The dynamics of a DC-AC self-oscillating LC resonant inverter with a zero current switching strategy is considered in this paper. A model that includes both the series and the parallel topologies and accounts for parasitic resistances in the energy storage components is used. It is found that only two reduced parameters are needed to unfold the bifurcation set of this extended system: one is related to the quality factor of the LC resonant tank, and the other accounts for the balance between serial and parallel losses. Through a rigorous mathematical study, a complete description of the bifurcation set is obtained and the parameter regions where the inverter can work properly is emphasized.
This paper deals with the global dynamical analysis
of a self-oscillating resonant inverter which is based on switching
between two symmetrical circuit configurations. The analysis
predicts coexisting steady-state solutions, which are increasingly
relevant for low values of the quality factor of the resonant circuit,
thus likely driving to an improper system operation. A repelling
sliding region is found to be connected with the two unstable limit
cycles that split the phase plane in three basins of attraction. To
avoid the presence of multiple steady-state solutions, a viable
solution consisting of a modification of the control is proposed
and validated by numerical simulations.
In this paper, the dynamics of a dc-ac resonant self-oscillating LC series inverter is analyzed
from the point of view of piecewise smooth dynamical systems. Our system is defined by two
symmetric configurations and its bifurcation analysis can be given in a one dimensional param-
eter space, thus finding a non smooth transition between two strongly different dynamics. The
oscillating regime, which is the one useful for applications and involves a repetitive switching
action between those configurations, is given whenever their open loop equilibrium is a fo-
cus. Otherwise, the only attractors are equilibrium points of node type whose stable manifolds
preclude the appearance of oscillations.
In this paper, the switching dynamics of a dc-ac
resonant self-oscillating inverter is considered. Using bifurcation
analysis coexisting steady-state solutions are predicted, which
are increasingly relevant for low values of the quality factor
of the resonant circuit. A repelling sliding region is found to be
connected with the two unstable limit cycles that split the phase
plane in three basins of attraction. Simulation results obtained
from the switched model confirm the theoretical derivations.
Cristiano, R.; Pagano, D.; Benadero, L.; Ponce, E. International journal of bifurcation and chaos Vol. 26, num. 4, p. 1630010-1-1630010-18 DOI: 10.1142/S021812741630010X Data de publicació: 2016-04 Article en revista
Direct current (DC) microgrids (MGs) are an emergent option to satisfy new demands for power quality and integration of renewable resources in electrical distribution systems. This work addresses the large-signal stability analysis of a DC–DC bidirectional converter (DBC) connected to a storage device in an islanding MG. This converter is responsible for controlling the balance of power (load demand and generation) under constant power loads (CPLs). In order to control the DC bus voltage through a DBC, we propose a robust sliding mode control (SMC) based on a washout filter. Dynamical systems techniques are exploited to assess the quality of this switching control strategy. In this sense, a bifurcation analysis is performed to study the nonlinear stability of a reduced model of this system. The appearance of different bifurcations when load parameters and control gains are changed is studied in detail. In the specific case of Teixeira Singularity (TS) bifurcation, some experimental results are provided, confirming the mathematical predictions. Both a deeper insight in the dynamic behavior of the controlled system and valuable design criteria are obtained.
Benadero, L.; Cristiano, R.; Pagano, D.; Ponce, E. IEEE Journal on Emerging and Selected Topics in Circuits and Systems Vol. 5, num. 3, p. 326-335 DOI: 10.1109/JETCAS.2015.2462017 Data de publicació: 2015-09-01 Article en revista
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.
El Aroudi, A.; Ouakad, H.; Benadero, L.; Younis, M. International journal of bifurcation and chaos Vol. 24, num. 5, p. 1-20 DOI: 10.1142/S0218127414500667 Data de publicació: 2014-05 Article en revista
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, Poincar´e 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.
El Aroudi, A.; Benadero, L.; Ouakad, H.; Younis, M. International Conference on Structural Dynamics and Diagnosis p. 08001-1-08001-5 DOI: 10.1051/matecconf/20141608001 Data de presentació: 2014-05 Presentació treball a congrés
In this paper, we study the dynamical behavior of a Duffing-like piecewise linear (PWL) springmass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. From this PWL model, numerical simulations are carried out by computing frequency response and bifurcation diagram under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Fillipov method.
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.
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
A piecewise linear (PWL) continuous map is used to analyze the nonsmooth bifurcation phenomena in a single-inductor two-output (SITO) DC–DC converter under a PWM interleaved control scheme. This map models the dynamical behavior of the converter when the waveforms of the inductor current can be approximated by straight lines during each switching subinterval. The parameter space is constrained by the interleaving control and the physical restriction on the values of some parameters. The main focus of this paper is on the existence and stability conditions of the rich variety of k-periodic orbits and the different bifurcation patterns that can be exhibited in this system. The analytical results in the form of 1-D and 2-D bifurcation diagrams are compared with numerical simulations obtained from the circuit-based switched model getting a good agreement between the two approaches.
In this paper, continuous conduction mode (CCM) operation of a class of single inductor multiple output dc-dc converters is proposed. The power stage combines boost and buck-boost structures loading non-inverted and inverted outputs. The control strategy is based on current mode control under an interleaving scheme, in which each output is controlled by a specific channel. These channels use different dynamic references, which are obtained from a set of proportional-integral (PI) controllers associated to the voltage outputs. The dynamical behaviour of this system is described by means of a large signal averaged model and direct simulations of the switched circuit-based model. Small signal stability analysis of the slow scale dynamics is also carried out by using the averaged model. Finally, some experimental results are provided to validate the theoretical predictions and the numerical simulations.
In this paper, different discrete-time models in the form of maps are proposed and analyzed in order to describe the dynamics of single inductor multiple-input multiple-output (SIMIMO) switching DC–DC converters. These systems can be used to regulate generally multiple (positive and/or negative) outputs by means of individual switches associated to each of the outputs.
These switches are current mode controlled through corresponding channels. The discrete-time approach allows the dynamical behavior of these systems to be accurately predicted as well as to detect possible subharmonic oscillations and chaotic behavior. Under certain operating
conditions, for which the system can be modeled by a one-dimensional piecewise constant vector field, a simple one-dimensional and piecewise-linear (PWL) map can be obtained. Some closed form expressions for ensuring stability are derived from this map in terms of a stability index λ,which is, in turn, expressed in terms of system parameters. However, some discrepancies have been found between the switched model and this simpler map, and therefore a full order model is derived to obtain more accurate information about the actual dynamical behavior of these
converters. The theoretical results are confirmed by one-dimensional bifurcation diagrams and codimension 1 two-parameter bifurcation curves obtained by standard continuation methods applied to the derived discrete-time models as well as from computer simulations from the switched model.
Moreno-Font, V.; El Aroudi, A.; Calvente, J.; Giral, R.; Benadero, L. IEEE transactions on circuits and systems I: regular papers Vol. 57, num. 2, p. 415-426 DOI: 10.1109/TCSI.2009.2023769 Data de publicació: 2010-02 Article en revista
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.
This paper is concerned with the analysis of two-parameter bifurcation phenomena in the buck
power converter. It is shown that the complex dynamics of the converter can be unfolded by considering
higher codimension bifurcation points in two-parameter space. Specifically, standard smooth
bifurcations are shown to merge with discontinuity-induced bifurcation (DIB) curves, giving rise to
intricate bifurcation scenarios. The analytical results are compared with those obtained numerically,
showing excellent agreement between the analytical predictions and the numerical observations. The
existence of these two-parameter bifurcation phenomena involving DIBs and smooth bifurcations,
predicted in [P. Kowalczyk et al., Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), pp. 601–629;
A. Colombo and F. Dercole, SIAM J. Appl. Dyn. Syst., submitted], is confirmed in this important
class of systems.
Els convertidors commutats de potència són solucions apropiades per subministrar energia a dispositius electrònics per la seva elevada eficiència i reduït cost. El seu ús extensiu en les últimes dècades ha motivat els investigadors a millorar els seus dissenys i aprofundir en la comprensió del seu comportament el qual, com la majoria de dispositius electrònics de potència, presenta dinàmiques no lineals. Recentment, han aparegut equipaments electrònics que disposen de múltiples càrregues com són els PDA, telèfons mòbils, MP3... Freqüentment, aquestes aplicacions necessiten múltiples alimentacions amb doble polaritat. Els convertidors amb inductor únic i múltiples sortides, Single-Inductor Multiple-Input Multiple-Output (SIMIMO), han esdevingut solucions per subministrar energia a dispositius de baixa potència, com pantalles LCD, i per carregar bateries ja que l'ús d'un sol inductor redueix significativament la mida del convertidor. La inherent naturalesa commutada d'aquests sistemes classifica la seva dinàmica dins el camp de sistemes d'estructura variable, Variable Structure Systems (VSS), els quals també es coneixen com a sistemes suaus a trams, Piecewise Smooth (PWS) systems. Atès que la teoria clàssica per a sistemes suaus no pot explicar completament el seu comportament, en els últims anys s'han dirigit molts esforços cap a la recerca de les propietats de la dinàmica no suau en diferent camps d'aplicació. Aquesta tesi aprofundeix en la caracterització de convertidors SIMIMO, que ens permetrà provar la seva viabilitat. Es proposen dues estratègies de control basades en el conegut control PWM (Pulse Width Modulation). En la primera alternativa, el control ens permet regular un convertidor amb dues entrades i dues sortides (Two-Input Two-Output , SITITO), amb polaritats oposades. En aquest cas, les dues senyals moduladores necessàries són generades sincronitzadament i per aquest motiu, en aquesta tesi ens referirem a aquesta estratègia de control PWM com a SPC (Single Phase Control) en contraposició amb la segona alternativa, la qual serà anomenada IC (Interleaved Control), capaç de regular un número generalitzat de sortides. Aquest control està basat en l'ús de diverses senyals moduladores, tantes com a sortides, les quals s'han desfasat progressivament. La dinàmica dels convertidors SIMIMO, al igual que els convertidors bàsics contínua - contínua, exhibeix una rica varietat de fenòmens, els quals engloben des de bifurcacions suaus, com són les bifurcacions de doblament de període (period doubling bifurcation), Saddle-Node o Hopf, fins a bifurcacions no suaus. Un cop verificada l'existència de dinàmica estable quan els paràmetres s'han seleccionatapropiadament, aquesta tesi aborda la recerca de models amb els quals analitzar la complexa dinàmica dels convertidors en un rang ampli de paràmetres. Es proposen i analitzen alguns models que s'utilitzen complementàriament: els anomenats averaged models, amb els quals es pot analitzar la dinàmica lenta, i els models discrets, capaços de detectar les inestabilitats degudes a la dinàmica ràpida. A més a més, alguns d'aquest models seran definits i analitzats. La seva utilitat s'ha provat no només en la predicció de la estabilitat, sinó també en la caracterització de bifurcacions no suaus presents en el circuit. Es demostra que senzills sistemes lineals a trams de dimensió ú proporcionen expressions analítiques per a les condicions d'estabilitat y existència de punts fixos. Per finalitzar, es desenvolupen mapes de dimensió més elevada per tal d'incrementar la precisió de les prediccions obtingudes mitjançat els averaged models i els models discrets. L'anàlisi discreta del convertidor SITITO governat per cadascuna d'aquestes estratègies ha revelat que la dinàmica por ser modelada per un sistema lineal a trams en un rang específic de paràmetres. Fins on sabem, la bibliografia proporcionada sobre mapes PWL inclou tant mapes continus com discontinus, encara que limitats a dos trams. Per tant, aquesta tesi contribueix en el camp de la dinàmica no suau amb el desenvolupament de les propietats d'un mapa de tres trams. Respecte al control IC, s'ha obtingut una anàlisi general de la seva estabilitat per a un convertidor SIMIMO amb un nombre genèric de càrregues. L'estudi de l'estabilitat del model discret de dimensió ú ha revelat l'existència d'un tipus de bifurcació no suau la qual ha estat classificada con una non-smooth pitchfork atesa l'aparició de nous punts fixos després de produir-se la bifurcació. Una anàlisi més detallada de models discrets de dimensions més elevades, associa aquesta bifurcació a una Neimark-Sacker. Finalment, aquesta tesi també inclou alguns resultats experimentals obtinguts amb un prototip d'un convertidor SITITO, per tal de validar els escenaris trobats en l'anàlisi del comportament dinàmic del convertidor regulat per les dues estratègies de control.
Switching power converters are known to be appropriate solutions to supply energy to electronic devices owing to their high efficiency and low cost. Their extensive use in the last decades has motivated researches to improve their designs and to go deeply into the comprehension of their behavior which, like most power electronic devices, exhibit nonlinear dynamics. More recently, electronic equipments containing multiple loads have been arisen such as PDA, mobile phones, MP3... These applications frequently require multiple supplies with different polarities. Single-Inductor Multiple-Input Multiple-Output (SIMIMO) switching dc-dc converters are becoming as solutions to supply low power devices as LCD displays and to charge batteries due to the significant reduction of size because the use of a single inductor. The inherent switching nature of these systems classifies their dynamics into the field of Variable Structure Systems (VSS), which are also known as Piecewise Smooth (PWS) systems. Due to the fact that their dynamics cannot be completely explained with the classical smooth theory, in the last years a lot of effort has been addressed towards the research on a theory of non-smooth dynamics motivated by different fields of application. This dissertation deals with the dynamical characterization of SIMIMO converters, which can help us to prove their viability. Two strategies of control, both of them based on the widely used Pulse Width Modulation (PWM) control, are discussed. In the first alternative, the control is used to regulate a Two-Input Two-Output (SITITO) converter with opposite polarity. The two required modulate signals are generated synchronizely. This strategy of PWM control is called in this work Single Phase Control (SPC) in contrast to a second strategy, which is noted here as Interleaved Control (IC), capable of driving a generalized single inductor multiple-input multiple-output converters. This control is based on the use of various modulating signals, equal to the number of outputs, which are progressively time delayed. The dynamics of the SIMIMO converters, just like of the basic dc-dc converters, presents a rich variety of nonlinear phenomena, which covers from smooth bifurcations, such as period doubling, Saddle-Node or Hopf bifurcations, to non-smooth bifurcations. After proving the existence of stable dynamics if appropriate parameters are selected, this dissertation will deal with the investigation of models to analyze the complex dynamics of the converter in a wide range of parameters. Several models are proposed and analyzed in this work. Averaged models, from which slow scale instability condition can be determined, and discrete-time models, able to prove fast scale instabilities, are used in a complementary way. Besides this, several approaches of these models will be established and validated. Their usefulness will be proved not only in the prediction of the stability, but also in the characterization of the non-smooth bifurcations presents in this converter. It will be shown that simple one-dimensional Piecewise-Linear (PWL) models provide analytical expressions for the stability and existence conditions of fixed points of the discrete-time models. Furthermore, higher dimensional maps are developed to improve the accuracy of the predictions obtained by means of one-dimensional maps and averaged models. The discrete-time analysis of a SITITO converter driven by each of the two strategies of control has revealed that its dynamics can be modeled by a PWL with three trams in a specific range of parameters. To our best knowledge, the literature on PWL maps includes continuous and discontinuous maps but is limited to two trams. Therefore, this dissertation is a contribution in the field of non-smooth dynamics in base to the unfolding of specific dynamics of three-piece maps. Concerning the IC control, a generalized analysis of the stability is obtained for a SIMIMO converter with a generic number of loads. The stability analysis of the one-dimensional model has revealed the existence of a type of non-smooth bifurcation, which has been classified in this dissertation as a non-smooth pitchfork owing to the appearance of two new fixed points after undergoing the bifurcation. Detailed analysis in higher dimensional maps associates this bifurcation to a Neimark-Sacker, whose existence cannot be predicted by averaged models. This dissertation also includes some experimental results obtained with a SITITO dc-dc converter prototype, to validate some of the scenarios found in the analysis.
We study the dynamical behavior of a single inductor two inputs two outputs (SITITO) power electronics DC-DC converter under a current mode control in a PWM interleaved scheme. This system is able to regulate two, generally one positive and one negative, voltages (outputs). The regulation of the outputs is carried out by the modulation of two time intervals within a switching cycle. The value of the regulated voltages is related to both duty cycles (inputs). The stability of the whole nonlinear system is therefore studied without any decoupling. Under certain operating conditions, the dynamical behavior of the system can be modeled by a piecewise linear (PWL)map, which is used to investigate the stability in the parameter space and to detect possible
subharmonic oscillations and chaotic behavior. These results are confirmed by numerical one dimensional and two-dimensional bifurcation diagrams and some experimental measurements from a laboratory prototype.
In this paper we deal with the analysis of nonlinear dynamical behavior of a single inductor two inputs two outputs (SITITO) power electronics DC-DC converter. The
system can be used to regulate generally two outputs (one positive and one negative). Under certain operating conditions, the switching model can be approximated by a
one dimensional piecewise constant vector field and, as a consequence, the corresponding map is piecewise linear
(PWL). This model is derived and then it is used to study a nonsmooth pitchfork bifurcation in the system. Coexistence
of attractors are detected by using the same model. Intermittent chaotic behavior is also addressed. Analytical results are confirmed by one dimensional and twodimensional
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.
Benadero, L.; Giral, R.; Aroudi, A.; Calvente, J. Mathematics and computers in simulation Vol. 71, num. 4-6, p. 256-269 DOI: 10.1016/j.matcom.2006.02.009 Data de publicació: 2006-06-19 Article en revista
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.
El Aroudi, A.; Debbat, M. B.; Giral, R.; Olivar, G.; Benadero, L.; Toribio, E. International journal of bifurcation and chaos Vol. 15, num. 5, p. 1549-1578 DOI: 10.1142/S0218127405012946 Data de publicació: 2005-05 Article en revista
This paper presents, in a tutorial manner, nonlinear phenomena such as bifurcations and chaotic behavior in DC-DC switching converters. Our purpose is to present the different modeling approaches, the main results found in the last years and some possible practical applications. A comparison of the different models is given and their accuracy in predicting nonlinear behavior is discussed. A general Poincare map is considered to model any multiple configuration of DC-DC switching converters and its Jacobian matrix is derived for stability analysis. More emphasis is done in the discrete-time approach as it gives more accurate prediction of bifurcations. The results are reproduced for different examples of DC-DC switching converters studied in the literature. Some methods of controlling bifurcations are applied to stabilize Unstable Periodic Orbits (UPOs) embedded in the dynamics of the system. Statistical analysis of these systems working in the chaotic regime is discussed. An extensive list of references is included.
Toribio, E.; Llaquet, J.; Benadero, L.; Gomez, E.; Moreno-Font, V. Seminario Anual de Automática, Electrónica Industrial e Instrumentación p. 232 Data de presentació: 2003-09-11 Presentació treball a congrés
Benadero, L.; El Aroudi, A.; Olivar, G.; Toribio, E.; Gomez, E. International journal of bifurcation and chaos Vol. 13, num. 2, p. 427-451 DOI: 10.1142/S0218127403006728 Data de publicació: 2003-05 Article en revista
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 Poincaré 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.