Relationship of fast‐scale and slow‐scale instabilities in switching circuit with multiple inputs

Abstract

Power converter circuits, such as current-controlled or voltage-controlled converters and inverters often have multiple inputs in the controller. The multiple inputs cause high-frequency and low-frequency oscillations. In earlier studies, the characteristics of circuits in fast-scale and slow-scale dynamics have been investigated. However, in many cases, circuits with multiple inputs have three or more dimensional topology which makes detailed analysis difficult. In this paper, we analyze a simple interrupted electric circuit in order to understand essential characteristics of fast-scale and slow-scale dynamics. The advantage of this simple interrupted circuit is that it is possible to derive a 1-dimensional map, which facilitates rigorous studies. Based on the structure of the return map and the characteristic multiplier, we explain the characteristics of the system. We report the occurrence of pitchfork, period doubling, and border collision bifurcations in slow scale, and period doubling bifurcation in fast scale. We found that local bifurcation, which appears in fast-scale dynamics, does not significantly affect the global behavior of the system while instabilities in the slow-scale dynamics strongly affect the system behavior.

A Filippov method based analytical perspective on stability analysis of a DC‐AC H‐bridge inverter with nonlinear rectifier load

“…Four key differences between ( 20) and ( 33) should be noted: i) As ( 17) contains a term "V ref " which corresponds to phase angle, onset of localized instability can be detected by ( 20) unlike (33), where the Jacobian matrix is time-invariant in nature. Thus, (33) can detect instabilities which manifest throughout the whole fundamental 50 Hz line cycle only. ii) Equation ( 20) takes into account V D additionally through (17).…”

“…The Stability boundaries by (20) as well as (33) have been shown in Figure 2. Both methods display equal capability in predicting Hopf bifurcation, while only Filippov's method has predicted the boundary line of onset of PD bifurcation, as well as the zone exhibiting localized PD bifurcation, beyond which PD bifurcation throughout the whole line cycle develops.…”

“…A plethora of work on chaos control of inverter considers linear resistive load, 28 mostly in conjunction with an inductor 29–32 . Rigorous analysis of 1‐D map of a generalized switching circuit with multiple inputs revealed coexisting attractors, nonsmooth PD, and pitchfork bifurcations 33 …”

“…[29][30][31][32] Rigorous analysis of 1-D map of a generalized switching circuit with multiple inputs revealed coexisting attractors, nonsmooth PD, and pitchfork bifurcations. 33 Based on simulation results only, earlier onset of both Hopf and PD bifurcations in two parallel connected inverters with rectifier load when compared to resistive load were reported. 34 Linearized modeling of rectifier load results in an equivalent AC resistance load given by R ac ¼ 8 π 2 R e by its first harmonic approximation.…”

Application of the Filippov Method to PV‐fed DC‐DC converters modeled as hybrid‐DAEs

“…The desired behavior of DC‐DC converters is a stable periodic motion around a predefined value with a frequency that is equal to the external clock. However, as parameters vary, such systems exhibit a wide range of nonlinear behaviors such as subharmonic oscillations 7‐10 . These oscillations can manifest themselves through a series of bifurcations, which can increase the current/voltage ripple, add extra harmonics, and increase the switching losses 10 .…”