Control Design and Stability Analysis of Power Converters: The MIMO Generalized Bode Criterion

Abstract

Three-phase dynamic systems and multiphase generators are frequently modeled and controlled in the synchronous reference frame. To properly model the cross-coupling terms in this reference frame, complex vector theory and transfer function matrices are commonly applied, obtaining multipleinput multiple output (MIMO) dynamic models. The stability of MIMO systems can be assessed through the Nyquist Generalized Stability Criterion. However, the use of the Nyquist diagram complicates the controller design. The Bode diagram is a more intuitive tool for the controller design, however, the Bode Stability Criterion is not applicable to MIMO systems. In this paper, the MIMO Generalized Bode Criterion is proposed. Since this stability criterion is based on the Nyquist Generalized Stability Criterion it can be applied to any system. Furthermore, it is simple to use, as it only requires information contained in the open-loop transfer matrix and the Bode diagram. The proposed stability criterion thus offers an interesting tool for the controller design procedure in MIMO systems, as it is shown in in this paper for two common applications: the current control loop of a power converter, a 2×2 system, and the current control loop of two independent power converters in parallel, a 4×4 system.

Control Design and Stability Analysis of Power Converters: The Discrete Generalized Bode Criterion

“…When there are poles at z = −1, L(z) cannot be evaluated at ω = ω N = π/T s , and the original Nyquist contour z = exp(jωT s ) must be modified [17], [58]. In order to exclude these poles, the contour is indented around z = −1 as z = −1 + ε•exp(jϕ), with ε positive and sufficiently small, and ϕ increasing from π/2 to 3π/2.…”

Sub-Synchronous Resonance Damping Control Strategy for DFIG Wind Turbines

“…Impedance matrices are used to model the different components, as they are a straightforward approach to model the cross-coupling terms between both axis. Moreover, this modeling methodology can be used for symmetric and non-symmetric systems [20], [23], as the ones obtained when the power control loops and the PLL effect cannot be neglected for the stability analysis.…”

“…To analyze the system stability, the MIMO Generalized Bode Criterion (MIMO GBC) is applied [23]. According to this criterion…”

“…Exploiting the benefits of transfer function matrices, in this paper a systematic modeling methodology is presented for DFIG wind turbines, which can be easily adapted for different wind turbine topologies and other case studies. This modeling methodology is combined with the frequency response analysis and its various tools, which have not been fully exploited, as the Bode diagram, for instance, that can help to understand the system dynamics and in the design of controllers and resonance damping control strategies [23].…”

“…A model for the phase-locked loop (PLL), using transfer function matrices is also developed to evaluate the PLL influence on the overall system stability. The stability of the derived MIMO model is analyzed by means of the Bode diagram and the MIMO Generalized Bode Criterion (MIMO GBC) [23]. This stability criterion is based on the open-loop frequency response analysis, different to the conventional approach in SSR literature focused on the closed-loop system eigenvalues.…”