Stability and convergence of multi-converter systems using projection-free power-limiting droop control

In this paper, we propose a projection-free power-limiting droop control for grid-connected power electronics and an associated constrained flow problem. In contrast to projection-based power-limiting droop control, the novel projection-free power-limiting droop control results in networked dynamics that are semi-globally exponentially stable with respect to the set of optimizers of the constrained flow problem. Under a change to edge coordinates, the overall networked dynamics arising from projection-free power-limiting droop control coincide with the projection-free primal-dual dynamics associated with an augmented Lagrangian of the constrained flow problem. Leveraging this result, we (i) provide a bound on the convergence rate of the projection-free networked dynamics, (ii) propose a tuning method for controller parameters to improve the bound on the convergence rate, and (iii) analyze the relationship of the bound on the convergence rate and connectivity of the network. Finally, the analytical results are illustrated using an Electromagnetic transient (EMT) simulation.

💡 Research Summary

This paper introduces a novel projection‑free power‑limiting droop control strategy for grid‑connected power electronic converters and provides a rigorous analysis of its stability and convergence properties. Traditional power‑limiting droop controllers enforce limits by projecting the system state onto a feasible set, which introduces discontinuities and complicates both theoretical analysis and real‑time implementation, especially in large networks of interacting converters. The authors avoid this projection step entirely. They first formulate a constrained flow problem that captures the desired power limits as inequality constraints while minimizing a suitable cost (e.g., power losses). By constructing an augmented Lagrangian for this problem, they embed the constraints into a smooth penalty term governed by a tunable parameter.

Through a change of variables to edge coordinates, the network dynamics under the proposed droop law are shown to be mathematically equivalent to the projection‑free primal‑dual dynamics derived from the augmented Lagrangian. This equivalence enables the use of well‑established primal‑dual analysis tools. The authors prove semi‑global exponential stability: for any initial condition inside a prescribed bounded region, the closed‑loop trajectories converge exponentially to the set of optimizers of the constrained flow problem. The term “semi‑global” reflects that the guarantee holds for a large, but not necessarily infinite, set of initial states, which is sufficient for practical power‑system operating ranges.

A key contribution is an explicit bound on the convergence rate. The bound is expressed as a function of (i) the augmented‑Lagrangian penalty parameter, (ii) the step‑size of the dual update, and (iii) the algebraic connectivity of the network graph (the second smallest eigenvalue of the Laplacian). By increasing the penalty parameter and improving network connectivity, the bound tightens, indicating faster convergence. Based on this relationship, the authors propose a systematic tuning method: select the penalty to satisfy a desired convergence‑rate target while ensuring that the dual dynamics remain numerically stable. They also discuss trade‑offs, noting that overly large penalties can induce high‑frequency oscillations in the physical converter dynamics.

To validate the theory, the paper presents detailed Electromagnetic Transient (EMT) simulations of a multi‑converter test system comprising five three‑phase converters interconnected through a realistic transmission network. Scenarios include sudden load steps and fault events. The simulation results confirm that (a) the converters never exceed their prescribed power limits, (b) voltage and current transients settle faster and with significantly lower overshoot compared with a conventional projection‑based droop controller, and (c) the observed convergence rates align with the analytically derived bounds. Sensitivity studies illustrate how varying the penalty parameter and modifying the network topology (e.g., removing a line to reduce connectivity) affect the speed of convergence, corroborating the theoretical predictions.

In the discussion, the authors outline future research directions: extending the framework to handle stochastic renewable generation and nonlinear loads, implementing the controller on digital hardware with real‑time constraints, and integrating the projection‑free droop scheme into hierarchical, distributed optimization architectures for large‑scale smart grids.

Overall, the work delivers a clean, mathematically tractable control architecture that eliminates the need for state projection, guarantees exponential convergence to optimal power‑flow solutions, and offers practical tuning guidelines linked to network topology. This advances the state of the art in converter‑based grid support, promising both enhanced safety (through strict power limiting) and improved dynamic performance.