Journal of Control Science and Engineering

Volume 2016, Article ID 7354791, 14 pages

http://dx.doi.org/10.1155/2016/7354791

## Performance Analysis of a Peak-Current Mode Control with Compensation Ramp for a Boost-Flyback Power Converter

Facultad de Ingeniería y Arquitectura, Departamento de Ingeniería Eléctrica, Electrónica y Computación, Percepción y Control Inteligente, Bloque Q, Universidad Nacional de Colombia Sede Manizales, Campus La Nubia, 170003 Manizales, Colombia

Received 21 January 2016; Revised 13 March 2016; Accepted 22 March 2016

Academic Editor: Ayman S. Abdel-Khalik

Copyright © 2016 Juan-Guillermo Muñoz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

High voltage gain power converters are very important in photovoltaic applications mainly due to the low output voltage of photovoltaic arrays. This kind of power converters includes three or more semiconductor devices and four or more energy storage elements, making the dynamical analysis of the controlled system more difficult. In this paper, the boost-flyback power converter is controlled by peak-current mode with compensation ramp. The closed-loop analysis is performed to guarantee operation conditions such that a period-1 orbit is attained. The converter is considered as a piecewise linear system, and the closed-loop stability is determined by using the monodromy matrix, obtained by the composition of the saltation matrixes with the solutions of the dynamical equations in the linear intervals. The largest eigenvalue of the monodromy matrix gives the stability of the period-1 orbit, and a deep analysis using bifurcation diagrams let us reach a conclusion about the loss of the stability, which is experimentally verified. To avoid overcompensation effects, the minimum value required by the compensation ramp is obtained, and the minimum and maximum values of the load resistance are found too. The system has a good transient response under disturbances in the load and in the input voltage.

#### 1. Introduction

High gain power converters have attracted great interest thanks to the wide variety of new applications. Photovoltaic panels and fuel cells are application examples, in which high gain power converters play an important role in the development of new architectures aiming to improve the power systems efficiency and energy quality [1]. The classic boost converter is a simple structure but, in practice, only works for voltage gains near to two and needs extreme duty cycles for higher voltage gains [2]. Extreme duty cycles are undesirable, because the MOSFET is closed most of the time and the conduction power losses increase. To avoid extreme duty cycles, a slope compensation may be used. However, as the slope increases, the system exhibits overcompensation [3] and limitations for high gains are present again. To solve this problem, new converters have been designed and tested. Most of these new technologies use more semiconductor elements and more storage energy elements, and, consequently, the nonlinear behavior rises as well as the complexity of the model. As the complexity increases, there are new challenges regarding control strategies and stability analysis.

In this paper, we analyze the boost-flyback converter which was initially proposed in [2, 4, 5] and then studied in [6]. It is a high gain voltage converter with two capacitors, two coupled inductors, one controlled switch (MOSFET), and two uncontrolled switches (diodes). The main practical advantages of this converter are as follows: no extreme duty cycles are required to obtain high voltage gains and the rectifier reverse recovery problem is alleviated [2]. Starting from this basic structure, new and different configurations have been proposed. A summary list of applications of the coupled inductor in dc-dc converters can be found in [7]. Photovoltaic applications of boost-flyback converters have been studied in [8] and battery charging applications in [9]. In [10], a single-stage single-switch parallel boost-flyback-flyback converter with high gain and high efficiency is proposed and analyzed by linearization of its large signal equations. In [11], a coupled inductor is used in the boost-cell of boost-flyback converter to reduce the voltage stress across the output diode and obtain ripple free input current, increasing the complexity of the model. In [12], high voltage gain is obtained by using input-parallel output-series of two windings of coupled inductor; even though the objective is reached, the complexity of the model increases considerably, making it more difficult to analyze the system. In [13], a system based on coupled inductor transfer source energy is designed and analyzed in steady state. To obtain the desired results, four capacitors and four diodes are used. In [14], a small signal modelling for high voltage gain and high efficiency dc-dc converters is developed. This approach does not include nonlinear behavior and the results cannot be confirmed. In [15], a new high step-up dc-dc converter with three capacitors, three inductors, four diodes, and one MOSFET is designed (i.e., six dynamical equations and 32 possible topologies to analyze).

Despite the fact that a lot of work has been devoted to analyze and control systems similar to the boost-flyback converter, there are three main weaknesses associated: () all the reported analyses are performed by linear methods starting from a chosen steady-state solution. As the system is highly nonlinear, it is not possible to guarantee the steady state, since eventually the system state may evolve to a different limit solution. () The mathematical tools to analyze the dynamical behavior are based on linear systems, for which nonlinear phenomena, such as coexistence of attracting solutions, bifurcating phenomena, quasiperiodicity, and even chaos, cannot be investigated. () Most of the proposed systems have six or more energy storage elements and three or more semiconductor devices, increasing the complexity of dynamic equations to be analyzed which is probably the reason why rigorous mathematical analyses are not performed in any paper.

In this paper, we analyze and control the boost-flyback converter using nonlinear methods to guarantee operation in a defined oscillating steady state using bifurcation diagrams. The analysis uses numerical tools for piecewise smooth linear models [16], in order to derive analytic conditions for stability of solutions and to determine a suitable set of parameters. The study focusses on the stability analysis for periodic orbits by means of the spectrum of the fundamental matrix associated with an equivalent discrete map [17]. The analytical procedure implies the use of a saltation matrix [17, 18], in order to infer some implicit time derivatives related to the switching times. The system is controlled by a PI (Proportional Integral) peak-current controller with slope compensation. A peak-current controller with slope compensation has been studied with basic topologies [19–21], where the design parameters are determined in order to avoid overcompensations [3]. The limit value of the slope compensation as well as the range of the load resistances is computed analytically and confirmed by bifurcation diagrams obtained numerically. As far as we know, neither mathematical tools nor PI peak-current control with slope compensation has been used to analyze and control the boost-flyback converter. Finally, to complete the work, an experiment has been designed and experimental data are consistent with numerical results.

The paper is organized as follows. In Section 2, the mathematical model in open-loop and the PI peak-current controller with compensation ramp are introduced. In Section 3, the stability analysis and bifurcation diagrams are computed. The stability is performed based on the study of the monodromy matrix, using saltation matrixes. Bifurcation diagrams are computed by numerical methods, confirming and widening the results obtained by the monodromy matrix. With these results, limits on load resistance, input voltage, reference voltage, and compensation slope are obtained. In Section 4, the dynamical behavior is performed, when different disturbances are considered. In Section 5, experimental results showing the loss of the stability of the period-1 orbit and the transition to chaos are displayed. Conclusions are presented in the last section.

#### 2. The Controlled Boost-Flyback Converter

The peak-current control for a boost-flyback converter is schematized in Figure 1. The aim of this converter is to obtain high voltage gains without extreme duty cycle values. The duty cycle, noted as , is the ratio between the time that the MOSFET is closed and the period of an external clock.