Energy Markets and Power Grids

application

The domain where the physics and the economics are both networks of constraints, and where the standard method is already a factor graph under another name.

Sources: original to this vault (design and analysis; no single paper).

The relations

Two layers, both acausal, sharing variables:

Physical. Kirchhoff’s current law at every bus ( $\sum_{i} i_{i} = 0$ ), Kirchhoff’s voltage law around every loop, line-flow equations relating flows to voltage angles, and the generation–load balance. None has an input or an output — a bus does not “cause” its incident currents.

Economic. Market clearing, bid stacks, ramp limits, transmission constraints, and the locational prices that come out of them. Also relations rather than assignments.

The LinearConstraintFactor example in this project is a three-way Kirchhoff node, and the DC power-flow approximation is linear — so an entire useful class of grid model lands in the exact fragment.

The standard method is already this

Power-system state estimation is weighted least squares over a network of measurement residuals, run continuously on every transmission grid in the world. That is a Gaussian factor graph with a MAP estimator, built before anyone called it one.

Which means the framework does not have to argue its way in — it has to justify what it adds: a posterior instead of a point, learned components alongside the physics, and one model answering more than one question.

What is observed

SCADA measurements at some buses, PMUs at a few, meter readings at slow cadence, and nothing at all at most distribution-level nodes. Partial observation is structural — grids are chronically under-instrumented at the edges — and the observability analysis that grid operators run is precisely the question of whether a polarity is well posed.

What would be learned

Renewable generation forecasts — wind and solar, the dominant source of uncertainty.
Demand models, increasingly with behind-the-meter solar and storage that nobody measures.
Distribution-network topology and impedances, which are frequently wrong in the records.

All beside exact Kirchhoff constraints. That is the grey-box case in its clearest industrial form: physics you would never want to learn, next to quantities you can only learn.

What a residual means

Bad data. Residual analysis is the standard method for detecting failed sensors, and it is the same computation as the graded energy. Per-factor attribution answers “which measurement disagrees with the network”, which is what an operator acts on.

It also detects topology errors — a breaker whose recorded state is wrong makes a whole region of the graph inconsistent, and that shows up as a spatially structured pattern of residuals rather than a single outlier.

One model, several questions

State estimation: measurements clamped, state inferred.
Contingency analysis: hypothesise a line outage, propagate, ask what the state would be.
Observability: is this set of measurements enough to determine the state? — a structural question about the graph, not a numerical one.
Optimal dispatch: clamp costs and limits, solve for generation.

Same network, different clamps. Channels and Polarity.

What would be hard

AC power flow is nonlinear — the exact results here are linear-Gaussian, so full AC is approximate. DC is linear and widely used, which is a real foothold, but it is an approximation before you start.
Meshed grids are loopy, and Kirchhoff’s voltage law is a statement about loops. So the loopy-message-passing problem is not incidental to this domain, it is intrinsic: exact means, wrong variances (Loopy Message Passing) — and for a risk-constrained dispatch decision the variance is the point.
Scale: $1 0^{4}$ – $1 0^{5}$ buses for a transmission network, more with distribution. See Parallelism and Compilation.
The bootstrap problem is real here too — a loop of pure constraints with no priors cannot start propagating (constraint.md §4.2), and a grid model is exactly such a loop.

Lenticulum — theory vault

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