This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

Some asymptotic methods for strongly nonlinear equations

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

The dynamics of fractional-order systems have attracted increasing attentions in recent years. In this paper, we numerically study the chaotic behaviors in the fractional-order Rossler equations. We found that chaotic behaviors exist in the fractional-order Rossler equation with orders less than 3, and hyperchaos exists in the fractional-order Rossler hyperchaotic equation with order less than 4. The lowest orders we found for chaos and hyperchaos to exist in such systems are 2.4 and 3.8, respectively. Period doubling routes to chaos in the fractional-order Rossler equation are also found.

Chaos and hyperchaos in the fractional-order Rössler equations

Preservation of the environment has become the main motivation to integrate more renewable energy sources (RESs) in electrical networks. However, several technical issues are prevalent at high level RES penetration. The most important technical issue is the difficulty in achieving the frequency stability of these new systems, as they contain less generation units that provide reserve power. Moreover, new power systems have small inertia constant due to the decoupling of the RESs from the AC grid using power converters. Therefore, the RESs in normal operation cannot participate with other conventional generation sources in frequency regulation. This paper reviews several inertia and frequency control techniques proposed for variable speed wind turbines and solar PV generators. Generally, the inertia and frequency regulation techniques were divided into two main groups. The first group includes the deloading technique, which allow the RESs to keep a certain amount of reserve power, while the second group includes inertia emulation, fast power reserve, and droop techniques, which is used to release the RESs reserve power at under frequency events.

Inertia response and frequency control techniques for renewable energy sources: A review

Dynamical systems: stability, symbolic dynamics and chaos

Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying t...

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Bifurcation control: theories, methods, and applications

The Hopf bifurcation theorem continuation of bifurcation curves on the parameter plane degenerate bifurcations in the space of system parameters high-order Hopf bifurcation formulas Hopf bifurcation in nonlinear systems with time delays birth of multiple limit cycles appendix.

Hopf Bifurcation Analysis: A Frequency Domain Approach

Brief Paper: Feedback Control of Limit Cycle Amplitudes from A Frequency Domain Approach

In this article bifurcation analysis of the 9 bus power system model corresponding to the Western Systems Coordinating Council is performed. In order to use standard continuation packages like MATCONT, a full ordinary differential equations model, including the corresponding dynamics of the control loops and the transmission lines, is derived. Different loading conditions are studied by using the load demands as bifurcation parameters. For variations of one of the loads, it is shown that the equilibrium point undergoes Hopf and saddle-node bifurcations. Furthermore, the bifurcation analysis varying two loads simultaneously reveals the existence of a pair of double Hopf and a zero-Hopf bifurcations, acting as organizing centers of the dynamics. Finally, a power system stabilizer has been added in order to modify the location of a Hopf bifurcation curve.

Bifurcation Analysis on a Multimachine Power System Model

This article offers a brief review of the fundamental concepts of bifurcation and chaos in nonlinear dynamical and control systems. Both the time-domain and frequency-domain versions of the classical Hopf bifurcation theory are studied in detail. Generalized (or degenerate) Hopf bifurcation is also discussed. Theoretical analysis and potential applications of the bifurcation theory in power systems are introduced. Meanwhile, chaos and the route to chaos from period-doubling bifurcations are described. In particular, chaos and bifurcations in feedback control systems and adaptive control systems are addressed. Because a nonlinear control system is by nature a very complex nonautonomous dynamical system due essentially to the involving of the control input, understanding and utilizing the rich dynamics of nonlinear control systems have an important impact in the modern technology. It calls for new effort and endeavor devoted to this scientific and engineering challenge.

Jorge L. Moiola

Papers

Bifurcation control: theories, methods, and applications

Hopf Bifurcation Analysis: A Frequency Domain Approach

Brief Paper: Feedback Control of Limit Cycle Amplitudes from A Frequency Domain Approach

Bifurcation Analysis on a Multimachine Power System Model

An overview of bifurcation, chaos and nonlinear dynamics in control systems