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Convexity and Optimal Online Control of Grid-Interfacing Converters with Current Limits

Lauren Streitmatter, Trager Joswig-Jones, Baosen Zhang

Abstract

Converter-based generators and loads are growing in prevalence on power grids across the globe. The rise of these resources necessitates controllers that handle the power electronic devices' strict current limits without jeopardizing stability or overly constraining behavior. Existing controllers often employ complex, cascaded control loop architecture to saturate currents, but these controllers are challenging to tune properly and can destabilize following large disturbances. In this paper, we extend previous analysis to prove the feasible output region of a grid-connected converter is convex regardless of filter topology. We then formulate a convex optimal control problem from which we derive a projected gradient descent-based controller with convergence guarantees. This approach drives the converter toward optimality in real-time and differs from conventional control strategies that regulate converter outputs around predefined references regardless of surrounding grid conditions. Simulation results demonstrate safe and stabilizing behavior of the proposed controller, in both the single-converter-infinite-bus systems and multi-converter networks.

Convexity and Optimal Online Control of Grid-Interfacing Converters with Current Limits

Abstract

Converter-based generators and loads are growing in prevalence on power grids across the globe. The rise of these resources necessitates controllers that handle the power electronic devices' strict current limits without jeopardizing stability or overly constraining behavior. Existing controllers often employ complex, cascaded control loop architecture to saturate currents, but these controllers are challenging to tune properly and can destabilize following large disturbances. In this paper, we extend previous analysis to prove the feasible output region of a grid-connected converter is convex regardless of filter topology. We then formulate a convex optimal control problem from which we derive a projected gradient descent-based controller with convergence guarantees. This approach drives the converter toward optimality in real-time and differs from conventional control strategies that regulate converter outputs around predefined references regardless of surrounding grid conditions. Simulation results demonstrate safe and stabilizing behavior of the proposed controller, in both the single-converter-infinite-bus systems and multi-converter networks.
Paper Structure (19 sections, 3 theorems, 34 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 3 theorems, 34 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and Problem Formulation
  3. Converter Model
  4. Optimal Control Problem
  5. $P,Q$ Tracking
  6. $P,V^2$ Tracking
  7. $Q,V^2$ Tracking
  8. Relation to grid-forming control
  9. Convexity of the Feasible Region
  10. Dynamic Controller
  11. Semi-Definite Programming Formulation
  12. Extract $\mathbf{I}_{\mathrm{dq}}$ from high rank $\mathbf W$
  13. Modified Projected Gradient Descent of the SDP
  14. Proof of Theorem \ref{['thm:algo_convergence']}
  15. Simulation Studies
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(S_1,S_2)$ be a pair of points formed by choosing any two of the three quantities $P,Q,{V}_\mathrm{dq}^2$. Let $\mathcal{S} \in \mathbb{R}^2$ be the set of all achievable points $(S_1,S_2)$ by $\mathbf{I}_{\mathrm{dq}} \in \mathcal{I}$, where $\mathbf{I}_{\mathrm{dq}}$ is the current of a conve

Figures (11)

  • Figure 1: We model an inverter (though results hold if the inverter is replaced by a rectifier) as a controllable voltage source in the $\mathrm{dq}$ reference frame connected through an $RLC$ filter and an $RL$ line to an infinite bus.
  • Figure 2: We can instead represent the network using the equivalent impedance model of the filter and line impedance, $Z_{eq}=R_{eq} + j \omega L_{eq}$. Depending on the filter parameters, $L_{eq}$ may be positive (inductive) or negative (capacitive).
  • Figure 3: Limiting the current magnitude forms the safe set $\mathcal{I}$ (top left) which is a circle of radius $I_\mathrm{max}$. We transform the set of safe currents to the $\mathcal{S}=(P,Q)$ region (top right), $(P,{V}_\mathrm{dq}^2)$ region (bottom left), and $(Q,{V}_\mathrm{dq}^2)$ region (bottom right) for the extreme case of a converter network with capacitive reactance ($L_{eq}<0$), and we observe convexity is still preserved.
  • Figure 4: Comparisons between the optimal controller (blue) and droop (orange) inverters' $(P,V^2)$ trajectories (top) and current magnitude responses over time (bottom) following a setpoint change at $t_0=0.05s$ are shown above. While neither controller violates current magnitude constraints, the droop controller fails to converge.
  • Figure 5: The dashed blue line is the OC trajectory considering inner voltage and current loop dynamics compared to the OC trajectory of Figure \ref{['fig:compare_pv2_setpoint_change']} shown again in solid blue for reference. Inner loop dynamics introduce noise but maintain overall trajectory and convergence behavior, validating the assumption that these dynamics can be neglected with sufficient timescale separation.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof