The two-bus system, a slack source feeding a single load through one line, is the smallest model that still captures voltage stability. Because it has a closed-form solution, it is the standard teaching vehicle for the PV nose curve and the QV reactive-margin curve that underpin every large voltage-stability study. This calculator solves the system exactly, draws the single-line diagram with live values, and traces both curves as you change the load.
Set the slack voltage, the line impedance R + jX, and the Bus 2 load P + jQ (all in per-unit), or drag the sliders. The diagram, the operating point, and the two stability curves all update together.
Two-Bus Load Flow & Voltage Stability
Solve a slack-to-load system, then trace the PV nose curve and QV reactive margin
How the two-bus power flow is solved
With the slack voltage \(E\angle 0^\circ\) at Bus 1 and a load drawing \(P + jQ\) at Bus 2 through impedance \(Z = R + jX\), the complex power balance leads to a quadratic in the squared voltage magnitude \(u = |V_2|^2\):
\[ u^{2} + \left(2a-E^{2}\right) u + \left(a^{2} + b^{2}\right) = 0, \qquad a = PR+QX, \quad b = QR-PX \]
The two roots are the two mathematically valid voltages for the same load: the upper root is the normal, stable operating point, and the lower root is the unstable solution that lies on the bottom half of the PV curve. As load increases the two roots move toward each other; when the discriminant reaches zero they merge at the nose, the point of maximum loadability. There is no solution beyond it, which is voltage collapse.
Reading the PV nose curve
The PV curve plots Bus 2 voltage against the real power delivered to it, holding the power factor fixed. The solid upper branch is the stable operating region; the dashed lower branch is unstable. Your operating point rides on the upper branch, and the red marker is the nose. The horizontal distance from the operating point to the nose is the loading margin, the headroom before collapse. Weakening the line (larger X, as happens after a contingency) pulls the nose inward and shrinks that margin, which is why voltage stability is fundamentally a question of system strength.
Reading the QV curve
The QV curve fixes the real load and asks a different question: how much reactive power injection would Bus 2 need to hold a given voltage? The curve is a bowl. Its minimum is the critical point, and the vertical distance from that minimum up to the zero-injection line is the reactive power margin, how much reactive reserve stands between the present state and collapse. Operators use exactly this curve, computed at real buses, to size capacitor banks, SVCs, and STATCOMs.
Worked example
Load the example: E = 1.0, Z = 0.02 + j0.20, and a load of 0.8 + j0.4 pu. Bus 2 settles at about 0.875∠−10° pu, drawing 1.02 pu of line current with roughly 0.02 pu of real loss. The PV nose sits at P ≈ 1.45 pu, so the loading margin is about 81%. Now drag the P slider toward 1.45 and watch the operating point climb toward the nose while the voltage falls away steeply: that steepening slope, \(\left|dV/dP\right| \to \infty\), is the signature of approaching collapse.
When is the operating point acceptable?
- Healthy: voltage within about ±5% of nominal and a comfortable margin to the nose. Standard operation.
- Marginal: voltage drifting from nominal or the loading margin shrinking. Acceptable now, but a single contingency could erode it.
- Low voltage / thin margin / collapse: either the voltage is well off nominal, the operating point is near the tip of the nose where a small load increase triggers instability, or the load already exceeds the nose and no stable solution exists.
New to per-unit and impedance modeling?
The free Per-Unit System Cheat Sheet covers base conversions and change of base, the exact quantities this calculator expects, on one printable page.
Frequently asked questions
Why are there two voltage solutions?
The power flow equations are nonlinear, and for a given load below the nose they have two real solutions: a high-voltage, low-current point (stable) and a low-voltage, high-current point (unstable). Real systems operate on the upper root. The lower root matters because it defines the bottom of the PV curve and the collapse point where the two meet.
Does adding reactive support really help?
Yes. Reducing the reactive load Q, or injecting reactive power at Bus 2, raises the whole PV curve and pushes the nose outward, buying more real-power margin. That is exactly what capacitor banks and dynamic reactive devices do. Try lowering Q in the calculator and watch the nose move right.
How does this scale to real networks?
Real studies solve a multi-bus power flow and trace the nose with a continuation method (continuation power flow), plus QV curves at selected buses. The physics is identical to this two-bus model; only the bookkeeping grows. The two-bus case is where the intuition is built.
Is line resistance important?
For transmission, X dominates and R mainly sets the real losses. For distribution the R/X ratio is much higher and resistance materially affects both the voltage drop and the shape of the nose. The calculator keeps both R and X so you can explore either regime.
Related reading
- Voltage stability with high IBR penetration: how inverter-based resources change the classic picture.
- What is grid strength? Short-circuit ratio explained: the strength metric that sets how far the nose is.
- SCR Calculator: screen grid strength at a point of interconnection.
- Symmetrical Components Calculator: the companion tool for unbalanced conditions.
References
- P. Kundur, Power System Stability and Control, McGraw-Hill, 1994, chapter 14 (voltage stability).
- T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems, Springer, 1998.
- IEEE/CIGRE Joint Task Force, “Definition and Classification of Power System Stability,” IEEE Trans. Power Systems, 2004.