Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex, and monetary risk measures on L p -spaces for 1 ≤ p < ∞ and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior is real-valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg-norm Lipschitzcontinuous in the coherent case and locally Luxemburg-norm Lipschitz-continuous in the convex monetary case. In the second part of the paper we investigate cash-additive hulls of transformed Luxemburg-norms and expected transformed losses. They provide two general classes of coherent and convex monetary risk measures that include many of the currently known examples as special cases. Explicit formulas for their robust representations and the maximizing probability measures are given.

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We extend earlier representation results for monetary risk measures on Orlicz hearts. Then we give general conditions for such risk measures to be Gâteaux-differentiable, strictly monotone with respect to almost sure inequality, strictly convex modulo translation, strictly convex modulo comonotonicity, or monotone with respect to different stochastic orders. The theoretical results are used to analyze various specific examples of risk measures.

Using ultra-high-frequency (UHF) method in practical partial discharge (PD) detection can be affected by the positioning and placement of the UHF sensor. This in turn can affect the PD diagnosis. To ensure optimal performance of the sensor and understand the propagation process of electromagnetic (EM) wave, there is a need to fully analyze how the sensor's placement affects the output signal and the attenuation of EM wave in various positions and directions. As the previous researches are mainly concentrating on the radial component of the UHF signal, the propagation of the signal components in axial and radial directions and that perpendicular to the radial direction of the GIS tank are investigated in detail in this paper. Firstly, the attenuation of UHF signals at different radial positions of a GIS model is analyzed using the finite difference time domain (FDTD) method. Then, secondly, the coupled signals in the three directions are calculated respectively. By comparing the signal received for different directions and circumferential angles, the peak to peak value (Vpp) and cumulative energy of the coupled voltage in each case are considered both in the time and frequency domain. As well, the attenuation characteristics and rules are summarized, based on which a new method for circumferential and axial location is proposed. The investigation on the propagation and the detection mechanism of EM wave in GIS provides a significant understanding of the application of UHF sensor and actual PD detection.Index Terms -Gas insulated switchgear (GIS), partial discharge (PD), UHF sensor, electromagnetic (EM) wave propagation, finite difference time domain (FDTD).