One machine, four quadrants, eight zones
A pump running as a turbine (PAT) — or a reversible pump-turbine in a pumped-storage plant — does not live in one duty quadrant. Depending on the signs of rotational speed N and flow Q, and on the resulting signs of head H and torque T, the same hydraulic passage operates as a pump, a turbine, a brake or a pure energy dissipator. Plotting every regime on the dimensionless (α, ν) plane gives the classical four-quadrant, eight-zone map used for transient studies:
α = 0 means standstill, α = 1 rated pump speed; ν = 1 rated pump flow. Negative ν is back-flow through the machine; negative α is reverse rotation. The Bad Creek reference machine of this page has N₀ = 300 rpm, so α = 1.58 ↔ 475 rpm.
Dimensionless coefficients — from model test to prototype transients
These are the working variables of four-quadrant model tests: each guide-vane opening yields one Kcm1–Ku1 and one KM1–Ku1 curve traced through all four quadrants. Converting model data to prototype water-hammer / load-rejection studies needs both pairs (H–Q and T–Q data). The (α, ν) plane of this simulator is the per-unit sibling of the same representation.
| Symbol | Meaning | Unit |
|---|---|---|
| α, ν | Per-unit speed N/N₀ and per-unit flow Q/Q₀ | – |
| Ku1 | Unit (peripheral) speed coefficient, U = ωD/2 at impeller outer diameter | – |
| Kcm1 | Unit flow coefficient, based on the discharge area πD²/4 | – |
| KM1 | Unit torque coefficient | – |
| U, ω, D | Peripheral speed, angular speed, impeller diameter | m/s, rad/s, m |
| H, T | Head across the machine, hydraulic torque | m, N·m |
| ρ, g | Water density, gravitational acceleration | kg/m³, m/s² |
The eight zones — sign table
Each zone is fixed by the signs of the four quantities N, Q, H, T (energy flow follows directly: shaft power ∝ T·N, hydraulic power ∝ ρgQH):
| Quadrant | Zone | Operating mode | Energy | N | Q | H | T |
|---|---|---|---|---|---|---|---|
| I | A | Normal pumping | consumed | + | + | + | + |
| I | B | Energy dissipation | dissipated | + | + | − | + |
| I | C | Reverse turbining | produced | + | + | − | − |
| II | D | Energy dissipation | dissipated | − | + | + | − |
| II | E | Reverse pumping | consumed | − | + | + | − |
| III | F | Braking | dissipated | − | − | + | − |
| III | G | Normal turbining | produced | − | − | + | + |
| IV | H | Energy dissipation (back-flow) | dissipated | + | − | + | + |
Model test reports usually publish only the positive-head zones A / H / G / F / E — those are the ones a load-rejection or power-failure transient actually traverses.
Runaway — concept and screening criterion
When a turbine-mode unit rejects its load, or a pumping unit loses power with the vanes still open, the hydraulic torque no longer balances the electromagnetic or driven torque and the rotor accelerates (or, with back-flow, reverse-accelerates) until it reaches the speed where the net hydraulic torque is zero — the runaway speed, T = 0. It is a true equilibrium of the torque curves, not a failure of the machine, but the overspeed margin governs the mechanical and generator design:
- 🟢 ≤ 120 % of rated speed — normal transient band; standard generator overspeed margin (≥ 150 %) comfortably covers it.
- 🟡 120 – 150 % — approaching runaway; check the guide-vane closure law and the inertia of the rotating parts.
- 🔴 > 150 % of rated speed — runaway territory. The Bad Creek reference case reaches 475 rpm = 158 % of 300 rpm after full load rejection — a genuine runaway excursion that the unit must survive mechanically.
The load-rejection / power-failure walk-through
Press Play and follow the classic power-failure path of a pumping unit — the trajectory the replay animates, digitised in compressed time from the Bad Creek transient (rated 300 rpm → reverse runaway 475 rpm):
| Phase | Zones crossed | What happens |
|---|---|---|
| Trip at pump duty | A | Drive torque collapses; flow decays faster than speed while the pump still spins forward. |
| Back-flow sets in | A → H | ν crosses zero: the downward system head drives flow backwards through the forward-spinning impeller — pure dissipation (zone H). |
| Reverse rotation begins | H → G | Hydraulic torque spins the rotor backwards; the machine re-enters turbine action in reverse (zone G) and accelerates. |
| Reverse runaway | G, T = 0 | Reverse speed peaks at the zero-torque point — 475 rpm, 158 % of rated. This is the maximum the shaft ever sees. |
| Closure and decay | G → F | Guide vanes throttle the flow; torque turns braking (zone F) and speed bleeds away. |
If the vane servo fails, or a generate-to-pump changeover is mistimed, the trajectory can continue into zone E (reverse pumping). Model data for the full eight zones is exactly what a transient study of such malfunctions requires.
Anchors used by this page
| Check | Input (α, ν) | Zone returned |
|---|---|---|
| Pump mode preset | (+1.00, +1.00) | A — normal pumping |
| Forward braking preset | (+0.80, +1.75) | B — energy dissipation |
| Turbine mode preset | (−1.00, −1.00) | G — normal turbining |
| Reverse braking preset | (−0.45, −0.30) | F — braking |
| Runaway preset (475 rpm) | (−1.58, −1.10) | G/F edge, T = 0 — 158 % of N₀, red |
| Replay phase 2 (back-flow) | (+0.62, −0.72) | H — dissipation |
| Replay end (decay) | (−0.75, −0.25) | F — braking |