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$H_{\infty}$-Control of Grid-Connected Converters: Design, Objectives and Decentralized Stability Certificates

Linbin Huang, Huanhai Xin, Florian Dörfler

arXiv:1906.11331v2eess.SYcs.SY

TL;DR

Converter control design relies on single-converter analysis and engineering experience, while existing optimal designs do not guarantee multi-converter stability. The paper proposes a fixed-structure H∞ framework and a decentralized small-gain stability criterion; simulations and HIL results demonstrate the design's performance and stability in the studied systems.

Abstract

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The modern power system features high penetration of power converters due to the development of renewables, HVDC, etc. Currently, the controller design and parameter tuning of power converters heavily rely on rich engineering experience and extrapolation from a single converter system, which may lead to inferior performance or even instabilities under variable grid conditions. In this paper, we propose an $H_{\infty}$-control design framework to provide a systematic way for the robust and optimal control design of power converters. We discuss how to choose weighting functions to achieve anticipated and robust performance with regards to multiple control objectives. Further, we show that by a proper choice of the weighting functions, the converter can be conveniently specified as grid-forming or grid-following in terms of small-signal dynamics. Moreover, this paper first proposes a decentralized stability criterion based on the small gain theorem, which enables us to guarantee the global small-signal stability of a multi-converter system through local control design of the power converters. We provide high-fidelity nonlinear simulations and hardware-in-the-loop (HIL) real-time simulations to illustrate the effectiveness of our method.

I. INTRODUCTION

Converter control and tuning rely heavily on single-converter analysis and engineering experience, despite coupled dynamics that complicate robust, optimal design. The paper presents a fixed-structure H∞ framework for grid-connected converters and describes the converter model it uses.

I. INTRODUCTION

  • Coupled control loops complicate tuning, while conventional parameter searches may not yield optimal damping or stability margins and do not guarantee multi-converter stability.Existing H2/H∞ designs also omit stable synchronization under variable grid conditions.Source: §I · pp. 1–2
  • Prior H∞ converter studies address applications including microgrids, voltage control, current control, and harmonic suppression, but some use simplified models without current control loops.Dynamic synthesized controllers can have the same order as the system, further complicating its dynamics.Source: §I · p. 1
  • The proposed framework uses fixed-structure H∞ control to address multiple objectives, including grid synchronization, active-power regulation, and voltage regulation.The paper also proposes weighting-function choices for grid-forming or grid-following small-signal behavior and a decentralized criterion for multi-converter stability.Source: §I · p. 2

A. Converter system descriptions

  • The converter model uses a three-phase grid connection through an LCL filter, with signals represented in a synchronously rotating dq frame.Its converter admittance matrix maps terminal-voltage perturbations to current output.Source: A. Converter system descriptions · p. 2

B. H∞-control design setup

The design setup defines disturbances, performance outputs, and controller signals to encode converter objectives in an H∞ model. Its structure encompasses PLL-based and frequency-droop controllers as special cases.

B. H∞-control design setup

  • The disturbance inputs encode power and voltage-reference changes, admittance-related signals for decentralized certification, and angle disturbances for grid-synchronization performance.The performance outputs quantify tracking and disturbance rejection for voltage, power, and angle signals.Source: B. H∞-control design setup · pp. 2–3
  • The setup uses a static parameter matrix K and allows auxiliary loops to be incorporated by modifying references or the current-control loop.Integrals in the measured-output vector determine steady-state regulation; voltage and active power are regulated to their references.Source: B. H∞-control design setup · pp. 3–4

Two Noteworthy Cases

  • The controller structure includes a PLL-based grid-following controller and a frequency-droop grid-forming controller as special cases.Their respective settings use PLL PI gains and a frequency-droop coefficient.Source: Two Noteworthy Cases · p. 4

C. H∞-control formulation

The formulation minimizes a weighted H∞ objective, but standard output-based weighting cannot independently shape every input-to-output response. The paper uses fixed static gains and entrywise weighting to support multi-objective design.

C. H∞-control formulation

  • The standard objective seeks a stabilizing controller minimizing the H∞ norm of the weighted closed-loop transfer matrix, with user-defined weights and the largest singular value.The controller design affects the closed-loop performance represented by that matrix.Source: C. H∞-control formulation · p. 4
  • Standard H∞ weighting assigns the same weight to all inputs reaching a given output, limiting simultaneous tracking and disturbance-rejection objectives.Standard synthesis can also produce a high-dimensional dynamic controller with system-order complexity.Source: C. H∞-control formulation · p. 4
  • The proposed formulation uses fixed static gains and entrywise weighting functions to specify the shape of each closed-loop transfer-matrix entry.The weighting matrix need not be diagonal and has the same dimensions as the transfer matrix.Source: C. H∞-control formulation · p. 4

D. Grid-forming and grid-following dynamics

The paper compares grid-frequency disturbance responses under droop and PLL-based control. The PLL-based controller has higher bandwidth, with faster tracking but greater sensitivity to grid disturbances.

D. Grid-forming and grid-following dynamics

  • The PLL-based controller has higher bandwidth than the droop controller, indicating faster tracking and greater sensitivity to grid disturbances.The comparison uses the closed-loop sensitivity P77(K)(s) for angle disturbances from the grid.Source: D. Grid-forming and grid-following… · p. 4

III. CONTROL OBJECTIVES AND WEIGHTING FUNCTIONS

The section explains how H∞ weighting functions shape tracking and disturbance rejection, then specifies weights for synchronization, power, voltage, and admittance objectives. Different synchronization weights can produce grid-forming or grid-following small-signal dynamics.

A. Brief review on weighting functions for H∞-control

  • For tracking, shape the inverse weight like complementary sensitivity and set its bandwidth to specify tracking speed; for disturbance rejection, use sensitivity-function shape.The weighting matrix entry Wij(s) shapes the transfer from input j to output i.Source: §III · p. 4, A. Brief review on weighting functions… · p. 5

B. Specification of the weighting functions for converters

  • W77(s) shapes synchronization disturbance rejection; its low-frequency-only high gain yields grid-forming small-signal dynamics, while different weights can specify grid-following dynamics.The selected weight limits the bandwidth of P77(K)(s).Source: B. Specification of the weighting… · p. 5
  • Power weights target fast active-power reference tracking and disturbance rejection, while voltage weights target fast d- and q-axis reference tracking.The power weighting also accounts for reactive-power effects and reduces emphasis near the LCL resonance frequency.Source: B. Specification of the weighting… · pp. 5–6
  • Admittance weights emphasize shaping G(s)Y(s) over 0–100 Hz while placing less emphasis on its high-frequency characteristics.The other entries of W(s) are set to zero so the design focuses on the specified objectives.Source: B. Specification of the weighting… · p. 6

C. Constant reactive power control mode

The paper distinguishes PV mode, which regulates active power and internal voltage magnitude, from PQ mode, which directly regulates active and reactive power.

C. Constant reactive power control mode

  • In PV mode, the converter regulates active power and internal voltage magnitude to their references, with reactive power controlled indirectly through the internal voltage reference.The internal voltage magnitude is measured before the virtual impedance.Source: C. Constant reactive power control mode · p. 6
  • PQ mode directly regulates active and reactive power to their references by changing the reactive-power tracking signal and associated weights.The design removes voltage-magnitude tracking and configures reactive-power tracking and disturbance rejection analogously to active-power regulation.Source: C. Constant reactive power control mode · p. 6

IV. DECENTRALIZED H∞STABILITY CERTIFICATES

The section presents a decentralized H∞ stability criterion and incorporates it into the control framework to ensure multi-converter stability through local design.

IV. DECENTRALIZED H∞STABILITY CERTIFICATES

  • The proposed decentralized H∞ stability criterion is incorporated into the control framework to ensure multi-converter stability through local design.Source: §IV · p. 6

A. Multi-device system descriptions

The system description models devices by their admittances and the network by a Kron-reduced admittance structure. This separation provides the structure used for decentralized stability certificates.

A. Multi-device system descriptions

  • The model considers a Kron-reduced network interconnecting n devices, with interior nodes eliminated under the assumption that loads are constant current sources.Source: A. Multi-device system descriptions · p. 6
  • The network lines are assumed to have identical R/L ratios, and Qred encodes topology and susceptances; Qred⊗F(s) represents the network admittance.The grounded node represents the infinite bus in the small-signal model.Source: A. Multi-device system descriptions · pp. 6–7
  • The closed-loop model separates block-diagonal device admittances from the network admittance, a structure that enables decentralized stability certificates.The illustrated three-converter system is reduced to a three-node network by eliminating interior nodes.Source: A. Multi-device system descriptions · p. 7

B. Decentralized H∞stability criterion

The small-gain criterion evaluates multi-device stability using each device’s local dynamics and the network structure. Stability is guaranteed when every device’s weighted admittance norm is below the network’s smallest reduced-Laplacian eigenvalue.

B. Decentralized H∞stability criterion

  • The multi-device system is stable if each device’s ∥F^-1(s)Yi(s)∥∞ is below λ1, the smallest eigenvalue of Qred.The criterion partitions the system into network and device parts and gives a decentralized stability test.Source: B. Decentralized H∞stability criterion · p. 7
  • F^-1(s)Yi(s) captures device i’s local frequency-domain dynamics, while λ1 reflects network connectivity and increases as the network becomes denser.The network can satisfy the condition by becoming sufficiently dense, but the paper notes that building more transmission lines is impractical and costly.Source: B. Decentralized H∞stability criterion · pp. 7–8
  • Reducing each device’s ∥F^-1(s)Yi(s)∥∞ below λ1 can guarantee system stability without changing the network structure.Choosing G(s)=F^-1(s) lets the proposed H∞ design reduce this norm.Source: B. Decentralized H∞stability criterion · p. 8

C. H∞-design with decentralized stability certificates

The design incorporates grid impedance into the device model and seeks robustness across a set of grid inductances. With the resulting controller, the reported weighted-admittance singular values remain bounded across the tested inductances.

C. H∞-design with decentralized stability certificates

  • Including grid impedance Lg in the device part of the partition increases λ1 of the network part and reduces conservativeness of the stability condition.Lg includes the grid-side LCL inductor and line inductance, so it need not be known exactly.Source: C. H∞-design with decentralized… · p. 8
  • The H∞-optimal control problem is formulated for L in {0.05x: x ∈ {1, ..., 10}} to ensure robustness against varying grid inductances.The paper uses hinfstruct in MATLAB to solve the problem and obtain a static gain matrix.Source: C. H∞-design with decentralized… · p. 8
  • For Lg = 0.2, 0.35, and 0.5 p.u., σ1(jω) stays below 10 with no unexpected resonance peak.The plotted singular values are for F^-1(jω)Y(jω) with the H∞-optimal controller.Source: C. H∞-design with decentralized… · p. 8, Figure 6 · p. 8
  • Below 100 Hz, σ1(jω) has lower magnitude, which the authors say enhances robustness and prevents sub-synchronous oscillations.The lower magnitude is attributed to the weighting-function choice in (16).Source: C. H∞-design with decentralized… · p. 8

V. SIMULATION RESULTS

Nonlinear simulations test the H∞-optimal controller on single- and three-converter systems under grid changes and disturbances. The single-converter results report similar power tracking across grid inductances, anticipated responses from weighting-function choices, and acceptable transients under severe disturbances.

A. Single-converter system

  • With different grid inductances, active power has nearly the same fast response with no overshoot.The internal frequency also responds quickly, and the authors report anticipated performance from the weighting functions.Source: A. Single-converter system · pp. 8–9
  • Increasing the bandwidths of W83(s) and W53(s) leads to faster active-power responses, with all tested responses showing no overshoot.The weighting functions are adjusted to change tracking speed rather than directly changing control parameters.Source: A. Single-converter system · p. 9, Figure 8 · p. 9
  • The controller has acceptable transient performance under the tested severe disturbances, without additional control switching or auxiliary loops.Tests include a grid-voltage drop and recovery, grid disconnection and reconnection with a local load, and a stepped internal voltage reference.Source: A. Single-converter system · p. 9

V. SIMULATION RESULTS

  • The three-converter simulation steps the converters’ active-power references at 1, 2, and 3 s, then changes their grid inductances at 4, 5, and 6 s.The inductances change from 0.2 p.u. to 0.4, 0.3, and 0.5 p.u., respectively.Source: Figure 10 · p. 9

B. Three-converter system

The H∞-optimal controller outperformed droop and PLL-based controllers in the three-converter study, while a decentralized stability condition was close to the observed stability boundary. HIL simulations showed fast, smooth responses to reference changes and mode switching.

B. Three-converter system

  • At Lg1=Lg2=Lg3=0.5 p.u., the sufficient condition gives stability for λ1>6.17 (15.8 dB), near the observed boundary of about λ1=5.The authors describe the sufficient decentralized condition as remarkably tight for this case study; its conservativeness provides a robustness margin for model uncertainties.Source: B. Three-converter system · p. 10
  • The H∞-optimal controller had fast dynamics without overshoot; droop was slower with overshoots, while the PLL-based system became unstable when the grid weakened.The PLL-based system became unstable when Lg1 stepped from 0.2 p.u. to 0.4 p.u.Source: B. Three-converter system · p. 10

C. HIL real-time simulation results

  • HIL responses tracked active and reactive power references with rise times below 0.1 s and no overshoot, and mode switching was fast and smooth.The FPGA simulated the three-converter power system in real time with a step size below 0.5 µs; Converter 1’s three-phase voltages remained well maintained under disturbances.Source: C. HIL real-time simulation results · p. 10, Figure 12 · p. 10, Figure 13 · p. 11

VI. CONCLUSIONS

  • The framework uses weighting functions to specify grid-forming or grid-following small-signal dynamics and incorporates a decentralized stability criterion into H∞ design for multi-converter systems.The paper reports testing the resulting controller through detailed simulations and HIL implementation of a three-converter system.Source: §VI · p. 11
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