calculating
direct Newton route: calculating
angle: calculating
A small grid with nowhere for the algebra to hide
The Grid Has a Fold
Turn up real-power demand at the far end of one ideal AC line. Two valid voltages approach, meet, and vanish at the exact fold; change load power factor or local capacitive support and watch the margin move. Analytic roots, direct Newton solutions, residuals, the Jacobian, and a deliberately false operating point are checked live.
The source voltage and line reactance are each fixed at 1.000000 per unit. Positive P and Q mean power consumed by a constant-power load. This is a lossless two-bus steady-state model, deliberately too small to impersonate a real network.
Push through the fold marker. The page will not invent a real equilibrium after the discriminant turns negative.
Right is lagging consumption. Left is leading load power factor. Capacitor injection is handled separately.
The shunt contributes B V², so its help weakens as voltage falls. Constant Q is shown as an ideal comparison.
Crossing this line makes the ideal regulated source infeasible. A real generator would change control mode; this two-bus page marks that boundary but does not pretend to solve the missing network.
two real equilibria
waiting for the first calculation
calculating
direct Newton route: calculating
angle: calculating
fold Pcalculating
quadratic Dcalculating
det J, highcalculating
smallest |λ|calculating
Jacobian κ₂calculating
phasor residualcalculating
source Q, highcalculating
distance to foldcalculating
What folds
The receiving bus is not assigned a voltage. It asks for real and reactive power, and the network must find a magnitude V and angle δ that deliver both. Eliminating the angle leaves a quadratic in y = V². Its two roots are the two branches on the plot.
P = E V sin(δ) / X
Qnet = (E V cos(δ) - V²) / X
(P X)² + (Qnet X + V²)² = (E V)²
Qnet = (E V cos(δ) - V²) / X
(P X)² + (Qnet X + V²)² = (E V)²
The exact hinge
With no shunt and unity power factor, the discriminant is 1 - 4P². It reaches zero at P = 0.500000 pu, where both roots become V = 0.707107 pu. Past that demand, this stated model has no real steady solution.
That statement is not inferred from Newton iteration failing. The analytic branches continue right to their meeting point. At the same point, the original two-equation Jacobian loses rank: its determinant and one eigenvalue reach zero.
Vars have a location
Move the load-var slider toward lagging consumption and the fold comes left. Move it toward leading behavior and the fold moves right. Then switch in local support. The ideal constant-Q option keeps injecting the same vars at low voltage; the physical shunt produces B V² and fades with the very voltage it is trying to hold up.
The source reactive ceiling adds a second boundary. A mathematical high-voltage root can remain after the regulated source has exceeded its allowed Q output. That root is drawn but flagged as infeasible under the chosen ceiling. Voltage magnitude alone cannot tell you which side of either boundary you occupy.
The check
calculating
Independent routes have not run yet.
Free choices and uncertainties. Per-unit bases cancel out of this normalized example. The model fixes a lossless line, a stiff source with E = 1, X = 1, and a static constant-power load. The capacitor sizes and reactive ceiling are reader choices, not measured grid values. Static P-V analysis omits resistance, motors, tap changers, protection, load shedding, inverter controls, dynamics, and angle stability. The offline verifier is at research/the-grid-has-a-fold/verify-the-grid-has-a-fold.mjs.
Why this is not a miniature blackout reconstruction
The two-bus fold is an exact mechanism inside its assumptions, not a claim that every failed large power flow proves physical collapse. NERC says practical P-V studies model contingencies, controls, reactive reserves, voltage criteria, and security margins. A major outage report used tens of thousands of buses and explicitly warned that voltage magnitude alone is a poor stability indicator.
The North American and South Australian reports below are here because grid operators use P-V and Q-V studies. This page does not attribute either event solely to this toy mechanism.