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Constructive Port-Hamiltonian Energy Shaping Design of Dispatchable Virtual Oscillators in Grid-Forming Converters

Lu Gao, Lihui Yang, Feng Ji, Dong Liu, Longze Kou

arXiv:2609.01349v1eess.SY

TL;DR

dVOC lacks a unified dissipative energy interpretation for its coupled amplitude, synchronization, and dispatch functions. The paper develops a Logistic-VOC dissipative port-Hamiltonian outer loop with a shaped Hamiltonian, radial gain matching, and explicit parameter inequalities. The resulting design guarantees almost-global asymptotic stability and local exponential convergence, while power-error weighting can eliminate the low-voltage stationary solution and ensure a unique high-voltage equilibrium.

  • Problem

    dVOC couples amplitude regulation, synchronization, and power dispatch without a unified analytical energy structure.

  • Method

    The paper formulates Logistic-VOC outer-loop dynamics as a dissipative port-Hamiltonian system with a shaped Hamiltonian and radial gain-matching dissipation design.

  • Results

    The design guarantees almost-global asymptotic stability and local exponential convergence, while suitable power-error weighting excludes undesired stationary points and yields a unique high-voltage equilibrium.

  • Takeaways & Limitations

    Power-error weighting shapes the energy landscape so the closed loop converges to the desired high-voltage operating point within the certified domain.

Abstract

from arXiv · show

Port-Hamiltonian (PH) theory offers a passivity-based framework for grid-forming control, yet conventional dispatchable virtual oscillator control (dVOC) does not naturally admit a dissipative PH realization, since its amplitude egulation, synchronization, and power dispatch are inherently coupled without a unified energy interpretation. This paper formulates the outer-loop dynamics as a dissipative PH system, thereby unifying amplitude regulation, synchronization, and power dispatch within one energy structure. The formulation rests on the key property that logistic-type radial regulation, characterized by an inherent saturation-like nonlinearity, permits an exact gradient decomposition compatible with the quadratic energy storage. On this basis, a unified shaped Hamiltonian is constructed, which encapsulates both amplitude restoration and power dispatch. Radial gain matching derived from this Hamiltonian yields explicit closed-form arameter inequalities that guarantee almost-global asymptotic stability and local exponential convergence. Moreover, tuning the power-error weighting coefficient actively shapes the energy landscape, thereby eliminating the undesirable low-voltage power-flow solution from the stationary set and ensuring convergence to the desired high-voltage equilibrium point. The Hessian singularity condition further provides the critical weight threshold that guarantees uniqueness of the high-voltage equilibrium. Numerical simulations validate the proposed method.

I. INTRODUCTION

Grid-forming converters need voltage, frequency, and power-control capabilities, while dVOC combines synchronization, amplitude regulation, and dispatch without a unified energy structure. The paper addresses this gap by formulating a dissipative port-Hamiltonian outer loop with constructive energy-shaping design.

  • Grid-forming control supports voltage establishment, frequency support, and power regulation in power-electronics-dominated systems.
  • VOC enables parallel converters to self-synchronize and share power using local electrical quantities.
  • dVOC directly embeds power-tracking errors into oscillator dynamics, but its coupled functions lack a unified analytical energy structure.
  • Port-Hamiltonian theory separates interconnection and dissipation structures, supporting passivity-based analysis and energy-shaping design.
  • The proposed formulation yields closed-form design inequalities linking convergence rate and bandwidth to independently selectable controller gains.
  • The paper constructs a dissipative PH outer loop whose shaped Hamiltonian unifies amplitude restoration and power dispatch, with logistic radial regulation embedded through an exact gradient decomposition.

B. Grid-Connected Outer-Loop Model

The grid-connected model considers a converter linked to an ideal grid through an inductive line and describes outer-loop voltage generation and power dispatch in synchronous coordinates. Inner voltage and current loops are assumed to track outer-loop references ideally, while the state domain excludes the origin.

  • The converter is connected to an ideal grid through a line inductance, with converter and grid voltage vectors expressed in the synchronous rotating frame.
  • The analysis focuses on outer-loop voltage generation and power dispatch, assuming ideal tracking by the inner voltage and current loops.
  • The model defines active and reactive power as quantities injected by the converter into the grid in port-voltage coordinates.
  • The outer-loop state domain is R^2 \ {0} because the polar-coordinate transformation is singular at the origin.

III. PORT-HAMILTONIAN FRAMEWORK-BASED LOGISTIC-VOC GRID-FORMING CONTROL

The proposed Logistic-VOC design starts from an energy function and represents the outer-loop dynamics as a dissipative port-Hamiltonian system. Logistic radial regulation is expressed as state-dependent dissipation multiplied by the negative gradient of an amplitude potential.

  • The design begins with the energy function and formulates a Logistic-VOC outer loop through port-Hamiltonian energy shaping.
  • A. Logistic Radial Dynamics and Amplitude Potential Energy: The radial voltage-restoration mechanism evolves the voltage magnitude toward its reference under a positive radial regulation gain.
  • A. Logistic Radial Dynamics and Amplitude Potential Energy: Logistic-type radial regulation has an exact decomposition into a state-dependent positive dissipation gain and the negative gradient of amplitude potential energy.

B. Shaped Hamiltonian and Gradient Structure

The shaped Hamiltonian combines amplitude and power-error potentials, while its gradient exposes power-error coupling in the radial direction. This coupling motivates radial gain matching and subsequent stability-oriented parameter design.

  • The shaped Hamiltonian is defined from a positive definite weighting matrix over amplitude, active-power, and reactive-power error coordinates.
  • With k_ρ=1 and k_P=k_Q=α, the Hamiltonian decomposes into weighted amplitude, active-power, and reactive-power potential terms.
  • The energy-gradient construction applies the chain rule and the Jacobian of the power mapping with respect to the port-voltage state.
  • The radial component of the shaped energy gradient contains both amplitude error and a power-error-generated coupling term C(v).
  • The Hessian and radial projection structure support analysis of stationary points and parameter conditions around the desired high-voltage equilibrium.

D. Radial Gain-Matching and Port-Hamiltonian Control

The controller adds radial gain matching to the shaped-energy gradient feedback, producing an unforced dissipative port-Hamiltonian outer-loop representation.

  • Cross-coupling problem: The indefinite-sign cross term C(v)ρ̃ in the Hamiltonian derivative prevents direct certification of nonpositive energy change.The cross coupling arises from power-error contributions to the radial energy gradient.
  • Gain matching: A radial correction term matches the power-error channel’s radial gain to the Logistic amplitude channel.This correction is added to the control law to remove the sign-indefinite cross coupling.
  • Dissipation structure: The gain-matching correction modifies gradient feedback through a state-dependent radial dissipation term.The control law is rearranged to expose this dissipation structure.
  • Port-Hamiltonian realization: The resulting closed-loop outer-loop dynamics admit an unforced dissipative port-Hamiltonian representation.The associated dissipativity and stability properties are established subsequently.

E. Closed-Loop Dissipativity and Stability

The closed-loop Hamiltonian is non-increasing and supports bounded trajectories, almost-global asymptotic stability relative to the admissible domain, and local exponential convergence under Hessian conditions.

  • Dissipativity: For η>0, k>0, and ρ>0, the state-dependent dissipation matrix R(v) is positive definite.This condition supports the dissipativity calculation along closed-loop trajectories.
  • Dissipativity: The Hamiltonian is monotonically non-increasing, while the admissible domain D is forward invariant for the designed parameters.The radial component points outward near, but not at, the origin.
  • Boundedness: Radial unboundedness makes every Hamiltonian sublevel set compact, so closed-loop trajectories are bounded.The reactive-power term grows sufficiently with voltage magnitude to make H tend to infinity.
  • Asymptotic stability: The desired equilibrium is almost globally asymptotically stable when it is the unique stationary point of H on D.LaSalle’s invariance principle gives convergence throughout D, which excludes only the origin.
  • Exponential convergence: The desired equilibrium is locally exponentially stable when the Hessian of H is locally bounded below by a positive multiple of the identity.The resulting estimate is H(t) ≤ e^-2ηmt H(0).

F. Parameter Design

Parameter design first excludes undesired stationary points through a critical power-error weight, then selects dissipation gains to meet convergence-rate and bandwidth specifications.

  • Power-error weighting: The stationary-point analysis identifies equilibria through ∇H = 0, combining the amplitude gradient with active- and reactive-power error gradients.Additional stationary points determine candidate weights through the radial balance condition.
  • Power-error weighting: The critical power-error weight is obtained from candidate bifurcation points satisfying the stationary-point and Hessian-singularity conditions.Real positive roots are retained, and the minimum defines the critical threshold.
  • Power-error weighting: For 0<α<α_cr, all undesired stationary points are excluded from D, leaving the desired high-voltage equilibrium as its unique stationary point.The threshold is selected from α_cr = min α_hv over the retained candidates.
  • Gain selection: The gradient dissipation gain η and radial dissipation gain k are selected according to the desired convergence rate and outer-loop bandwidth.The admissible inequalities use the desired decay rate, Hessian bounds, maximum voltage magnitude, and allowable bandwidth.
  • Gain selection: The first gain inequality guarantees the prescribed local convergence rate, while the second limits radial gain and enforces the prescribed bandwidth.These constraints provide direct controller-synthesis conditions.

G. Energy Landscape Analysis

The shaped Hamiltonian is interpreted through separate amplitude and power energy landscapes, then their superposition reveals how energy shaping organizes the combined dynamics.

  • Landscape decomposition: The energy-landscape analysis separately presents Logistic amplitude potential energy, PQ power potential energy, and their shaped superposition.This decomposition illustrates the geometric composition of the Hamiltonian.
  • PQ power landscape: The PQ power potential has a double-valley landscape whose high-voltage and low-voltage valleys represent distinct steady-state power-flow solutions.A saddle point separates the two valleys, and convergence depends on the basin of attraction when power dispatch alone is used.
  • Energy shaping: The power-only landscape therefore requires reshaping through the amplitude potential.The amplitude contribution is introduced to alter the stationary-point structure.

H. Comparison with dVOC and PH-Based GFM Control

The proposed Logistic-VOC unifies amplitude regulation, synchronization, and power dispatch within one shaped Hamiltonian and dissipative port-Hamiltonian flow, unlike the separated dVOC design logic.

  • The controller combines amplitude regulation, synchronization, and power dispatch within a single shaped Hamiltonian.
  • Its stability follows from a direct energy-based argument rather than dVOC’s hierarchical phase-first, magnitude-second analysis.
  • Potential-weight design excludes undesired stationary points from the certified operating domain.
  • The decomposition yields a direct dissipative port-Hamiltonian form without an auxiliary current loop or switching pumping-or-damping action.

IV. SIMULATION VERIFICATION

The simulations examine energy landscapes, convergence trajectories, and disturbance responses for the proposed port-Hamiltonian Logistic-VOC outer loop. Results show that the power-error weighting coefficient determines whether the desired high-voltage equilibrium is unique and globally selected.

  • Simulation Verification: The case study evaluates shaped-Hamiltonian energy landscapes, closed-loop trajectories, and time evolution under different power-error weighting coefficients.The system is a grid-forming converter connected to the grid through a tie-line inductance.
  • Equilibrium Solutions: The power-flow equation has high- and low-voltage solutions vh = 0.687+j0.182 kV and vl = 0.002+j0.182 kV for Pref = 2 MW and Qref = 0.5 Mvar.The target-point amplitude is ρh = 0.711 kV.
  • Weight Threshold: The critical power-error weight is αcr = 0.0537, separating energy-landscape regimes with different stationary-point structures.This value is calculated using the design method proposed in Section III-F.
  • Energy-Landscape Shaping: For α = 0.06, a low-voltage local minimum and separatrix appear, and some trajectories converge to the low-voltage equilibrium instead of the desired high-voltage point.For α = 0.04, the shaped Hamiltonian admits the desired high-voltage equilibrium as the unique stationary point.
  • Energy-Landscape Shaping: Once α exceeds αcr, the low-voltage attraction region reappears, so convergence to the desired high-voltage equilibrium is no longer guaranteed.Thus, α reshapes both the energy landscape and the set of stationary points.

B. P/Q Dispatch-Step Response

The proposed PH-Logistic-VOC controller tracks simultaneous active- and reactive-power reference steps smoothly while maintaining bounded frequency behavior. Under weak-grid voltage sag and phase-jump disturbances, voltage and power responses recover without sustained oscillations.

  • P/Q Dispatch-Step Response: Two consecutive simultaneous active- and reactive-power reference steps are applied to assess the controller’s P/Q dispatch capability.The converter is evaluated during changes in both active- and reactive-power references.
  • P/Q Dispatch-Step Response: At t = 1 s, references change from Pref = 2 MW and Qref = 0.5 Mvar to Pref = 2.5 MW and Qref = 0 Mvar, then return at t = 2.5 s.These are the two dispatch conditions used in the simulation.
  • P/Q Dispatch-Step Response: Following the reference change, active and reactive powers converge smoothly to their new set-points without sustained oscillation.The terminal-voltage magnitude moves continuously between high-voltage power-flow solutions, while frequency deviation remains bounded and rapidly decays to zero.
  • Transient Recovery under Large Disturbances in a Weak Grid: Under a 0.8 pu grid-voltage sag lasting 200 ms at SCR = 1.26, terminal voltage has a bounded deviation and recovers after the disturbance is cleared.Active power, reactive power, and frequency deviation show only short-duration deviations without sustained oscillations.
  • Transient Recovery under Large Disturbances in a Weak Grid: Under a 15° grid-voltage phase jump at SCR = 1.26, the terminal-voltage angle rapidly adjusts, while power and frequency return to steady-state values.The disturbance-rejection scenario maintains Pref = 2.0 MW and Qref = 0.5 Mvar.
  • Controller Contribution: The proposed controller is formulated as a dissipative port-Hamiltonian outer loop that unifies amplitude restoration and P/Q power dispatch through energy shaping.The formulation uses the port-voltage vector as the state variable and enables gradient-based power dispatch control.
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