Parameter estimation (calibration)

Kind: Problem specification · Maturity: prototype · Direction: inverse · Temporal: inter-temporal (shared parameters)

solve_parameter_estimation calibrates uncertain network parameters — line lengths and transformer/regulator tap ratios — from smart-meter data. It is the shared-parameter dual of state estimation, and a multi-time-step extension of it:

The formulation follows Vanin, Geth, Heidari & Van Hertem, Distribution System State and Impedance Estimation Augmented with Carson's Equations (arXiv:2506.04949): a weighted measurement-residual objective over a time series, the multiconductor IVR (rectangular current–voltage) power flow as the physics, time-invariant impedances written as nominal · length, and the smart-meter data as noisy (P, Q, |V|) triples per user.

Why multiple time steps

A single snapshot cannot separate a long line from a heavy load: both pull the downstream voltage down by the same amount. With T snapshots of diverse loads the shared parameters become over-determined — the one length (or tap) that explains every snapshot at once is pinned. Calibration therefore needs a time series; this is the essential difference from a one-shot state estimate.

Measurements

Each metered user contributes noisy (P, Q, |V|) readings as Measurements — the same struct as state estimation:

• :vmag — phase-to-neutral voltage magnitude [V].
• :pinj, :qinj — active/reactive power injected into the network [W]/[var] (negative for a load). A free injection current is added at each measured bus and fit to these; buses with no injection measurement are zero-injection.

sigma is the standard deviation (WLS weight 1/σ²). Choose the objective with objective=:wls (smooth least squares) or objective=:wlav (weighted least absolute value, which is less sensitive to isolated large residuals and is used in Vanin et al.). WLAV does not identify or remove bad measurements; inspect the fitted residuals before attributing a discrepancy to a particular meter.

How the uncertain elements are stamped

The two element classes are handled differently, reflecting how the BMOPFTools engine exposes them:

• Transformer taps use the engine's native free-tap variable. A transformer carrying tap_min/tap_max gets a :tap decision variable that the engine threads through its per-unit-correct, base-referred winding constraints. CalibTap keeps the transformer in every snapshot net, sets those bounds, and the solver couples the tap equal across snapshots — the engine's transformer physics (leakage, losses, referral) is reused unchanged.
• Line lengths have no native free variable, so CalibLines are omitted from the physics nets and re-stamped here with a shared free length ℓ: V_f − V_t = (r₀+jx₀)·ℓ·I. This is the one genuine "replace Ohm's law" step; per-unit scaling by z_base is applied so SI and per-unit solves agree.

Worked example

using PowerOptLab
using BMOPFTools: parse_bmopf, solve_pf

# Physics net for one snapshot: known source; the uncertain line l1 (src→b1) is
# OMITTED (its length is the unknown); NO loads (injections come from the meters).
calnet() = parse_bmopf("""
{"bus":{
   "src":{"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"]},
   "b1": {"terminal_names":["1","n"],"perfectly_grounded_terminals":["n"]}},
 "voltage_source":{"vs":{"bus":"src","terminal_map":["1"],"v_magnitude":[230.0],"v_angle":[0.0]}}}
"""; from_string=true)

# Per-snapshot meter readings (P, Q, |V|) at b1 — here for four diverse loads.
meas = [
  [Measurement(kind=:vmag, bus="b1", value=v,  sigma=0.3),
   Measurement(kind=:pinj, bus="b1", value=-p, sigma=50.0),   # negative = load
   Measurement(kind=:qinj, bus="b1", value=-q, sigma=50.0)]
  for (v, p, q) in [(229.3, 1000.0, 200.0), (226.8, 4000.0, 800.0),
                    (224.1, 7000.0, 1400.0), (228.0, 2500.0, 500.0)]
]
nets = [calnet() for _ in meas]

r = solve_parameter_estimation(nets, meas;
        lines=[CalibLine(id="l1", bus_from="src", bus_to="b1",
                         r_per_length=0.4, x_per_length=0.25)],
        objective=:wls)      # or :wlav

r.line_length["l1"]   # estimated length of l1
r.residual_rms        # RMS voltage misfit (V) — near the meter noise floor if good
r.snapshots           # per-snapshot fitted state (SI)

Add CalibTaps (naming transformers present in the nets) to the taps keyword to calibrate taps jointly. Both per_unit=true and per_unit=false give identical estimates. A residual_rms well above the known meter noise is the diagnostic that something structural is still wrong (a misidentified tap, a missing line, an unmetered load).

Scope

This is a prototype. It handles single-phase elements (one terminal per element end — add one CalibLine per phase for a multi-phase feeder). Estimating the series-impedance type via Carson's equations (conductor geometry/material) as in Vanin et al., regularising toward GIS priors, and four-wire mutual coupling are natural extensions left for later.

See the API reference for CalibLine, CalibTap, solve_parameter_estimation, and ParameterEstimationResult.