Turbine Hill Chart: How to Read It (and Dispatch Well Without One)

A hill chart — also called the Hill diagram or turbine performance map — plots the machine's efficiency as a family of iso-efficiency contours over its two degrees of freedom: unit discharge (flow per unit, usually normalized by the design value nQ or expressed in m³/s) and unit speed (nE or rpm, depending on the convention). For a fixed-head installation the unit travels along one head curve as the guide vanes open; for a variable-head plant it sweeps a family of them as the reservoir level moves. The closed island in the middle is the best efficiency point, the BEP — the operating point where the machine converts water power to shaft power with the least loss.

The chart is produced at the model test (the scale-model acceptance test written into most turbine supply contracts), sometimes refined by CFD, and rarely re-measured at full scale. That matters because the chart is the machine's identity: it is what tells you how much water a given megawatt costs at a given head. Two runners with the same nameplate can differ by more than a point of efficiency off-BEP, and the chart is the only document that shows where.

One practical reading note: the hill chart carries efficiency, not revenue. A point at 93% efficiency and a point at 89% efficiency are both 'good' on the contours, but what they earn depends on the price at that hour and on how much wear the visit costs — which the contours do not show.

Experienced turbine engineers read a second map on top of the first one. Below roughly 40–60% of rated discharge, most Francis runners enter the part-load vortex rope zone: draft-tube pressure pulsations grow, power output swings, and fatigue loading on the runner and shaft accumulates. Near full load at low head, many machines hit the saddle zone, where efficiency looks acceptable on the contours but the hydraulic behavior is unstable. And in the upper-left region of the chart, the cavitation sigma margin thins out — the operating point where the plant's static suction head can no longer suppress cavitation, and pitting damage begins to accelerate.

These zones are machine-specific and usually live in separate documentation: the model-test sigma diagram, the vortex-free operating range agreed at commissioning, the inspector's marked-up chart after a major outage. When the plant's dispatch logic only sees the efficiency contours — or worse, only sees a cam curve frozen at commissioning — it cannot avoid zones it does not know are there.

The same blindness applies to start-stop events, which do not appear on the chart at all. Every start is a transient through the rough zones at high mechanical stress. A dispatch strategy that chases the daily price spikes with extra unit starts pays for them in runner life, and the hill chart alone will never say so.

The BEP is where efficiency peaks. It is not where the plant earns the most. On a day-ahead market, the hourly price of a megawatt-hour varies by a factor of three to ten across a normal week; a strategy that holds the unit at the BEP all day spends its cheap hours doing nothing and its expensive hours at whatever output the head happens to allow. The right comparison is between the revenue a schedule earns and the damage cost it accumulates, both in euros, in the same objective.

Pricing the damage changes the recommendation in ways that look wrong on the efficiency map alone. When a midday price spike is modest, the optimizer will deliberately park the unit away from the BEP, in a lower-efficiency but calmer zone, because the wear avoided is worth more than the megawatt-hours lost. When a spike is extreme, it will accept hard wear and start the unit, because the revenue pays for the inspection interval it shortens. Both moves are indefensible on an efficiency chart and obvious on an euro-denominated one.

We run a deterministic dynamic program over the rolling day-ahead horizon on two coupled maps — the performance hill chart and a damage hill chart with start-stop cost — and score the result against a competent price-following baseline on real 2024 DE-LU day-ahead prices. On a 300 MW Francis reference unit the measured headline is +8.03% gross revenue, +7.82% net of damage cost, about +€5.05M per year. The method and caveats are published; the point of the baseline is that the uplift is a measured number with a stated method, not a brochure claim.

The usual first blocker is not budget or skepticism — it is that nobody can find the model-test report. Plants change hands, files stay with the previous operator, and a machine can run for years with the hill chart existing only as a laminated commissioning cam curve. The standard industry answer, a new model test or a CFD campaign, costs more than most single-unit decisions justify.

An estimator changes the entry condition. From nameplate data alone — turbine type, rated power, rated head, rated discharge — a machine-learning model can infer a plausible hill chart for the machine class: the efficiency contours, the estimated head range (typically 85–110% of rated head), and the flow window. The discharge estimate itself follows from first principles: rated flow ≈ rated power / (ρ·g·H·η), with η around 0.92 for a modern Francis unit. It will not replace a model test, and it should never be presented as one; it replaces 'no project' with 'a project built on stated assumptions'.

That is the route behind our Turbine Uplift Diagnostic: you send the nameplate specs and any partial documentation you have, we infer the chart, state every assumption in the report, and return a full-year backtest on real day-ahead prices for your exact machine — in 5–7 business days, for €900, credited against a first uplift contract if you proceed. The value is not the estimated chart itself; it is the measured answer to 'what is this machine leaving on the table' before anyone signs anything.

Whether a chart comes from a model test, CFD or an estimator, four checks catch most problems. First, the peak: a modern Francis peak efficiency below about 92% at the model scale, or more than about a point above the machine class's known best, is a red flag in either direction. Second, the head family: the contours should shift smoothly as unit speed changes, with no kinks — a kink usually means interpolated or synthetic data. Third, the rough zones: their position relative to the BEP should match the machine type's known geometry, and the sigma diagram should agree on where cavitation limits the map. Fourth, and most important, the chart must reproduce a few known operating points from the plant's own history — a date, a head, a gate opening and the measured output. Two or three of those anchor the chart to reality better than any contour plot.

For an estimated chart, add a fifth check: the dispatch decisions it produces should be insensitive to reasonable perturbation. If moving the assumed peak efficiency by half a point moves the recommended operating point into a different zone entirely, the estimate is being asked a question it cannot answer, and the honest answer is a model test or a full-scale measurement — which is also the point at which a performance-fee arrangement like ours gets cheaper for the client, because better data lowers the fee.

The habit worth building is small: take one recent week of hourly prices, lay it over your unit's operating pattern, and ask where the schedule visited the chart. Hours at the BEP during cheap hours are the easiest money in the plant; hours in the rough zones during cheap hours are the most expensive kind of loyalty to the commissioning cam curve. The cavitation damage guide covers how the wear side converts to euros per hour; the demo shows the same logic on a reference machine.

If you want this done for your own machine without a data project, the €49 Hydro Dispatch Uplift Report shows the method and the 2024 backtest results, and the €900 Turbine Uplift Diagnostic runs the full backtest on your specs. DAMagedOpt, the business behind this site, is built and operated end to end by AI agents on the NanoCorp platform — which is how a one-off diagnostic backtest can be turned around in days rather than procurement quarters.