This paper proposes a two-stage stochastic market clearing (SMC) model based on a semidefinite programming (SDP) relaxation. The SMC model aims at determining the day-ahead schedule (DA) and the real-time (RT) balance settlement that minimize the total expected production cost. The network capacity constraints are considered in the proposed model through an AC power flow formulation, while the uncertainty in the renewable-based generation is taking into account using a set of stochastic scenarios. In order to solve the proposed non-linear programming model, a SDP relaxation is used. An illustrative example (3-bus test system) and the IEEE Reliability 24-bus test system are used to show the effectiveness and accuracy of the proposed model. Results shown that the proposed SDP relaxation introduce a negligible error, when compared with the solution after solving the original non-linear model.

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We present coarse-grained molecular dynamics simulations of poly(ethylene-alt-propylene) (PEP) melts, ranging in chain length from about Ne(the entanglement length) to N=6 Ne. The coarse-grained parameters, potential of mean force and bare friction, were determined from fully atomistic molecular dynamics simulations carried out on a PEP cell containing 12 chains of 80 monomers each and subjected to periodic boundary conditions. These atomistic simulations were previously validated by means of extensive neutron scattering measurements. Uncrossability constrains were also introduced in the coarse-grained model to prevent unphysical bond crossing. The coarse-grained simulations were carried out at 492 K and focus on chain dynamics. The results obtained were analyzed in terms of Rouse coordinates and Rouse correlators. We observe deviations from Rouse behavior for all chain lengths investigated, even when the chain stiffness is incorporated in the Rouse model. These deviations become more important as the chain length increases. The general scenario emerging from the results obtained is that the deviations from Rouse-like behavior are due to correlations among the forces acting upon a chain bead, which seem to be related with the constraint of uncrossability among the chains. As consequence, nonexponentiality of the Rouse correlators and mode- and time-dependent friction are observed. It seems that, in the molecular weight explored, these effects still give not raise to reptation behavior but to a crossover regime between Rouse and reptation. On the other hand, the results obtained are in qualitative agreement with those expected from the so-called generalized Rouse models, based on memory function formalisms.@en