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Global phase and magnitude synchronization of coupled oscillators with application to the control of grid-forming power inverters

Marcello Colombino, Dominic Groß, Jean-Sébastien Brouillon, Florian Dörfler

arXiv:1710.00694v3math.OC

TL;DR

Inverter-based grids can lose synchronous machines’ inertia and self-synchronization, while machine-emulation controls may have complex dynamics subject to delays. This paper develops decentralized oscillator control and proves almost-global asymptotic stability under a stated condition.

  • Problem

    Inverter-based grids may lack synchronous machines’ rotational inertia and self-synchronization, while machine-emulation controllers can produce complex dynamics subject to control and signal-processing delays.

    Source: §I · p. 1

  • Method

    The paper designs decentralized control laws using each oscillator’s local state and relative output to achieve harmonic oscillations with prescribed frequency, relative phases, and magnitudes.

    Source: §II · p. 2

  • Results

    Under Condition 1, the proposed dynamics are almost globally asymptotically stable with respect to the target sets for the control objectives.

    Source: C. Main result: Almost global… · p. 8

  • Takeaways & Limitations

    For grid-connected inverters, the method ensures synchronization and almost-global stability of a solution with desired power injections.

    Source: §VI · p. 13

    The analysis neglects transmission-line dynamics and considers networks containing only inverters acting as sources or loads.

    Source: §VI · p. 13

Abstract

from arXiv · show

In this work, we explore a new approach to synchronization of coupled oscillators. In contrast to the celebrated Kuramoto model we do not work in polar coordinates and do not consider oscillations of fixed magnitude. We propose a synchronizing feedback based on relative state information and local measurements that induces consensus-like dynamics. We show that, under a mild stability condition, the combination of the synchronizing feedback with a decentralized magnitude control law renders the oscillators' almost globally asymptotically stable with respect to set-points for the phase shift, frequency, and magnitude. We apply these result to rigorously solve an open problem in control of inverter-based AC power systems. In this context, the proposed control strategy can be implemented using purely local information, induces a grid-forming behavior, and ensures that a network of AC power inverters is almost globally asymptotically stable with respect to a pre-specified solution of the AC power-flow equations. Moreover, we show that the controller exhibits a droop-like behavior around the standard operating point thus making it backward-compatible with the existing power system operation.

I. INTRODUCTION … A. Synchronization: a geometric characterization

The paper develops a decentralized approach to synchronizing oscillators with prescribed phase, frequency, and magnitude, then applies it to inverter-based power systems. It frames the objectives geometrically as stabilization of a set defined by those steady-state specifications.

I. INTRODUCTION

  • The loss of synchronous-machine inertia and self-synchronization challenges future grids, while inverter machine-emulation controls can produce complex dynamics subject to control and signal-processing delays.Existing droop and Kuramoto-based analyses often neglect voltage dynamics and provide only local guarantees; virtual oscillator control in its original form cannot track power-injection references.Source: §I · p. 1
  • Under a mild stability condition, the proposed controller achieves almost-global stability at prescribed phase, frequency, and magnitude set-points, and applies to inverter networks tracking AC power-flow solutions.The inverter controller uses local measurements and set-points and behaves similarly to conventional droop control around the synchronous harmonic steady state.Source: §I · p. 2

Notation

  • The notation section establishes conventions for real numbers, the circle, matrix and vector operations, norms, solution flows, neighborhoods, and the ceiling operator.These conventions include the Euclidean distance from a point to a set and the largest singular value of a matrix.Source: Notation · p. 2

Preliminaries from graph theory

  • The graph preliminaries define a simple connected undirected weighted graph, its oriented incidence matrix, and its Laplacian for representing network interconnections and node differences.The incidence matrix maps node values to edge differences.Source: Preliminaries from graph theory · p. 2

II. PROBLEM SETUP AND CONTROL STRATEGY

  • The abstract problem considers N coupled two-dimensional integrators and seeks decentralized controls that synchronize them almost globally to harmonic oscillations with prescribed frequency, relative phases, and magnitudes.Decentralized controllers use local state and relative-output information, with set-points that may be updated infrequently by a centralized decision maker.Source: §II · p. 2

A. Synchronization: a geometric characterization

  • Synchronization and magnitude regulation are expressed as set-stabilization: the target set combines prescribed relative phases, harmonic motion at the target frequency, and the required oscillator magnitudes.The objective is for the intersection of these specification sets to be invariant and almost globally asymptotically stable.Source: A. Synchronization: a geometric… · p. 3

B. Synchronizing controllers and closed-loop dynamics

The proposed decentralized feedback combines phase synchronization with magnitude regulation while retaining harmonic oscillation at frequency ω0. Under a mild stability condition, it drives the oscillators to synchrony and satisfies the stated objectives from almost all initial conditions.

B. Synchronizing controllers and closed-loop dynamics

  • The controllers use only local state and output measurements to pursue global control objectives, yielding decentralized closed-loop dynamics.The phase error depends on neighboring oscillator states, whereas the magnitude error depends only on the local oscillator state.Source: B. Synchronizing controllers and… · p. 3
  • The closed-loop feedback combines harmonic rotation at frequency ω0 with phase-error feedback and radial magnitude regulation.The phase term synchronizes relative angles to their set-points, while the magnitude term regulates each oscillator’s amplitude.Source: B. Synchronizing controllers and… · pp. 3–4
  • Under a mild stability condition, controller (9) drives the system to synchrony and satisfies the stated objectives from almost all initial conditions.When phase and magnitude errors are zero, the residual dynamics are N decoupled harmonic oscillators with synchronous frequency ω0.Source: B. Synchronizing controllers and… · p. 4

C. Relationship to the Kuramoto oscillator · III. DEFINING ENGINEERING APPLICATION: INVERTER-BASED POWER GRIDS · A. Modelling of inverter-based power grids

The paper relates its oscillator dynamics to the Kuramoto model through a unit-circle projection, then formulates inverter-based grids as coupled systems using local measurements. The grid model assumes quasi-steady transmission lines and a uniform inductance-resistance ratio for the theoretical analysis.

C. Relationship to the Kuramoto oscillator

  • Projecting the dynamics (10) onto the unit circle yields dynamics equivalent to the Kuramoto model (16).The paper identifies the linear part of (10) as a “linear reformulation of the Kuramoto model”.Source: C. Relationship to the Kuramoto… · p. 4
  • Unlike the Kuramoto model, the closed loop (10) does not fix state magnitudes during transients and can converge to the torus asymptotically for small amplitude-control gain α.Large α makes the magnitude approximately track its set-point; the paper says this difference enables conditions for almost-global rather than local synchronization.Source: C. Relationship to the Kuramoto… · p. 4

III. DEFINING ENGINEERING APPLICATION: INVERTER-BASED POWER GRIDS

  • In inverter-based grids, controllers rely on local measurements even though frequency synchronization and convergence to a desired power-flow solution are global objectives.The section connects this control problem to the paper’s abstract setup.Source: §III · p. 4

A. Modelling of inverter-based power grids

  • The model treats each of N balanced three-phase inverters as a controllable voltage source and uses the Clarke transformation to represent voltages and currents as two-dimensional αβ vectors.Each inverter has controllable voltage v_k and output current i_o,k.Source: A. Modelling of inverter-based power… · p. 5
  • Assuming transmission-line time constants are much faster than terminal-voltage time constants lets the model treat lines as reaching steady state almost instantaneously and coupling inverters algebraically.The paper notes that electromagnetic transients can compromise stability; extending the analysis to include line dynamics is outside its scope, though such guarantees are available in [44].Source: A. Modelling of inverter-based power… · p. 5
  • The theoretical results require a uniform inductance-resistance ratio across transmission lines, an assumption usually satisfied for lines at the same voltage level but not across different voltage levels.Simulations indicate that heterogeneous ratios do not compromise stability.Source: A. Modelling of inverter-based power… · p. 5
  • The grid is represented by a simple, connected, undirected graph, and a relative output signal obtainable from local current measurements is used in the model.The transmission lines are associated with resistance and inductance parameters.Source: A. Modelling of inverter-based power… · p. 5

B. Decentralized control of inverter-based power grids

The section frames inverter-based grid control as decentralized coupled-oscillator regulation: under asymptotic stability of T_f,ω0, voltages converge to a pre-specified power-flow solution and evolve at synchronous frequency ω0.

B. Decentralized control of inverter-based power grids

  • The controller uses local terminal-voltage measurements, and each network variable component can be computed locally from the inverter’s output current, placing the problem in the coupled-oscillator class.The controller structure is depicted in Figure 2.Source: B. Decentralized control of… · p. 6
  • The control objective is to steer inverter voltages to a synchronous harmonic solution with prescribed voltage magnitudes and active and reactive power injections.The active and reactive power set-points are related to angle and voltage set-points by the power-flow equations.Source: B. Decentralized control of… · p. 6
  • Equation (24), obtained by expanding standard power-flow equations in line resistances and inductances, ensures a corresponding inverter-grid steady state exists [46, 47].Source: B. Decentralized control of… · p. 6
  • When T_f,ω0 is asymptotically stable, the network converges to a pre-specified power-flow solution, with state variables evolving as sinusoids at synchronous frequency ω0.The target solution lies in F_f,ω0 ∩ A ∩ P and corresponds to prescribed voltage magnitudes and power injections.Source: B. Decentralized control of… · p. 6

C. Decentralized implementation for networks of inverters · D. Droop-like behavior

The paper makes the inverter controller implementable without angle setpoints or network line parameters, then characterizes its droop-like behavior and the conditions under which that resemblance holds. Its droop-like response is local to the nominal operating point; the controller also supports almost-global stability for consistent setpoints.

C. Decentralized implementation for networks of inverters

  • The controller matrices K_k can be constructed independently of angle setpoints and line parameters, using power and voltage setpoints and the inductance-resistance ratio ρ.This addresses the practical obstacle that inverter operators may not know network line parameters and receive setpoints as powers and voltage magnitudes rather than relative angles.Source: C. Decentralized implementation for… · p. 6
  • Stability analysis treats setpoints as constant and consistent with the network power-flow equations, since a centralized decision maker is assumed to update them relatively infrequently.The passage notes that this fixed-setpoint assumption is also used in analysis of established inverter control algorithms such as droop control.Source: C. Decentralized implementation for… · p. 6

D. Droop-like behavior

  • Classic power droop adjusts inverter frequency and voltage in proportion to local active- and reactive-power imbalance, and its dynamics can be modeled as coupled Kuramoto oscillators.The paper notes that projecting its closed-loop dynamics onto the circle recovers the nonlinear Kuramoto model.Source: D. Droop-like behavior · p. 7
  • Near the nominal operating point, the controller is equivalent to standard active-power droop with a rescaled power setpoint only for inductive lines and constant voltage magnitude.The passage also restricts the established synchronization and active-power-sharing properties to operation near that point.Source: D. Droop-like behavior · p. 7
  • Unlike standard droop during large transients, the proposed controller permits an almost globally asymptotically stable grid for consistent setpoints from almost all initial conditions.The passage attributes this guarantee to Theorem 1 and Proposition 8.Source: D. Droop-like behavior · p. 7
  • The controller is expected to inherit robustness to inconsistent setpoints similar to droop-like methods, but this remains unproven; preliminary simulations support its droop characteristics.A rigorous analysis is identified as future work, while preliminary simulation results are cited in [53].Source: D. Droop-like behavior · p. 7

IV. ANALYSIS OF THE CLOSED-LOOP DYNAMICS · A. Basic definitions · B. Stability condition

The section defines almost global asymptotic stability with respect to a set and gives a condition under which the closed-loop dynamics achieve it. In the power-systems setting, the condition supports almost-global stability of a pre-specified power-flow solution.

IV. ANALYSIS OF THE CLOSED-LOOP DYNAMICS

  • Under a tractable stability condition on angle setpoints and grid parameters, the closed-loop dynamics are almost globally asymptotically stable from almost all initial conditions.The result concerns the control objectives of the closed-loop dynamics (13).Source: §IV · p. 7

A. Basic definitions

  • Almost global asymptotic stability with respect to a set combines stability with attraction from all initial conditions except a zero-measure set.The paper defines this for a set that need not be compact.Source: A. Basic definitions · pp. 7–8

B. Stability condition

  • The stability condition requires a connected graph and constraints on the angle setpoints and design parameters.For a connected graph, the inequality can be satisfied with a proper choice of control gains and sufficiently small angle setpoints.Source: B. Stability condition · p. 8
  • The condition expresses a heterogeneity-connectivity trade-off for the desired steady state and is closely related to stability conditions for Kuramoto oscillators.The graph’s second-smallest Laplacian eigenvalue is identified as a relevant connectivity quantity.Source: B. Stability condition · p. 8
  • With purely inductive transmission lines, the stability condition specializes to a condition involving reactive branch powers at the steady-state power-flow solution.The supplied passage introduces this specialization but does not provide the full condition.Source: B. Stability condition · p. 8
  • Together with Theorem 1, Condition (3) ensures almost-global stability of a pre-specified power-flow solution, including solutions with nonzero power flows.Virtual oscillator control can achieve global synchronization on connected graphs but cannot be dispatched to track pre-specified power and voltage setpoints –,.Source: B. Stability condition · p. 8

C. Main result: Almost global asymptotic stability

Under Condition 1, the closed-loop system is almost globally asymptotically stable with respect to the target sets for phase, frequency, and magnitude. The proof combines convergence to the correct relative phases and magnitudes with stability of the target set.

C. Main result: Almost global asymptotic stability

  • The controller drives the closed-loop trajectories to the synchronous steady-state with the correct angle set-points and amplitudes.For the power-systems application, the angle set-points correspond to power flows.Source: C. Main result: Almost global… · p. 8
  • Under Condition 1, system (13) is almost globally asymptotically stable with respect to the target set T_f,ω0.This target set specifies the control objectives (3), (4), and (5).Source: C. Main result: Almost global… · p. 8
  • In a rotating frame, the relative-magnitude and phase set S is globally exponentially stable, while magnitude set A is asymptotically stable on S.The origin is the only equilibrium outside the target set and has a zero-measure region of attraction; trajectories remain bounded, so almost all approach S and then A.Source: C. Main result: Almost global… · p. 8
  • A family of nested invariant sets is constructed to establish stability of the target set.These sets are used in Theorem 4.Source: C. Main result: Almost global… · p. 8

1) Proof of attractivity:

Under Condition 1, the rotating-frame dynamics are almost globally uniformly attractive to the target set. The proof combines exponential convergence to the correct relative phases with magnitude convergence, except from a zero-measure set.

1) Proof of attractivity:

  • Under Condition 1, V(v̄) establishes global exponential stability of the subspace S for dynamics (30).The proof identifies V(v̄) as a Lyapunov function for S.Source: 1) Proof of attractivity · p. 10
  • The equilibrium v̄ = 0 is unstable, and its attractive set Z has zero Lebesgue measure.Thus, the excluded initial conditions in the almost-global attractivity result form a zero-measure set.Source: 1) Proof of attractivity · p. 10
  • Under Condition 1, every trajectory remains in an invariant compact set for each oscillator.The bound holds for every initial condition.Source: 1) Proof of attractivity · p. 11
  • For every initial condition in S \ {0}, dynamics (30) are asymptotically stable with respect to T̄_f,0.The proof establishes that S \ {0} is invariant under the dynamics.Source: 1) Proof of attractivity · p. 11
  • All trajectories converge exponentially to S, setting the correct relative phase angles; outside Z, magnitudes converge once states are sufficiently close to S.The proof uses this two-stage convergence to establish almost global attractivity.Source: 1) Proof of attractivity · p. 11
  • For every ε > 0, (42) holds for all initial conditions v̄₀ ∈ Rⁿ\Z with T = TεH + TμS; since Z has zero measure, convergence holds for almost every initial condition.Source: 1) Proof of attractivity · p. 12

2) Proof of stability:

Under Condition 1, the target set Tf,ω0 is stable, completing the proof of Theorem 1 when combined with Theorem 3’s attractivity result. The stability argument uses invariant neighborhoods around the target set.

Proof of stability

  • For Condition 1, each M(γA) is non-empty and invariant, contains T ¯ f,0 in its interior for positive γA, and is nested as γA increases.Proposition 11 states these properties for γA ∈ R(0, 1/2 v⋆ min) and γA ∈ R[0, 1/2 v⋆ min).Source: 2) Proof of stability · p. 12
  • Under Condition 1, the dynamical system (13) is stable with respect to Tf,ω0.Theorem 4 establishes stability of the target set.Source: 2) Proof of stability · p. 12
  • Compact invariant neighborhoods containing T ¯ f,0 can be chosen inside any ε-neighborhood, ensuring trajectories starting sufficiently close remain within that ε-neighborhood.The proof uses the non-empty interior and invariance of M(γA) to establish stability in the rotating coordinate frame.Source: 2) Proof of stability · p. 12
  • Theorem 1 follows directly from Definition 5 and the results of Theorems 3 and 4.Source: 2) Proof of stability · p. 12

V. POWER SYSTEMS CASE STUDY

In a three-inverter grid case study, feasible dispatches achieve the prescribed relative phase, frequency, voltage magnitudes, and power injections. After an inverter changes its active-power set-point so the dispatch is infeasible, the system remains stable and synchronous, and other inverters share the adjustment.

V. POWER SYSTEMS CASE STUDY

  • The case study uses three grid-connected inverters with controllers whose matrices are constructed using only power and voltage-magnitude set-points.The simulations include a black-start from near-zero voltages with zero power set-points, followed by dispatches to feasible and infeasible set-points.Source: §V · pp. 12–13
  • Despite an infeasible set-point change by inverter 3, the system remains stable and synchronous, with inverters 2 and 3 sharing the power reduction needed to accommodate the reduced load.The change removes the guarantees of Theorem 1; the observed response is consistent with the closed-loop droop-like behavior described in Proposition 4.Source: Figure 6 · p. 13, §V · p. 13
  • After feasible dispatch, the inverters synchronize to the prescribed relative phase and frequency and reach their voltage-magnitude and power-injection set-points.The selected parameters, η = 0.0015 and α = 0.01, satisfy condition (29).Source: §V · p. 13

VI. CONCLUSION AND OUTLOOK

The paper proposes decentralized control for coupled oscillators and grid-connected inverters, with almost-global stability guarantees for synchronization and desired power injections. The conclusion identifies restrictive assumptions and open questions in extending, analyzing, and experimentally validating the controller.

VI. CONCLUSION AND OUTLOOK

  • The decentralized method ensures synchronization and almost-global stability of coupled oscillators and inverter systems at solutions with desired power injections.The paper also notes almost-global asymptotic stability guarantees for the proposed method.Source: §VI · p. 13
  • The analysis assumes inverter-only networks with source or load roles and neglects transmission-line dynamics, motivating study of dynamic lines, load-bus models, and synchronous generators.The paper characterizes some technical assumptions as restrictive, particularly for the power-system application.Source: §VI · p. 13
  • Robustness to inconsistent power or magnitude setpoints remains unanalyzed, and analytical conditions guaranteeing both stability and robustness are still lacking.The authors expect beneficial power-sharing behavior based on droop-like behavior and preliminary simulations, but lack analytical guarantees.Source: §VI · p. 13
  • Experimental validation is ongoing, with preliminary results reported in [53].Source: §VI · p. 13

APPENDIX A PROOFS OF TECHNICAL RESULTS

Appendix A provides proofs of technical results from Sections II-B and IV. The proofs establish instability and measure-zero convergence at the origin, boundedness, and invariant sets for the dynamics.

Proofs of Propositions 3 and 4

  • Equation (25) immediately follows from (7); Proposition 4 expresses each vector as a complex number, rewrites (13) using K from (25), then separates (51)'s real and imaginary parts and applies trigonometric manipulations; similar reformulations yield (26).Source: Appendix A · p. 14

Proof of Proposition 8

  • The origin is unstable, so the set of initial conditions whose trajectories converge to it has zero Lebesgue measure.The Jacobian has eigenvalues with positive real part, and the vector field is continuously differentiable.Source: Appendix A · p. 14

Proof of Proposition 9

  • For every initial condition, each W_k remains bounded for all time under the dynamics (30).The proof shows dW_k/dt < 0 above a threshold, yielding a constant upper bound on W_k.Source: Appendix A · p. 14

Proof of Proposition 11

  • For γ_A in [0, 1/2 v⋆_min), M(γ_A) is invariant, contains a neighborhood of T̄_f,0 when γ_A > 0, and is nested as γ_A increases.The proof constructs γ_S(γ_A) as a strictly increasing function and establishes invariance of the component sets.Source: Appendix A · pp. 14–15

APPENDIX B CLARKE TRANSFORMATION

The appendix defines an amplitude-preserving Clarke transformation for three-phase signals and states that the resulting two-dimensional signal contains the same information as the original signal.

APPENDIX B CLARKE TRANSFORMATION

  • The amplitude-preserving Clarke transformation is defined for a three-phase signal vabc = (va, vb, vc).Source: Appendix B · p. 15
  • The two-dimensional signal vαβ := (vα, vβ) contains the same information as the original three-phase signal vabc.Source: Appendix B · p. 15

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