Numerical estimation of the lock-in domain of a DC/AC inverter

TL;DR

The paper addresses the transient stability of a grid-tied DC/AC inverter control loop by estimating the lock-in domain for a cascade of a 4D linear current controller and a 2D nonlinear PLL. It introduces a vector Lyapunov framework: a quadratic Lyapunov function for the CC and a PLL Lyapunov function built from limit cycles of a CC‑influenced PLL comparison system, yielding a forward-invariant region whose LaSalle-based convergence to the origin is established. An improved estimation uses a nonincreasing bound Φ that ties the PLL and CC Lyapunov functions, derived via a differential equation dependent on a bound function F; the result is a larger, practically computable lock-in domain. The approach is implemented numerically using DAEs to construct the comparison system and to approximate the PLL’s Lyapunov function, yielding actionable domain estimates in the CC–PLL state space with observable differences when PLL parameters are altered. This framework extends to other cascaded linear-nonlinear systems and offers a robust tool for ensuring favorable transient performance in power-electronics control loops.

Abstract

We estimate the lock-in domain of the origin of a current control system which is used in common DC/AC inverter designs. The system is a cascade connection of a 4-dimensional linear system (current controller, CC) followed by a two-dimensional nonlinear system (phase-locked loop, PLL). For the PLL, we construct a Lyapunov function via numerical approximation of its level curves. In combination with the quadratic Lyapunov function of the CC, it forms a vector Lyapunov function (VLF) for the overall system. A forward-invariant set of the VLF is found via numerical application of the comparison principle. By LaSalle's invariance principle, convergence to the origin is established.

Figures (4)

• Figure 1: Direction of the $V$-comparison system \ref{['eq: comparison']} is leftmost in the range of possible directions of the PLL subsystem \ref{['eq: pll']} under the constraint $V^\mathrm{CC}(x) \leq V$. If a clockwise limit cycle of \ref{['eq: comparison']} bounds a region $\Lambda(V)$ then its boundary $\partial\Lambda(V)$ is crossed by \ref{['eq: pll']} inward.
• Figure 2: Estimations of the lock-in domain provided by Theorems \ref{['th: trivial estimation']} and \ref{['th: main']} are given in the $(V^\mathrm{CC}, V^\mathrm{PLL})$-plane. The comparison inequality \ref{['eq: domain border tangent']} prescribes that the boundary should only be crossed inward, as shown by small black arrows.
• Figure 3: Surface plot and level curves of the Lyapunov function $V^\mathrm{PLL}$ of the PLL subsystem \ref{['eq: pll']} with version I of parameters $k_p$ and $k_i$ in the numerical example.
• Figure 4: Estimation of the lock-in domain of system \ref{['eq: system']} by Theorem \ref{['th: main']} in the numerical example with two versions of the PLL parameters. The hatched part is the trivial estimation by Theorem \ref{['th: trivial estimation']} (almost the same for both versions).

Theorems & Definitions (2)

• Definition 2
• Definition 3