The Counterfactual Problem

Our reference Panamax bulker, about 75,000 dwt, which carries the worked examples through most of this specialisation and appears in them as Ship A as well, needs 6,320 kW of delivered power to hold 14.0 knots in calm water. A device sold on a 4 per cent saving is therefore worth about 253 kW. The day-to-day scatter of that ship's normalised daily consumption, after the obvious rubbish has been filtered out, sits somewhere between 8 and 15 per cent, which on the same ship is 500 to 950 kW. The quantity you are trying to see is between a quarter and a half the size of the quantity you cannot control. Everything in this course follows from that ratio.

One caution on those kilowatts. Our reference ship is a teaching hull with round numbers: her 577 kN of calm-water resistance at 14.0 knots, and the 6,320 kW that follows from it, sit low for her size, where a real Panamax needs nearer 680 kN and so roughly 7,450 kW. Every kilowatt quoted for her runs about 15 per cent below a real ship's, so the simplification makes each saving look smaller in kilowatts than it would be; the ratios the course turns on, a 4 per cent effect inside 8 to 15 per cent of scatter, are the same.

A small difference between two large numbers

To say that a device saved 4 per cent you must know what the ship would have burned without it: same voyage, same day, same draught, same weather, same current, same hull condition. That reference voyage is the counterfactual, the comparison case that did not happen, and it never will. Every method in the seven lessons that follow is a way of manufacturing a substitute for it, and every substitute is bought at a price.

The structural difficulty is that a saving is not a measured quantity. It is the difference between two measured quantities of almost the same size. Give each leg an independent relative uncertainty of 1 per cent: the two combine to about 1.4 per cent of the whole, and a difference worth 4 per cent of the whole inherits an uncertainty of 1.4 divided by 4, or 35 per cent. Nothing has gone wrong with the instruments; the arithmetic of differencing did it.

There is one escape, and every good method here uses it: a bias common to both legs cancels in the difference. A torsionmeter reading 2 per cent high reads 2 per cent high on both legs; a hull fouled since the baseline was measured is equally fouled for both. What does not cancel is anything that moves with the comparison itself, and that is what confound means here.

Four substitutes, four prices

There are only four counterfactuals available, and the whole subject is a comparison of what each one costs.

The same ship before the retrofit. Cheap, and available for any device. It is attacked by everything that drifts over the same window: hull and propeller fouling, which drifts monotonically and is confounded with the calendar, so that a before-and-after window cannot tell it from the device, and whose reset at a docking moves in the same direction as a claimed saving; the mix of draughts and the trade pattern; seasonal weather; engine tune; fuel quality. The worst case is the commonest: a device fitted at a dry-docking is credited with the docking's own cleaning gain unless the analysis handles it.

A sister ship without the device. This holds time constant rather than the ship. Sisters are never identical in build tolerance, coating history and age, engine condition or crew practice, and the two ships sail different weather on different routes, which is usually the larger term.

The same ship, minutes later, with the device switched off. The strongest of the four, because the device state can be made to vary fast while every confound varies slowly. Its price is that the device must be switchable, that the environment drifts between the two legs of a pair, and that the schedule must be fixed in advance. If the crew switches the device on when they judge it will help, the device state becomes correlated with favourable wind, and the estimate is inflated by an amount nobody can afterwards recover.

A model test or a computation. Here the counterfactual is calculated rather than sailed, which makes it perfectly controlled and answerable at any condition you like. It is bought with scale and modelling error of unknown magnitude, and with whatever boundary conditions you fed it.

Leverage, and why a modest claim is the hard one

A theoretical worked example of an ITTC-style wind-propulsion trial shows how the price is actually paid. A small ship of about 900 kW ran paired legs, rig off and rig on. Off: 8.82 knots at 957.5 kW. On: 10.10 knots at 899.3 kW. The ship went 1.28 knots faster on 58 kW less power, so the saving is plainly not 58 kW: it must include the speed gained.

The procedure values that speed gain on a baseline curve, the ship's speed-power relationship without the device, taken from an earlier speed trial, a model test or a short speed-variation test on the day. The example used a pure cubic, f(V) = k V³, and formed

ΔP = − [ f(V1) − f(V0) + P0 − P1 ]

where V0 and P0 are the mean speed and shaft power on the device-off leg, V1 and P1 the same with the device on, f the fitted baseline curve, and a negative ΔP means power saved. Read it physically: you do not compare the device-on power against the curve, you take the device-off leg exactly as measured, wherever it sits, and transport it along the curve from V0 to V1. Any constant offset between the curve and the day is common to both legs and cancels; only the shape of the curve between the two speeds survives.

The example reports the two legs, the saving and its band, and leaves the curve's coefficient for the reader to back out: k of about 1.432 kW per knot cubed, so f(8.82) = 983 kW and f(10.10) = 1,475 kW. That puts the curve 25 kW above the measured device-off leg at 8.82 knots, an offset the differential construction cancels. The transport term is 493 kW, the measured power drop is 58 kW, and ΔP is −551 kW, which is 37 per cent of the baseline power at the higher speed.

Now look at where those 551 kW came from. 493 of them, 89 per cent, are a speed gain valued on a curve measured on another day. Only 58 kW was watched on a meter. Define the leverage factor as the transported term divided by the saving, because it is what multiplies the baseline's own uncertainty:

δ(ΔP)/|ΔP| = (δk/k) · [ f(V1) − f(V0) ] / |ΔP|

where δk/k is the relative uncertainty of the baseline coefficient. Here the leverage is 493/551 = 0.89, and the reported band of ±10.7 per cent on the saving implies about 12 per cent uncertainty on k. Had the analyst compared the device-on power directly against the curve at 10.10 knots, the comparison would have credited 576 kW, since the curve sits above the day, with a leverage of 1475/576 = 2.56, and the same baseline uncertainty would have given about ±31 per cent. The differential construction is worth a factor of about three.

Then the sting. Repeat that arithmetic for a device claiming 3 per cent rather than 37. For the naive comparison the leverage is the reciprocal of the fractional saving, 1/0.03 = 33, so 12 per cent on the baseline becomes 400 per cent: an error bar four times the claim. The differential construction rescues the baseline term, but not the rest: the torsionmeter's scatter, a wind mismatch between the legs and a current that moves are the same number of kilowatts whatever the device claims, so they weigh ten times as much against 3 per cent as against 30. Verifiability falls with the modesty of the claim. A 30 per cent device can be proved in an afternoon; a 3 per cent device sits inside the band a real trial carries once instruments, wind and current are counted, as lesson 5 shows, which is why most of this course is spent on methods that take months.

Be careful with those figures. They are one theoretical example, one small ship in 33 to 35 knots of apparent wind at an exceptionally favourable operating point, and k, the curve values and the 12 per cent on k are backed out of the reported results rather than stated. Suppose the report also gives a second speed for the same runs, 9.3 knots over the ground against 8.82 through the water, without saying which its results use: carrying that 0.48 knot offset through both legs raises the saving from 551 to 602 kW, about 9 per cent. A trial report that does not say which speed it used has not told you its answer. The example illustrates the algebra; it is not a number to quote.

Monday morning

Take the most recent saving claim on your desk and answer three questions before reading another line of it: which of the four counterfactuals is it built on, what confound attacks that one, and does the report mention it. Then find the date of the baseline and the hull condition it was measured at. If it predates the last hull cleaning, the number is uninterpretable and you can stop there. The next lesson turns these four into a proper ladder, from a brochure figure to a two-year in-service study, with an uncertainty band on every rung.