Safety Filter for Limiting the Current of Grid-Forming Matrix Modular Multilevel Converters

TL;DR

The paper tackles current-limiting challenges in grid-forming converters by introducing a modular safety filter based on Control Barrier Functions (CBFs) that enforces current constraints via a real-time Quadratic Program (QP). An offline Sum-of-Squares (SOS) optimization provides a formal safety certificate by synthesizing a CBF $B(x,z)$ and a Control Lyapunov Function $V(x,z)$, with an optional CLF facilitating fast return to nominal operation after grid events. The method is demonstrated with two GFM implementations (Virtual Synchronous Machine and Enhanced Direct Power Control) on a Matrix Modular Multilevel Converter (M3C) in both high- and low-inertia grids, showing effective current limitation, preserved GFM behavior, and smooth recovery. The approach offers a certifiable, modular, and practically deployable solution that reduces tuning burden and enhances reliability under transient grid faults, with potential for broad applicability in modern power systems.

Abstract

Grid-forming (GFM) converters face significant challenges in limiting current during transient grid events while preserving their grid-forming behavior. This paper offers an elegant solution to the problem with a priori guarantees, presenting a safety filter approach based on Control Barrier Functions (CBFs) to enforce current constraints with minimal deviation from the nominal voltage reference. The safety filter is implemented as a Quadratic Program, enabling real-time computation of safe voltage adjustments that ensure smooth transitions and maintain the GFM behavior during nominal operation. To provide formal safety certificate, the CBF is synthesized offline using a Sum-of-Squares optimization framework, ensuring that the converter remains within its allowable operating limits under all conditions. Additionally, a Control Lyapunov Function is incorporated to facilitate a smooth return to the nominal operating region following grid events. The proposed method is modular and can be integrated into many of the GFM control architectures, as demonstrated with two different GFM implementations. High-fidelity simulations conducted with an enhanced matrix modular multilevel converter connected to both high-inertia and low-inertia grid scenarios validate the effectiveness of the safety filter, showing that it successfully limits current during faults, preserves GFM behavior, and ensures a seamless recovery to nominal operation.

Figures (14)

• Figure 1: The GFM converter setup consists of a Voltage Source Converter (VSC) connected to the PCC through a transformer. The grid filter suppresses high-frequency harmonics generated by the modulated converter voltage.
• Figure 2: Two GFM control strategies: (a) the VSM, as defined in \ref{['def:vsm']}, emulates the physical behavior of a SM by dynamically integrating the converter's output power, and (b) the EDPC, as defined in \ref{['def:edpc']}, combines the phase of the PLL with the output of an active power loop. For both the power reference is determined by the inverse frequency droop in \ref{['def:inverse_frequency_droop']}.
• Figure 3: illustrates the implementation of voltage reference limitation using a voltage behind an impedance model. This approach ensures that a current reference amplitude remains below the current threshold, i.e. $\|i_r\| \leq i_\text{th}$.
• Figure 4: Current-limiting control strategies: (a) Switched Current Control (SCC), (b) Reference-Limited Proportional Current Control (RL-CC), and (c) Adaptive Virtual Impedance (AVI).
• Figure 5: The safety filter is a modular component that can be integrated into the existing control hierarchy, ideally without compromising the performance of the legacy controller during nominal operation. State with zero time derivative, referred to as stationary states, are grouped into a separate variable $z$.