Theory

This section gives the mathematical account of each stage of VANE, in the order the pipeline applies them. It starts from the linearized turbine model that OpenFAST writes, passes through mode identification and the Campbell diagram, and ends with the reduced, scheduled state-space model and the Kalman filter built on it. Each page covers one stage and can be read on its own. The equations name the functions that compute them, so each result can be traced to the code.

Readers who need only the identified modes and the Campbell diagram can stop after Campbell diagram, resonance, and uncertainty. The pages on reduction, observability, LPV models and estimation describe the state-space export built on those results. Evidence that the implementation reproduces analytical and reference results is collected in Validation.

The pipeline in brief

  1. Linearization and the multi-blade transform (Linearized dynamics and the multi-blade transform). OpenFAST linearizes the turbine about a periodic steady operating point at several rotor azimuths. The multi-blade (Coleman) transform moves the blade degrees of freedom into the fixed frame, and the transformed matrices are averaged over the azimuths. For a rotor with identical blades on a symmetric support, the result is a time-invariant model.

  2. Mode identification (Mode identification). The eigenvalues of the averaged model give the natural frequencies and damping ratios. The eigenvectors give the mode shapes. Left and right eigenvectors together give a participation measure that does not depend on the units of the states, and that measure names each mode (tower, platform, blade flap or edge, collective, regressive or progressive). Modes are then linked across operating points into the lines of a Campbell diagram, and crossings and veerings are reported for review.

  3. Floquet analysis (Floquet analysis of the periodic system). When the rotor or its support is not symmetric, the transformed model stays periodic in azimuth, and averaging becomes an approximation. Floquet theory treats the periodic model exactly and provides the reference against which the averaging error is measured.

  4. Campbell diagram and uncertainty (Campbell diagram, resonance, and uncertainty). Mode lines are compared with the rotor harmonics to locate resonances. Each mode receives a confidence built from its numerical conditioning, its azimuth sensitivity and its tracking quality.

  5. Realization, reduction and observability (Realization, reduction, and observability). The model is written in real modal coordinates and reduced to the modes that matter for the chosen inputs and outputs, by modal selection or balanced truncation. The observability of the retained states from a sensor set is quantified, and sensors and process noise are chosen from it.

  6. LPV models and estimation (LPV export, load maps, and estimation). Reduced models at many operating points are joined into a family scheduled on wind speed or rotor speed. A Kalman filter runs on that family, and its consistency is checked on recorded or simulated data.

Notation

The symbols are shared across the pages. \(N_b\) is the number of blades (three unless stated), \(\Omega\) the rotor speed (rad/s), \(\psi\) the rotor azimuth, \(T = 2\pi/\Omega\) the rotor period, and \(j=\sqrt{-1}\). The state-space model is \((\mathbf A, \mathbf B, \mathbf C, \mathbf D)\) with state \(\mathbf x\), input \(\mathbf u\), and output \(\mathbf y\). VANE assembles the state in the layout \(\mathbf x = [\,\mathbf q_2;\ \dot{\mathbf q}_2;\ \mathbf q_1\,]\) (second-order displacements, their rates, then first-order states); see Coordinate systems, origins and datums. An eigenvalue is written \(\lambda = \sigma + j\omega\) with \(\sigma = \operatorname{Re}\lambda\). The natural frequency is \(\omega_n = |\lambda|\) and the damping ratio is \(\zeta = -\sigma/|\lambda|\). Continuous-time quantities carry no subscript; a discrete-time model with step \(\Delta t\) is marked \((\cdot)_d\).