The Taylor series method of order $p$ and Adams-Bashforth method on time scales

A recent study on the Taylor series method of second order and the trapezoidal rule for dynamic equations on time scales has been continued by introducing a derivation of the Taylor series method of arbitrary order $p$ on time scales. The error and convergence analysis of the method is also obtained. The 2 step Adams-Bashforth method for dynamic equations on time scales is concluded and applied to examples of initial value problems for nonlinear dynamic equations. Numerical results are presented and discussed.

Resiliency-Oriented Optimization of Critical Parameters in Multi Inverter-Fed Distributed Generation Systems

In the modern power grid, with the growing penetration of renewable and distributed energy systems, the use of parallel inverters has significantly increased. It is essential to achieve stable parallel operation and reasonable power sharing between these parallel inverters. Droop controllers are commonly used to control the power sharing between parallel inverters in an inverter-based microgrid. In this paper, a small signal model of droop controllers with secondary loop control and an internal model-based voltage and current controller is proposed to improve the stability, resiliency, and power sharing of inverter-based distributed generation systems. The distributed generation system’s nonlinear dynamic equations are derived by incorporating the appropriate and accurate models of the network, load, phase locked loop and filters. The obtained model is then trimmed and linearized around its operating point to find the distributed generation system’s state space representation. Moreover, we optimize the critical control parameters of the model, which are found using eigenvalue analysis, and Grey Wolf optimization technique. Through time-domain simulations, we show that the proposed method improves the system’s resiliency, stability, and power sharing characteristics.

New Criteria on Oscillatory and Asymptotic Behavior of Third-Order Nonlinear Dynamic Equations with Nonlinear Neutral Terms

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples.

Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.