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Small-Signal Analysis of the Microgrid Secondary Control Considering a Communication Time Delay
Ernane A. Alves Coelho, Dan Wu, Josep M. Guerrero, Juan C. Vasquez, Tomislav Dragicevic, Cedomir Stefanovic, Petar Popovski
TL;DR
The paper addresses small-signal stability analysis of an islanded microgrid whose secondary frequency restoration uses consensus over delayed communication. It builds and validates a delay-differential model combining inverter controls, network topology, and time delay, finding stable behavior over the considered delay range while acknowledging a constant-delay scope boundary.
Problem
The problem is to analyze stability and frequency restoration in islanded parallel-inverter microgrids when secondary-control communication has a time delay.
Method
The method builds a DDE small-signal model combining droop-controlled inverter dynamics, consensus-based secondary control, directed communication topology, and a single constant delay.
Results
The model agrees well with simulations and experiments, and the system remains stable across the considered communication-delay variation.
Takeaways & Limitations
The analysis supports frequency restoration while maintaining equitable active-power sharing through the distributed secondary controller.
Takeaways & Limitations
The communication analysis assumes a single constant delay; realistic sampling rates and packet loss were reported not to affect secondary-control performance in the studied microgrid.
Abstract
from arXiv · showhide
This paper presents a small-signal analysis of an islanded microgrid composed of two or more voltage source inverters connected in parallel. The primary control of each inverter is integrated through internal current and voltage loops using PR compensators, a virtual impedance, and an external power controller based on frequency and voltage droops. The frequency restoration function is implemented at the secondary control level, which executes a consensus algorithm that consists of a load-frequency control and a single time delay communication network. The consensus network consists of a time-invariant directed graph and the output power of each inverter is the information shared among the units, which is affected by the time delay. The proposed small-signal model is validated through simulation results and experimental results. A root locus analysis is presented that shows the behavior of the system considering control parameters and time delay variation.
I. INTRODUCTION
The paper develops a distributed small-signal framework for islanded microgrids, extending droop-based primary control with consensus-based secondary frequency restoration over delayed communication. It targets stability analysis involving controller parameters, network topology, and communication delay.
- Islanded microgrids use hierarchical primary, secondary, and tertiary control, with primary droop control sharing power while allowing voltage and frequency deviations.
- Secondary control restores frequency and regulates voltage, requiring communication and supporting decentralized cooperative operation among inverter agents.
- The proposed model represents secondary frequency restoration with a consensus algorithm, a graph-described data network, and a single communication time delay.
- The approach is organized around a DDE model that supports stability studies over primary and secondary control parameters, data-network topology, and communication delay.
- Primary inverter control combines PR-based current and voltage loops, virtual impedance, and frequency and voltage droops using local measurements.
III. SMALL-SIGNAL ANALYSIS
The model treats communication timing as a constant delay in the small-signal analysis and compares its predictions with simulations under realistic communication impairments. The supplied results indicate negligible effects from realistic packet loss on the analyzed behavior.
- The communication model assumes a constant delay for all data links, representing buffered real-time digital communication and the upper bound of allowed delay.
- For realistic packet-loss probabilities, simulations show no significant difference from the small-signal model that omits packet loss.
A. Small-Signal Model for each Inverter Under The Primary Control
The inverter-level model linearizes the primary-control dynamics around an equilibrium point while treating the power reference as a secondary-control input. It produces state equations for frequency and inverter-voltage components.
- The primary-control equations are linearized around equilibrium values for frequency, voltage amplitude, active power, and reactive power.
- Because Pref is variable, the linearized model includes both the power-reference deviation and its derivative.
- The inverter voltage is represented in direct-axis and quadrature-axis coordinates and linearized around equilibrium voltage components.
- The voltage phase δ is the absolute inverter voltage phase, so the formulation includes a redundant state and an eigenvalue at the origin.
- The resulting state equation describes deviations in frequency and inverter-voltage components near equilibrium, with inputs from apparent-power and reference-power deviations.
B. Small-Signal Model for The Entire Microgrid Under The Primary Control
The microgrid-level model combines inverter state equations with a linearized admittance representation of three parallel inverters connected to a common load bus. It then forms the complete primary-control state model around an equilibrium point.
- The analyzed network contains three inverters connected in parallel to a common load bus, although the modeling principle applies to an arbitrary number of nodes.
- The network model neglects frequency variation in frequency-dependent loads and treats network reactances as constant, with lower precision expected over wider frequency ranges.
- The islanded nodal admittance equation is obtained by relating the microgrid to a regular networked microgrid with zeroed gray admittances and no inverter at the load bus.
- Because the load-bus node has no power injection, it is eliminated, yielding a reduced admittance equation for the three-inverter system.
- The complex network equation is converted to real form and combined with linearized inverter power relations to obtain the system state equation.
- The complete model describes near-equilibrium behavior for initial conditions and active-power-reference deviations, reducing to the primary-control case when those inputs are null.
C. Small-Signal Model for The Entire Microgrid Under The Secondary Control
The secondary controller modifies each inverter’s power reference to restore nominal frequency while maintaining equitable active-power sharing. Its consensus law is determined by the directed communication graph and integral gain.
- Secondary control modifies each inverter’s power reference to restore nominal frequency despite load variation while maintaining equitable active-power sharing.
- The islanded microgrid’s primary droop controllers provide power sharing, but frequency and voltage may deviate from nominal values under load changes.
- Average consensus preserves equitable active-power sharing under load variation but does not guarantee nominal-frequency operation.
- The implemented secondary law combines the consensus objective with nominal-frequency restoration through an integral gain k_pri in each inverter.
- The summation terms in the distributed law are selected by the directed graph’s edges, with each vertex assumed to have at least one incoming edge.
- Changing the graph requires changing its degree and adjacency matrices, while self-loops are excluded from the network model.
D. Time Delay on The Secondary Control
The secondary-control communication delay affects only exchanged inverter output-power measurements, so the delay is inserted into the distributed controller before forming the complete small-signal model.
- A constant delay t_d is assigned to every data-communication edge and replaces the no-delay distributed-controller equation.
- Substituting the delayed controller into the primary-control model eliminates the input-derivative term before algebraic reduction.
- The local state vectors use feedback internal to each inverter, whereas only measured output power is transmitted between vertices and delayed by t_d.
- The network relations connect deviations from average active power with each inverter’s instantaneous power.
E. Small-Signal Model for The Entire System - a DDE Model
The complete small-signal model is expressed as a single-delay delay differential equation, whose characteristic equation has infinitely many eigenvalues. Because Lambert W does not apply to the model matrices, the spectrum is computed numerically.
- The complete state vector combines inverter, average-power, and power-reference deviations in the whole-system small-signal model.
- The model has the single-delay DDE form ∆Ẋ(t) = A∆X(t) + A_d∆X(t − t_d), with initial history function φ(t).
- The characteristic equation has infinitely many solutions, corresponding to an infinite number of eigenvalues.
- Lambert W analysis is unavailable because the system matrices A and A_d are not simultaneously triangularizable.
- The DDE spectrum is therefore obtained using a numerical approach implemented with Matlab’s dde23 function.
IV. SIMULATION AND EXPERIMENTAL RESULTS
The proposed small-signal model is evaluated against simulations and laboratory experiments during a load-transient scenario with communication delay. Results show close agreement across model, simulation, and experiment, while root-locus analysis examines delay-dependent stability.
- Validation setup: The evaluation compares the DDE model with ideal-inverter simulation, detailed-inverter simulation, and experimental results.Model, Sim1, Sim2, and Exp represent progressively more detailed validation stages, including a laboratory prototype.
- Validation setup: The transient starts from the Load 1 equilibrium and moves to a new equilibrium after Load 2 is connected in parallel.Initial conditions and equilibrium constants are obtained through load-flow calculations.
- Frequency response: The model and ideal-inverter simulation agree perfectly, while agreement remains very good with detailed inverter dynamics and experimental measurements.These results support neglecting internal inverter dynamics when studying interactions between nodes in the stability analysis.
- Frequency response: At td = 200 ms, the system nearly reaches its new equilibrium frequency before secondary control begins restoring frequency toward the nominal value.Primary control responds rapidly after the load change and provides load sharing, while secondary control performs restoration.
- Root-locus analysis: The root locus is numerically approximated for communication delays from 0 to 200 ms and focuses on the rightmost eigenvalues.The no-delay spectrum is a finite set of eigenvalues, whereas the delayed system requires numerical treatment.
- Root-locus analysis: The system remains stable across the considered delay range, although larger delays move low-frequency modes toward the imaginary axis and reduce exponential decay.The eigenvalues do not cross the imaginary axis over the analyzed range.
V. EXTENSION OF THE PROPOSED MODEL
The proposed model extends straightforwardly from three to twelve inverters, preserving the analysis for larger systems. A twelve-inverter example with a 200 ms communication delay shows frequency restoration under a regular communication network.
- Model extension: The model extends to additional inverters by increasing the model order by 5 for each new inverter.The mathematical development and validation use a three-inverter system, while larger systems are represented by direct extension.
- Twelve-inverter example: A twelve-inverter example uses the same droop gains and distinct transmission-line inductances ranging from 0.95 to 3.6 mH.The example considers a communication time delay td of 200 ms.
- Twelve-inverter example: The twelve-inverter frequency-restoration result is obtained from a 60th-order model after connecting Load2(40Ω) in parallel with Load1(40Ω).The regular communication network includes all edges of the twelve-vertex graph, implying fast consensus convergence.
VI. CONSTANT TIME DELAY AND PACKET LOSS IN A COMMUNICATION SYSTEM
The communication analysis models practical digital links using buffering and packet ordering, then evaluates a twelve-inverter system at 50 Hz with packet loss. Its frequency response agrees well with the idealized delayed-link result.
- Communication implementation: Buffering and sequence-number or timestamp inspection can enforce equal packet delay and ordered processing across communication links.Packets may carry measurement and control information, including sequence numbers or timestamps.
- Communication implementation: Off-the-shelf WiFi experiments indicate secondary-control frequencies of approximately 50–100 Hz for about 10 stations with all-to-all scheduled access.Measurement packets last less than 1 ms, while packet generation occurs at roughly 1–5 ms intervals.
- Realistic-link evaluation: The twelve-inverter system was simulated with a 50 Hz secondary-control sampling rate and packet-loss probability of 10−2.These settings were chosen to represent a communication setup supported by off-the-shelf equipment and 2 Mb/s WiFi links in rural scenarios.
- Realistic-link evaluation: The realistic communication result shows good agreement with the 200 ms delayed-link result and no significant difference in system behavior.The comparison concerns the angular frequency of each inverter in the twelve-inverter system.
VII. CONCLUSION
The paper develops a small-signal model for distributed secondary frequency restoration with a single constant communication delay and configurable network topology. The model supports stability analysis and finds no instability from the stated delay in the presented system.
- Conclusion: The work analyzes an islanded microgrid using primary droop control and distributed secondary frequency restoration over a single constant-delay data link.The secondary controller uses a consensus algorithm.
- Conclusion: The proposed small-signal model allows stability analysis while accommodating different data-network configurations.The network configurations can be set into the model directly.
- Conclusion: A single constant communication delay does not cause instability over the presented system.This is the paper's stated stability conclusion for the studied microgrid system.
- Conclusion: The study is presented as a starting point for future research on time delays in secondary control with more realistic communication links.The constant-delay assumption is described as reasonable for the studied actual communication system.
- Conclusion: Typical sampling rates and packet loss in the studied communication systems do not affect secondary-control performance in the studied microgrid.This scope is limited to the communication conditions examined in the work.