Self-propelled particles with hydrodynamic interactions (microswimmers) have previously been shown to produce long-range ordering phenomena. Many theoretical explanations for these collective phenomena are connected to instabilities in the hydrodynamic or kinetic equations. By incorporating stochastic fluxes into the mean field kinetic equation, we quantify the dynamics of a suspension of microswimmers in the parameter regime where the deterministic equation is stable. We can thereby compute nontrivial collective phenomena concerning spatial correlations of orientation and stress as well as the enhanced diffusion of tracer particles. Our analysis here focuses primarily on twodimensional systems, though we also show how superdiffusion of tracers in three dimensions can occur by our framework.
Application of sensors in the smart grid has promoted the development of demand side management (DSM). However, the incentives of DSM such as peak–valley time-of-use (TOU) price will change the load pattern in the future; the substation capacity sizing will be further influenced accordingly. This paper proposes a substation capacity sizing method in distribution network considering DSM and establishes a peak-valley TOU pricing method based on the cost–benefit analysis of each participant in the TOU price. Compared with the conventional fixed peak–valley ratio, a dynamic division method is proposed to calculate the optimal pull-off ratio for the TOU pricing. By considering the proposed TOU pricing method, a substation capacity sizing model for the distribution network is further proposed. Finally, the economic benefits of the two substation capacity sizing schemes are compared and evaluated according to the selected economic indicators. The results of the case study demonstrate that under the premise of reasonable pricing, considering the impact of TOU on substation capacity sizing, the construction investment and the user cost of power supply companies can be saved while meeting the power demand. The economy and rationality of the planning scheme have been significantly improved.
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