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Uncovering Droop Control Laws Embedded Within the Nonlinear Dynamics of Van der Pol Oscillators

Mohit Sinha, Florian Dorfler, Brian B. Johnson, Sairaj V. Dhople

arXiv:1411.6973v2eess.SYmath.DS

TL;DR

The paper addresses how nonlinear-oscillator control can relate to conventional droop control for islanded inverter microgrids. Using periodic averaging and network stability analysis, it shows that droop laws emerge within Van der Pol oscillator dynamics on slower time scales, while averaged amplitude and phase dynamics converge in resistive networks.

  • Problem

    Droop control operates on phasor quantities and presumes quasi-stationary sinusoidal steady state, motivating analysis of a time-domain oscillator-based alternative for inverter control.

  • Method

    The paper analyzes Van der Pol oscillator-controlled inverters using periodic averaging, polar-coordinate voltage dynamics, and a gradient-system formulation with LaSalle’s invariance principle.

  • Results

    Droop laws are recovered from averaged Van der Pol oscillator dynamics on slow AC-cycle time scales, while amplitude and phase dynamics converge globally in resistive networks.

  • Takeaways & Limitations

    Van der Pol oscillator control can be parameterized to mimic droop behavior near sinusoidal steady state and remain compatible with droop-based secondary and tertiary strategies.

Abstract

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This paper examines the dynamics of power-electronic inverters in islanded microgrids that are controlled to emulate the dynamics of Van der Pol oscillators. The general strategy of controlling inverters to emulate the behavior of nonlinear oscillators presents a compelling time-domain alternative to ubiquitous droop control methods which presume the existence of a quasi-stationary sinusoidal steady state and operate on phasor quantities. We present two main results in this work. First, by leveraging the method of periodic averaging, we demonstrate that droop laws are intrinsically embedded within a slower time scale in the nonlinear dynamics of Van der Pol oscillators. Second, we establish the global convergence of amplitude and phase dynamics in a resistive network interconnecting inverters controlled as Van der Pol oscillators. Furthermore, under a set of non-restrictive decoupling approximations, we derive sufficient conditions for local exponential stability of desirable equilibria of the linearized amplitude and phase dynamics.

I. INTRODUCTION

The paper studies decentralized control of islanded inverter microgrids using Van der Pol oscillator dynamics as a time-domain alternative to droop control. It derives their correspondence near sinusoidal steady state and analyzes convergence and stability in resistive networks.

  • Islanded inverter microgrids regulate terminal-voltage amplitude and frequency to stabilize the system while sharing network load fairly and economically.
  • VOC controls inverters by emulating nonlinear limit-cycle oscillators, using sinusoidally varying oscillator states to construct the PWM signal.
  • Unlike droop control, VOC operates in the time domain and stabilizes arbitrary initial conditions toward a sinusoidal steady state, whereas droop uses phasor quantities near quasi-stationary AC steady state.
  • Van der Pol oscillators provide the paper’s VOC implementation, using smooth cubic polynomial dynamics derived from deadzone oscillators.
  • Periodic averaging establishes parameter conditions under which VOC and droop-control voltage dynamics are identical near sinusoidal steady state.
  • The analysis establishes convergence for averaged amplitude and phase dynamics in resistive networks and sufficient conditions for local exponential stability under decoupling assumptions.

B. VOC implemented with a Van der Pol Oscillator

The paper models inverter control with Van der Pol oscillator dynamics, transforming the oscillator into polar coordinates and comparing its averaged behavior with droop-controlled inverters. Under stated assumptions, the two strategies can be matched up to O(ε) differences over specified time intervals.

  • Oscillator model: The Van der Pol virtual oscillator uses a parallel RLC circuit with a cubic voltage-dependent current source and current gain κ.The cubic source is g(v) = v − β(v^3/3), and the oscillator is analyzed in the quasi-harmonic limit ε → 0.
  • Oscillator model: Polar coordinates represent the oscillator voltage using amplitude r and phase offset θ, with v = r cos(φ).The paper uses these variables to compare oscillator dynamics with droop-control amplitude and phase dynamics.
  • VOC–droop comparison: Under matched setpoints, droop coefficients, steady-state operation, and O(ε) initial differences, VOC and droop trajectories remain O(ε) apart over t ∈ [0,t∗].The comparison assumes identical microgrids and sufficiently small ε.
  • VOC–droop comparison: Averaging the VOC dynamics bridges real-time oscillator behavior and droop control based on quasi-stationary sinusoidal behavior.The averaged VOC system provides the basis for comparing phase dynamics and steady-state voltage amplitudes.
  • Scope and assumptions: The correspondence is an asymptotic result: small ε weakens nonlinear damping, slows convergence, and formally limits the comparison to bounded time horizons unless exponential stability is established.The paper states that the correspondence can extend to an unbounded horizon when the averaged system is exponentially stable.

1) Averaging the VOC dynamics:

Periodic averaging converts the rapidly varying VOC equations into averaged amplitude and phase dynamics that can be compared with droop control. Standard averaging results bound the difference between the averaged and original VOC solutions by O(ε) on an appropriate time interval.

  • Averaging procedure: In the quasi-harmonic limit ε → 0, periodicity in the scaled time coordinate permits standard averaging of the VOC dynamics.The averaged equations describe slower amplitude and phase behavior across AC cycles.
  • Averaging procedure: The oscillator current is related to the Van der Pol input through i(t) = −u(t), linking the circuit model to inverter quantities.The averaged power expressions are then used to obtain the amplitude and phase dynamics.
  • Averaged model: The averaged dynamics in (20) recover the amplitude and phase system in (14) from the instantaneous and average power definitions.This establishes the averaged model used for subsequent comparison with droop control.
  • Averaged model: For sufficiently small ε, the averaged VOC solution is O(ε) close to the original VOC solution over t ∈ [0,t∗/ε].The result requires the stated existence and boundedness assumptions on the relevant solutions and average active power.

2) Correspondence of phase dynamics:

The averaged phase dynamics of Van der Pol oscillator-controlled inverters correspond to droop phase laws when reactive-power setpoints and frequency-droop coefficients are selected appropriately. The amplitude analysis similarly connects the oscillator’s steady-state voltage profile to droop-controlled amplitude behavior up to O(ε).

  • Phase correspondence: For sufficiently small ε, the VOC phase trajectory is O(ε) close to the corresponding harmonic-oscillator phase trajectory on t ∈ [0,t∗].The comparison uses a fixed equilibrium amplitude and neglects amplitude dynamics when analyzing phase correspondence.
  • Phase correspondence: Choosing the reactive-power setpoint and frequency-droop coefficient appropriately makes droop phase dynamics match AC-cycle-averaged VOC phase dynamics up to O(ε).The matching is stated for the phase dynamics of each inverter.
  • Amplitude correspondence: The VOC steady-state voltage profile is obtained from nonlinear amplitude-balance equations for the inverter network.The analysis examines positive roots, their reality condition, and the high-voltage solution.
  • Amplitude correspondence: The high-voltage amplitude solution is exponentially stable and can be placed in correspondence with droop amplitude dynamics.The averaged amplitude trajectory satisfies the stationary-solution conditions up to O(ε) for sufficiently small ε.
  • Amplitude correspondence: Selecting the active-power setpoint and voltage-droop coefficient appropriately makes droop amplitude dynamics correspond to VOC amplitude dynamics up to O(ε).The correspondence is formulated around the equilibrium terminal-voltage amplitude.

IV. STABILITY OF VOC AMPLITUDE & PHASE DYNAMICS

For resistive networks, averaged VOC dynamics are analyzed using network reduction and a gradient formulation. Under conditions maintaining positive amplitudes, all trajectories converge to equilibria, while decoupling assumptions yield local exponential-stability conditions.

  • Network model: The analysis applies to connected microgrids with resistive lines, arbitrary network topology, and loads modeled as resistances and current sources.Electrical coupling is represented through conductance-based network equations and Kron reduction.
  • Network model: Kron reduction eliminates interior-node voltages and produces an effective conductance network coupling inverter terminals.The reduced model defines effective shunt and line conductances for the inverter nodes.
  • Global convergence: Under Theorem 2’s amplitude-bound condition, radii remain positive and trajectories of the averaged VOC dynamics converge to a nonempty set of equilibria.Positive invariance excludes zero-amplitude states, allowing the phase dynamics to remain well defined.
  • Global convergence: The convergence proof rewrites the dynamics in gradient form and applies LaSalle’s invariance principle to compact, forward-invariant sublevel sets.The zero-derivative set identifies equilibria after excluding points with zero amplitude.
  • Decoupled stability: Under standard decoupling assumptions, fixing phase offsets for amplitude analysis and amplitudes for phase analysis yields sufficient conditions for local exponential stability.These approximations are associated with unstressed networks having nearly uniform voltages and approximately equal phase angles.

B. Amplitude Dynamics in Decoupled Settings

With phase offsets fixed at equilibrium, the amplitude subsystem is linearized around its equilibrium. Strict diagonal dominance makes the relevant Jacobian negative definite, establishing local exponential stability under the stated conditions.

  • Amplitude formulation: The decoupled amplitude dynamics fix phase offsets at their equilibrium values before analyzing terminal-voltage radii.This produces the amplitude system used for linearization around the equilibrium solution.
  • Stability result: Theorem 3 states that an equilibrium of the decoupled amplitude dynamics is locally exponentially stable when its stated equilibrium conditions hold.The theorem concerns terminal-voltage amplitude dynamics under the specified loading model.
  • Stability proof: Linearization around the equilibrium yields an error system with Jacobian KΓ, where K is the positive diagonal matrix of oscillator parameters.Connectivity makes Γ irreducible and symmetric, enabling an inertia-based definiteness argument.
  • Stability proof: Strict irreducible diagonal dominance makes Γ, and consequently KΓ, negative definite under the stated condition.The resulting bounds include the open-circuit voltage as the upper bound.

C. Phase Dynamics in Decoupled Settings

With amplitudes fixed at equilibrium, the phase subsystem admits locally exponentially stable equilibria under a conductance and phase-angle condition. Without constant current loads, the stable object becomes a phase-synchronized equilibrium manifold.

  • Phase formulation: The decoupled phase dynamics fix terminal-voltage amplitudes at their equilibrium values before analyzing phase offsets.The resulting system is studied using coupled-oscillator theory.
  • Stability result: Theorem 4 establishes local exponential stability of a phase equilibrium when the stated equilibrium exists and at least one constant current load is present.The condition corresponds to the nonsingular, strictly diagonally dominant case.
  • Equilibrium conditions: Condition (39) identifies equilibria with small reactive-power flows and requires local current sources to inject reactive power.The paper relates these requirements to phase-angle inequalities involving line and source offsets.
  • Stability proof: Negative definiteness of the phase Jacobian follows from connectivity, diagonal dominance, and the positive amplitude and oscillator-parameter matrices.The inertia argument transfers definiteness from the symmetric matrix Θ to the linearized phase dynamics.
  • Stability result: Without constant current loads, the phase-synchronized equilibrium manifold is locally exponentially stable rather than an isolated equilibrium.The negative Jacobian becomes a Laplacian associated with an undirected connected graph.

V. REVERSE ENGINEERING DROOP CONTROL, CONVERGENCE RATES, AND NUMERICAL VALIDATION

The simulations validate the averaging-based correspondence between virtual-oscillator control and droop control, including voltage regulation, load sharing, and reactive-power dispatch. They also show proportional sharing after a load step and connect convergence-rate analysis to the oscillator model.

  • Numerical validation: The voltage-regulation curve from the averaged VOC expression is validated against steady-state simulations of the original non-averaged Van der Pol model.Figure 3 compares the analytical characteristic with time-domain simulations run to steady state.
  • VOC–droop correspondence: VOC and droop control are compared through equilibrium-voltage and phase-offset differences for a 15 kW inverter supplying specified active and reactive power.The test uses a lagging power-factor load and derives the corresponding droop controller from the established expressions.
  • Load sharing and optimality: Droop control provides proportional reactive-power sharing in resistive networks, while suitable VO-controller current gains can reproduce this dispatch objective.Theorem 1 translates droop-design insights into current-gain choices for VO-controlled inverters.
  • Load sharing and optimality: Three identical VO-controlled inverters with current gains [2 2 1]T share 25%, 25%, and 50% of the load, respectively, including after the active-power demand doubles.The observed sharing follows the inverters’ ratings after the load step.
  • Load sharing and optimality: The load-sharing objectives and associated droop gains coincide when Rj/λj = Rℓ/λℓ for all inverter pairs.This condition links the two stated optimization objectives under the paper’s parameterization.

C. Convergence Rate of a Van der Pol Oscillator

The convergence-rate analysis examines an unforced Van der Pol oscillator in the quasi-harmonic limit. It derives an inverse dependence of the transition arc length, and thus a convergence-time proxy, on ε and validates it with simulations.

  • Quasi-harmonic limit: The analysis studies convergence to the limit cycle of an open-circuited Van der Pol oscillator with driving term u = 0 as ε approaches zero.The dynamics are expanded and averaged in the quasi-harmonic regime.
  • Quasi-harmonic limit: The locally stable equilibrium of the averaged dynamics is the open-circuit voltage.This equilibrium supplies the endpoint used in the transition analysis.
  • Convergence rate: φs ≈ 6 (εα)^−1, so the transition arc length is inversely proportional to ε.The arc length is proportional to a notion of convergence time to O(ε).
  • Convergence rate: Figure 6 compares the predicted convergence-rate relationship with simulations of the original unforced nonlinear dynamics.The simulations are superimposed to demonstrate validity of the analysis.

APPENDIX

The appendix derives the averaged amplitude and phase dynamics from the original VOC equations using standard averaging, integration by parts, periodicity, and controlled O(ε^2) truncations.

  • Amplitude dynamics: Standard averaging establishes that the averaged VOC solution remains O(ε) close to the original dynamics over the relevant scaled-time interval.The appendix begins by applying this result to the amplitude dynamics.
  • Phase dynamics: The phase derivation uses integration by parts and periodicity to transform the averaged phase expression.The integration limits can be shifted because the integrand is 2π-periodic.
  • Phase dynamics: Terms of order O(ε^2) are discarded while cosine and sine expressions are expanded to obtain phase dynamics in terms of average real power.The derivation transitions from scaled-time variables to the original time coordinate before reaching the target equation.
  • Derivation endpoint: The appendix concludes by recovering equation (14) from the preceding transformed expressions.The final step follows after substituting the intermediate relation and changing the dummy integration variable.

r(s)i(s) sin(s +

The derivation continues by reformulating the phase dynamics through intermediate sine expressions and time-coordinate changes, ultimately recovering the target averaged equation.

  • Phase reformulation: The right-hand side is reformulated using an intermediate relation before expanding the sine function.The derivation retains only terms needed through the stated approximation order.
  • Phase reformulation: After ignoring O(ε^2) terms, the expression is converted from τ to t coordinates.This produces the phase-dynamics form used in the main derivation.
  • Derivation endpoint: The resulting expression recovers equation (14).This completes the appendix’s phase-dynamics derivation.
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