The power flow has two answers
Solve x² = 4 and nobody is surprised there are two answers. Solve a power flow and the same thing happens, but here the answers are operating points of a real network, and one of them is a place the network cannot actually stay.
Open the interactive tutorial →
Source, and the Dash application it was ported from: NR_ROC_Dashboard
What the page does
Five sections, each interactive:
- Two answers. Both solutions drawn as phasors beside a table comparing voltage, current, and line losses. Both deliver identical P and Q; the alternative needs several times the current, and on a strong line burns well over half the transferred power in the conductor itself.
- Existence. The nose curve. Sweep the loading multiplier and watch the two branches approach and merge. Collapse is not voltage falling to zero, it is the two solutions running out of room to differ.
- Which one. Two magnets under a table and a ball that rolls to whichever pulls harder, as an analogy for basins of attraction — with the caveat that the rolling ball is the solver searching, not the network behaving.
- Explore. A convergence map over 45 000 starting points per impedance. Click anywhere and the page runs Newton-Raphson from that guess and names which of the two exact solutions it reached, or reports that it failed.
The solver is deliberately undamped: it does not restart, clamp, or quietly rescue a diverging run, because seeing the failure is the point. Converge onto the low-voltage root and the mismatch really is zero, so nothing looks wrong, and a tipping point gets handed over as an operating plan.