Speed Optimization for Sustainable Shipping

Abstract

Among the spectrum of logistics – based measures for sustainable shipping – this chapter focuses on speed optimization. This involves the selection of an appropriate speed by the vessel, so as to optimize a certain objective. As ship speed is not fixed, depressed shipping markets and/or high fuel prices induce slow steaming which is being practiced in many sectors of the shipping industry. In recent years the environmental dimension of slow steaming has also become important, as ship emissions are directly proportional to fuel burned. Win-win solutions are sought, but they will not necessarily be possible. The chapter presents some basics, discusses the main trade-offs and also examines combined speed and route optimization problems. Some examples are presented so as to highlight the main issues that are at play, and the regulatory dimension of speed reduction via speed limits is also discussed.

Appendix A

1.1 Comparison Between a Bunker Levy and Speed Limits

With all the discussion on speed limits at the IMO, the purpose of this appendix is to investigate the issue of a bunker levy vs speed limits. Both measures would cause speed reduction and hence a reduction in CO₂ and other emissions (GHG and non-GHG). A bunker levy would induce speed reduction, and a speed limit would mandate it. Below we attempt to compare the two measures, in terms of emissions reduction and other attributes.

To investigate the issue, we use a rudimentary scenario in the container sector. A generalization to more realistic scenarios or other shipping markets is straightforward. The example is taken from the ‘cart before the horse’ paper (Psaraftis 2017).

A containership of capacity Q (TEU) shuttles between port A and port B, whose interport distance is L (nautical miles, nm). The ship’s speed is v (nm/day) ⁴ which is within the bracket v _min and v _max.

Assume that the ship is semi-full in both directions and that the freight rate received by the ship owner is R (USD/TEU), assumed the same in both directions. The assumed load factor of the ship is u (0 ≤ u ≤1), again assumed the same in both directions. R is assumed to be on a per loaded TEU basis, meaning that if the ship is 75% full (u = 0.75), its per (one way) trip income will be 0.75RQ. Assume that the fuel price is p (USD/tonne) and that the fuel consumption function is FC = kv ³ (tonnes/day) with k being a constant. Assume finally that miscellaneous/other operating expenses are X (USD/day) and that port turnaround times are ignored. Q, L, R, u, p, k, v _min, v _max and X are assumed known inputs (see Table 10.5 above for an example), and the sole decision variable is the ship’s speed v.

Table 10.5 Assumed inputs

We also note that this analysis assumes that R is an exogenous variable outside the line’s control, and we do not attempt to estimate R as a function of container capacity supply and demand. In that sense, it is expected that slow steaming or speed reduction, if applied for all ships sailing the given route, will generally increase R; however this is not captured in our model.

Finally k is such that FC = 144 tonnes/day when v = 22 knots. The value of k for which this is the case is 9.7827*10⁻⁷ (again, v in the formulas is in nm/day).

In this scenario, we can compute various attributes of the round trip, such as:

• Round trip time T = 2 L/v (days)

• Round trip TEU throughput H = 2uQ (TEU)

• Round trip cost C = T(pkv ³+X) = 2(pkLv ²+LX/v) (USD)

• Round trip income I = 2uRQ (USD)

• Round trip profit P = I – C = 2(uRQ – pkLv ² – LX/v) (USD)

• Average per day profit P ^′ = P/T = uRQv/L – pkv ³ – X (USD/day)

• Average per day TEU throughput H ^′ = 2uQ/T = uQv/L (TEU/day)

If the objective of the line is to maximize average per day profit, that is, P′, the optimal speed can be shown to be as follows.

$$ {v}_{\mathrm{opt}}={v}_{\mathrm{min}}\kern0.5em \mathrm{if}\ {v}_{\mathrm{min}}>{v}_0 $$

$$ {v}_{\mathrm{opt}}={v}_0\kern0.5em \mathrm{if}\ {v}_{\mathrm{min}}\le {v}_0\le {v}_{\mathrm{max}} $$

$$ {v}_{\mathrm{opt}}={v}_{\mathrm{max}}\kern0.5em \mathrm{if}\ {v}_{\mathrm{max}}<{v}_0 $$

with v ₀ = (uRQ/3pkL)^1/2

Then CO₂ emissions per unit time (tonnes/day) for this ship are equal to

$$ {\mathrm{CO}}_2\kern0.5em =\kern0.5em {fkv_{\mathrm{opt}}}^3 $$

with f being the carbon coefficient (assumed here equal to 3.11).

For an individual ship, Table 10.6 above shows the optimal speed and other solution attributes for the above inputs and for selected values of the freight rate R ranging between 500 USD/TEU to 2000 USD/TEU, with a base case value of 1500 USD/tonne.

Table 10.6 Optimal speed as a function of freight rate R, individual ship

One can see in general that a higher state of the market (higher R) induces a higher speed and hence higher CO₂ emissions for the ship, and vice versa. It should also be noted that in this particular example and for the two extreme cases R = 500 and 2000 USD/TEU, the optimal speed hits the speed’s lower and upper bounds, respectively.

To lower CO₂ emissions, one contemplates either a levy q on fuel or a speed limit equal to V, with q and V being user inputs. Either of those would generally result in a lower speed. The question is: Which of these alternatives achieves lower CO₂ emissions? The answer of course depends on the values of q and V. Depending on these values, a levy can achieve lower, the same or higher CO₂ emissions reductions vis-à-vis those achieved by a speed limit.

Note also that for this comparison to make sense, constant average per day TEU throughput should be maintained, even though speed is reduced. This would necessitate deploying additional ships.

If the initial speed before the levy or the speed limit is v ₁ and the final speed after the levy or the speed limit is v ₂ (<v ₁), we define as the ‘throughput factor’ the ratio r = v ₁/v ₂ (>1). A ratio r = 1.20 means that r-1 (in this case 20%) more ships should be deployed on the route so as to maintain the same average per day TEU throughput. These additional ships would generate additional profit and additional CO₂, both of which should be taken into account. To do so, the average per day profit and the average per day CO₂ emissions should be multiplied by r, vis-à -vis those for an individual ship.

To further investigate the issue, we assume that v _min ≤ V ≤ v _max because if V is outside that range, then either the speed limit is superfluous (V > v _max) or the problem is infeasible (V < v _min).

The two cases are compared in Table 10.7 above as follows.

Table 10.7 Comparison between the speed limit and levy cases, individual ship

The superfluous speed limit case occurs if V ≥ (uRQ/3pkL)^1/2 _, which for our case and for the base case for R means V ≥ 23.07 knots. If this is the case, v _opt is also 23.07 knots.

The non-superfluous (binding) speed limit case occurs if V < (uRQ/3pkL)^1/2 = 23.07 knots.

Table 10.8 shows the results for the base case R and for selected values of the speed limit V ranging from 18 to 22 knots.

Table 10.8 Reductions of CO₂ and other attributes as a function of the speed limit V, constant throughput

The last row in the table shows the reductions of CO₂ (in tonnes/day) that can be achieved as a function of the speed limit, vis-à-vis the ‘no speed limit’ case (516.67 tonnes/day). Note that the figures for P′, CO₂ and ΔCO₂ have factored in the effect of the throughput factor r.

In turn, we can investigate what happens if we impose a levy q on bunker fuel. Table 10.9 shows these results (again base case for R) for selected values of q ranging from 100 to 500 USD/tonne.

Table 10.9 Reductions of CO₂ and other attributes as a function of the levy q, constant throughput

Again, the last row in the table shows the reductions of CO₂ (tonnes/day) that can be achieved as a function of the levy, vis-à-vis the ‘no levy’ case (516.67 tonnes/day). As before, the figures for P′, CO₂ and ΔCO₂ have factored in the effect of the throughput factor r.

Tables 10.8 and 10.9 are not directly comparable, in the sense that from these tables no direct conclusions can be drawn as to what is preferrable, a speed limit or a levy. To draw such conclusions, we ask the following question: For a given levy q, what is the value of the speed limit V so that the results are the same in terms of CO₂? And once this happens, what are the other differences between the two cases?

It turns out that the speed limit V for which the optimal speed is the same as that with a levy q is as follows.

$$ V={\left( uRQ/3\left(p+q\right) kL\right)}^{1/2} $$

Then the optimal speed is equal to V in both cases.

In this case, and for an individual ship, daily CO₂ is also the same and equal to fkV ³ = 2fk(uRQ/3(p+q)kL)^3/2

However, daily profit P′ is different. With a levy q, it is P^′ = uRQV/L − (p+q)kV³ – X.

With an equivalent speed limit V, and no levy, it is P ^″ = uRQV/L − pkV ³ – X (>P ^′).

The difference in daily profit is ΔP ^′ = qkV ³.

The above are for an individual ship. To maintain the same TEU throughput, the effect of the throughput factor r has also to be taken into account.

This means that for the ship owner, and if the same speed (and hence the same CO₂ emissions) reduction are to be achieved, a speed limit is more profitable than a bunker levy. The ship owner will sail the ship at the same speed as that with a levy, but without paying the levy.

Table 10.10 shows the values of V, CO₂ and ΔP’ for values of q between 100 and 500 USD/tonne.

Table 10.10 Equivalent speed limit V, CO₂ and ΔP′ as functions of levy q, constant throughput

This cuts both ways. The difference in daily profit ΔP′, which is positive for the ship owner and which possibly reflects an external cost of CO₂ pollution that is not internalized, is a net cost to society. It is money not collected which could be used to achieve out-of-sector emissions reductions⁵ or for other noble causes (e.g. financial aid to developing countries, research and development, etc.). In that sense, and from a societal point of view, a levy is better than an equivalent speed limit.

An equally serious problem with a speed limit is that for ships of different size, a common and uniform levy q will result in different optimal speeds. A larger ship would in general imply a higher optimal speed, everything else being equal. Therefore, achieving equivalence such as the above by a common and uniform speed limit V will be impossible. To do so, one would have to set size-specific (or maybe even ship type-specific or route-specific) speed limits, which will make the whole exercise an administrative nightmare.

Conversely, if a common and uniform speed limit V is imposed, the limit may be superfluous for some ship sizes and binding for some others, depending on the state of the market, the price of fuel and a host of other parameters. Having the same speed limit in boom market periods and in depressed market periods could create all sorts of distortions. In depressed market periods, the limit may be superfluous, and in boom market periods, the limit would force some ships (likely at the high end of the scale) to slow down, whereas others do not. A speed limit may also be superfluous in one route direction (e.g. from Europe to the Far East, where ships go slower anyway) and binding in the other direction (ships go faster from the Far East to Europe).

Last but not least, a speed limit would be difficult or impossible to enforce, even if it is the same for all ship sizes or types, and it would hardly serve as an incentive to economize and improve the energy efficiency of ships.

For at least the above reasons, a conjecture that we can safely make is that a bunker levy is a preferable instrument (as compared to a speed limit) if one wants to reduce maritime emissions.