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Grid-forming Control for Power Converters based on Matching of Synchronous Machines
Catalin Arghir, Taouba Jouini, Florian Dorfler
TL;DR
The paper addresses grid-forming control for converters in weak, low-inertia grids where synchronous-machine-like synchronization is needed. It matches converter dynamics to a synchronous-machine model using a DC-voltage-driven virtual oscillator, then develops stability, passivity, droop, power-sharing, and regulation results. The framework replaces AC frequency measurement with DC-link voltage measurement and is reported as compatible with synchronous machines and conventional power-system requirements.
Problem
Low-inertia and weak-grid systems require inverter controls that provide grid-forming, self-synchronizing behavior without relying on a stiff-grid frequency measurement.
Method
The paper augments an average-switch converter with a virtual oscillator whose DC-voltage-driven frequency sets PWM, matching the converter’s closed-loop dynamics to a synchronous-machine model.
Results
The framework provides a sufficient condition for unique globally asymptotically stable equilibria and preserves strict incremental passivity, droop, and proportional power-sharing properties.
Takeaways & Limitations
DC-link voltage can replace AC frequency measurement for the matching controller, while additional loops regulate DC voltage, AC frequency, and AC amplitude.
Takeaways & Limitations
The incremental-passivity analysis uses a dq frame attached to one converter angle and therefore does not cover networks containing multiple grid-forming inverters.
Abstract
from arXiv · showhide
We consider the problem of grid-forming control of power converters in low-inertia power systems. Starting from an average-switch three-phase inverter model, we draw parallels to a synchronous machine (SM) model and propose a novel grid-forming converter control strategy which dwells upon the main characteristic of a SM: the presence of an internal rotating magnetic field. In particular, we augment the converter system with a virtual oscillator whose frequency is driven by the DC-side voltage measurement and which sets the converter pulse-width-modulation signal, thereby achieving exact matching between the converter in closed-loop and the SM dynamics. We then provide a sufficient condition assuring existence, uniqueness, and global asymptotic stability of equilibria in a coordinate frame attached to the virtual oscillator angle. By actuating the DC-side input of the converter we are able to enforce this sufficient condition. In the same setting, we highlight strict incremental passivity, droop, and power-sharing properties of the proposed framework, which are compatible with conventional requirements of power system operation. We subsequently adopt disturbance decoupling techniques to design additional control loops that regulate the DC-side voltage, as well as AC-side frequency and amplitude, while in the end validating them with numerical experiments.
1 Introduction
Low-inertia systems lack the kinetic inertia and self-synchronizing physics that synchronous machines provide, making inverter control a central challenge. The paper proposes a voltage-driven virtual-oscillator controller that matches synchronous-machine dynamics and supports stability, passivity, droop, and regulation objectives.
- Motivation: Low-inertia systems with many inverter-interfaced renewable sources lack synchronous machines’ kinetic-energy storage and self-synchronizing safeguards.Inverters have little or no built-in energy storage but operate at much faster time scales than synchronous machines.
- Background: Grid-following inverters rely on stiff-grid frequency measurements, whereas grid-forming inverters interact with non-stiff grids similarly to synchronous machines.A low-inertia system cannot be operated with only grid-following units.
- Related work: Existing droop and virtual-synchronous-machine controllers emulate synchronous-machine behavior using AC measurements, whose processing delays can reduce control effectiveness.Examples measure injected power, frequency, or amplitude, often through a phase-locked loop.
- Contributions: The proposed controller augments converter dynamics with a harmonic oscillator whose frequency tracks DC-side voltage and drives PWM, exactly matching synchronous-machine dynamics.The paper treats DC voltage as the key control and imbalance signal analogous to synchronous-machine angular velocity.
- Contributions: The framework preserves strict incremental passivity, droop, and power-sharing properties while adding regulation loops for DC voltage, AC frequency, and AC amplitude.The paper also presents a numerical case study validating the resulting control design.
2 The Three-Phase Converter Model, Synchronous Machine Model, & their Analogies
The paper formulates an average-switch three-phase converter and a synchronous-machine model in compatible coordinates, then uses their energy-exchange analogy to define grid-forming objectives. These objectives include dynamic matching, voltage and amplitude regulation, droop-based power sharing, and strict incremental passivity.
- 2.1 Preliminaries and coordinate transformations: The balanced three-phase model separates zero-sequence and αβ components, with αβ quantities subsequently transformed into dq coordinates using an angle.A sinusoidal steady state becomes an equilibrium in the dq frame when the transformation angle rotates at the steady-state frequency.
- 2.2 Three-Phase DC/AC Converter Model: The converter is represented by a continuous-time average-switch model whose principal nonlinearity is the modulation block.The DC side includes a controllable current source, capacitance, and conductance; the AC side includes inductive, resistive, capacitive, and conductive elements.
- 2.2 Three-Phase DC/AC Converter Model: The switching block uses a complementary PWM carrier and a bounded modulation signal while remaining lossless to preserve energy conservation.The modulation signal lies in the unit ball, ∥mαβ∥≤1.
- 2.3 Control objectives: Grid-forming control aims to match synchronous-machine electromechanical interaction rather than force converter frequency to the grid through a PLL.Additional objectives are exact DC-voltage and AC-amplitude regulation, local frequency-power droop, and strict incremental passivity at AC and DC ports.
- 2.4 The Synchronous Machine Model: The converter’s DC capacitor is analogous to synchronous-machine rotor inertia, while switching current and voltage correspond to electrical torque and electromotive force.Their exchange of kinetic and electrical energy is identified as the source of self-synchronizing behavior.
3 Grid-Forming SM Matching Control
The paper matches converter dynamics to a synchronous-machine model using a voltage-driven internal oscillator, then establishes passivity, stability, droop, and power-sharing properties under suitable control conditions.
- 3 Grid-Forming SM Matching Control: A sinusoidal modulation scheme with an internal oscillator is matched to synchronous-machine dynamics through dynamic feedback.The oscillator angle drives the modulation, while its frequency is linked to the DC-side voltage.
- 3 Grid-Forming SM Matching Control: The equivalent synchronous-machine interpretation identifies DC-side capacitance, conductance, and current with mechanical inertia, damping, and driving torque.The switching-node voltage corresponds to the machine EMF, and the DC-side current scaled by η corresponds to electrical torque.
- 3.1 Closed-Loop Incremental Passivity: In the oscillator-attached dq frame, the model-matched inverter is analyzed relative to a steady state using incremental passivity and a physical storage function.The analysis removes dependence on the angle state and characterizes passivity with DC and AC port variables.
- 3.1 Closed-Loop Incremental Passivity: Under condition (11), the dq-frame system is strictly passive, and constant source and load currents yield a unique globally asymptotically stable equilibrium.Positive definiteness of the dissipation matrix establishes strict passivity; radial unboundedness of the storage function gives global stability.
- 3.2 Closed-Loop Incremental Stability: A DC-side proportional controller enforces the required damping condition when parasitic converter resistances and conductances are small.The sufficient inequality is 4R < Gdc + Kp, and it can be satisfied by selecting Kp appropriately.
- 3.2 Closed-Loop Incremental Stability: With the oscillator-based load model and DC-side P-control, the closed-loop system has a unique globally asymptotically stable steady state under the stated assumptions.The load model represents a passive shunt impedance in parallel with a synchronized sinusoidal current source.
- 3.3 Droop properties of matching control: At equilibrium, matching control gives a virtual-frequency relation to amplitude, a local frequency droop, and no direct relation between reactive power and frequency or amplitude.The virtual frequency satisfies ωx = 2ηµrx, while amplitude droop follows analogously from rx = µ 2ηωx.
- 3.3 Droop properties of matching control: Two converters achieve proportional power sharing when their droop and power quantities satisfy the ratio ρ, while operation beyond the nose-curve bifurcation has no stationary solution.The practically relevant equilibrium is the high-voltage solution, and the maximum switching-node active power occurs at the nose-curve tip.
4 Voltage and frequency regulation
The paper adds outer control loops to the matching controller for regulating DC voltage, AC frequency, and AC voltage amplitude. Integral frequency control yields a unique globally asymptotically stable equilibrium, while amplitude regulation and disturbance-rejection properties require explicit feasibility or plant-information conditions.
- Outer-loop objectives: The outer-loop design actuates the DC current source and modulation amplitude to track DC-capacitor voltage and AC-voltage-amplitude references.The design first regulates the DC-side voltage and then extends amplitude control through the modulation input.
- Exact frequency regulation: The proposed PID frequency controller pairs DC-voltage error feedback with a reference frequency ω0 = ηvdc,ref and integral action.The controller uses idc,ref together with positive gains Kp, Ki, and Kd.
- Exact frequency regulation: Choosing η = ω0/vdc,ref enables simultaneous DC-voltage and frequency specifications at the desired steady state.The resulting closed-loop system is analyzed for a steady state satisfying vdc = vdc,ref and ω = ω0.
- Exact frequency regulation: Theorem 6 establishes a unique steady state at the origin, and under the stated condition, that equilibrium is globally asymptotically stable.The stability proof uses a storage function augmented for the integral state and DC-voltage gain, followed by a LaSalle-type argument.
- Frequency-control interpretation: The PID gains provide additional effective inertia and damping in the DC circuit, with Kp and Kd inducing synthetic droop and inertia when Ki = 0.The frequency-error dynamics then resemble standard swing equations.
- Secondary regulation: Secondary frequency regulation requires sufficiently large equivalent DC energy storage to handle a given power imbalance.Distributed integral control can be adapted for multiple inverters to support robust power sharing.
- Disturbance rejection and droop: Disturbance-feedback and PI-PBC reject constant, or eventually constant state-independent, load-current disturbances, while voltage droop is globally stable for sufficiently small droop coefficient dv.The disturbance-feedback design requires exact plant knowledge and load measurement; the droop result assumes condition (11) and an admitted steady state.
5 Numerical case study
The numerical case study evaluates regulation under a 55% load step and proportional power sharing in a two-inverter network with resistive load steps.
- 5 Numerical case study: The case study uses a 104 W inverter with specified DC, filter, and line parameters, nominal DC voltage 1000, and nominal frequency 2π50.The selected gains target open-circuit amplitude 165 and nominal frequency 2π50.
- 5.1 Single-converter regulation: A 55% load step at t = 0.5s tests matching control, frequency regulation, and three amplitude controllers.The resulting amplitudes and power waveforms are shown in Figure 4, while Figure 6 magnifies the output capacitor voltage transient.
- 5.1 Single-converter regulation: All considered voltage controllers make the DC voltage exactly track vdc,ref = 1000.The compared controllers are feedforward, PI-PBC, and droop control.
- 5.2 Multi-Converter Case Study: The multi-converter case study connects two inverters in parallel to a conductance load through a Π-transmission line model.The line parameters are Rnet = 0.5, Lnet = 2.5·10^-5, and Cnet = 2 · 10^-7.
- 5.2 Multi-Converter Case Study: A prescribed power sharing ratio of 3:1 is obtained under resistive load steps at t = 0.3 and t = 0.7.The matching controller uses gains selected to demonstrate proportional power sharing.
6 Conclusions
The paper proposes matching-based grid-forming control for weak grids by coupling DC-side voltage to AC-side frequency, then adds outer loops for voltage regulation. The framework provides droop, proportional power sharing, passivity, and exact tracking of output frequency and amplitude.
- 6 Conclusions: The proposed controller matches inverter dynamics to a synchronous machine by coupling DC-side voltage feedback to AC-side frequency.This replaces AC grid-frequency measurement with DC-link voltage measurement and avoids the conventional time-scale separation approach.
- 6 Conclusions: The matching control yields droop and proportional power-sharing characteristics while preserving inverter passivity and allowing straightforward synthetic damping and inertia.These properties are presented as compatible with weak-grid operation and synchronous-machine interaction.
- 6 Conclusions: Additional passivity-based and disturbance-decoupling outer loops achieve exact tracking of output voltage frequency and amplitude under measurable disturbances.The paper validates the proposed controllers through numerical experiments.