This 2D companion solves the exact same nonlinear feeder problem as the 3D scene — the ZIP load model and the backward-forward power-flow sweep — but shows something the 3D row of houses never can: the sweep's own iteration-by-iteration convergence, laid out as a heatmap instead of a spatial scene.
P(V) = P₀ · (V / V_nom)^k
k = 0 → constant power (I = P₀/V, no CVR benefit)
k = 1 → constant current
k = 2 → constant impedance (I ∝ V, largest CVR benefit)
Backward-forward sweep, repeated to convergence:
1. Backward: I_i = P(V_i) / V_i at every house;
sum downstream currents into each span current.
2. Forward: walk from the substation outward,
V_i = V_(i-1) − I_span · R_span.
The top panel is a grid: rows are sweep iterations (row 0 = the flat initial guess, every house at V₀), columns are houses 1…24, and cell colour is that house's voltage at that iteration. Watching the rows settle from a flat green band into the final profile is the numerical method converging — for a single resistive radial feeder like this one it typically settles within 4–6 iterations, which the grid makes visible directly instead of hiding it inside a single final render.
The lower-left panel plots the converged voltage profile against house index, shaded to the ANSI Range A band (114–126 V) — the last house is always the worst case, because current (and hence voltage drop) only accumulates moving outward from the substation. The lower-right panel plots the ZIP demand curve P(V) itself for k = 0, 1 and 2, with a marker at the feeder's present average operating voltage, showing directly why only k > 0 loads respond to CVR.
Utilities that run CVR programs typically report 0.5–3% total energy reduction feeder-wide — small per house, large across a grid.
Note on units: with 24 houses strung on one radial span, cumulative downstream current near the substation reaches several hundred amps — at that current, a span resistance quoted in whole ohms drives the feeder to a nonphysical voltage collapse (verified numerically while building this model: the naive ohm scale diverges to roughly −900 V even at plain default settings). This model uses milliohms per span instead, which is the physically correct order of magnitude for a short low-voltage secondary run and keeps every slider combination inside a sane operating range.