Background: The key roles of epistatic interactions between multiple genetic variants in the pathogenesis of complex diseases notwithstanding, the detection of such interactions remains a great challenge in genome-wide association studies. Although some existing multi-locus approaches have shown their successes in small-scale case-control data, the "combination explosion" course prohibits their applications to genome-wide analysis. It is therefore indispensable to develop new methods that are able to reduce the search space for epistatic interactions from an astronomic number of all possible combinations of genetic variants to a manageable set of candidates.

The detection of epistatic interactive effects of multiple genetic variants on the susceptibility of human complex diseases is a great challenge in genome-wide association studies (GWAS). Although methods have been proposed to identify such interactions, the lack of an explicit definition of epistatic effects, together with computational difficulties, makes the development of new methods indispensable. In this paper, we introduce epistatic modules to describe epistatic interactive effects of multiple loci on diseases. On the basis of this notion, we put forward a Bayesian marker partition model to explain observed case-control data, and we develop a Gibbs sampling strategy to facilitate the detection of epistatic modules. Comparisons of the proposed approach with three existing methods on seven simulated disease models demonstrate the superior performance of our approach. When applied to a genome-wide case-control data set for Age-related Macular Degeneration (AMD), the proposed approach successfully identifies two known susceptible loci and suggests that a combination of two other loci—one in the gene SGCD and the other in SCAPER—is associated with the disease. Further functional analysis supports the speculation that the interaction of these two genetic variants may be responsible for the susceptibility of AMD. When applied to a genome-wide case-control data set for Parkinson's disease, the proposed method identifies seven suspicious loci that may contribute independently to the disease.

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As an aggregator involved in various renewable energy sources, energy storage systems, and loads, a virtual power plant (VPP) plays a key role as a prosumer. A VPP may enable itself to supply energy and ancillary services to the utility grid. This paper proposes a novel scheme for optimizing the operation and bidding strategy of VPPs. By scheduling the energy storage systems, demand response, and renewable energy sources, VPPs can join bidding markets to achieve maximum benefits. The potential uncertainties caused by renewable energy sources and the demand response are considered in a robust optimization model. Moreover, the robust VPP optimization accounts for its influence on markets to ensure optimal energy and reserve capacity bidding transactions in the day-ahead market and deals balancing in the real-time market. To demonstrate the performance of the proposed scheme, markets comprising various participants and managed by the system operator are implemented using mathematical models. The proposed method is evaluated using an illustrative system and the practical Taiwan power (Taipower) system with diverse uncertainty levels. The numerical results demonstrate the promising performance and the efficiency of the proposed method. The results also verify the effectiveness of the proposed method VPP with various combinations of renewable energy sources, energy storage systems, and loads. INDEX TERMSVirtual power plant, demand response model, ancillary service, energy storage system, electricity markets, renewable energy source, robust optimization, game theory, mixed integer programming.