Source-linked AI summary
The effect of transmission-line dynamics on grid-forming dispatchable virtual oscillator control
Dominic Groß, Marcello Colombino, Jean-Sébastien Brouillon, Florian Dörfler
TL;DR
Transmission-line dynamics can undermine dVOC stability when inverter-based grids are analyzed with a dynamic network rather than an algebraic approximation. The paper combines Lyapunov and singular-perturbation tools to derive explicit conditions that restore almost global asymptotic stability for prescribed power-flow solutions.
Problem
The paper addresses whether dVOC remains almost globally asymptotically stable when electromagnetic transmission-line dynamics, usually neglected in algebraic network models, are included.
Method
The authors combine a Lyapunov characterization with singular perturbation theory to construct an explicit stability condition for dVOC coupled to dynamic transmission lines.
Results
The derived bounds on controller gains, setpoints, and power transfer guarantee almost global asymptotic stability relative to a synchronous steady state with prescribed power flows.
Takeaways & Limitations
The results provide a theoretical foundation for applying dVOC under non-standard conditions such as black starts and islanded operation.
Abstract
from arXiv · showhide
In this work, we analyze the effect of transmission line dynamics on grid-forming control for inverter-based AC power systems. In particular, we investigate a dispatchable virtual oscillator control (dVOC) strategy that was recently proposed in the literature. When the dynamics of the transmission lines are neglected, i.e., if an algebraic model of the transmission network is used, dVOC ensures almost global asymptotic stability of a network of AC power inverters with respect to a pre-specified solution of the AC power-flow equations. While this approximation is typically justified for conventional power systems, the electromagnetic transients of the transmission lines can compromise the stability of an inverter-based power system. In this work, we establish explicit bounds on the controller setpoints, branch powers, and control gains that guarantee almost global asymptotic stability of dVOC in combination with a dynamic model of the transmission network.
I. INTRODUCTION
The paper studies dVOC grid-forming control when transmission-line dynamics are retained, addressing stability challenges created by replacing synchronous machines with power-electronic generation. It develops explicit stability conditions for inverter-based AC systems with dynamic transmission networks.
- Replacing synchronous machines with renewable generation interfaced through power electronics removes self-synchronizing dynamics, rotational inertia, and resilient controls relied upon by conventional grids.
- dVOC extends virtual oscillator control by enabling dispatch of nominal power injections while targeting a desired AC power-flow solution.
- Transmission-line dynamics are often neglected through an algebraic network model, but inverter controls operate on faster time scales than conventional synchronous-machine dynamics.
- The paper combines Lyapunov analysis with singular perturbation ideas to derive explicit stability conditions for the full inverter-network model.The conditions quantify the time-scale separation required between inverter and network dynamics.
- The resulting conditions account for network topology, nonzero relative angles, control gains, and power-injection setpoints, while identifying voltage collapse as an exponentially unstable steady state.
B. Quasi-steady-state network model
The paper replaces the dynamic transmission network with a quasi-steady-state model to formulate inverter control objectives and consistent power-flow setpoints. It emphasizes that this approximation is conventional for synchronous-machine systems but can affect stability in inverter-based systems.
- The quasi-steady-state network model replaces transmission-line current dynamics with their exponentially stable steady-state map.
- The approximation is typically justified in conventional systems by time-scale separation between fast line transients and slow synchronous-machine dynamics.
- For inverter-based systems, electromagnetic line transients can significantly influence stability boundaries, motivating analysis beyond the quasi-steady-state approximation.
- The model considers balanced three-phase inverters connected through a resistive-inductive transmission network represented by a weighted graph.
- Controller objectives require synchronous frequency, prescribed voltage magnitudes, steady-state currents, and active and reactive power injections consistent with the power-flow equations.
D. Dispatchable virtual oscillator control
dVOC uses local voltage and power feedback to regulate inverter magnitudes, track prescribed power injections, and synchronize phase information across the network. Line dynamics delay this information relative to the quasi-steady-state model.
- dVOC is a decentralized controller for each inverter that combines voltage regulation with feedback related to power setpoint errors and network synchronization.
- The controller regulates voltage magnitude through a feedback term based on the normalized quadratic voltage error.
- The dVOC feedback can be interpreted either as tracking active and reactive power setpoints or as synchronizing voltage phases.
- Under the quasi-steady-state approximation, the local feedback becomes a distributed synchronizing feedback based on weighted phase errors.
- Transmission-line dynamics delay propagation of the phase and power information used by dVOC, creating stability concerns absent from the algebraic approximation.
III. ALMOST GLOBAL ASYMPTOTIC STABILITY OF SET-VALUED CONTROL SPECIFICATIONS
The paper defines almost global asymptotic stability relative to a compact target set and introduces Lyapunov-function conditions for establishing it. The definition combines attraction for almost all initial conditions with Lyapunov stability.
- Almost global asymptotic stability is defined with respect to a compact set rather than necessarily a single equilibrium.
- The definition requires almost global attractivity, excluding only an initial-condition set of zero Lebesgue measure.
- It also requires Lyapunov stability with respect to the target set.
- The paper introduces class K and class K∞ comparison functions for expressing Lyapunov bounds.
- A Lyapunov-function theorem supplies a characterization used to establish almost global asymptotic stability for the system.
IV. STABILITY ANALYSIS OF THE INVERTER-BASED AC
The analysis derives sufficient conditions under which dVOC remains almost globally asymptotically stable when transmission-line dynamics are included. These conditions constrain operating points, network connectivity, and controller gains while accounting for electromagnetic transients.
- Stability condition: If the transmission graph is connected, suitable control gains and set-points can always be chosen to satisfy Condition 2.Connectivity is characterized by λ2(L) > 0.
- Main result: Theorem 2 establishes almost global asymptotic stability of the desired equilibrium set T when Condition 2 holds, while the origin is exponentially unstable.The equilibria correspond to harmonic trajectories with desired power flows in the static frame.
- Scope of result: Unlike results restricted to identical angles and voltages or local stability near trivial power flows, the theorem covers nonzero power-flow solutions and electromagnetic line transients.The condition also accounts for network topology and provides explicit bounds on gains and power-injection set-points.
- Stability condition: The stability conditions require moderate network loading and sufficient time-scale separation between inverter and line dynamics.They quantify the separation needed for the singular-perturbation argument to guarantee stability.
- Power-transfer bound: The achievable steady-state power transfer depends on network connectivity λ2(L), stability margin c, control gain α, and voltage set-points v⋆.Power transfer can increase with higher voltage set-points, lower voltage-regulator gain α, or stronger transmission connectivity.
- Gain bound: The second inequality bounds the control gain η so that transmission-line dynamics do not destabilize the system.Because line time constants represent phase-information propagation delays, longer time constants require lower η.
C. Singular perturbation theory
The analysis separates inverter voltage dynamics from transmission-line current dynamics, then combines Lyapunov functions for the reduced-order and boundary-layer systems. This singular-perturbation construction yields conditions for stability of the full system relative to the target set.
- C. Singular perturbation theory: The transmission-line dynamics are replaced by their steady-state map to obtain a reduced-order voltage system.The resulting model assumes line currents are at quasi-steady state.
- C. Singular perturbation theory: The analysis studies stability relative to the set T0 rather than a single equilibrium.The set T0 combines the target synchronous trajectories with the origin.
- C. Singular perturbation theory: The reduced-order construction uses voltage set-points, controller parameters, and matrix bounds to establish the required Lyapunov inequalities.The proof derives these inequalities through coefficient matching, projection properties, and a positive-definiteness condition.
- C. Singular perturbation theory: A Lyapunov function V is constructed for the reduced-order system and shown to decrease along its trajectories.The function is positive definite and radially unbounded with respect to the relevant target set.
E. Lyapunov function for the Boundary layer system
The boundary-layer analysis isolates deviations between actual and steady-state transmission-line currents while holding voltages fixed. A separate Lyapunov function W is then shown to decrease for this current-error dynamics.
- E. Lyapunov function for the Boundary layer system: The boundary-layer variable is the difference between actual transmission-line currents and their steady-state values.The analysis defines yo := io − is(v) and treats the voltage as fixed in the boundary-layer model.
- E. Lyapunov function for the Boundary layer system: Graph connectivity determines the dimension of the cycle-space component used in the boundary-layer construction.For a connected graph, rank(B)=N−1 and the nullspace dimension is M−N+1.
- E. Lyapunov function for the Boundary layer system: The Lyapunov function W combines the edge-current error with its cycle-space component.Its quadratic form uses B and the nullspace basis Bn.
- E. Lyapunov function for the Boundary layer system: W is positive definite and decreases along the boundary-layer trajectories for every fixed voltage state.The result follows from the full-rank cycle-space term and the skew-symmetry properties used in the proof.
F. Proof of the main result
The full-system proof combines the reduced-order and boundary-layer Lyapunov functions into a single candidate. Bounds on coupling terms then provide a sufficient decrease condition, while linearization establishes that the origin has a measure-zero attraction region.
- F. Proof of the main result: The full-system Lyapunov candidate is a convex combination of the reduced-order function V and boundary-layer function W.The weighting parameter d lies in (0,1).
- F. Proof of the main result: The derivative bounds introduce constants β1, β2, and γ that quantify coupling between inverter and transmission-network dynamics.These constants depend on K−L, the network admittance norm, the line time constant, and the controller gain.
- F. Proof of the main result: A sufficient condition requiring β1+β2∈(0,1) ensures decrease of the full-system Lyapunov function.Under the stated condition, the derivative is bounded negatively away from the target set T0.
- F. Proof of the main result: The proof establishes almost global asymptotic stability relative to T0 by combining convergence and an unstable-origin argument.The origin has an attraction region of zero Lebesgue measure because the linearized system has an eigenvalue with positive real part.
V. POWER SYSTEMS IMPLICATIONS AND TEST-CASES
The test cases examine how transmission-line parameters and controller gains affect stability in three-bus and IEEE 9-bus systems. Increasing line admittances can reduce damping and eventually destabilize the inverter-based system.
- V. POWER SYSTEMS IMPLICATIONS AND TEST-CASES: Two test cases illustrate the theoretical results: a three-bus system for line-parameter effects and an IEEE 9-bus system for broader numerical behavior.The IEEE 9-bus case also tests behavior when some technical assumptions do not hold.
- V. POWER SYSTEMS IMPLICATIONS AND TEST-CASES: The three-bus study varies individual line admittances, recomputes the steady state, and evaluates the minimum damping ratio from the linearized system.The damping ratio excludes the zero eigenvalue associated with rotational invariance.
- V. POWER SYSTEMS IMPLICATIONS AND TEST-CASES: The minimum damping ratio is insensitive to small changes in the 1–2 line admittance until that admittance becomes large enough to affect dmax.This dependence reflects the maximum summed admittance of lines connected to an inverter.
B. Stability boundaries of a three-bus transmission system
The three-inverter study maps control-gain regions to guaranteed stability, instability, inverter-current violations, and local stability. IEEE 9-bus simulations show that small gains handle contingencies, whereas larger gains destabilize the system through line-dynamics interactions.
- Region (a) satisfies Theorem 2’s guarantee of almost global asymptotic stability, while region (b) is unstable.
- Region (c) exceeds inverter operational current limits, while region (d) is locally asymptotically stable but may contain unstable solutions.Theorem 2 excludes poorly damped regions, and its bound is conservative by approximately an order of magnitude.
- The IEEE 9-bus model replaces generators with same-rated inverters and uses passive RLC load dynamics, although the test case does not satisfy Assumption 1.
- With η = 10^-3 p.u., dVOC black-starts the grid, tracks desired power injections, survives a 20% load increase, and remains synchronous after losing inverter 1.The remaining inverters increase their power injections, and current transients avoid undesirable overshoots.
- Increasing the gain to η = 10^-2 p.u. destabilizes the system because high-gain control interferes with transmission-line dynamics.
- The theoretical conditions require appropriate gains, setpoints, loading, and time-scale separation, but apply only to transmission lines with constant inductance-to-resistance ratio.The analysis does not cover heterogeneous line ratios, transformers, or other commonly omitted network dynamics.
APPENDIX
The appendix supplies proof steps for stability, geometric bounds, and singular-perturbation estimates. These results establish Lyapunov stability and almost global attractivity under the stated sufficient conditions.
- Under Assumption 1, the recursively defined quantities P_m remain nonnegative on the nonnegative orthant.
- A Lyapunov function with nonpositive derivative establishes stability of the invariant set C.
- Almost global attractivity follows because trajectories outside the exceptional zero-measure set decrease the Lyapunov function toward zero.
- The appendix bounds phase-angle expressions and quadratic forms to show that Condition 2 is sufficient for the required inequality.