Nonstandard characterization of convergence in law for -valued random variables

We prove for random variables with values in the space of cadlag functions - endowed with the supremum metric - that convergence in law is equivalent to nonstandard constructions of internal -cadlag processes, which represent up to an infinitesimal error the limit process. It is not required that the limit process is concentrated on the space , so that the theory is applicable to a wider class of limit processes as e.g. to Poisson processes or Gaussian processes. If we consider in the Skorokhod metric - instead of the supremum metric - we obtain a corresponding equivalence to constructions of internal processes with -separated jumps. We apply these results to functional central limit theorems.

     

Power supply–demand balance in a Smart Grid: An information sharing model for a market mechanism

In the future, global energy balance of a Smart Grid system can be achieved by its agents deciding on their own power demand and production (locally) and the exchange of these decisions. In this paper, we develop a network model that describes how the information of power imbalance of individual agents can be exchanged in the system. Compared to existing network models with hierarchical structures, our developed model, together with a market mechanism, achieve the power balance in the system in a completely distributed way. Additionally, dynamics, constraints and forecasts of each agent can be conveniently involved.