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Reshaping Converter-Network Interactions in Microgrids: From Virtual Impedance to Virtual Two-Port Control

Liaoyuan Yang, Peng Yang, Yang Wu, Feng Liu

arXiv:2609.27223v1eess.SYcs.SY

TL;DR

Converter terminal characteristics shape both dynamic interactions and steady-state power sharing, but additive virtual impedance uses a single series-like action. The paper introduces a four-channel virtual two-port impedance transformation and reports advantages over virtual impedance across passivation, uncertainty compression, and power-flow regulation applications.

  • Problem

    Additive virtual impedance applies the same series-like branch across impedance variations, coupling uncertainty accommodation and power-flow correction to a single action.

    Source: §I · p. 1

  • Method

    Four coordinated voltage-current channels reconstruct the converter terminal relation through a virtual two-port, with virtual impedance as a degenerate case.

    Source: §I · pp. 1–2

  • Results

    The paper reports lower controller coefficient gain for prescribed passivity, compressed uncertainty-induced terminal variation, and improved power-sharing accuracy with smaller voltage drops.

    Source: §I · p. 2, §VI · p. 10

  • Takeaways & Limitations

    The same port transformation is presented for dynamic interaction shaping and steady-state network regulation in inverter-based microgrids.

    Source: §I · p. 2

Abstract

from arXiv · show

Converter terminal characteristics are central to both dynamic interactions with the network and steady-state power sharing in inverter-based microgrids. Conventional additive virtual impedance (VI) shapes these characteristics through a single virtual branch. This paper proposes virtual two-port control, which uses four coordinated transfer channels to reconstruct the terminal impedance. The connected network consequently observes the original converter through a virtually inserted twoport interface, with conventional virtual impedance recovered as a degenerate case. The proposed control transforms the original impedance through a matrix linear-fractional map. We also derive a necessary and sufficient condition for the reconstruction to be stable and proper. Among its various potential applications in microgrids, we develop three representative ones in detail: passivation with reduced control effort, uncertainty compression, and seriesshunt power-flow regulation. Simulations of VSG-controlled converters demonstrate the advantages of the proposed method over conventional VI in these applications.

I. INTRODUCTION

Converter terminal voltage-current relations shape both dynamic network interactions and steady-state power sharing. The paper generalizes additive virtual impedance into a four-channel virtual two-port interface for reshaping that relation, with three applications spanning interaction shaping and network regulation.

I. INTRODUCTION

  • The terminal relation governs both converter-network interactions and power sharing, making it a local shaping target without redesigning the entire microgrid controller.The paper frames these dynamic and steady-state viewpoints as the same design question: how to reshape the converter terminal relation.Source: §I · p. 1
  • Fixed additive virtual impedance can couple uncertainty accommodation and power-flow correction to one series-like action, constraining reliably realizable terminal behavior.The passage notes that current limits, saturation, parameter variation, delays, and operating conditions affect the implemented behavior or the impedance family.Source: §I · p. 1

II. PROPOSED VIRTUAL TWO-PORT CONTROL

  • The four-block control inserts a virtual two-port between the original converter and network, reconstructing the external terminal relation while recovering virtual impedance as a special case.The reconstructed impedance is a matrix linear-fractional transformation; stable-proper conditions are also established.Source: §I · pp. 1–2, A. Control Structure · pp. 2–3

I. INTRODUCTION

  • Three applications use the same port transformation for reduced-gain passivation, compression of uncertainty-induced impedance variation, and series-shunt power-flow regulation.The paper reports lower controller coefficient gain for a prescribed passivity requirement, reduced terminal variation, and improved power-sharing accuracy with smaller voltage drops.Source: §I · p. 2

A. Control Structure

  • Unlike additive virtual impedance’s single series-like action, coordinated voltage and current channels can reshape both transfers between converter and network ports.The channels include current-dependent voltage injection, voltage transfer, current transfer, and voltage-dependent current injection.Source: §I · p. 1, A. Control Structure · p. 2

B. Linear-Fractional Impedance Map

The four control blocks transform the original terminal impedance through a matrix linear-fractional map. Its return inverse determines whether the reconstructed impedance is stable and proper.

B. Linear-Fractional Impedance Map

  • The reconstructed impedance is a matrix linear-fractional transformation, with A + BZ as its forward term, D in the return denominator, and C postprocessing the current.The map is eZ = (A + BZ)[I − D(A + BZ)]^-1C.Source: B. Linear-Fractional Impedance Map · p. 3
  • Stable and proper control blocks alone do not ensure a stable and proper reconstructed impedance; the return inverse must also satisfy the stated conditions.The additional finite poles are governed by zeros of det[I − D(s)H(s)], where H = A + BZ; ∥DH∥∞ < 1 is a sufficient condition.Source: B. Linear-Fractional Impedance Map · p. 3
  • When D = 0, the impedance map becomes affine and introduces no return poles; choosing A = Zv and B = C = I recovers additive virtual impedance.That specialization gives eZ = Z + Zv.Source: B. Linear-Fractional Impedance Map · p. 3

C. Reciprocal Π-Network Realization

A reciprocal subclass of the virtual two-port admits a Π-network interpretation, including a series-shunt form that adds a shunt degree of freedom beyond virtual impedance. The section then frames passivation as a common-margin comparison and establishes the gain burden of the virtual-impedance benchmark.

C. Reciprocal Π-Network Realization

  • A reciprocal subclass of the general four-block control has a direct virtual Π-network interpretation, although the general construction need not admit a circuit realization.The Π network comprises converter-side shunt, series, and line-side shunt impedances.Source: C. Reciprocal Π-Network Realization · p. 3, Figure 3 · p. 3
  • With the converter-side shunt open, the series branch reconstructs terminal voltage and the line-side shunt reconstructs current, while setting the shunt to zero recovers virtual impedance.This series-shunt specialization is used for network-level voltage and current coordination; Applications 1 and 2 instead use unrestricted reconstruction for dynamic impedance shaping.Source: C. Reciprocal Π-Network Realization · p. 4

A. Design Objective

  • The passivation comparison requires virtual impedance and virtual two-port control to achieve the same prescribed margin over the same frequency set, using a common coefficient-magnitude measure.For virtual impedance, the gain measure reduces to ∥Zv∥∞.Source: A. Design Objective · p. 4

B. VI Benchmark: Minimum Passivation Gain

  • For additive virtual impedance, the passivity shortage ε − µ0 appears directly as the minimum coefficient gain, so a larger shortage raises the implementation burden.The passage connects larger coefficients with larger inner-loop commands that can activate converter limits.Source: B. VI Benchmark: Minimum Passivation… · p. 4

C. Two-Port Design with Reduced Coefficient Gain

The two-port can meet the same prescribed passivity margin as VI with lower coefficient gain when the original passivity shortage is sufficiently large. In the case study, its lower command demand avoids VSG 1 saturation and is associated with smaller PCC-voltage and neighboring-VSG frequency excursions.

C. Two-Port Design with Reduced Coefficient Gain

  • The two-port realization uses A = (ε − ρµ0)I, B = ρI, C = I, and D = 0 to produce the target margin with adjustable scaling 0 < ρ < 1.The boundary choice ρ = 1 recovers the minimum-gain VI realization.Source: C. Two-Port Design with Reduced… · p. 4
  • Under the theorem conditions, the two-port meets margin ε with coefficient gain below the minimum VI gain when the original passivity shortage is sufficiently large.As the original worst-case passivity margin becomes strongly negative, the two-port gain approaches a fraction ρ of the minimum required by VI.Source: C. Two-Port Design with Reduced… · p. 4

D. Case Study

  • In the 0.4-Hz test, two-port peaks of 0.46658 pu current reference and 1.17846 pu converter voltage activate neither limit, unlike VI.VI limits are active for 1.67% and 5.09% of the disturbance interval; its limited and unlimited responses separate by up to 0.01684 pu in PCC voltage and 0.16957 Hz in VSG 2 frequency, while the two-port responses coincide.Source: Figure 6 · p. 5, D. Case Study · p. 6
  • Under the same disturbance, minimum PCC voltage is 0.99168 pu with two-port control versus 0.97874 pu with VI, and peak absolute VSG 2-frequency deviation is 0.22592 Hz versus 0.30823 Hz.VSG 2 remains unsaturated under both strategies; the paired limited and unlimited simulations isolate the effect of VSG 1's limits.Source: Figure 6 · p. 5, D. Case Study · p. 6

IV. APPLICATION 2: UNCERTAINTY COMPRESSION

The section measures how converter-parameter uncertainty appears in the network-facing passivity-margin domain. Fixed VI can shift that domain, but its additive structure preserves the family's spread.

A. Design Objective

  • The motivating design question is whether one fixed terminal controller can reduce the external interaction characteristic's sensitivity to internal converter uncertainty.The paper considers a scalar parameter varying continuously over a closed interval.Source: A. Design Objective · p. 6
  • The passivity-margin interval width measures how strongly converter uncertainty is reflected in the selected terminal-interaction measure.A smaller spread means the observable is less sensitive to the underlying parameter uncertainty.Source: A. Design Objective · p. 6

B. VI Control: Translation Without Compression

  • A fixed VI can raise the family's passivity margin to a desired lower bound, but its additive translation preserves every pairwise impedance difference.For isotropic VI, every family member receives the same scalar margin shift, so the passivity-margin spread remains unchanged.Source: B. VI Control: Translation Without… · p. 6

C. Two-Port Design: Uncertainty-Domain Compression

The virtual two-port's linear-fractional map can reshape both the location and diameter of an uncertainty domain, unlike fixed isotropic VI, which preserves its spread. In the case study, it reduces the passivity-margin spread from 67.436 pu to 0.321 pu while also using substantially smaller coefficient gain than VI.

C. Two-Port Design: Uncertainty-Domain Compression

  • The two-port map can reshape both the location and diameter of the mapped uncertainty domain, whereas fixed isotropic VI preserves its spread.This provides a way to reduce how strongly internal converter uncertainty appears in the selected network-facing interaction characteristic.Source: C. Two-Port Design: Uncertainty-Domain… · p. 7
  • For any lower bound m > 0 and diameter bound Δ > 0, the theorem gives a stable, proper two-port reconstruction with m ≤ ηTP(ω, p) ≤ m + Δ across the parameter and frequency sets.Thus m places the compressed domain in the passive region, while Δ independently bounds its dispersion.Source: C. Two-Port Design: Uncertainty-Domain… · p. 7

D. Case Study

  • The two-port reduces passivity-margin spread from 67.436 pu to 0.321 pu, while VI leaves it at 67.436 pu.The two-port's minimum passivity margin is 1.679 pu, below neither the prescribed lower target m = 4/3 nor the spread bound Δ = 1.667 pu.Source: D. Case Study · p. 7
  • In the case study, the two-port combines substantial uncertainty-domain compression with a considerably smaller coefficient gain than VI, whose coefficient gain is 237.17105.Source: D. Case Study · p. 7

V. APPLICATION 3: FLEXIBLE POWER-FLOW REGULATION

Series-only VI changes the controlled voltage through a series drop, while series–shunt two-port control adds a shunt path and expands algebraic voltage reachability. In the asymmetric-ring case, that added freedom reshapes transfer terms and improves reactive-power sharing with a smaller voltage drop than VI.

A. Design Objective

  • In high-R/X networks or with unequal feeders, stronger VI-based sharing correction generally requires a larger series voltage drop, limiting regulation flexibility.The application tests whether the two-port’s additional degree of freedom can address this limitation.Source: A. Design Objective · p. 7

B. VI Control: Series-Only Compensation

  • Series VI reaches only controlled voltages satisfying its nonnegative-resistance condition; points outside that set require a negative-resistance series branch.The point vm = vp is included with zero virtual impedance.Source: B. VI Control: Series-Only Compensation · p. 8

C. Two-Port Control: Series-Shunt Compensation

  • Unlike VI’s series-only voltage drop, the two-port coordinates series and line-side shunt branches to extend power-flow regulation.The series branch acts through vm − vp, while the shunt branch acts directly at vm.Source: C. Two-Port Control: Series-Shunt… · p. 8
  • Series-shunt compensation reaches all evaluated voltages in the stated annulus except a one-dimensional collinear segment, while series VI reaches 94.03%.The comparison assumes positive branch resistance; practical voltage/current commands, modulation headroom, and bandwidth further restrict usable operating points.Source: C. Two-Port Control: Series-Shunt… · p. 8, Figure 9 · p. 8

D. Case Study

  • The case-study objective is to equalize source self and source-to-load transfer terms, making the network invariant to interchange of the identical source ports.Series VI changes the source self terms but retains the transfer terms, leaving a finite mirror residual; the series-shunt realization can modify both.Source: D. Case Study · p. 9
  • The two-port reduces the maximum PCC reactive-power increment mismatch to 0.0172 pu, versus 0.0645 pu with VI and 0.115 pu in the original network.Network asymmetry εsym falls to 1.88 × 10−16 with the two-port, compared with 0.239 with VI; minimum load-bus voltage is 0.957 pu with the two-port and 0.950 pu with VI.Source: Table III · p. 9, Figure 11 · p. 9, D. Case Study · pp. 9–10

VI. CONCLUSION

The conclusion presents virtual two-port control as a four-channel way to reshape converter terminal characteristics, with VI as a degenerate case. It reports benefits across passivation, uncertainty compression, and power-flow regulation, alongside a stable-properness condition.

VI. CONCLUSION

  • Four coordinated control channels reconstruct terminal voltage and current, yielding a matrix linear-fractional impedance map with VI as a degenerate case.The paper derives a return-inverse criterion for stability and properness of the reconstructed impedance.Source: §VI · p. 10
  • The paper reports lower coefficient gain for prescribed passivity, compressed uncertainty-induced terminal variation, and improved power sharing with less voltage drop than VI.The conclusion describes these advantages in theoretical analyses and case studies.Source: §VI · p. 10
  • The reconstructed impedance must satisfy the return-inverse properness and stability conditions stated in the paper’s criterion.The inverse is proper when I − D(∞)H(∞) is nonsingular and stable when det[I − D(s)H(s)] has no zeros for Re(s) ≥ 0.Source: §VI · p. 10

APPENDIX B FINITE RECIPROCAL Π-EQUIVALENT CONDITIONS

The appendix material describes when reciprocal Π-network equivalents are well defined and unique, and records related passivity and uncertainty-margin results. The algebraic equivalence requires existing inverses, while stable proper dynamic branches require their implemented inverse blocks to belong to RH∞.

APPENDIX B FINITE RECIPROCAL Π-EQUIVALENT CONDITIONS

  • Every finite reciprocal Π network induces reciprocal impedance parameters, provided the indicated algebraic inverses exist.A stable proper dynamic branch realization additionally requires implemented inverse blocks in RH∞.Source: Appendix B · p. 10
  • A finite reciprocal Π equivalent is well defined only when the required determinants are nonzero, and its uniqueness follows when the defining relations uniquely determine the branch inverses.The listed conditions include det ΔZ, det Z12, det(Z22 − Z12), and det(Z11 − Z12) being nonzero.Source: Appendix B · p. 10
  • Applying the same virtual impedance to every family member leaves pairwise differences unchanged, while the selected shift can achieve the passivity bound.The proof passage states that Zv = (ε − μ0)I attains the bound and gives µ(Z + Zv; Ω) = ε.Source: Appendix B · pp. 10–11
  • For isotropic VI, the same scalar shift is added to each passivity-margin endpoint, so the uncertainty-induced margin spread is unchanged.The passage states Δµ(eZVI; Ω) = Δµ(Z; Ω).Source: Appendix B · p. 11

APPENDIX F PROOF OF THEOREM 5

The proof bounds the passivity-margin spread of a transformed impedance family using a chosen reference and normalization radius. Its stated application evaluates the construction over the specified converter-uncertainty interval.

APPENDIX F PROOF OF THEOREM 5

  • The construction defines a reference impedance Zc := Z(·, pc) and bounds the family’s maximum deviation from that reference.The proof chooses a normalization radius ρ using α, with α ≤ ρ < 1.Source: Appendix F · p. 11
  • The transformed family’s passivity-margin intervals lie within [m, M], with width at most Δ = M − m.Taking the supremum over Ω gives Δµ(eZTP; Ω) ≤ Δ.Source: Appendix F · p. 11
  • For Application 2 over Dω/Dω0 ∈ [0.8, 1.2], the evaluation gives α = 539.846 pu and the coefficient-gain minimization gives h = 0.546835.These auxiliary quantities are substituted into the realization blocks identified in the passage.Source: Appendix F · p. 11

APPENDIX G PROOF OF THEOREM 6

Theorem 6 establishes reachability conditions for target voltages, while network-level tests show that two-port control compresses operating-point variation in terminal impedance and bus-voltage response.

APPENDIX G PROOF OF THEOREM 6

  • Theorem 6's proof shows every noncollinear target voltage is reachable; collinear targets require the stated sign condition on ϕ(t).For 0 < t < 1, positive conductances can realize any value of ϕ(t); outside that interval feasibility requires its sign to match the stated condition.Source: Appendix G · p. 11

APPENDIX H NETWORK-LEVEL VALIDATION ON THE MODIFIED IEEE

  • The network tests use eight 1-MVA VSGs on a modified IEEE 33-bus feeder, islanded for uncertainty tests and grid-connected for series–shunt coordination.The same feeder layout, distributed loads, and VSG locations are used in both tests.Source: Appendix H · pp. 11–12, Figure 12 · p. 12

A. Operating-Point Uncertainty and Bus-Voltage Response

  • The two-port reduces all four terminal-impedance-entry diameters by approximately 67.8% at 0.010 Hz, while fixed VI leaves the operating-point-set diameters unchanged.The two-port maintains a comparable reduction over the tested frequency band.Source: Figure 13 · p. 12, A. Operating-Point Uncertainty and… · p. 12
  • Bus-voltage envelope widths at buses 6 and 33 fall from 0.14762 and 0.14763 pu with VI to 0.007618 and 0.007598 pu with two-port control, reductions of 94.8% and 94.9%.The intervals span 120 operating-point realizations in the islanded eight-VSG microgrid.Source: A. Operating-Point Uncertainty and… · p. 12, Figure 14 · p. 13
  • At bus 18, the maximum separation between endpoint voltage-deviation trajectories falls from 0.01267 pu with VI to 0.004077 pu with two-port control, a 67.8% reduction.The comparison uses the 0.010-Hz perturbation and endpoint realizations at 0.5 and 0.7 pu active-power setpoints.Source: A. Operating-Point Uncertainty and… · p. 12, Figure 15 · p. 13

B. Multi-Scenario Series–Shunt Coordination

Across six loading and feeder configurations, reciprocal series–shunt two-port control reduces reactive-power spread compared with series VI. Under heavy loading, voltage and converter-current improvements accompany higher feeder current and network losses.

B. Multi-Scenario Series–Shunt Coordination

  • VI uses series reactances from 0 to 0.18 pu, while the reciprocal two-port adds network-side shunt susceptances from −0.30 to 0.30 pu; each strategy uses fixed parameters across six conditions.The conditions include uniform and downstream-concentrated loading and two feeder reconfigurations.Source: B. Multi-Scenario Series–Shunt… · p. 13
  • Under heavy loading, the two-port raises minimum bus voltage from 0.99308 pu with VI to 0.99901 pu and confines VSG outputs to 0.4367–0.6556 MVAr, versus 0.1507–0.9539 MVAr with VI.The corresponding reactive-power spread is reduced by 72.8%.Source: B. Multi-Scenario Series–Shunt… · p. 13
  • Under heavy loading, maximum VSG current falls from 45.83 to 32.66 A, while maximum feeder current rises from 139.02 to 193.11 A and network loss from 0.09653 to 0.15775 MW.The higher feeder current and network loss constrain practical parameter selection.Source: B. Multi-Scenario Series–Shunt… · p. 13
  • Across all six operating conditions, two-port control reduces reactive-power spread by 60.4% to 94.2%, with clustering persisting across load patterns and feeder reconfigurations.The comparison is shown for heavy-load voltage profiles and VSG reactive-power outputs across the six conditions.Source: Figure 16 · p. 13, B. Multi-Scenario Series–Shunt… · p. 13
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