Measurements and What They Mean

Everything on this page is about turning sensors into the rows of \(z\) and \(H\). It is the part of state estimation with the least mathematics and the most opportunity for a silent, confident, wrong answer — a flipped sign or a misattributed quantity does not fail, it converges to something plausible.

One measurement is one scalar

A Measurement is a single scalar observation, not a sensor. Newton-Raphson WLS treats \(\sigma_P\) and \(\sigma_Q\) as independent, so a power sensor becomes two rows; a voltage sensor becomes one or two depending on whether it carries an angle. A voltage sensor that does carry an angle is a phasor (PMU) measurement, and it is what makes the global phase determinate.

Aggregation runs in two opposite directions

Redundant sensors are merged before estimation rather than passed through as extra rows, and the two cases move opposite ways:

Several sensors observing one quantity merge by inverse variance:

\[ z = \frac{\sum_k z_k / \sigma_k^2}{\sum_k 1 / \sigma_k^2}, \qquad \sigma^2 = \frac{1}{\sum_k 1 / \sigma_k^2} \]

The result is more certain than any input. A quantity you measure twice is one you know better.

Several appliances making up one bus injection sum, and their variances sum with them:

\[ S = \sum_k S_k, \qquad \sigma^2 = \sum_k \sigma_k^2 \]

The result is less certain than its parts, because independent errors accumulate.

Conflating these is easy and costly. An early version of gridoxide's aggregation merged per bus rather than per appliance, which turned two sensors watching one load into two loads and doubled the injection. The ordering that works is: merge per appliance first, then sum across appliances at the bus.

Sign conventions are not uniform

Taken from power-grid-model's reference-direction rules, which differ by sensor type:

Branch measurements pass through unchanged, since that is already the convention branch_flow::terminal_flow uses. Loads and shunts are negated to become injections.

Where gridoxide's model differs from power-grid-model's

This is the subtlety that most affects correctness, and it is not a sign issue — it is a difference in what the quantity is.

power-grid-model treats a source and a shunt as appliances at a node, so their power counts toward that node's injection. gridoxide models both structurally: a source becomes a virtual slack bus feeding through an impedance branch, and a shunt becomes a Y-bus diagonal entry. Neither therefore appears in network::power_injections at that bus at all — by Kirchhoff's current law, the net injection at a source node with no load is zero, because the source's power arrives through a branch that is part of the network.

Each of the three needs its own measurement function:

Using the plain bus injection for any of them is wrong by construction. This was found rather than predicted: tests/measurement_residual_test.rs evaluates every measurement function at the state power-grid-model published and reported a 63-sigma disagreement on exactly that quantity, with the model saying 0 and the sensor saying 2.4 p.u.

Zero-injection buses

A bus with no load and no generator injects exactly nothing. That is a property of the network, not an observation of it: no sensor, no noise, no uncertainty.

The common shortcut is to feed it in as a pseudo-measurement of zero with a very small \(\sigma\). It works, and it is why so many estimators are described as ill-conditioned — the weight matrix then spans many orders of magnitude and \(G\) squares that spread. power-grid-model ships fixtures named ill-conditioned-by-line-meshed and ill-conditioned-by-link-meshed for precisely this failure.

gridoxide enforces it as a hard equality constraint instead, via the Lagrangian stationarity conditions:

\[ \begin{bmatrix} G & C^{T} \\ C & 0 \end{bmatrix} \begin{bmatrix} \Delta x \\ \lambda \end{bmatrix} = \begin{bmatrix} H^{T} W r \\ -c(x) \end{bmatrix} \]

The augmented matrix is symmetric indefinite rather than positive definite, which rules out a Cholesky-style solver — but every gridoxide backend is a general sparse LU, so this costs assembly work only. A constraint row turns out to be the same bus-injection partials an injection measurement would produce; the difference between a constraint and a measurement is entirely in how the system consumes the row.

Two details worth knowing:

• Which buses qualify is read off the input document, not off Bus::p_spec. A state-estimation document leaves p_specified unset, so an unmeasured load looks exactly like zero injection in the converted network while being nothing of the sort. Getting this backwards would constrain real loads to zero.
• Sources and shunts do not disqualify a bus, because gridoxide models both structurally and neither appears in that bus's injection. The virtual slack buses are excluded, since that is where a source's unknown power enters.

The fixture that pins this down is node-injection-sensor-and-zero-injection: an injection sensor reading 0.1 p.u. on a node with no appliance attached. power-grid-model requires at least one appliance for such a sensor to mean anything, so it overrides the reading and reports both buses at exactly 1.0∠0. Without the constraint, weighted least squares fits the sensor perfectly by driving that node to \(\sqrt{2} \angle -45°\) — a 41% overvoltage on a bus with nothing connected to it, and an objective of \(4 \times 10^{-28}\). By its own criterion, a flawless answer.