Beating WLS: state estimation with the SDK

Measurements in, state out, better than weighted least squares on the public test cases. Runs off the default timeline download (data release v0.8.3); pip install "fdia-graph[se]".

Baseline first

import fdia_graph as fg
from fdia_graph.se import WLS

train, test = fg.load("ieee14", split="train"), fg.load("ieee14", split="test")
wls = WLS().fit(train)
xhat = wls.estimate(test)  # [n, 2N-1] = [theta rad (non-slack) | V pu (all buses)]
print(wls.score(test).geo)  # angle_mae_deg, voltage_mae_pu (docs/se/README.md has the table)

fit() learns	estimate() returns
per-meter error scales: RMS of benign residuals at the dataset's clean truth (the benchmark default, calibrate="truth"), or each meter's accuracy class from measurements alone (calibrate="measured", what every detector path uses); the reference angle theta_ref	the classical 2N-1 state, every voltage magnitude and every non-slack angle
the chord Jacobian at the benign mean state	in one angle frame: the slack angle is fixed to theta_ref, taken from the training truth at fit time (the case's reference angle on the released pools)

The better estimator

from fdia_graph.se import SubspacePrior

est = SubspacePrior(rank_frac=0.2, reweight="huber", c=1.5).fit(train)
print(est.score(test).geo)  # about half WLS's angle error on IEEE-14 (the table below)

piece	does	why it helps
prior	restricts the estimate to the low-rank subspace benign operation occupies (SVD of the training states)	learns generator voltage setpoints from history instead of re-estimating them from noisy meters every scan
Huber	down-weights measurements the physics cannot explain	rejects in-place meter corruption
together on IEEE-14 test	45% lower angle error, 81% lower voltage error than WLS	geometric mean over the eight record classes (benign and seven families), v0.8.3 timeline

Validation-selected hyperparameters from the estimation paper:

system	rank_frac	Huber c	ResidualRemoval threshold
ieee14	0.20	1.5	4.0
ieee118	0.50	2.5	5.0
ieee300	0.50	6.0	none helps

PMU pseudo-measurements

On a hybrid-meter file (new generation) a SCADA voltmeter reads no angle, so the angle comes from the PMUs and the power meters alone. pmu_pseudo=True (every estimator, JacobianWeighting included, which reads the previous frame's currents from prev_pmu_i) adds the pseudo-measurements of [WU26, eq. (3)]: a PMU at the near end n of a branch reads V_n and I_n, and the branch's pi model gives the voltage at the far end f,

V_f = (I_n - y_nn V_n) / y_nf

with y_nn, y_nf the branch's Y_f or Y_t entries (taps and phase shifts included), averaged over the branches that reach f (formulas.estimation.pmu_pseudo_voltages). The pseudo |V| and angle fill the slots of the reached buses that no meter reads; a SCADA |V| is kept where one exists. Their first-order error, with independent PMU errors, is the covariance of

dV_f = R(1/y_nf) dI_n - R(y_nn/y_nf) P_n (d|V_n|, dtheta_n)

(R(c) the real 2x2 matrix of multiplying by c, P_n the polar-to-rectangular Jacobian at V_n), summed over the links with the voltage terms grouped by PMU bus, then carried to |V_f| and theta_f through the rectangular-to-polar gradients; for a round error of complex variance s^2 this is var|V_f| = s^2 / 2 and var theta_f = s^2 / (2 |V_f|^2). calibrate="measured" weights the pseudo slots by that variance, calibrate="truth" by their benign residual like every other slot. The formula is exact without noise (to 1e-16 on IEEE-14 and 118) and its variance matches a Monte Carlo of 20,000 draws to within 3% (tests/test_pmu_channels.py).

Measured with WLS on the test split, with only the meter model changed: taking the angles away from the SCADA voltmeters raises the benign angle error, and the eq. (3) pseudo-measurements win part of it back. The comparison is rerun into the results store with the v0.9.0 data; its earlier numbers had no stored run behind them and are not repeated here.

What to expect per family

records	behaviour	why
At, Am (stealthy, and Aq, Al in older releases)	robust weighting sees nothing; the prior moves them part of the way	a consistent AC state leaves no residual, and it sits off the benign operating subspace, so the prior pulls the estimate part of the way back
Ad bias, As scaling, Ar replay (older releases only)	improve a lot	meters corrupted in place, which robust weighting exists to reject
benign	improves on the larger systems, can lose slightly on ieee14	the baseline is already at the noise floor there

Full per-family table and figures, on data release v0.8.x: ../se/README.md.

Your own estimator

Subclass SEBase and override one hook; the rest is shared, so comparisons are estimator against estimator.

hook	does
_fit_states(x_benign)	learn anything from the benign training states
_subspace()	a [2N-1, K] basis that restricts the state space, or None for the full state
_solve(z, w)	the per-batch solve; w is per-record weights [n, m], or None for the shared ones

Inside _solve: self._w_solve(z, w) is the divergence-guarded weighted iteration, self._nres(x, z) the normalized residuals, and formulas.estimation holds every equation (huber_weights, wls_step, normalized_residual, ...).