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Secondary Frequency and Voltage Control of Islanded Microgrids via Distributed Averaging

John W. Simpson-Porco, Qobad Shafiee, Florian Dorfler, Juan C. Vasquez, Josep M. Guerrero, Francesco Bullo

arXiv:1504.06784v1math.OC

TL;DR

Islanded microgrids need secondary control that regulates frequency and voltage while sharing active and reactive power without centralized coordination. The paper develops DAPI controllers using distributed averaging and neighbor communication, achieving frequency regulation with active-power sharing and a tunable voltage–reactive-power trade-off. Stability analysis and experiments validate the designs, including communication failures and plug-and-play operation.

  • Problem

    Existing strategies do not jointly provide flexible frequency and voltage regulation with precise active and reactive power sharing among non-identical DGs without centralized control.

  • Method

    The paper combines decentralized proportional droop and integral control with distributed averaging over sparse neighbor communication to form DAPI frequency and voltage controllers.

  • Results

    The controllers regulate frequency while sharing active power, and voltage-controller tuning provides a compromise between voltage regulation and reactive power sharing, supported by stability analysis and experiments.

  • Takeaways & Limitations

    The distributed controllers support plug-and-play microgrid control without requiring a central controller or precise voltage and reactive-power objectives simultaneously.

  • Takeaways & Limitations

    Precise voltage regulation and reactive power sharing are generally conflicting objectives, with achievable sharing depending on voltage limits and line-reactance homogeneity.

Abstract

from arXiv · show

In this work we present new distributed controllers for secondary frequency and voltage control in islanded microgrids. Inspired by techniques from cooperative control, the proposed controllers use localized information and nearest-neighbor communication to collectively perform secondary control actions. The frequency controller rapidly regulates the microgrid frequency to its nominal value while maintaining active power sharing among the distributed generators. Tuning of the voltage controller provides a simple and intuitive trade-off between the conflicting goals of voltage regulation and reactive power sharing. Our designs require no knowledge of the microgrid topology, impedances or loads. The distributed architecture allows for flexibility and redundancy, and eliminates the need for a central microgrid controller. We provide a voltage stability analysis and present extensive experimental results validating our designs, verifying robust performance under communication failure and during plug-and-play operation.

I. INTRODUCTION

The paper introduces distributed secondary controllers for islanded microgrids, addressing the challenge of regulating frequency and voltage while sharing power without centralized control. The approach combines decentralized droop and integral control with sparse neighbor communication, and is validated analytically and experimentally.

  • Primary droop control stabilizes the microgrid and establishes power sharing but causes steady-state frequency and bus-voltage deviations.Secondary control is introduced to remove these deviations.
  • Existing secondary-control strategies do not jointly guarantee frequency and voltage regulation with precise active and reactive power sharing among non-identical DGs.Communication between DGs is identified as a key ingredient for achieving these goals without a centralized architecture.
  • The proposed DAPI controllers combine decentralized proportional droop and integral control with distributed averaging over neighboring DG units.The architecture uses sparse communication and decentralized control actions.
  • The frequency controller targets precise frequency regulation and active power sharing, while the voltage controller offers a tunable compromise between voltage regulation and reactive power sharing.The paper explicitly frames voltage regulation and reactive power sharing as conflicting objectives.
  • The paper provides voltage stability analysis and extensive experiments, including tests under communication-link failures and plug-and-play operation.These results extend the theoretical analysis to practical operating conditions.
  • The paper organizes its contributions around reviews of primary and secondary control, controller design, stability analysis, experiments, and concluding remarks.The main contributions are presented in Sections III–VI.

A. Problem Setup and Review of Power Flow

This section reviews inverter-based microgrid power flow and primary droop control before motivating secondary integral control. Droop control provides stabilization and proportional sharing, but its steady-state frequency differs from nominal and related secondary strategies vary in communication and tuning requirements.

  • A. Problem Setup and Review of Power Flow: Microgrids contain buses that are distributed generators or loads, with active and reactive injections determined by voltage magnitudes, phase angles, and line reactances.The setup considers inductive connections between buses.
  • B. Review of Primary Droop Control: Primary droop control stabilizes islanded microgrids and establishes proportional load sharing through local inverter control loops.The inverters operate as grid-forming voltage-source inverters with controlled frequencies and voltage magnitudes.
  • B. Review of Primary Droop Control: The steady-state network frequency under primary droop control differs from the nominal frequency because it depends on the total active power load.The cited expression uses P0 as the total active power load in the microgrid.
  • B. Review of Primary Droop Control: Secondary integral controllers remove the steady-state frequency and voltage deviations generated by primary droop control.Secondary control methods span centralized and decentralized architectures.
  • B. Review of Primary Droop Control: Some distributed secondary strategies require all inverters to communicate with one another and finely tuned gains to maintain active power sharing.These requirements create dense communication and tuning burdens.

2) Voltage Regulation:

Secondary voltage control faces a fundamental trade-off: precise voltage regulation can impair reactive power sharing, while enforcing sharing can worsen voltage disparities. The achievable compromise depends on voltage limits and line-reactance homogeneity.

  • Voltage Regulation: Line impedance differences prevent even identical parallel inverters from sharing reactive power accurately under primary droop control.The example considers two identical DGs connected to a common bus through unequal reactances, with X01 > X02.
  • Voltage Regulation: Standard voltage-regulating secondary control restores both inverter voltages to the common rating but worsens their reactive-power-sharing error.The figure shows Q′1 < Q1 and Q′2 > Q2 after voltage regulation.
  • Reactive Power Sharing: Power-sharing secondary control can make identical inverters inject equal reactive power, but their resulting voltages become more dissimilar than under primary control.The shifting of droop characteristics is not unique, so multiple actions can achieve power sharing.
  • Trade-off: Except under special circumstances, precise voltage regulation produces large reactive-power-sharing errors, whereas reactive-power sharing alone does not uniquely determine DG bus voltages.Sharing accuracy depends on upper and lower DG-voltage limits and on the homogeneity of line reactances.

IV. DISTRIBUTED AVERAGING PROPORTIONAL INTEGRAL (DAPI) CONTROLLERS FOR MICROGRIDS

The paper introduces distributed averaging as the communication mechanism for DAPI microgrid controllers. Continuous-time averaging lets each DG update its variable from neighboring values and converge collectively when the communication network is connected.

  • DAPI Controllers: DAPI controllers combine droop and integral control with distributed averaging from multi-agent systems for microgrid secondary control.The approach is introduced as a distributed alternative built around communication among DGs.
  • Communication Network: The communication layer is modeled as a weighted graph whose adjacency matrix encodes which DGs exchange information.The graph uses DGs as nodes, communication links as edges, and weighted adjacency entries to represent connectivity.
  • Distributed Averaging: In continuous-time distributed averaging, each node adjusts its value using neighboring values and associated convex weights.The update evolves toward a weighted average of neighbors, with a time constant determined by the sum of incident communication weights.
  • Distributed Averaging: A connected communication network causes all node variables to converge to a common value.The paper applies this convergence property to microgrid control after reviewing the averaging dynamics.

B. Frequency Regulation and Active Power Sharing

The frequency DAPI controller combines droop and integral control with distributed averaging so DGs regulate frequency while agreeing on a common droop shift for active power sharing. The voltage DAPI controller uses communication and tunable gains to trade off voltage regulation against reactive power sharing.

  • Frequency control: The frequency DAPI controller adds a secondary control variable to standard droop control and uses neighboring-DG communication for distributed averaging.The secondary variable integrates local frequency error, while averaging terms coordinate the DGs.
  • Frequency control: Without communication, each DG can reach nominal frequency but different secondary-control values, producing unequal droop shifts and poor active power sharing.The outcome depends on initial conditions and controller gains.
  • Frequency control: With nonzero diffusive averaging, all DGs agree on the droop shift ω∗−ωss, maintaining active power sharing independently of controller gains.The gains affect transient behavior rather than this steady-state performance.
  • Voltage control: The voltage DAPI controller combines voltage-regulation and reactive-power-sharing terms, with gains βi and bij setting their relative influence.Its communication matrix is an adjacency matrix among the DGs.
  • Voltage control: Pure reactive-power sharing can drive DG voltages far from nominal, whereas pure voltage regulation shares reactive power poorly.A mixed-gain regime provides a compromise between these objectives.
  • Implementation and performance: The communication architecture is customizable for redundancy and need not mirror the electrical topology, while gain time constants tune secondary-control speed.Experiments indicate that primary and secondary control can operate on similar time scales without stability or performance degradation.

V. STABILITY & PERFORMANCE OF DAPI CONTROL

The paper establishes stability results for the DAPI controllers but treats frequency and voltage control differently. Frequency DAPI stability is supported by prior large-signal analysis, whereas voltage DAPI receives small-signal sufficient conditions because a complete nonlinear analysis is beyond scope.

  • Stability analysis: The frequency DAPI controller has a prior large-signal nonlinear stability analysis, and it is stabilizing for any choice of gains ki.The cited prior result concerns the frequency controller equations.
  • Stability analysis: The secondary frequency controller will not destabilize the primary controller, but the secondary voltage controller can potentially destabilize its primary voltage controller.The voltage risk arises from the conflict between reactive power sharing and voltage regulation.
  • Stability analysis: The paper presents a small-signal stability analysis with sufficient conditions for the voltage DAPI controller and examines how controller gains affect transient performance.A complete nonlinear stability analysis of the voltage/reactive-power controller is beyond the article’s scope.

A. Small-Signal Stability of Voltage DAPI Control

The voltage DAPI dynamics are linearized around nominal voltages, yielding sufficient matrix conditions for exponential stability. These conditions relate DG heterogeneity, voltage-regulation gains, power-sharing gains, electrical stiffness, and communication averaging.

  • Model assumptions: The model assumes impedance loads collocated with DGs and uses the standard decoupling relation between reactive power and voltage-magnitude differences.The text states that these assumptions can be relaxed with more complicated formulas.
  • Model and linearization: The nonlinear voltage-control model is linearized by approximating [E] ≃ [E∗] near nominal voltages.The resulting state includes voltage magnitudes and secondary control variables.
  • Stability analysis: The linearized system’s characteristic equation reduces to det(sI2n − W) = det(s2In + sW1 + W2) = 0.The reduction enables stability analysis through a matrix quadratic polynomial.
  • Stability conditions: If conditions (12a) and (12b) hold, the linearized system is exponentially stable.The proof uses positive quadratic coefficients and the Routh-Hurwitz criterion.
  • Physical interpretation: Condition (12a) restricts DG dissimilarity, while (12b) captures the interaction among electrical stiffness, voltage regulation, power sharing, and communication averaging.Voltage-regulation gains stabilize the pure-voltage-regulation case, while sufficiently small power-sharing gains preserve stability by eigenvalue continuity.

B. Transient Performance of DAPI Control

The transient study varies DAPI controller gains around a nominal four-DG ring-network configuration and tracks the resulting closed-loop eigenvalues. The traces connect gain choices to frequency and voltage-response speed and damping, supporting stable high-performance tuning.

  • Study setup: The study uses a four-DG case with a ring communication network and nominal gains k = 1.7 s, κ = 1 s, β = 1.2, and b = 180 V.The nominal values match Study 1c.
  • Eigenvalue analysis: Each gain sweep numerically finds the operating point, linearizes the closed loop, and plots the eigenvalues of that linearization.Black crosses mark nominal eigenvalues, while arrows show the direction of increasing gain.
  • Frequency dynamics: Increasing the frequency time-constant k moves real eigenvalues toward the origin, producing slower, smoother frequency and active-power responses.Decreasing k instead gives faster but still overdamped frequency regulation.
  • Voltage dynamics: Increasing the voltage time-constant κ collapses complex-conjugate eigenvalues onto the real axis, producing an overdamped voltage/reactive-power response for sufficiently slow secondary control.Increasing feedback gains b or β makes that response increasingly underdamped.
  • Tuning implications: The stability conditions, gain-impact table, and eigenvalue traces together support tuning the DAPI controllers for stability and high performance.This conclusion is based on the analysis despite its simplifying assumptions.

VI. EXPERIMENTAL RESULTS

Experiments validate the DAPI controllers on a four-DG microgrid with heterogeneous impedances, local loads, and neighboring communication links. Four studies assess baseline performance, communication failure, heterogeneous gains, and plug-and-play operation.

  • Experimental setup: The experimental setup contains four DGs interconnected through impedances, with loads locally placed at units 1 and 4.The setup is implemented in the Intelligent Microgrid Laboratory at Aalborg University.
  • Experimental setup: Units 1 and 4 are rated for twice as much power as units 1 and 3, creating heterogeneous generator ratings.The controllers were implemented in Simulink with measurements recorded through a dSP system.
  • Study design: The experiments are organized into four studies covering controller characterization, communication-link failure, heterogeneous controller gains, and plug-and-play operation.This sequence broadens validation beyond nominal controller performance.
  • Analysis reference: Figure 8 shows eigenvalue traces as controller gains vary, with nominal-gain locations marked by black crosses and several fast eigenvalues omitted.The traces are used to examine transient behavior for the experimental system parameters.
  • Communication and parameters: The communication structure uses adjacency matrices A = [aij] and B = [bij], with b varied across studies while other parameters remain fixed.Plots use consistent colors for DG 1 through DG 4.

A. Study 1: Controller Performance

Study 1 shows that frequency DAPI control rapidly restores nominal frequency and maintains active-power sharing, while voltage DAPI tuning trades voltage regulation against reactive-power sharing. A smart asymmetric tuning improves both objectives relative to the compromise case and reduces transient ringing.

  • Study 1: Controller Performance: The frequency DAPI controller quickly eliminates primary-droop frequency deviation, maintains regulation through load changes, and accurately shares active power among heterogeneous DGs.The same robust frequency and active-power behavior occurs in all four sub-studies.
  • Study 1a: Reactive power sharing: Pure reactive-power-sharing tuning accurately shares reactive power but produces poor voltage regulation.Study 1a uses βi = 0 and b = 50 V, with voltage magnitudes deviating from E∗ = 325.3 V.
  • Study 1b: Voltage regulation: Pure voltage-regulation tuning tightly regulates voltage but yields poor reactive-power sharing.Study 1b uses βi = 2.2 and b = 0 V; the text identifies this trade-off as a general property of exact voltage regulation.
  • Study 1c: Compromised tuning: Compromise tuning with b = 180 V and βi = 1.2 roughly clusters voltages around E∗ = 325.3 V while approximately sharing reactive power.This corresponds to Study 1c.
  • Study 1d: Smart tuning: Study 1d slightly improves voltage regulation and noticeably improves reactive-power sharing compared with Study 1c.It uses b = 100 V and β2 = 4, with β1 = β3 = β4 = 0; sharing remains accurate through load changes and transients, with reduced ringing.

B. Study 2: Communication Link Failure

The DAPI controllers retain high performance under communication-link failure, heterogeneous controller gains, and plug-and-play operation. The broader design regulates frequency and voltage while supporting proportional active-power sharing and tunable reactive-power sharing, although large-signal voltage stability remains unresolved.

  • Communication Link Failure: The DAPI controllers maintain high performance after the communication link between DG units 3 and 4 fails.The link fails at t = 2 s; a local load is detached at t = 7 s and reattached at t = 18 s.
  • Heterogeneous Controller Gains: Heterogeneous integral gains change DG transient responses but leave steady-state behavior and system stability unchanged.The tested gains were k1 = 1.5 s, k2 = 1 s, k3 = 2 s, and k4 = 0.5 s.
  • Plug-and-Play Operation: During plug-and-play operation, the DAPI controllers maintain accurate power sharing and frequency and voltage regulation before, during, and after unit 3 is disconnected and reconnected.Unit 3 was disconnected at t = 10 s and reconnected at t = 30 s after synchronization with the remaining microgrid.
  • Controller Capabilities: The controllers regulate frequency while sharing active power proportionally and can be tuned for voltage regulation, reactive power sharing, or a compromise between them.The methodology uses distributed averaging algorithms for primary and secondary control in islanded microgrids.
  • Open Problems: Large-signal stability of the voltage controller remains an open analysis problem, and non-collocated load-bus voltages are not addressed by the stated control goals.A further open problem is guaranteeing voltage tolerances at non-collocated load buses while maintaining high performance.
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