ABSTRACT. Let φ : [0, ∞) → R and let y 0 be a non-negative constant. Let (λ n ) n∈N be a nondecreasing sequence of positive numbers which tends to infinity, let (r n ) n∈N be a complex sequence, and c a real number. Assume that φ is square-integrable on [0, y 0 ] and for y ≥ y 0 , φ can be expressed as φ(y) = c + ℜ λn≤X r n e iλny + E(y, X),We prove that, under certain assumptions on the exponents λ n and the coefficients r n , φ(y) is a B 2 -almost periodic function and thus possesses a limiting distribution. Also if {λ n } n∈N is linearly independent over Q, we explicitly calculate the Fourier transform of the limiting distribution measure. Moreover, we prove general versions of the above results for vector-valued functions. Finally, we illustrate some applications of our general theorems by applying them to several classical error terms which occur in prime number theory. Examples include the error term in the prime number theorem for an automorphic L-function, weighted sums of the Möbius function, weighted sums of the Liouville function, the sum of the Möbius function in an arithmetic progression, and the error term in Chebotarev's density theorem.

This paper describes a control approach applied on a doubly fed induction generator (DFIG) to provide both voltage and frequency regulation capabilities, and hence, an improvement in the dynamic behavior of a microgrid system. The microgrid system is assumed to be a portion of a medium voltage distribution feeder and is supplied by two distributed generation (DG) units, i.e., a gasturbine synchronous generator and a variable-speed wind turbine with DFIG. A control approach algorithm is proposed for the DFIG unit to improve both voltage and primary frequency controls. Two distinct operation modes, i.e., grid-connected and islanding mode, are used in the proposed approach for proper transfer from normal to islanding operation. Case studies are simulated based on both planned and unplanned islanding scenarios to evaluate the performance of the control approach. The study results show that the proposed control approach for DGs in the microgrid increase the microgrid system's dynamic performance, reduce frequency changes, and improve bus voltages regulation during islanding and autonomous operations.Index Terms-Distributed generation (DG), doubly fed induction generator (DFIG), frequency and voltage controls, islanding, microgrid.

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