L. Benadero

Hopf bifurcation and chaos from torus breakdown in a PWM voltage-controlled DC-DC boost converter

The behavior of a closed loop dc-dc boost converter is investigated when the pulse width modulation (PWM) period is varied. The dynamics are analyzed both by using analytical solutions of the state equations and by the stroboscopic map. This analysis shows that Hopf bifurcation occurs at a certain value of the parameters. Phase-locking periodic windows, the period adding sequence, and the transition from quasi-periodicity to period doubling via torus breakdown are also obtained. An experimental prototype was built to check the numerical results. Parasitic elements, such as the equivalent series resistance of the inductor and the conducting voltage of the diode, are included in the model to obtain better concordance with experiments.

QUASIPERIODICITY AND CHAOS IN THE DC–DC BUCK–BOOST CONVERTER

This paper is concerned with the study of nonlinear phenomena in a closed loop voltage-controlled DC–DC Buck–Boost converter when suitable parameters are varied. The dynamics is analyzed using both the continuous-time model and the numerically computed stroboscopic map. The analysis of the one-dimensional bifurcation diagram shows that Neimarck–Sacker bifurcation occurs at certain values of the parameters. Phase-locking periodic windows, the period-adding sequence, and transition from quasiperiodicity to period-doubling via torus breakdown are also obtained. The two-dimensional bifurcation diagram is carefully computed. This shows that phase-locking orbits produce so-called Arnold tongues in the parameter space. It is shown that the winding number plotted as a function of the bifurcation parameter is a devils staircase. As typically occurs in general circle maps, the fine structures of the Arnold tongues and the devils staircase show self-similarity.

Dynamics and Stability Issues of a Single-Inductor Dual-Switching DC–DC Converter

A single-inductor two-input two-output power electronic dc-dc converter can be used to regulate two generally nonsymmetric positive and negative outputs by means of a pulsewidth modulation with a double voltage feedback. This paper studies the dynamic behavior of this system. First, the operation modes and the steady-state properties of the converter are addressed, and, then, a stability analysis that includes both the power stage and control parameters is carried out. Different bifurcations are determined from the averaged model and from the discrete-time model. The Routh-Hurwitz criterion is used to obtain the stability regions of the averaged (slow-scale) dynamics in the design parameter space, and a discrete-time approach is used to obtain more accurate results and to detect possible (fast-scale) subharmonic oscillations. Experimental measurements were taken from a system prototype to confirm the analytical results and numerical simulations. Some possible nonsmooth bifurcations due to the change in the switching patterns are also illustrated.

TWO-DIMENSIONAL BIFURCATION DIAGRAMS: BACKGROUND PATTERN OF FUNDAMENTAL DC–DC CONVERTERS WITH PWM CONTROL

One of the usual ways to build up mathematical models corresponding to a wide class of DC–DC converters is by means of piecewise linear differential equations. These models belong to a class of dynamical systems called Variable Structure Systems (VSS). From a classical design point of view, it is of interest to know the dynamical behavior of the system when some parameters are varied. Usually, Pulse Width Modulation (PWM) is adopted to control a DC–DC converter. When this kind of control is used, the resulting mathematical model is nonautonomous and periodic. In this case, the global Poincare map (stroboscopic map) gives all the information about the system. The classical design in these electronic circuits is based on a stable periodic orbit which has some desired characteristics. In this paper, the main bifurcations which may undergo this orbit, when the parameters of the circuit change, are described. Moreover, it will be shown that in the three basic power electronic converters Buck, Boost and Buck–Boost, very similar scenarios are obtained. Also, some kinds of secondary bifurcations which are of interest for the global dynamical behavior are presented. From a dynamical systems point of view, VSS analyzed in this work present some kinds of bifurcations which are typical in nonsmooth systems and it is impossible to find them in smooth systems.

Numerical and experimental study of the region of period-one operation of a PWM boost converter

A study is presented of a DC-DC boost converter whose output voltage is controlled by naturally sampled constant-frequency PWM operating in both continuous and discontinuous mode. For certain values of the circuit parameters, instability occurs. Moreover, nonlinearities may produce bifurcations and chaos when parameters are varied. The goal of this paper is to delimit in the parameter space the region of period-one operation of the converter which is of practical interest for engineering design. Linear analysis and other conventional methods are sometimes not strong enough to delimit this region, and one must rely on experimental results and careful numerical simulation. The regions are located using two different methods: a prototype which allows variation of all circuit parameters is built up and the results are checked by means of special-purpose C-code simulations, resulting in a good agreement with the experiments. Some families of curves characterize the operational regions, which border on quasiperiodic and subharmonic behavior; these bifurcation phenomena may be related to resonances and to the change of conduction mode.

Stability analysis of a single inductor dual switching dc-dc converter

This paper deals with the analysis of a single inductor switching dc-dc power electronics converter which is used to regulate two, in general non-symmetric, positive and negative outputs. A PWM control with a double PI feedback loop is used for the regulation of both output voltages. The steady state properties of this converter are first discussed and then stability is studied in terms of both power stage and control parameters.

Analysis of Bifurcation Behavior of a Piecewise Linear Vibrator with Electromagnetic Coupling for Energy Harvesting Applications

Recently, nonlinearities have been shown to play an important role in increasing the extracted energy of vibration-based energy harvesting systems. In this paper, we study the dynamical behavior of a piecewise linear (PWL) spring-mass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. Different configurations of the PWL model and their corresponding state-space regions are derived. Then, from this PWL model, extensive numerical simulations are carried out by computing time-domain waveforms, state-space trajectories and frequency responses under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Filippov method, Poincare map modeling and finite difference method (FDM). The Floquet multipliers are calculated using these three approaches and a good concordance is obtained among them. The performance of the system in terms of the harvested energy is studied by considering both purely harmonic excitation and a noisy vibrational source. A frequency-domain analysis shows that the harvested energy could be larger at low frequencies as compared to an equivalent linear system, in particular, for relatively low excitation intensities. This could be an advantage for potential use of this system in low frequency ambient vibrational-based energy harvesting applications.

Single inductor multiple outputs interleaved converters operating in CCM

Feasibility of single inductor multiple outputs (SIMO) DC-DC converters, able to operate in continuous-conduction mode (CCM), is shown in this paper. The power stage combines simple structures based on boost for non-inverted outputs and buck-boost for inverted outputs. Individual switches associated to each of the outputs are current mode controlled through respective stages as called channels. The set of dynamical references for every channel is obtained by means of a matrix arrangement whose input is the set of signals provided by proportional-integral (PI) blocks applied to each of the input-output errors. Finally, phase-shifted (interleaved) compensating ramps are added to those references. Analysis of dynamics stability is provided by means of an averaged model and direct simulation.

Nonlinear Analysis of Interconnected Power Converters: A Case Study

In this paper the nonlinear dynamics of interconnected power converters in an islanded direct current (DC) microgrid is analyzed. By using a simplified scheme based on two cascaded converters we analyze the dynamical behavior that can arise from the interconnection of these devices on a common DC bus. Furthermore, in order to address the bus voltage control problem, we propose a Sliding Mode Controller for a DC-DC bidirectional power converter to control the DC bus voltage under instantaneous constant power loads. This class of loads introduces a destabilizing nonlinear effect on the converter through an inverse voltage term that can lead to significant oscillations in the DC bus voltage. Simulation results are shown to illustrate the nonlinear analysis.

Sliding mode control of interconnected power electronic converters in DC microgrids

In this work, a typical structure of cascaded converters present in DC microgrids is analyzed. From the control point of view, converters acting as loads add a nonlinear effect to the supply bus, caused by its constant power load behavior. A realistic analysis of the equivalent load is presented. The open loop dynamics of the system is discussed, taking into account the equivalent load profile. A nonlinear sliding mode controller based on a proportional-integral controller is proposed to regulate the bus voltage under unknown load variations. The proposed methodology is validated trough simulation and experimental results.