Voltage Stability Part 2: The Two-Bus Model — Setting Up the Voltage Stability Problem
The Two-Bus Model: Setting Up the Voltage Stability Problem
- Part 1 of 5 — What is Voltage Stability? Definition, Classification and Physical Intuition
- Part 2 of 5 — The Two-Bus Model: Setting Up the Voltage Stability Problem
- Part 3 of 5 — Deriving the PV Curve: The Nose Curve from First Principles
- Part 4 of 5 — The Nose Point: Maximum Loadability, Critical Voltage and the Jacobian
- Part 5 of 5 — Voltage Collapse: QV Curves, OLTC Feedback and Stability Indices
1. The two-bus model
The two-bus system is the canonical analytical model for voltage stability. A source bus (Bus 1, infinite bus V₁∠δ) supplies a load bus (Bus 2, V₂∠0°) through line impedance Z = R + jX. The load draws complex power S = P + jQ.
Notation
| Symbol | Meaning | Units |
|---|---|---|
| V₁ | Sending-end voltage magnitude | pu or kV |
| V₂ | Receiving-end (load bus) voltage magnitude | pu or kV |
| δ | Angle between V₁ and V₂ | rad or ° |
| Z = R + jX | Line impedance | Ω or pu |
| P, Q | Active and reactive power at load bus | MW, MVAr |
2. Power balance equations
Taking Bus 2 as the reference (V₂∠0°), the complex power at the load bus is:
Line current from Bus 1 to Bus 2:
Substituting (2) into (1) and separating real and imaginary parts:
Equations (3) and (4) are the two-bus power flow equations. With Q = P·tanφ (fixed power factor) and V₁ fixed, we have two equations in two unknowns (V₂ and δ).
3. The quadratic in V₂
Eliminating δ entirely via the voltage phasor geometry gives:
Multiplying through by V₂² and rearranging gives a quadratic in V₂². Setting u = V₂²:
This quadratic has two roots — the mathematical origin of the PV curve’s two branches, derived in full in Part 3.
4. Why two solutions exist
Both roots are real under normal loading: one is the high-voltage (stable) operating point, the other is an abnormal low-voltage state. As loading P increases, the two roots approach each other. When they coincide, the discriminant equals zero — that is the nose point, the subject of Part 4.
Part 3 solves the quadratic explicitly, plots the complete nose curve for three power factors, and shows how reactive compensation shifts the stability boundary.
References
- P. Kundur, Power System Stability and Control, McGraw-Hill, 1994 — Chapters 14–15.
- T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems, Springer, 1998.
- A. Bergen and V. Vittal, Power Systems Analysis, 2nd ed., Prentice Hall, 2000.
- J. Grainger and W. Stevenson, Power Systems Analysis, McGraw-Hill, 1994.