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

• Grid-forming control supports voltage establishment, frequency support, and power regulation in power-electronics-dominated systems.Source: §I · p. 1
• VOC enables parallel converters to self-synchronize and share power using local electrical quantities.Source: §I · p. 1
• dVOC directly embeds power-tracking errors into oscillator dynamics, but its coupled functions lack a unified analytical energy structure.Source: §I · p. 1
• Port-Hamiltonian theory separates interconnection and dissipation structures, supporting passivity-based analysis and energy-shaping design.Source: §I · p. 1
• The proposed formulation yields closed-form design inequalities linking convergence rate and bandwidth to independently selectable controller gains.Source: §I · p. 2
• 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.Source: §I · p. 2

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

• The design begins with the energy function and formulates a Logistic-VOC outer loop through port-Hamiltonian energy shaping.Source: §III · p. 2

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.Source: A. Logistic Radial Dynamics and… · p. 3
• Logistic-type radial regulation has an exact decomposition into a state-dependent positive dissipation gain and the negative gradient of amplitude potential energy.Source: A. Logistic Radial Dynamics and… · p. 3

• The shaped Hamiltonian is defined from a positive definite weighting matrix over amplitude, active-power, and reactive-power error coordinates.Source: B. Shaped Hamiltonian and Gradient… · p. 3
• With k_ρ=1 and k_P=k_Q=α, the Hamiltonian decomposes into weighted amplitude, active-power, and reactive-power potential terms.Source: B. Shaped Hamiltonian and Gradient… · p. 3
• The energy-gradient construction applies the chain rule and the Jacobian of the power mapping with respect to the port-voltage state.Source: C. Energy Gradient, Hessian Matrix,… · p. 3
• The radial component of the shaped energy gradient contains both amplitude error and a power-error-generated coupling term C(v).Source: C. Energy Gradient, Hessian Matrix,… · p. 3
• The Hessian and radial projection structure support analysis of stationary points and parameter conditions around the desired high-voltage equilibrium.Source: C. Energy Gradient, Hessian Matrix,… · p. 3

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.Source: D. Radial Gain-Matching and… · p. 4

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.Source: D. Radial Gain-Matching and… · p. 4

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.Source: D. Radial Gain-Matching and… · p. 4

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.Source: D. Radial Gain-Matching and… · p. 4

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.Source: E. Closed-Loop Dissipativity and… · p. 4
• 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.Source: E. Closed-Loop Dissipativity and… · p. 4

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.Source: E. Closed-Loop Dissipativity and… · p. 4

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.Source: E. Closed-Loop Dissipativity and… · p. 4

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).Source: E. Closed-Loop Dissipativity and… · p. 4

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.Source: F. Parameter Design · p. 4
• 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.Source: F. Parameter Design · p. 4
• 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.Source: F. Parameter Design · p. 4

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.Source: F. Parameter Design · pp. 4–5
• 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.Source: F. Parameter Design · p. 5

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.Source: G. Energy Landscape Analysis · p. 5, Figure 1 · p. 5, Figure 2 · p. 5

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.Source: G. Energy Landscape Analysis · p. 5, Figure 2 · p. 5

Energy shaping

• The power-only landscape therefore requires reshaping through the amplitude potential.The amplitude contribution is introduced to alter the stationary-point structure.Source: G. Energy Landscape Analysis · p. 5

• The controller combines amplitude regulation, synchronization, and power dispatch within a single shaped Hamiltonian.Source: H. Comparison with dVOC and PH-Based… · p. 5
• Its stability follows from a direct energy-based argument rather than dVOC’s hierarchical phase-first, magnitude-second analysis.Source: H. Comparison with dVOC and PH-Based… · p. 5
• Potential-weight design excludes undesired stationary points from the certified operating domain.Source: H. Comparison with dVOC and PH-Based… · p. 5
• The decomposition yields a direct dissipative port-Hamiltonian form without an auxiliary current loop or switching pumping-or-damping action.Source: H. Comparison with dVOC and PH-Based… · p. 5

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.Source: §IV · p. 6

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.Source: §IV · p. 6

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.Source: §IV · p. 6

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.Source: A. High-Voltage Equilibrium Point… · p. 6
• 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.Source: A. High-Voltage Equilibrium Point… · p. 6

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.Source: B. P/Q Dispatch-Step Response · p. 7
• 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.Source: B. P/Q Dispatch-Step Response · p. 7
• 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.Source: B. P/Q Dispatch-Step Response · p. 7

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.Source: C. Transient Recovery under Large… · pp. 7–8
• 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.Source: C. Transient Recovery under Large… · pp. 7–8

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.Source: §V · p. 8