6 Supplementary Material
6.1
Biofuel pathway cost component estimation: method detail
6.1.1 Feedstock production costs
The assumed nominal price is that required to cover additional costs of collecting the residue after crop harvest plus an
additional sum to cover the replacement of nutrients that the residue would otherwise provide. A minimum assumed
cost just covers nutrient replacement, where marginal harvesting costs are assumed zero for the case where the residue
is removed irrespective of sale options. A maximum price is derived by assuming that the collection and harvest costs
are double that of the nominal.
It is assumed here that all the potentially available agricultural residue is accessible at a unique price. Other literature
has identified a dependence of cost on feedstock concentration (1), but here we assume that local competition results
in a single farm-gate price (at least within a given collection area). Hence the feedstock supply curve is modelled as
perfectly elastic up to the theoretical maximum amount of residue potentially available for harvest (and inelastic
beyond).
In practice, the relationship of residue availability to price is likely to be more sophisticated, as prices required by
specific producers will depend on annual yield and other individual circumstances. For example, under the Australia
no-till systems, where there are high residue loads that interfere with subsequent planting, there is a need for reduction
either via burning or mechanical removal. In low yielding years, however, residue cover is required for environmental
reasons, so farmers are much less likely to part with it. Secondly, there are also other uses for crop residue, including
animal bedding, mushroom composting and even export (2). Thus it is realistic to expect a range of prices over which
feedstock would be available. In other words, feedstock supply is likely to be somewhat elastic over a range of prices
that includes nominal values within the range identified here.
6.1.2 Feedstock transport costs
Page 1 of 18
The transport model (3) is a “bottom-up” cost model composed from a detailed breakdown of capital, operating and
labour costs for three alternative truck types (Semi-trailer, B-double and Road Train).
The modelled relationship of cost to mass transported is linear, and to distance is piecewise linear (plus offset).
𝑇(𝑉, 𝐹, 𝑄𝑓 , 𝑑) = {
[𝑎𝑁 (𝑉, 𝐹) + 𝑑 ∙ 𝑏𝑁 (𝑉, 𝐹)]𝑄𝑓 ,
[𝑎𝐹 (𝑉, 𝐹) + 𝑑 ∙ 𝑏𝐹 (𝑉, 𝐹)]𝑄𝑓 ,
𝑑<𝐷
𝑑≥𝐷
In the above equation, 𝑇 is the total cost of transporting feedstock of quantity 𝑄𝑓 a transport distance 𝑑, parametrised
by vehicle and feedstock type 𝑉, 𝐹. See Error! Reference source not found. below. The reason for this shape is
explained as follows. Full truck loads are assumed, with maximum sizes constrained by mass or volume limits. Fuel
and vehicle wear costs are each assumed to be proportional to transport distance, but labour time includes a fixed
component per trip for loading and unloading in addition to a component proportional to distance for driving. The
only other minor subtlety is that the model assumes a maximum number of trips (loads) per day, so that for trips of
very short distances, labour hours become a fixed cost and the distance dependent component of the total includes
only fuel and vehicle wear. The net result is that the total cost is proportional to quantity (mass) but is a piecewise
linear function of distance, with only two segments (data appears in Error! Reference source not found. in
Supplementary Material 6.2).
The first segment, for transport distances less than a threshold that depends on the maximum number of allowable
trips per day and average driving speed, has a greater offset and lesser slope than the second segment. If each truck
load is mass-limited, the constants of mass-proportionality are independent of feedstock. The constants of massproportionality depend on feedstock type only if each load is volume limited, as is crop residue. All parameters
depend on the vehicle type. For calculations in this paper, the higher, conservative, transport costs of the semi-trailer
are assumed.
Average transport distance from the farm to factory gate is dependent upon the size and shape of the biomass
collection area. In the absence of detailed road map information, it is common in transport cost studies to assume that
the road distance differs from the direct point-to-point distance by a fixed factor called the tortuosity factor, which we
select as 1.5 as in Wright and Brown (4). For a collection area of fixed shape but variable scale, this allows the
average transport distance to be expressed as the square root of the area multiplied by a proportionality constant.
Page 2 of 18
The area weighted average of the straight line distance between points within a circle and its centre is 2/3 x radius. For
a square (or rectangle) of area 𝐴 the area weighted average straight line distance to the centre is [ln(1 + √2) +
√2]/6√A ≈ 0.765 × 12√A (5). Thus, assuming a tortuosity factor of 1.5 and that the collection area is rectangular (or
square), the area weighted average road distance as a function of are 𝑑 = 0.573√𝐴.
6.1.3 Biorefinery processing costs
For elemental biofuel processing cost data (capital costs, variable operations and maintenance costs, and fixed O&M
costs), a literature survey was undertaken to summarise a range of published values for each of the five selected
biorefining processes. There are many cost case-studies of a specific bioenergy production technology (6), although
some of them apply only within quite specific contexts (7). This paper relies preferentially on survey papers for
biofuel processing costs, which has the advantage that similar cost estimation methodologies are used for each of the
processes that a given survey covers.
The sources shown in
Elec
Lange (15)
(S&T)2 Consultants Inc. (25)
Wright and Brown (12)
Stucley, Schuck (6)
Tao and Aden (26)
Foust, Aden (27)
Hamelinck and Faaij (24)
Wright and Brown (4)
Anex, Aden (28)
Graham, Reedman (29)
Obernberger and Thek (7)
CSIRO (30)
Phillips, Aden (21)
Aden, Ruth (19)
Kazi, Fortman (31)
Wallace, Ibsen (17)
Hamelinck, van Hooijdonk (32)
Stephen, Mabee (33)
Boerrigter (11)
Tijmensen, Faaij (20)
Kreutz, Larson (34)
Page 3 of 18
*
Eth
(The)
*
*
*
*
*
*
Eth
(Bio)
*
*
*
*
*
*
*
*
*
Biosyndiesel
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Biosynjet
*
*
*
International Energy Agency (35)
*
National Energy Technology Laboratories (14)
*
Rodrigues, Faaij (18)
*
Hatch Engineering (13)
Table 1 were used, many of which individually cover several of the processes of interest. The table shows which
conversion processes have cost data provided within each reference. Economic (cost) data were converted into a
common currency (2009 USD) using exchange rates and inflation rates from (8). Physical quantities were converted
into SI units. Other required technical and economic data, such as process conversion efficiencies, were either taken
directly from the references or inferred if possible from the other reported data, such as nominal output production rate
and feedstock requirements.
6.1.4 Data population summaries: Confidence interval estimates
The data collected from the literature provide, for each parameter of interest, a population of data values. Nominal
representative values and confidence intervals were derived as described following. For this process, if the data
population were to be randomly sampled from a normally distributed population, it would provide an unbiased
estimate of both the mean and the standard error.
First, in order to limit the effect of outliers, values below the 15th and above the 85th percentiles were removed (this is
possible only if the number of samples exceeds six). The range between those two percentiles is then taken as an
estimate of the standard error. These error estimates are not to be taken as precise, but are merely indicative of the
degree of confidence in the nominal value.
A nominal representative value is then selected as a weighted average (either arithmetic or geometric as appropriate)
of the remaining values. The weighting used for each data point was such that the total weight for the data points in
each identified paper was equal (that is the weight of each datum was inversely proportional to the number of relevant
data points from each paper). This weighting ensures that any potential systematic bias estimates within each paper are
not over-represented in the sample population. There was no attempt to identify any correlation of data values by
reference source and therefore no attempt to correct for corresponding bias.
Page 4 of 18
6.1.5 Processing plant capital costs: scale economies
A simple (standard) model (9) of the relationship between scale (rate of production) and processing plant capital costs
was verified using data identified for each type of process considered. At larger scales of production output, industrial
processing plant will often become more efficient (in terms of structural dimensions, material quantities and economic
cost). In particular, many individual components of industrial plant will have a relationship, parameterised by plant
component 𝑖, between processing production scale 𝑄 and cost 𝑘𝑖 that approximates a power law: c = s(I)Qg(I) for
some constant 𝑠𝑖 , where 𝑔𝑖 is a scaling exponent that takes on a value between zero and unity. This property can often
be deduced from dimensional scaling relationships in the material realisation of the process, often based on
fundamental physical principles. Given several items, the costs of each following a power law relationship with
different, but approximately similar, exponents, their total combined cost will also approximate a power law
relationship (See the Supplementary Material 6.3). However, at the largest scales, components with larger exponents
(those closer to unity, that is those that do not show strong economies of scale) will tend to dominate: see also (10).
Accordingly, the model for total capital costs takes the form K = SQg , that is
ln(𝐾) = 𝑔 ln(𝑄) + ln(𝑆).
Equation 1: Capital cost scaling
This also can be expressed in terms of an arbitrarily selected nominal scale 𝑄0 with capital cost 𝐾0 .
𝑔
𝐾0 = 𝑆𝑄0
𝑔
𝐾 = 𝐾0 (𝑄⁄𝑄 )
0
Parameters 𝑆 and 𝑔 can be estimated from a (weighted) least squares linear regression of ln(K) versus ln(Q). This
regression was performed with weights selected as described in Section 6.1.4. A regression estimate over data from
several papers reduces the expected error, assuming that sources of error are independent across the selected papers.
Since only a handful of references were used for this estimation, the estimates are subject to high variance, and so
alternative approaches to estimating the scaling exponent were also investigated. Some of the literature explicitly
referred to a particular scaling exponent for the conversion processes described (4, 6, 11, 12). Other references
provided capital cost estimates for the complete process at two or more scales of production, enabling an implied
Page 5 of 18
scaling exponent to be inferred from aggregate data (sometimes known as a “top down” approach: (13-17). Still other
references provided costs and scaling factor exponents for individual components of plant (18-22). This allows an
aggregate exponent to be derived as a weighted average of the components in such a manner that the aggregate cost
curve approximates the sum of the components (a “bottom up” approach, see the Supplementary Material 6.3 for
further details). Estimates that were either directly reported in the references considered, or that were implicit and
identified as a weighted average as described above, cover a range of values.
6.1.6 Processing plant ongoing costs
The model of ongoing costs selected separates them into a component proportional to output (non-feedstock materials,
co-product sales) and a component proportional to plant capital cost (e.g. fixed operations and maintenance costs,).
Recall from the main text
𝑁 = (𝑤 + 𝑓𝑜 )𝑄 + 𝑚𝐾
where 𝑁 is the ongoing cost, 𝑄 is production quantity and 𝐾 is plant capital cost (see also (23)). Here fo is the factory
gate price of feedstock per unit output and, 𝑤 is non-feedstock per-unit (variable) cost and 𝑚 is a fixed operations and
maintenance cost, as a percentage of plant capital cost. Unfortunately, only very few of the references examined made
a clear distinction between fixed and variable ongoing costs. Furthermore there was a large range of costs, in terms of
both capital cost percentage and per unit quantity output, reported for different instances of the same process class. In
some cases (24) net ongoing costs were reported as negative, owing to co-product sales.
In order to develop a unified, consistent, cost model, parameters for each of two extreme alternative models of nonfeedstock ongoing costs were estimated, one assuming direct proportionality to scale (that is, all variable), the other
assuming proportionality to modelled capital cost (that is, all following the power law). Nominal representative values
of 𝑤 and 𝑚 and confidence intervals were determined as described in Section 6.1.4. A combined model was then
derived via a weighted averaging procedure. The weights were selected such that non-feedstock ongoing costs would
be attributed in the ratio rv : rf , to variable (proportional to output) and fixed (proportional to plant capital cost)
components, at a selected scale Q 0 , that is
𝑚𝑆𝑄0 𝑔
= 𝑟𝑓 : 𝑟𝑣
𝑤𝑄0
Page 6 of 18
1−𝑔
𝑚𝐾 𝑟𝑓 𝑄
= 𝑟 ( ⁄𝑄 )
0
𝑣
𝑤𝑄
1−𝑔
𝑟 𝑄
𝑁 = [ 𝑟𝑓 ( ⁄𝑄 )
+ 1] 𝑚𝐾 + 𝑓𝑜 𝑄
0
𝑣
1−𝑔
𝑟
= [1 + 𝑟𝑣 (𝑄⁄𝑄 ) ] 𝑤𝑄 + 𝑓𝑜 𝑄
0
𝑓
For the purposes of calculations here, we took Q 0 to be the (weighted, geometric) average (see Section 6.1.4) of the
range of scales identified. As a compromise between the two extreme alternative models, we assume that nonfeedstock variable and fixed ongoing costs are equal at this scale – that is rv : rf = 1: 1.
6.2
Transport Model Parameters
The values in
Elec
Lange (15)
(S&T)2 Consultants Inc. (25)
Wright and Brown (12)
Stucley, Schuck (6)
Tao and Aden (26)
Foust, Aden (27)
Hamelinck and Faaij (24)
Wright and Brown (4)
Anex, Aden (28)
Graham, Reedman (29)
Obernberger and Thek (7)
CSIRO (30)
Phillips, Aden (21)
Aden, Ruth (19)
Kazi, Fortman (31)
Wallace, Ibsen (17)
Hamelinck, van Hooijdonk (32)
Stephen, Mabee (33)
Boerrigter (11)
Tijmensen, Faaij (20)
Kreutz, Larson (34)
International Energy Agency (35)
National Energy Technology Laboratories (14)
Rodrigues, Faaij (18)
Hatch Engineering (13)
Table 1 were taken from data calculated for (3).
Page 7 of 18
*
Eth
(The)
*
*
*
*
*
*
Eth
(Bio)
*
*
*
*
*
*
*
*
*
Biosyndiesel
*
*
*
*
*
*
*
Biosynjet
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
6.3
Sum of powers
Q gi
Qo
Let the cost 𝑘𝑖 of individual plant components strictly follow an exponential scaling law so that k i = si ( ) where
𝑄 is the plant output scale, with nominal value 𝑄𝑜 , and 𝑔𝑖 is the corresponding exponent, so that the total plant cost is
then
𝑛
𝐾(𝑄) = ∑ 𝑘𝑖 ,
𝑖=1
𝑛
𝑄 𝑔𝑖
= ∑ 𝑠𝑖 ( ) .
𝑄𝑜
𝑖=1
We want to consider a scaled-up plant with scale Q = √RQ o compared to 𝑄𝑜 . Define nominal weights for plant scale
𝑄𝑜 as
𝑛
𝑤𝒊 = 𝑠𝑖 ⁄(∑ 𝑠𝑖 )
𝑖=1
so that we can write, taking 0 ≤ 𝛼 ≤ 1 for scales between 𝑄𝑜 and Q = √RQ o ,
𝑛
𝐾(𝑅 𝛼 𝑄𝑜 )
= ∑ 𝑤𝑖 𝑅 𝛼𝑔𝑖 .
𝐾(𝑄𝑜 )
𝑖=1
Define a weighted average scaling exponent 𝑔̂ over the range from 𝑄𝑜 to 𝑅𝑄𝑜 by
𝑛
𝑔̂ ≡ ln (∑ 𝑤𝑖 𝑅 𝑔𝑖 )⁄ln(𝑅)
𝑖=1
so that
𝑛
𝑛
𝑛
𝑔̂
𝑅 = (∑ 𝑠𝑖 𝑅 𝑔𝑖 )⁄(∑ 𝑠𝑖 ) = ∑ 𝑤𝑖 𝑅 𝑔𝑖 ,
𝑖=1
𝑖=1
and
𝑛
∑ 𝑤𝑖 𝑅 (𝑔𝑖−𝑔̂) = 1.
𝑖=1
We can now define an approximate plant cost as
Page 8 of 18
𝑖=1
̂ (𝑅 𝛼 𝑄𝑜 ) = 𝐾(𝑄𝑜 )𝑅 𝛼𝑔̂ ,
𝐾
𝛼
𝑛
𝑔𝑖
= 𝐾(𝑄𝑜 ) (∑ 𝑤𝑖 𝑅 ) .
𝑖=1
̃ (𝑅 𝛼 𝑄𝑜 ) as
Attempt to quantify the approximation error by defining a multiplicative error function 𝐾
̃ (𝑅𝛼 𝑄𝑜 ) =
𝐾
=
𝐾(𝑅 𝛼 𝑄𝑜 )
,
̂ (𝑅 𝛼 𝑄𝑜 )
𝐾
∑𝑛𝑖=1 𝑤𝑖 𝑅 𝛼𝑔𝑖
𝑅 𝛼𝑔̂
,
𝑛
= ∑ 𝑤𝑖 𝑅 𝛼(𝑔𝑖−𝑔̂) ,
𝑖=1
̃ (𝑄𝑜 ) = 𝐾
̃ (𝑅𝑄𝑜 ) = 1.
which has the property that 𝐾
6.3.1 Bounds on the relative error
Now consider various derivatives of the relative error with respect to the scaling factor exponent 𝛼
𝑛
𝑑𝑘
𝑑𝛼 𝑘
̃ (𝑅𝛼
𝐾
𝑄𝑜 ) = ∑ 𝑤𝑖 [(𝑔𝑖 − 𝑔̂) ln(𝑅)]𝑘 𝑅𝛼(𝑔𝑖−𝑔̂) .
𝑖=1
It is easy to see that the fourth derivative is strictly positive, so that the second derivative is a convex function. Convex
functions have global minima and on a bounded interval their maxima are realised at the bounds of the interval. It then
follows that the maximum of the second derivative on the bounded interval 0 ≤ 𝛼 ≤ 1 is equal to
𝑛
𝑛
𝐵𝑢 = max {∑ 𝑤𝑖 [(𝑔𝑖 − 𝑔̂) ln(𝑅)]2 , ∑ 𝑤𝑖 [(𝑔𝑖 − 𝑔̂) ln(𝑅)]2 𝑅(𝑔𝑖−𝑔̂) }.
𝑖=1
𝑖=1
Since the second derivative of the relative error is also strictly positive, on the unit interval
𝑑2
̃ (𝑅𝛼 𝑄𝑜 ) ≤ 𝐵𝑢 .
0 ≤ 𝑑𝛼2 𝐾
̃ (𝑅 𝛼 𝑄𝑜 )is convex with a bounded second derivative. Since it is convex, on the bounded interval 0 ≤
It follows that 𝐾
𝛼 ≤ 1 it achieves its maximum on the interval bounds and so it is bounded above by
̃ (𝑅 𝛼 𝑄𝑜 ) ≤ 𝐾
̃ (𝑄𝑜 ) = 𝐾
̃ (𝑅𝑄𝑜 ) = 1.
𝐾
Page 9 of 18
̃ (𝑅 𝛼 𝑄𝑜 ), consider the function
Now to derive a lower bound on 𝐾
̃ (𝑅 𝛼 𝑄𝑜 ) − 1 𝐵𝑢 𝛼(𝛼 − 1),
𝐿(𝛼) = 𝐾
2
so that
𝐿(0) = 𝐿(1) = 1,
𝑑2
𝐿(𝛼)
𝑑𝛼 2
2
𝑑
̃ (𝑅𝛼 𝑄𝑜 ) − 𝐵𝑢 ,
= 𝑑𝛼2 𝐾
𝑑2
−𝐵𝑢 ≤ 𝑑𝛼2 𝐿(𝛼) ≤ 0.
Since the second derivative of 𝐿(𝛼) is less than zero, it is concave and therefore on a bounded interval achieves its
minimum at one of the interval bounds. On the bounded interval 0 ≤ 𝛼 ≤ 1
𝐿(𝛼) ≥ 𝐿(0) = 𝐿(1) = 1,
and so
̃ (𝑅 𝛼 𝑄𝑜 ) ≥ 1 + 1 𝐵𝑢 𝛼(𝛼 − 1).
𝐾
2
1
But this achieves its minimum at 𝛼 = 2 which bounds the multiplicative error by
̃ (𝑅 𝛼 𝑄𝑜 ) ≥ 1 − 1 𝐵𝑢 .
𝐾
8
Together the upper and lower bounds imply that
1
̃ (𝑅 𝛼 𝑄𝑜 ) ≤ 1.
1 − 8 𝐵𝑢 ≤ 𝐾
𝛼
𝑛
1
8
𝑔𝑖
𝛼
𝑛
𝛼
𝑔𝑖
(1 − 𝐵𝑢 ) 𝐾(𝑄𝑜 ) (∑ 𝑤𝑖 𝑅 ) ≤ 𝐾(𝑅 𝑄𝑜 ) ≤ 𝐾(𝑄𝑜 ) (∑ 𝑤𝑖 𝑅 ) .
𝑖=1
𝑖=1
where 𝑅 (𝑔max −𝑔min ) is small 𝑔̂ ≈ 𝑔̅ ≡ ∑𝑛𝑖=1 𝑤𝑖 𝑔𝑖 which is the weighted average of 𝑔𝑖 , and
𝑛
𝐵𝑢 ≈ ∑ 𝑤𝑖 [(𝑔𝑖 − 𝑔̂) ln(𝑅)]2 ,
𝑖=1
which is the weighted variance.
Page 10 of 18
For given scaling exponents 𝑔𝑖 and scaling factor 𝑅, calculus of variations methods on weights 𝑤𝑖 can show that as the
weights are varied, the maximum multiplicative error
̃ (𝑅 𝛼 𝑄𝑜 )]
max[1 − 𝐾
𝛼
occurs when there is maximum dispersion in the scaling exponents. That is, it occurs when the weights are zero on all
but the maximum and minimum scaling exponents, that is, for n=2.
6.3.2 Maximum relative error for n=2
With 𝑛 = 2, taking the two exponents 𝑔 and 𝐺 ≥ 𝑔, we can write the multiplicative error function as
̃ (𝑅 𝛼 𝑄𝑜 ) =
𝐾
Perform the substitutions 𝛼 = 12 + 𝛼̃, 𝑅̃ 2 = 𝑅 𝐺−𝑔 , 𝑤
̃=
𝑤𝑙 𝑅𝛼𝑔 + 𝑤ℎ 𝑅 𝛼𝐺
.
(𝑤𝑙 𝑅 𝑔 + 𝑤ℎ 𝑅𝐺 )𝛼
(𝐺−𝑔)
𝑤ℎ
√𝑅
𝑤𝑙
observing that together with 𝑤𝑙 + 𝑤ℎ = 1 this
−1
implies 𝑤𝑙 = (1 + 𝑅̃ −1 𝑤
̃) which allows us to rewrite
1
̃ (𝑅 𝛼 𝑄𝑜 ) =
𝐾
𝑤𝑙 2−𝛼̃ (1 + 𝑅̃ 2𝛼̃ 𝑤
̃)
1
̃
+𝛼
̃)2
(1 + 𝑅̃ 𝑤
,
1 + 𝑅̃ 2𝛼̃ 𝑤
̃
=
1
̃
−𝛼
1
̃
+𝛼
,
̃)2 (1 + 𝑅̃ 𝑤
̃)2
(1 + 𝑅̃ −1 𝑤
̃
𝛼
1 + 𝑅̃ −1 𝑤
̃
=
) .
1(
̃ ̃
2 1+ 𝑅 𝑤
−1
2
̃
̃
[𝑅
]𝑤
̃ )
(1 + + 𝑅 ̃ + 𝑤
1 + 𝑅̃ 2𝛼̃ 𝑤
̃
To find the worst case approximation error we can perform the following analysis.
1
1
̃ = ln(1 + 𝑅̃ 2𝛼̃ 𝑤
ln 𝐾
̃) − (2 − 𝛼̃) ln(1 + 𝑅̃ −1 𝑤
̃) − (2 + 𝛼̃) ln(1 + 𝑅̃ 𝑤
̃).
𝜕
𝑅̃ 2𝛼̃
̃=
ln 𝐾
𝜕𝑤
̃
1+𝑤
̃𝑅̃ 2𝛼̃
−
(12 − 𝛼̃)𝑅̃ −1
1 + 𝑅̃ −1 𝑤
̃
−
(12 + 𝛼̃)𝑅̃
,
1 + 𝑅̃ 𝑤
̃
𝜕
2𝑅̃ 2𝛼̃ 𝑤
̃ ln 𝑅̃
̃=
ln 𝐾
+ ln(1 + 𝑅̃ −1 𝑤
̃) − ln(1 + 𝑅̃ 𝑤
̃).
𝜕𝛼̃
1+𝑤
̃𝑅̃ 2𝛼̃
𝜕
̃ to zero and solving for 𝑤
Setting the derivative 𝜕𝑤̃ ln 𝐾
̃ gives
Page 11 of 18
𝑤
̃=
𝑅̃ −𝛼̃ (𝑅̃ + 𝑅̃ −1 ) + 2𝛼̃𝑅̃−𝛼̃ (𝑅̃ − 𝑅̃ −1 ) − 2𝑅̃ 𝛼̃
.
𝑅̃ 𝛼̃ (𝑅̃ + 𝑅̃ −1 ) − 2𝛼̃𝑅̃ 𝛼̃ (𝑅̃ − 𝑅̃ −1 ) − 2𝑅̃ −𝛼̃
𝜕
̃ to zero and solving for 𝛼̃ gives
Setting the derivative 𝜕𝛼̃ ln 𝐾
𝑅̃ 2𝛼̃ =
̃
1+𝑅̃𝑤
}
̃
1+𝑅̃−1 𝑤
.
̃ −1 𝑤
̃
𝑤
̃ ln {𝑅̃ 2 (1+𝑅
)
}
̃
1+𝑅̃𝑤
ln {
These equations have a simultaneous solution at 𝑤
̃ = 1, 𝛼̃ = 0, resulting in maximum relative error with weights
𝑤𝑙 =
√𝑅
(𝐺−𝑔)
1+√𝑅
(𝐺−𝑔)
, 𝑤ℎ =
1
1+√𝑅
(𝐺−𝑔)
. For the worst case fixed weights, the multiplicative error varies with scale as
√𝑅̃
𝑅̃𝛼
√𝑅̃ (
̃ (𝑅 𝛼 𝑄𝑜 ) =
𝐾
+
𝑅̃𝛼
)
√𝑅̃
.
(1 + 𝑅̃ )
The error function is minimised (the error maximised) at 𝛼 = 12, scaling factor √𝑅, where
2√𝑅̃
,
1 + 𝑅̃
̃ (√𝑅𝑄𝑜 ) =
𝐾
4
4
−1
= 2 ( √𝑅 (𝐺−𝑔) + √𝑅 (𝑔−𝐺) ) .
4
̃ = 𝑀−1 we need 𝑅 (𝐺−𝑔) = (𝑀 + √𝑀2 − 1) . This maximum
For an error of relative magnitude(1 − 𝑀−1 ) so that 𝐾
relative error remains quite small across a wide range of scales, see Table 3.
6.4
Biofuel Production Cost function form
First we consider those components (𝑀) of costs that are proportional to production quantity (for example the cost of
materials including feedstock and other variable operations and maintenance).
𝑓𝑖
𝑀 = 𝑣𝑄 = (𝑤 + 𝑓𝑜 )𝑄 = (𝑤 + ) 𝑄
𝜂
In the above, fo and fi represent the costs of feedstock per unit product and per unit input respectively, and 𝜂 is
conversion efficiency. Next we consider those aspects (𝑃) of cost, mostly of processing equipment, that follow a
power law function, such as amortised capital costs and fixed maintenance.
Page 12 of 18
𝑃 = 𝑠 𝑄 𝑔 = 𝑠 𝑄1−𝛾 , 0 < 𝑔 < 1,
𝛾 =1−𝑔
Up front capital costs K = SQg can be converted into amortised (annualised) ongoing costs over a lifetime 𝐿 at
discount rate 𝑟 as
𝐾𝑟
𝑟𝑆𝑄 𝑔
=
1 − (1 + 𝑟)−𝐿 1 − (1 + 𝑟)−𝐿
Including maintenance costs proportional to plant capital costs at a percentage 𝑚 gives
𝑟
(
+ 𝑚) 𝑆𝑄 𝑔
1 − (1 + 𝑟)−𝐿
𝑠=(
𝑟
+ 𝑚) 𝑆
1 − (1 + 𝑟)−𝐿
We assume that transport costs per unit of output are essentially a linear function of distance, and that distance is
proportional to the square root of the collection area 𝐴. So total transport costs 𝑇 are given by
𝑇 = (𝑎 + 𝑏𝑑)𝑄𝑓
where Q f is the quantity of feedstock transported, but
𝑑 = ℎ √𝐴
and
𝑄𝑓 = 𝑄⁄𝜂
where 𝜂 is conversion efficiency so that
𝑇 = [𝑎 + 𝑏ℎ √𝐴 ] 𝑄⁄𝜂
that is
𝑇 = (𝑢 + 𝜏 √𝐴)Q
where u = a⁄η, τ = bh⁄η and we note that the quantity of available feedstock in a given area will depend on spatial
concentration 𝜌 given in GJ/ha-yr whereQ f = ρA, Q = ηρA.
The total cost C(Q) is given by
𝐶(𝑄) = 𝑀 + 𝑃 + 𝑇
Page 13 of 18
= 𝑣Q + 𝑠𝑄 𝑔 + (𝑢 + 𝜏√𝐴)𝑄
It follows that
𝐶⁄ = 𝑣 + 𝑠𝑄 𝑔−1 + (𝑢 + 𝜏 √𝐴),
𝑄
= 𝑣 + 𝑢 + 𝑠𝑄 −𝛾 +
𝜏
√𝜂𝜌
∙ 𝑄 0.5 ,
= 𝑣 + 𝑢 + 𝑠𝜂 −𝛾 𝜌−𝛾 𝐴−𝛾 + 𝜏𝐴0.5
6.5
Minimisation of a two-term sum of power law relations
Consider the function φ(x) = κx α + λx β . This has a first derivative
dφ
dx
= ακx α-1 + βλx β-1. The condition for an
ακ
dφ
extremum is dx = 0, and this occurs when - λβ = x β-α .
ακ
1
Let x ⋆ = (- λβ)β-α . Substituting into φ(x)gives
𝛼
𝛽
𝛼𝑘 𝛽−𝛼
𝛼𝑘 𝛽−𝛼
𝜑(𝑥 ⋆ ) = 𝜅 (− )
+ 𝜆 (− )
,
𝜆𝛽
𝜆𝛽
𝛼
𝛼𝜅 𝛽−𝛼
𝛼𝜅
= (− )
(𝑘 − ) ,
𝜆𝛽
𝛽
𝛼
𝜅
𝛼𝜅 𝛽−𝛼
(𝛽 − 𝛼),
= (− )
𝛽
𝜆𝛽
1
𝜅 −𝛽
𝜆 𝛼 𝛼−𝛽
= −(𝛼 − 𝛽) {( ) (− ) }
,
𝛽
𝛼
1
𝜅 −𝛽
𝜆 𝛼 𝛼−𝛽
= −(𝛼 − 𝛽) {( ) (− ) }
,
𝛽
𝛼
1
𝜅 𝛽
𝜆 −𝛼 𝛽−𝛼
= (𝛽 − 𝛼) {( ) (− ) }
.
𝛽
𝛼
Page 14 of 18
6.6
Technical efficiency improvement
We can show the explicit dependence of Error! Reference source not found. on the efficiency 𝜂 by substituting 𝑢 =
𝑎
𝜂
=
𝑠𝜂0
𝜂
𝑢𝜂0
𝜂
and 𝜏 =
𝑏ℎ
𝜂
=
𝜏𝜂0
𝜂
where 𝜂0 is a nominal efficiency for the base case. Note that 𝑣 =
𝑤𝜂0
𝜂
+
𝑓𝑖
𝜂
and 𝑠 becomes
in the case where variable costs and plant capital costs are reduced in proportion to the efficiency improvement,
but 𝑣 = 𝑤 +
𝑓𝑖
𝜂
and 𝑠 remains unchanged in the conservative case where variable operations and maintenance costs
and per unit plant costs are unaffected by the technical efficiency improvement.
Page 15 of 18
7 Tables and Figures
Elec
Lange (15)
(S&T)2 Consultants Inc. (25)
Wright and Brown (12)
Stucley, Schuck (6)
Tao and Aden (26)
Foust, Aden (27)
Hamelinck and Faaij (24)
Wright and Brown (4)
Anex, Aden (28)
Graham, Reedman (29)
Obernberger and Thek (7)
CSIRO (30)
Phillips, Aden (21)
Aden, Ruth (19)
Kazi, Fortman (31)
Wallace, Ibsen (17)
Hamelinck, van Hooijdonk (32)
Stephen, Mabee (33)
Boerrigter (11)
Tijmensen, Faaij (20)
Kreutz, Larson (34)
International Energy Agency (35)
National Energy Technology Laboratories (14)
Rodrigues, Faaij (18)
Hatch Engineering (13)
Table 1: Processes covered by each reference
Page 16 of 18
*
Eth
(The)
*
*
*
*
*
*
Eth
(Bio)
*
*
*
*
*
*
*
*
*
Biosyndiesel
*
*
*
*
*
*
*
Biosynjet
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Near
Near
Far Fixed
Far
Fixed
Variable ($/t)
Variable Threshold
($/t)
($/t-km)
($/t-km)
(km)
$8.35
$0.095
$3.84
$0.210
39.3
$6.55
$0.095
$1.82
$0.215
39.3
$12.20
$0.082
$9.51
$0.185
26.0
Semi Trailer
(Softwood)
Logs
Chips
Residues
Semi Trailer
(Crops)
Stubble
$10.70
$0.099
$5.89
$0.226
38.1
Logs
Chips
Residues
Chips
Bark
Green
Sawdust
Shavings
Logs
Chips
Residues
$7.56
$5.86
$11.93
$6.86
$4.61
$0.112
$0.112
$0.103
$0.131
$0.088
$3.68
$2.17
$9.79
$2.54
$1.71
$0.199
$0.202
$0.185
$0.236
$0.159
44.7
41.1
26.0
41.1
41.1
$7.17
$0.137
$2.65
$0.247
41.1
$3.67
$9.45
$7.33
$13.01
$0.070
$0.140
$0.140
$0.112
$1.36
$4.60
$2.72
$10.68
$0.126
$0.249
$0.252
$0.202
41.1
44.7
41.1
26.0
B-double
(Crops)
Stubble
$7.41
$0.095
$5.34
$0.171
27.2
Road Train
Logs
Chips
Residues
$5.96
$4.19
$12.67
$0.088
$0.088
$0.088
$3.80
$1.75
$11.14
$0.160
$0.162
$0.162
29.8
32.7
20.4
Road Train
(Crops)
Stubble
$8.31
$0.098
$6.60
$0.182
20.4
Semi Trailer
(Hardwood)
Semi Trailer
(Sawmill
Residues)
B-double
(softwood)
Table 2: Transport Cost Parameters
Page 17 of 18
Maximum Error (%)
𝑅(𝐺−𝑔)
𝑅 for (𝐺 − 𝑔) = 0.3
0.50
1.00
1.48
2.00
5.00
7.50
7.58
10.0
15.0
20.0
22.7
33.3
50.0
1.39
1.76
2.00
2.24
3.64
4.95
5.00
6.48
10.4
16.0
20
47
194
3.80
6.64
10.1
14.7
74.2
207
214
507
3
2.46 × 10
3
10.3 × 10
3
21.7 × 10
3
374 × 10
6
4.23 × 10
Table 3: Relative error on sum of powers approximation
Figure S1: Agricultural residue transport costs
Page 18 of 18