Nonlinear adaptive backstepping controller design for controlling bidirectional power flow of BESSs in DC microgrids

In this paper, a nonlinear adaptive backstepping controlleris designed to control the bidirectional power flow (charging/discharging) of battery energy storage systems (BESSs) in a DCmicrogrid under different operating conditions. The controller isdesigned in such a manner that the BESSs can store the excess energyfrom the renewable energy sources (RESs) in a DC microgrid aftersatisfying the load demand and also feeding back the stored energyto the load when RESs are not sufficient. The proposed controller isalso designed to maintain a constant voltage at the DC bus, whereall components of DC microgrids are connected, while controllingthe power flow of BESSs. This paper considers solar photovoltaic(PV) systems as the RES whereas a diesel generator equipped witha rectifier is used as a backup supply to maintain the continuity ofpower supply in the case of emergency situations. The controller isdesigned recursively based on the Lyapunov control theory whereall parameters within the model of BESSs are assumed to beunknown. These unknown parameters are then estimated throughthe adaptation laws and whose stability is ensured by formulatingsuitable control Lyapunov functions (CLFs) at different stages ofthe design process. Moreover, a scheme is also presented to monitorthe state of charge (SOC) of the BESS. Finally, the performanceof the proposed controller is verified on a test DC microgrid undervarious operating conditions. The proposed controller ensures the DCbus voltage regulation within the acceptable limits under differentoperating conditions.

Systems of coupled clamped beams equations with full nonlinear terms: Existence and location results

This work gives sufficient conditions for the solvability of the fourth order coupled system┊ u⁽⁴⁾(t)=f(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) v⁽⁴⁾(t)=h(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) with f,h: [0,1]×ℝ⁸→ℝ some L¹- Carathéodory functions, and the boundary conditions {┊ u(0)=u′(0)=u′′(0)=u′′(1)=0 v(0)=v′(0)=v′′(0)=v′′(1)=0. To the best of our knowledge, it is the first time in the literature where two beam equations are considered with full nonlinearities, that is, with dependence on all derivatives of u and v. An application to the study of the bending of two elastic coupled campled beams is considered.

On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this paper, we investigated a class of Liénard systems of the form x'=y, y'=f(x)+y g(x) with deg f=5 and deg g=4. We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.