for stations at the base of cliffs. From these considerations, we strongly recommend installing future seismological stations a few tens of meters away from the bases and ridges of even small cliffs or escarpments.
