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Revisiting Grid-Forming and Grid-Following Inverters: A Duality Theory

Yitong Li, Yunjie Gu, Timothy C. Green

arXiv:2105.13094v3eess.SY

TL;DR

The paper examines the subtle relationship between grid-forming and grid-following inverters and proposes a duality-based framework. It derives dual synchronization and swing formulations, showing corresponding grid-interfacing and stability characteristics across inverter and network cases.

  • Problem

    Grid-forming and grid-following inverters have distinct characteristics and theories, while their reported similarities leave their relationship subtle and incompletely unified.

  • Method

    The paper develops a technology-neutral duality theory and derives dual swing equations to analyze synchronization, grid interfacing, and stability.

  • Results

    The analysis identifies duals in synchronization controllers, grid interfacing, swing characteristics, inner-loop control, and grid-strength compatibility, with stability interactions illustrated in inverter network cases.

  • Takeaways & Limitations

    The unified view supports multi-inverter analysis and motivates concepts including current- and voltage-forming/following, voltage/current strength, and grid-following islanding.

Abstract

from arXiv · show

Power electronic converters for integrating renewable energy resources into power systems can be divided into grid-forming and grid-following inverters. They possess certain similarities, but several important differences, which means that the relationship between them is quite subtle and sometimes obscure. In this article, a new perspective based on duality is proposed to create new insights. It successfully unifies the grid interfacing and synchronization characteristics of the two inverter types in a symmetric, elegant, and technology-neutral form. Analysis shows that the grid-forming and grid-following inverters are duals of each other in several ways including a) synchronization controllers: frequency droop control and phase-locked loop (PLL); b) grid-interfacing characteristics: current-following voltage-forming and voltage-following current-forming; c) swing characteristics: current-angle swing and voltage-angle swing; d) inner-loop controllers: output impedance shaping and output admittance shaping; and e) grid strength compatibility: strong-grid instability and weak-grid instability. The swing equations are also derived in dual form, which reveal the dynamic interaction between the grid strength, the synchronization controllers, and the inner-loop controllers. Insights are generated into cases of poor stability in both small-signal and transient/large-signal. The theoretical analysis and simulation results are used to illustrate cases for simple single-inverter-infinite-bus systems and a multi-inverter power network.

I. INTRODUCTION

The paper addresses the subtle relationship between grid-forming and grid-following inverters by developing a technology-neutral duality perspective that unifies their synchronization and grid-interfacing characteristics.

  • Renewable integration relies on inverter-based resources whose control algorithms introduce flexibility alongside new control interactions and instability problems.
  • Grid-forming inverters control ac-side voltage and synchronize through frequency droop, whereas grid-following inverters control current and track grid voltage phase through a PLL.
  • Prior work identified structural resemblance between frequency droop and PLL behavior when grid impedance is included, but the inverter types remained largely distinct in analysis and theory.
  • The proposed duality perspective unifies the two inverter types symmetrically and technology-neutrally across synchronization, interfacing, swing dynamics, and stability questions.
  • Frequency droop and PLL synchronization methods are illustrated in synchronous dq-frame control structures, with droop for grid-forming and PLL for grid-following inverters.
  • The paper derives dual swing characteristics and studies synchronization stability through theoretical analysis, simulations, and case studies of inverter-infinite-bus and multi-inverter systems.
  • Under stated operating conditions, P-ω droop can reduce to id-ω droop, while PLL behavior can be related to Q-PLL through q-axis voltage and reactive-power relationships.
  • The synchronization duality further interprets id-ω droop as an id-PLL with proportional phase locking and PLL control as vq-ω droop with PI gain.

B. Duality of Grid-Forming and Grid-Following

The paper frames grid-forming and grid-following inverters as more precisely voltage-forming and voltage-following, then develops dual swing characteristics and models their synchronization and impedance interactions.

  • B. Duality of Grid-Forming and Grid-Following: Grid-forming and grid-following inverters are more precisely voltage-forming and voltage-following because grid sources and loads connect in parallel.The id-ω droop or id-PLL additionally identifies the grid-forming inverter as current-following.
  • C. Duality of Swing Characteristics: Frequency-droop grid-forming inverters synchronize through swing dynamics similarly to, or in some cases equivalently to, virtual synchronous generators.The corresponding swing of PLL grid-following inverters remains ill-defined and under-researched in the cited discussion.
  • 1) Frequency droop grid-forming inverter:: The single-machine-infinite-bus system provides a simple setting for investigating inverter swing dynamics.Figure 3 depicts the synchronization loops for frequency-droop grid-forming and PLL grid-following inverters.
  • 1) Frequency droop grid-forming inverter:: Figure 4 recasts the two single-inverter-infinite-bus models in small-signal complex dq± form, with Gs as the synchronization controller and T as the frame-transformation matrix.The models distinguish frequency-droop grid-forming and PLL grid-following cases.
  • 1) Frequency droop grid-forming inverter:: Figure 5 distinguishes large-signal angular frequency ω and phase angle θ from their small-signal counterparts and steady-state operating points.The swing and steady frames are also called controller/system or mechanical/electrical frames in cited literature.
  • 1) Frequency droop grid-forming inverter:: The grid-forming inverter is represented as a controlled voltage source whose current-following voltage-forming behavior includes grid and inverter impedances.Zg represents grid impedance, while Zc incorporates ac-filter and inner-voltage-loop dynamics.
  • 1) Frequency droop grid-forming inverter:: The physical plant, grid, and inverter use a steady frame rotating at Ω0, while synchronization and voltage reference use a local swing frame rotating at droop-controlled frequency ω.The transformation between frames reflects synchronization dynamics; the article uses complex dq± variables for concise small-signal analysis.
  • 1) Frequency droop grid-forming inverter:: The current-angle swing model begins with the frequency-droop equation, linearizes it in dq± variables, and derives a virtual impedance and whole-system characteristic equation.The modified characteristic captures interaction among the synchronization loop, inner voltage loop, and external grid impedance, supporting later multi-inverter impedance analysis.

2) PLL grid-following inverter:

The PLL grid-following inverter is modeled as a controlled current source with voltage-following, current-forming behavior. Its dual swing description is a voltage-angle characteristic derived from PLL dynamics and virtual admittance modeling.

  • The grid-following inverter is represented by a controlled current source, with Yc modeling the filter and inner current-loop dynamics.
  • Fig. 6’s right-hand column derives the PLL swing equation and its linearized form for the grid-following inverter.
  • The grid-following inverter has voltage-angle swing characteristics, dual to the grid-forming inverter’s current-angle swing.
  • A virtual admittance YPLL represents PLL dynamics, while S′vθ incorporates grid interaction in the whole-system model.
  • Within stated operating conditions, voltage-angle swing is equivalent to Q-angle swing when id is constant and iq is zero.

D. Summary of Duality

The paper summarizes a dual stability framework linking synchronization loops, grid strength, and inner-loop dynamics for grid-forming and grid-following inverters. Root-locus analysis and time-domain simulations identify corresponding instability boundaries and confirm their interaction-based explanation.

  • Summary of Duality: The two inverter types exhibit duality in synchronization loops, voltage/current forming and following, and swing dynamics.
  • Summary of Duality: The grid-forming and grid-following swing characteristics include interaction terms involving grid impedance or admittance, synchronization controllers, and inner-loop controllers.
  • Summary of Duality: Root loci vary grid impedance, synchronization gain, and inner-loop gain, with corresponding subfigures assigned to grid-forming and grid-following cases.
  • Summary of Duality: Reducing grid-forming grid impedance or grid-following grid admittance produces right-half-plane roots and instability, corresponding to strong-grid and weak-grid vulnerability.
  • Summary of Duality: Increasing droop gain or PLL bandwidth leads to instability, indicating that larger synchronization-controller gains increase instability susceptibility.
  • Summary of Duality: Reducing voltage-loop or current-loop bandwidth leads to unstable interaction, while simulations confirm predicted frequencies and re-stabilization after parameters return to initial values.
  • Summary of Duality: The unstable modes arise from insufficient damping rather than insufficient synchronizing coefficient, with tuning constrained by grid-code and hardware limits.
  • Summary of Duality: Stable synchronization imposes upper limits on synchronization gains, lower limits on inner-loop gains, and dual grid-strength limits for the two inverter types.

C. Rethinking Grid Strength

The paper distinguishes grid voltage strength from grid current strength to clarify the dual grid-strength vulnerabilities of grid-forming and grid-following inverters.

  • Rethinking Grid Strength: Conventional grid strength is more precisely grid voltage strength, while its dual is grid current strength.
  • Rethinking Grid Strength: A voltage grid becomes ideally strong as Zg approaches zero, whereas a current grid becomes ideally strong as Yg approaches zero.
  • Rethinking Grid Strength: Grid-forming inverters are voltage-forming and current-following, while grid-following inverters are current-forming and voltage-following.
  • Rethinking Grid Strength: Grid-forming inverters are vulnerable to strong grids and short circuits, whereas grid-following inverters are vulnerable to weak grids and open circuits.

D. A Multi-Inverter Power System

A modified IEEE 14-bus network replaces synchronous generators with grid-forming or grid-following inverters and evaluates stability using a whole-system impedance model.

  • A Multi-Inverter Power System: The modified IEEE 14-bus system replaces all synchronous generators with grid-forming or grid-following inverters while retaining the standard layout and line impedances.
  • A Multi-Inverter Power System: Whole-system stability is evaluated from poles obtained by connecting inverter impedance models to the transmission-line dynamic nodal admittance matrix.
  • A Multi-Inverter Power System: 17.3 Hz unstable oscillation results when lines connecting GFM1 are reduced to one fifth of their standard impedances, demonstrating strong-grid instability.
  • A Multi-Inverter Power System: Increasing line impedances can produce the dual weak-grid instability in GFL2 or GFL8.

IV. DUALITY OF TRANSIENT STABILITY

The paper extends its duality analysis from small-signal instability to transient stability under island operation and severe disturbances. It treats grid-following inverters as capable of forming an island grid through current, with transient assessment paralleling synchronous-generator analysis.

  • IV. DUALITY OF TRANSIENT STABILITY: Transient stability concerns robustness under abnormal operation and severe disturbances, beyond perturbations around a fixed operating point.The paper distinguishes this large-signal perspective from its preceding small-signal analysis.
  • IV. DUALITY OF TRANSIENT STABILITY: Duality suggests that grid-following inverters can operate an island with a formed grid current, complementing grid-forming inverters' formed grid voltage.The distinction follows the paper's reclassification of grid-following operation in islanded conditions.
  • IV. DUALITY OF TRANSIENT STABILITY: Transient stability of both inverter types can be assessed similarly to synchronous generators using virtual inertia and virtual damping instead of physical counterparts.The comparison retains the synchronous-generator assessment framework while substituting control-based quantities.

A. Single-Inverter System: Island Operation and Frequency Stability

The single-inverter islanding study tests whether a PLL grid-following inverter can maintain stable frequency while forming grid current. It finds that frequency can be re-stabilized by removing PLL integrator action, although voltage variation and equilibrium constraints remain important.

  • A. Single-Inverter System: Island Operation and Frequency Stability: When q = 0.09 pu, v_d is approximately 0.5 pu because inverter current and load impedance determine voltage through Ohm's law.The reported voltage is therefore below 1 pu for this operating condition.
  • A. Single-Inverter System: Island Operation and Frequency Stability: With the PLL integrator enabled, some current references produce no equilibrium, making system frequency unstable.The selected load and current references create a contradiction that prevents equilibrium.
  • A. Single-Inverter System: Island Operation and Frequency Stability: Figure 13 varies Z12 and Z15 between standard values and one fifth while recording pole loci, frequency, current, and voltage at five IBR buses.The impedance reduction occurs at 0.4 s and reverts at 0.8 s.
  • A. Single-Inverter System: Island Operation and Frequency Stability: Figure 14 presents island-operation layouts for single-inverter and two-inverter grid-following cases.
  • A. Single-Inverter System: Island Operation and Frequency Stability: At 0.8 s, setting k_i,PLL = 0 removes the contradiction and re-stabilizes frequency by converting the PLL into proportional v_q-ω droop.This behavior further illustrates the duality between PLL control and frequency droop, including the role of integrator action.
  • A. Single-Inverter System: Island Operation and Frequency Stability: A PLL grid-following inverter can operate in an islanding mode with stable frequency while forming grid current, but its grid voltage may vary widely.The voltage variation is attributed to the chosen i*_dq setting and load impedance.

B. Two-Inverter System: Angle Stability

The paper analyzes angle stability in two-inverter systems through dual swing characteristics. Two GFLs can synchronize in island operation, while mixed GFM-GFL synchronization depends on their virtual-inertia ratio.

  • Two-GFM system: Two grid-forming inverters exhibit a sine P_∆-θ_∆ relation with stable equilibrium for θ_∆ below 90°.A smaller angle difference provides a larger maximum decelerating area and a wider first-swing transient-stability region.
  • Two-GFL system: Two grid-following inverters exhibit a sine Q_∆-θ_∆ relation when the interconnection is resistance-dominated.The resulting transient characteristics can resemble those of two grid-forming inverters.
  • Two-GFL system: Two grid-following inverters can synchronize and operate stably in island operation without voltage sources when passive loads make the system resistance-dominated.A fault test showed the inverters re-established currents, re-synchronized through their PLLs, and returned the bus voltage after fault clearing.
  • GFM-GFL system: In a mixed GFM-GFL system, the inverter with smaller virtual inertia has the larger angular acceleration and synchronizes to the inverter with larger virtual inertia.For J1 ≫ J2, the GFM synchronizes to the GFL; for J1 ≪ J2, the GFL synchronizes to the GFM.
  • GFM-GFL system: Changing the virtual-inertia ratio changes the system’s power-angle relation, swing curves, and stable equilibrium points.Increasing the GFL virtual inertia to infinity shifted the system from SEP1 to SEP2 in the reported test.
  • GFM-GFL system: The specific multi-inverter swing dynamics depend on inverter type, virtual inertia, network impedance, and steady-state operating point.These dependencies make concise analytical solutions for transient swing dynamics challenging, motivating time-domain simulations.

C. Multi-Inverter System: Virtual Inertia and Virtual Damping

The multi-inverter tests examine how virtual inertia and damping parameters affect transient stability. Larger droop or PLL integral gains reduce virtual inertia and damping, increasing the chance of post-fault instability.

  • Grid-forming inverter: A large grid-forming droop gain caused large-signal instability after a fault despite small-signal stability before the fault.The reported 0.08 pu droop gain created small virtual inertia and damping, increasing transient-instability risk.
  • Grid-following inverter: A large grid-following PLL integral gain caused large-signal instability after a fault despite small-signal stability before the fault.The paper relates large ki,PLL to small virtual inertia and damping.
  • Comparison: The fault tests produced different loss-of-synchronism outcomes: frequency collapse occurred in the GFM case, whereas the GFL lost synchronism in the GFL case.The comparison indicates that the main source of transient instability differs between grid-forming and grid-following cases.

V. CONCLUSIONS

The paper presents GFM and GFL inverters as duals across synchronization, grid interfacing, swing behavior, controller gains, grid-strength compatibility, and transient stability. This duality unifies their analysis while identifying boundaries where the correspondence does not fully hold.

  • Synchronization duality: GFM and GFL synchronization loops form converse pairs linking frequency droop control and PLL behavior under stated signal constraints.GFM P-ω droop corresponds to id-ω droop and an id-based PLL, while GFL PLL behavior corresponds to vq-ω droop and, under constraints, Q-ω droop.
  • Grid interfacing: GFM is current-following and voltage-forming, whereas GFL is voltage-following and current-forming.
  • Swing characteristics: Their swing characteristics are also converse: GFM has current-angle or active-power-angle swing, while GFL has voltage-angle or reactive-power-angle swing.
  • Controller gains: Higher synchronization gains or lower inner-loop gains increase the likelihood of interaction and synchronization instability for both inverter types.The relevant gains are frequency-droop gain and PLL bandwidth, while inner-loop gains include voltage- and current-loop bandwidths.
  • Grid strength: GFM is vulnerable to strong grids, whereas GFL is vulnerable to weak grids, reflecting converse grid-strength compatibility.The stated conditions involve low grid impedance for GFM and low grid admittance for GFL.
  • Transient stability: Transient stability in both inverter types depends on swing equations, virtual inertia, and virtual damping, and GFL can island or synchronize without a voltage source.
  • Scope of duality: The duality framework supports multi-inverter analysis and new concepts, but some properties remain non-dual because of grid and synchronization-loop asymmetries.The paper identifies voltage-dominated sources, parallel-connected apparatuses, inductor-dominated lines, and integral action as reasons for these limits.

APPENDIX A FUNDAMENTALS OF COMPLEX dq FRAME

The appendix introduces the complex dq frame for small-signal analysis and explains that it provides a simpler, more symmetric model structure for power-system applications. Rated parameters are used for single-inverter results unless otherwise specified.

  • Frame definition: The complex dq± frame is used for small-signal analysis in the article.
  • Frame transformation: Forward and backward complex space vectors are defined as u+ = ud +juq and u− = ud −juq, with transformation matrix Tj relating them to dq-frame signals.
  • Analytical benefit: The transformed Gdq± model normally has a simpler and more symmetric mathematical structure than Gdq for power-system applications.This structure facilitates the analysis.
  • Parameter settings: Rated parameters from Table II are used for single-inverter cases, while parameters for the 14-bus multi-inverter case are available online.
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