Where a Physical Constraint Belongs: Constraint-Exact Generative Scenario Generation for Power Grids

Generative models are increasingly used to produce operational scenarios for power systems, and such scenarios must satisfy the physical laws they describe: an injection pattern that violates Kirchhoff’s laws cannot occur on any real net- work. Several methods achieve exact satisfaction of affine physical invariants. The question the field has not answered is where the constraint belongs: in the hypothesis class (train-time projection), at inference (zero-shot correction), in the loss (penalty), or in the coordinates (nullspace or completion). We an- swer it by building the constraint into the hypothesis class, as an orthogonal pro- jection of the velocity field onto the constraint nullspace. This is exact under any Runge–Kutta scheme, excludes no minimiser of the flow-matching ob- jective, and costs zero additional function evaluations (Theorem 1). On the transmission-network suite it sets the state of the art: the best variogram score of 16 methods, leading the strongest baseline by 3.8% (t = −6.13), completion- based constraint handling by 7.9% (t = −8.19) and inference-time correction by 10.5% (t = −12.40); the best CRPS of 16; and a worst-case constraint vi- olation 727,790× smaller than unconstrained flow matching, at -1.96% energy score (t = −6.29). Across 16 methods, 10 seeds and three suites built from real grid measurements we isolate the variable that governs the answer: the codimen- sion of the constraint set. On a controlled contrast in which the only change is how much of the state the physics determines, train-time projection moves from neutral to decisive. Soft penalties, the field’s default, lose on both axes at once. Raising λ from 1 to 1000 moves the violation only from 219 MW to 187 MW while degrading the energy score by +92%. We further contribute the first decision-level evaluation of constraint-exact scenario generation, through two-stage stochastic unit commitment. Code, the benchmark and the generators that produce every table and figure here from the raw run files are at https: //github.com/Nishant27-2006/constraint-placement, with the project page at https://hamiltonian-network.github.io.