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

Amirhossein Iraniparast, Dominic Groß

Abstract

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.

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

Abstract

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.
Paper Structure (18 sections, 13 theorems, 52 equations, 7 figures, 1 table)

This paper contains 18 sections, 13 theorems, 52 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Network model and preliminaries
  3. Power network and converter model in nodal coordinates and control objectives
  4. Power network and converter model in edge coordinates
  5. Graph of nodes with active constraints
  6. Projection-free power limiting droop control
  7. Review of projection-based network dynamics
  8. Projection-free networked dynamics
  9. Stability of projection-free networked dynamics in edge coordinate
  10. Convergence rate
  11. Preliminaries
  12. Convergence rate
  13. Control tuning
  14. Impact of network topology
  15. Numerical Case Study
  16. ...and 3 more sections

Key Result

Proposition 1

There exists $\theta \in \mathbb{R}^n$ such that $P_{\ell} < L\theta + P_{L} < P_{u}$ if and only if $P_\ell$, $P_u$, and $P_L$ satisfy Assumption assum:feas.

Figures (7)

  • Figure 1: The graph $\mathcal{G}$ and the active graph $\mathcal{G}_{\mathcal{I}}$ corresponding to $L_{\mathcal{I}}$: inactive constraint nodes (blue), active constraint nodes (red), edges connecting active constraint nodes (orange), and edges connecting active and inactive constraint nodes (green).
  • Figure 2: Projection-based power-limiting droop control
  • Figure 3: Projection-free power-limiting droop control
  • Figure 4: Representative different cases of bounds \ref{['eq:beta_constant']} and \ref{['eq:beta_cubic']} indicating the decaying rate $\beta$, $f(\alpha,\beta)$ and $f_2$ are left and righthand side of the bound \ref{['eq:beta_cubic']} respectively, where $r_1 = \arg_{\beta}(f(\alpha,\beta) = f_2)$, $r_2 = \sqrt[3]{\frac{\kappa \alpha}{16}}$, and $r_3 = \frac{\kappa \delta_{\min}}{46\rho L_{g}^2}$.
  • Figure 5: IEEE 9-bus test case system with three two-level voltage source converters and constant impedance (black) and constant power loads (red).
  • ...and 2 more figures

Theorems & Definitions (26)

  • Proposition 1: Feasibility in nodal coordinates IG2025
  • Definition 1: Active constraint sets
  • Definition 2: Graph of nodes with active constraint
  • Proposition 2
  • proof
  • Corollary 1
  • Definition 3: Projection
  • Theorem 1: Coinciding dynamics
  • proof
  • Definition 4: Semi-global exponential stability
  • ...and 16 more