Voltage Stability Part 3: Deriving the PV Curve from First Principles
Deriving the PV Curve: The Nose Curve from First Principles
- 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. Solving the quadratic
From Part 2, equation (6), we have a quadratic in u = V₂². The power balance gives:
Eliminating δ via the voltage phasor geometry and collecting terms in V₂² gives the quadratic whose solution is:
2. The discriminant and nose point condition
The discriminant inside the square root is:
The ‘+’ sign gives the upper (stable) branch; the ‘−’ sign gives the lower (unstable) branch. As P increases, Δ decreases. The nose point is where Δ = 0 — the two solutions coincide.
At any point on the lower branch, a small increase in load causes V₂ to fall further. This reduces delivered power (at fixed impedance), which increases current demand — a positive feedback loop that drives the system to collapse. The lower branch is drawn dashed in all nose curve plots.
3. The nose curve: upper and lower branches
4. Effect of power factor on the nose curve
From Figure 2, three critical observations:
Lagging load (consuming reactive power) shifts the nose curve inward: the maximum loadability P_max decreases, and collapse occurs at a higher voltage. This is why reactive-hungry loads (induction motors, arc furnaces) are so dangerous to system stability.
Leading load (capacitive, generating reactive power) shifts the nose curve outward: P_max increases. This is the fundamental justification for shunt capacitor banks, synchronous condensers, and SVCs at load buses.
Planning implication: grid codes in Europe (ENTSO-E) and North America (NERC) typically require each connection point to operate within a power factor band (e.g. 0.95 lagging to 0.95 leading) to avoid excessive reactive import.
5. Voltage stability margin
The voltage stability margin is the horizontal distance from the current operating point to the nose point, measured in MW (or pu). It answers the question: how much more load can the system absorb before collapse?
Most TSOs require a minimum voltage stability margin of 5–10% of the total load after credible N−1 contingencies. Some apply a 5% margin for normal conditions and a 2.5% margin for double contingencies. The exact criterion depends on the grid code and system operator guidelines.
Part 4 derives the closed-form expression for P_max and the critical voltage V₂_cr at the nose point, and explains why the power flow Jacobian becomes singular at exactly that point.
References
- P. Kundur, Power System Stability and Control, McGraw-Hill, 1994 — Chapters 14–15 give the complete derivation with lossy lines.
- T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems, Springer, 1998.
- P. Kessel and H. Glavitsch, “Estimating the voltage stability of a power system,” IEEE Trans. Power Delivery, vol. 1, no. 3, pp. 346–354, 1986.
- ENTSO-E, Network Code on Requirements for Grid Connection of Generators, 2016.
- NERC, Voltage Stability Criteria and Reactive Power Reserve Practices, Sep-2010.