An arXiv preprint proposes a control-theoretic way to define and assess harmonic stability in nonlinear, converter-based power systems. In a numerical case study of one averaged three-phase converter, the largest real part of the harmonic state-space (HSS) eigenvalues was negative at every tested truncation order, and the associated linear matrix inequality (LMI) was feasible at each order.
The paper treats that combination as a local harmonic-stability certification around the converter’s nominal periodic operating trajectory. The theorem is framed as a sufficient condition under its assumptions, and its conclusion is relative to that periodic motion.
A definition built around repeating motion
At the center of the proposal is a two-part definition. Small input errors must produce bounded output errors, known in control engineering as small-signal bounded-input, bounded-output (BIBO) stability. When the input error is zero, the internal error state must return locally and uniformly toward zero.
The definition assumes a nominal periodic motion: a continuous periodic input and bounded periodic state and output trajectories, with commensurable periods so that a common least-common-multiple period exists.
Turning the definition into a calculation
For sufficiently small state and input errors, the nonlinear error dynamics admit a local linear time-periodic (LTP) approximation. The authors calculate Fourier coefficients from periodic trajectories using FFT-based harmonic analysis over a finite harmonic set, then use those coefficients to build the HSS model.
The computational test uses a time-invariant LMI, or a set of matrix inequalities checked for feasibility. The HSS theorem says that if the inequalities hold for all sufficiently large truncation orders, they are sufficient for harmonic stability relative to the nominal periodic motion.
Under finite-harmonic assumptions, a corollary says the finite-dimensional linear time-invariant (LTI) approximation can be exact when the truncation order is set to the maximum of the relevant input, state and trajectory harmonic orders.
One converter case
The numerical example was an averaged nonlinear model of a three-phase grid-following voltage-source converter with an LCL filter, a dq-frame PI current controller and a synchronous-reference-frame phase-locked loop.
In the reported setup, the model had 14 states and 3 inputs. The main calculation used HSS truncation order 7 and ran from 0 to 40 seconds. The grid-voltage input included third-, fifth- and seventh-order harmonics at amplitudes of 0.03, 0.05 and 0.05, respectively.
Under the specified harmonic excitation, the simulated grid voltage and grid-side filter current were visibly and significantly distorted. The current spectrum also contained additional components consistent with frequency coupling.
Across truncation orders 1, 3, 5, 7, 9 and 11, the dominant real part—the highest real part among the HSS eigenvalues—ranged from −1.4209 × 10−5 at order 1 to −1.5893 × 10−5 at order 11. Every reported value was negative, and γ stayed at 0.4545 for every order.
The LMI was feasible at every considered truncation order with a strictness margin of ε = 2.6146 × 10−5. The paper says this met the theorem’s conditions and provided a local harmonic-stability certification for the modeled nonlinear converter.
Where the result stops
The scope is narrow. The evidence covers one averaged converter model and a local periodic trajectory; it does not include hardware, field or interconnected-network validation. The certification also depends on the smoothness and other assumptions in the mathematical setup.
At this stage, the result is best read as a demonstration that the proposed calculation can return a local certificate under its assumptions, rather than as a general verdict on converter-based power systems.
Paper data and sources
Original title: Harmonic Stability of Power Systems: A Control-Theoretic Definition and Assessment Criteria
Authors: N. Bhoir, A. Sarkar, J. E. Machado, J. Schiffer
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text