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There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost.
Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different.
Useful market simulations are key to the evaluation of diferent market designs existing of multiple market mechanisms or rules. Yet a simulation framework which has a comparison of diferent market mechanisms in mind was not found. The need to create an objective view on different sets of market rules while investigating meaningful agent strategies concludes that such a simulation framework is needed to advance the research on this subject. An overview of diferent existing market simulation models is given which also shows the research gap and the missing capabilities of those systems. Finally, a methodology is outlined how a novel market simulation which can answer the research questions can be developed.
Market abstraction of energy markets and policies - application in an agent-based modeling toolbox
(2023)
In light of emerging challenges in energy systems, markets are prone to changing dynamics and market design. Simulation models are commonly used to understand the changing dynamics of future electricity markets. However, existing market models were often created with specific use cases in mind, which limits their flexibility and usability. This can impose challenges for using a single model to compare different market designs. This paper introduces a new method of defining market designs for energy market simulations. The proposed concept makes it easy to incorporate different market designs into electricity market models by using relevant parameters derived from analyzing existing simulation tools, morphological categorization and ontologies. These parameters are then used to derive a market abstraction and integrate it into an agent-based simulation framework, allowing for a unified analysis of diverse market designs. Furthermore, we showcase the usability of integrating new types of long-term contracts and over-the-counter trading. To validate this approach, two case studies are demonstrated: a pay-as-clear market and a pay-as-bid long-term market. These examples demonstrate the capabilities of the proposed framework.
Transport through Redox-Active Ru-Terpyridine Complexes Integrated in Single Nanoparticle Devices
(2020)
Transition metal complexes are electrofunctional molecules due to their high conductivity and their intrinsic switching ability involving a metal-to-ligand charge transfer. Here, a method is presented to contact reliably a few to single redox-active Ru-terpyridine complexes in a CMOS compatible nanodevice and preserve their electrical functionality. Using hybrid materials from 14 nm gold nanoparticles (AuNP) and bis-{4′-[4-(mercaptophenyl)-2,2′:6′,2″-terpyridine]}-ruthenium(II) complexes a device size of 30² nm² inclusive nanoelectrodes is achieved. Moreover, this method bears the opportunity for further downscaling. The Ru-complex AuNP devices show symmetric and asymmetric current versus voltage curves with a hysteretic characteristic in two well separated conductance ranges. By theoretical approximations based on the single-channel Landauer model, the charge transport through the formed double-barrier tunnel junction is thoroughly analyzed and its sensibility to the molecule/metal contact is revealed. It can be verified that tunneling transport through the HOMO is the main transport mechanism while decoherent hopping transport is present to a minor extent.