Nonlinear Phenomena in Power Electronics

Abstract

A new class of phenomena has recently been discovered in nonlinear dynamics. New concepts and terms have entered the vocabulary to replace time functions and frequency spectra in describing their behavior, e.g. chaos, bifurcation, fractal, Lyapunov exponent, period doubling, Poincaré map, strange attractor etc. The main objective of the paper is to summarize the state of the art in the advanced theory of nonlinear dynamical systems and illustrate its application in power electronics by three examples.

Shrinking point bifurcations of resonance tongues for piecewise-smooth, continuous maps

“…Piecewise-smooth, continuous maps have been used as models in many areas, for example, economics [1,2], power electronics [3,4,5], and cellular neural networks [6]. Furthermore, they arise as Poincaré maps of piecewise-smooth systems of differential equations, particularly near sliding bifurcations [7,8] and near so-called corner collisions [9,10].…”

Harmonic Issues Assessment on PWM VSC-Based Controlled Microgrids Using Newton Methods

“…Assuming a stable limit cycle X and a good initial condition (x 0 P U ), the BF method has to be applied over a large number of cycles and with small time steps in poorly damped and stiff systems, respectively; both cases increase the computational effort. Detailed MG models are usually stiff and present stability problems and poorly damped scenarios if the controllers are not properly tuned, designed or implemented [17]. Therefore, even with a stable limit cycle and a good initial condition, the computation of the periodic steady state solution of MG's using the BF method may take a very large computational effort if the full harmonic interactions, the closed-loop controls and the harmonic crosscoupling need to be explicitly considered.…”

A new simple 2-D piecewise linear map

“…Contrary to this situation, there are many practical applications as in communication and spreading the spectrum of switch-mode power supplies Figure 9: Coexisting attractors for the map (3) at a = 1.2 and b = −0.6, where the large chaotic attractor at the center (in black) is surrounded by a period-3 chaotic attractor at its periphery (also in black), with their basins of attraction shown in yellow and magenta, respectively to avoid electromagnetic interference [19], where it is necessary to obtain reliable operation in the chaotic mode and robustness of chaos is required. A practical example can be found from electrical engineering to demonstrate robust chaos as shown in [13][14][15]. If both Lyapunov exponents are positive throughout the range, then the resulting attractors are called hyper-chaotic, and they are clearly robust.…”