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Synchronization and Power Sharing for Droop-Controlled Inverters in Islanded Microgrids
John W. Simpson-Porco, Florian Dörfler, Francesco Bullo
TL;DR
The paper addresses synchronization, power sharing, and secondary frequency control for droop-controlled inverters in islanded microgrids. It maps the network to a generalized Kuramoto model and introduces distributed control methods, obtaining synchronization conditions, proportional sharing, load-service bounds, and frequency regulation that preserves sharing.
Problem
The paper studies how droop-controlled inverters can achieve synchronization, power sharing, and frequency regulation in islanded microgrids.
Method
The authors cast the microgrid equations as a generalized Kuramoto model and combine coupled-oscillator theory with distributed averaging control.
Results
The paper derives necessary and sufficient conditions for a unique locally exponentially stable synchronized solution, proportional power-sharing designs, serviceable-load bounds, and distributed frequency regulation preserving sharing.
Takeaways & Limitations
Distributed averaging-based integral control regulates nominal frequency while retaining the proportional power-sharing properties of primary droop control.
Takeaways & Limitations
The results do not provide nonlinear reactive-power-sharing analysis and do not cover strongly mixed line conditions, where the adopted control laws are inappropriate.
Abstract
from arXiv · showhide
Motivated by the recent and growing interest in smart grid technology, we study the operation of DC/AC inverters in an inductive microgrid. We show that a network of loads and DC/AC inverters equipped with power-frequency droop controllers can be cast as a Kuramoto model of phase-coupled oscillators. This novel description, together with results from the theory of coupled oscillators, allows us to characterize the behavior of the network of inverters and loads. Specifically, we provide a necessary and sufficient condition for the existence of a synchronized solution that is unique and locally exponentially stable. We present a selection of controller gains leading to a desirable sharing of power among the inverters, and specify the set of loads which can be serviced without violating given actuation constraints. Moreover, we propose a distributed integral controller based on averaging algorithms which dynamically regulates the system frequency in the presence of a time-varying load. Remarkably, this distributed-averaging integral controller has the additional property that it maintains the power sharing properties of the primary droop controller. Our results hold without assumptions on identical line characteristics or voltage magnitudes.
1 Introduction
The paper frames islanded microgrid inverter control as a synchronization, power-sharing, and frequency-regulation problem. It recasts droop-controlled inverter networks as coupled oscillators and develops conditions and controllers addressing these goals.
- Motivation: Islanded microgrids use voltage-sourced inverters to interface heterogeneous generation and operate independently from the larger power system.Inverters must support synchronization, security, power balance, and load sharing during islanded operation.
- Droop control: Frequency-droop control changes each inverter’s frequency according to active-power injection, with n_i > 0 as the droop coefficient.For inductive lines, this controller balances active-power demands through an inverse frequency–power relation.
- Literature gap: The literature lacks nonlinear conditions for synchronous steady states, convergence guarantees, and performance guarantees for the conventional droop law.Existing stability results often concern two inverters, use linearization, or rely on additional assumptions.
- Core approach: The paper equivalently casts frequency-droop microgrid equations as a generalized Kuramoto model of phase-coupled oscillators.The oscillator framework describes phase dynamics through natural frequencies, time constants, and coupling strengths.
- Contributions: The paper provides necessary and sufficient synchronization conditions, proportional power-sharing controller choices, serviceable-load bounds, and distributed frequency-restoration control.The distributed integral controller regulates frequency while preserving proportional sharing, and the results extend to generic acyclic inverter–load interconnections.
2 Problem Setup for Microgrid Analysis
The paper models an islanded inductive microgrid as an inverter–load network with controlled voltage sources, phase angles, and power-flow constraints. The adopted inverter approximation and droop law rely on predominantly inductive network behavior.
- Inverter modeling: An inverter is modeled as a controlled voltage source behind a reactance, a standard approximation widely used in microgrid research.The model represents inverter output impedance through the reactance behind the controlled voltage source.
- Network model: The islanded microgrid is represented by a weighted graph containing inverter and load nodes connected through inductive lines.The bus admittance matrix is purely imaginary, and inverter output impedance is absorbed into line susceptances.
- Electrical variables: Each node receives a harmonic voltage signal with nominal angular frequency ω* > 0, amplitude E_i > 0, and phase angle θ_i.Inverters measure rolling time-averaged active-power injection and frequency, while inverter injections are constrained by ratings.
- Model scope: The frequency-droop control law is inappropriate when resistive effects are not dominated by highly inductive inverter output impedance.The paper identifies strongly mixed line conditions as outside the applicability of the adopted control laws.
3 Analysis of Frequency-Droop Control
The frequency-droop-controlled microgrid is equivalently represented as a generalized Kuramoto model, enabling nonlinear conditions for unique, locally exponentially stable synchronization. Synchronization is characterized by feasible edge flows, with robust extensions for uncertain voltages and line susceptances.
- Model equivalence: The droop-controlled network is equivalent to a generalized Kuramoto model with inverter time constants, natural frequencies, phases, and coupling weights.The equivalence uses inverse droop coefficients as oscillator time constants and fixed voltage magnitudes with couplings a_ij = E_iE_j|Y_ij|.
- Synchronization condition: A stable synchronized solution exists uniquely and locally exponentially when the network's active power flow is feasible.The theorem gives equivalent conditions between synchronized-solution existence and flow feasibility on an acyclic network.
- Synchronization condition: The synchronized frequency equals the scaled average power imbalance, while phase differences encode the feasible edge power flows.The solution has θ*(t) = θ_0 + ω_sync t1_n modulo 2π, with ω_sync = ω_avg and sin(B^Tθ*) determined by ξ and the coupling weights.
- Synchronization condition: Feasibility requires every edge power flow to remain below its physical maximum coupling a_ij = E_iE_j|Y_ij|.The condition is summarized by Γ = max_(i,j)∈E(ξ_ij/a_ij) < 1, with Γ = sin(γ).
- Stability: The analysis provides a lower bound on local exponential convergence using the weighted network Laplacian and inverse droop coefficients.The bound uses λ_2(D^-1L_red(θ*)) and further bounds λ_2(L(θ*)) by λ_2(L)cos(γ).
- Robustness: The stability condition is robust to bounded voltage and line-susceptance uncertainty when worst-case active power flows remain feasible.The robustified condition covers all admissible voltage magnitudes and line susceptances through worst-case flow feasibility.
- Robustness: The results assume purely inductive lines, although the stable synchronous solution persists under sufficiently small line conductances.The persistence follows from continuity of eigenvalues and is also illustrated in simulation.
4 Power Sharing and Actuation Constraints
The paper addresses how to select droop coefficients so synchronized inverters share power while respecting individual actuation limits. With proportional coefficients, feasible total-load conditions are equivalent to inverter injection constraints, and sharing follows inverter ratings.
- Controller Selection: The necessary-and-sufficient synchronization condition alone does not directly specify how to choose control parameters P*_i and D_i.The power-sharing construction addresses this controller-selection gap under actuation constraints.
- Controller Selection: Proportional droop coefficients are selected using the ratios P*_i/D_i and P*_i/P̄_i across inverters.The definition requires these ratios to agree for all inverter pairs.
- Power Flow Constraints: Theorem 7 makes inverter injection constraints equivalent to a corresponding bound on the total load when droop coefficients are selected proportionally.The constraints are 0 ≤ P_e,i ≤ P̄_i for every inverter, alongside the stated total-load inequality.
- Power Sharing: P_e,i/P̄_i = P_e,j/P̄_j, so inverters share the total load proportionally to their power ratings.This proportional-sharing result follows as a corollary for parallel inverters supplying a single load.
- Scope and Design: The power-sharing theorem does not depend on network voltage magnitudes or line admittances.However, selecting the droop coefficients requires global knowledge, making the design centralized despite decentralized implementation.
5 Distributed PI Control in Microgrids
The paper introduces distributed-averaging PI control to restore nominal frequency without relying on primary–secondary time-scale separation. The DAPI controller is locally exponentially stable under the droop stability condition and preserves proportional power sharing.
- Motivation: Conventional secondary integral control can be slow for large droop coefficients and may fail to regulate frequency dynamically under time-varying loads.The conventional approach also relies implicitly on time-scale separation between primary droop and secondary integral loops.
- DAPI Design: The DAPI controller uses inverter communication and distributed averaging to regulate frequency under large and rapid load variations.Its communication graph is weighted, undirected, connected, and represented by a Laplacian matrix.
- Scope: The presented results extend to discrete time and asynchronous communication.This extension is stated as an additional result associated with the distributed controller framework.
- Stability: The DAPI equilibrium is locally exponentially stable and unique exactly when the droop stability condition holds, under the stated acyclic-network and communication assumptions.The theorem assumes D_i > 0, k_i > 0, admissible nominal injections, and a connected communication Laplacian.
- Power Sharing: When droop coefficients are selected proportionally, DAPI preserves the primary droop controller’s proportional power-sharing property.Thus frequency restoration does not remove the established proportional-sharing behavior.
- Frequency Restoration: At the stable DAPI equilibrium, the network synchronizes to the nominal frequency ω* rather than retaining the primary droop frequency deviation.The equilibrium has zero synchronization frequency in the transformed coordinates.
6 Simulation Study
The simulation evaluates the proposed DAPI controller on two parallel inverters supplying a variable load. It demonstrates rapid frequency regulation and shows that gain selection can improve transient response.
- Simulation setup: The simulation models two parallel inverters supplying a variable load with DAPI-controlled closed-loop dynamics.The load changes at t ∈{2s, 4s}.
- Simulation parameters: Table 1 reports the parameter values used for the Figure 5 simulation, with resistances chosen at a resistance/reactance ratio of one half.
- Frequency regulation: The proposed DAPI controller quickly regulates system frequency despite rapid load changes.Figure 5 reports local-frequency spikes during the load transitions.
- Transient response: Increasing the gains k_i can further suppress frequency spikes caused by rapid load changes.The additional tuning freedom permits primary droop coefficients D_i of 10^3 W·s, compared with roughly 10^5 W·s typically used in the literature.
7 Conclusions
The paper applies coupled-oscillator, classical power-system, and multi-agent methods to synchronization, power sharing, and secondary control of droop-controlled inverters. It identifies reactive-power-sharing analysis and general line conditions as unresolved boundaries.
- The analysis does not address nonlinear reactive-power sharing because voltage-droop analysis does not yield simple physically meaningful algebraic existence conditions.
- For strongly mixed line conditions with Im(Y_ij) ≃ Re(Y_ij), neither analyzed control law is appropriate.
- A provably functional control strategy for general interconnections and line conditions remains an open problem.
A Proof of Theorem 8
The proof establishes equilibrium uniqueness and local exponential stability for the DAPI-controlled network by transforming its linearized dynamics into a generalized eigenvalue problem. It then shows that steady-state inverter injections preserve primary droop power sharing.
- Closed-loop formulation: DAPI closed-loop dynamics are written in error coordinates e_p,i(t) ≜ p_i(t) − D_iω_avg and vector form.
- Equilibrium characterization: The equilibrium conditions reduce to e_p = e_p* = 0 and P_e − P_e = 0, with a unique θ* modulo rotational symmetry when condition (8) holds.
- Stability analysis: Local exponential stability reduces to the generalized eigenvalue problem −X_1X_2v = λZv after factoring the reduced Jacobian as J(θ*,e_p*) = −Z^-1X.
- Perturbed problem: For ε > 0, the transformed generalized eigenvalue problem has real and negative eigenvalues modulo rotational symmetry.
- Unperturbed problem: In the unperturbed case, ker(X_1X_2) = ker(X_2) because X_2v never lies in ker(X_1).
- Power sharing: The DAPI steady-state inverter injection P_e,i = P*_i − ω_avgD_i matches the steady-state injection under primary droop control.