Transformer-Guided Symbolic Regression for Real-Time Phasor Estimation

In progress: a small transformer proposes the waveform model, and a physics-constrained fit estimates the phasor. For fault-distorted currents in protective relaying.

In progress   With: Sina Mohammadi, Van-Hai Bui, Prof. Wencong Su (Power Lab, University of Michigan–Dearborn)

The problem

Protective relays decide whether to trip based on the fundamental-frequency phasor (amplitude and phase) of the current, estimated from a short window, usually one cycle. Right after a fault, the current is not a clean 60 Hz sinusoid. It carries a decaying DC offset and low-order harmonics:

\[i(t) = A\cos(\omega t + \varphi) \;+\; \sum_{j} B_j\, e^{-t/\tau_j} \;+\; \sum_{h\in\{3,5\}} C_h \cos(h\omega t + \varphi_h) \;+\; \text{noise}.\]

A standard full-cycle DFT assumes none of that is there. Its estimate overshoots and oscillates exactly when the relay has to decide. Classical DC-offset-removal methods help, but they assume a fixed model (typically a single exponential). Symbolic-regression approaches can discover the model, but a search per window is far too slow for real-time protection.

The idea: the network proposes, physics decides

For each one-cycle window:

  1. A small transformer reads the samples and predicts which terms are present (decaying DC, 3rd harmonic, 5th harmonic) and a rough decay rate. It works like symbolic regression, but it is learned in advance, so there is no online search.
  2. A physics-constrained refinement fits all coefficients of that structure to the window using variable-projection Gauss–Newton.
  3. The fundamental phasor is read directly from the fitted model.

The expensive search moves into offline training. Online, it is a single forward pass plus a few Gauss–Newton steps, with no online training.

What I’m doing

  • Built the signal model and scenario generator for fault transients: short and long DC time constants, dual DC offsets, harmonics, off-nominal frequency and noise.
  • Designed and trained the transformer models, and implemented the variable-projection refinement.
  • Implemented the baselines for comparison: full-cycle DFT, a closed-form DC-offset-removal method (Abdoos et al., 2016), and a replication of a deep symbolic-regression phasor estimator.
  • Ran ablation studies on an HPC cluster (SLURM) and set up an evaluation of accuracy (total vector error, amplitude error, overshoot, settling time) and per-window runtime.

Status

The manuscript is in preparation, and the code will be released alongside it. Results will be posted here once it’s public.

Related work in the lab: Publications