Scale laws and learning curves Information about the costs of new technology may not always be in a.

Scale laws and learning curves Information about the
costs of new technology may not always be in a useful form. Some tools are
available to convert costs to the desired form. When equipment is imported,
conversion to the proper currency is needed using currency rates. Furthermore,
when cost of equipment is known for another year, a conversion is needed using
proper equipment price indexes. Here scale laws and learning curves will be
treated in more detail. Scale laws. When the available information for
equipment is for a scale different from the one required, one may use so-called
scale laws. Such a scale law has the form:

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where: C = cost of equipment C = (known) cost of
equipment with capacity Pref P ref = capacity of
equipment Pref = capacity of equipment for which the cost C is known
R = a constant, the scale factor

The scale factor depends on the type of equipment. Often
a scale factor of 0.7 (or 2/3) is used, since the capacity for many types of
equipment increases with the third power of the size (determined by volume)
whereas costs only increase in a quadratic way (determined by surface area).
The overall effect is that cost increases less than proportionally with scale.
This effect is often indicated as ‘economies-of scale’. One should be careful
using this approach and consider whether the characteristics of appliances
suggest a certain scale factor. For example, certain heat exchangers and
photovoltaic systems, being modular in nature, have a scale factor closer to
one as the costs increase in a more or less linear way with the throughput of
the equipment. Moreover, every production unit has a maximum scale above which units
need to be installed in parallel and the scaling equation does not hold.
Learning curves. A learning curve is a quantitative description of the process
of technological learning: after a product is brought on the market and sales
increase, the product improves, due to advances of technology, improvement of
production processes, scaling up of individual units, and an increase of the
scale of production. In many cases this leads to a steady decrease of the costs
per unit of performance. This effect is often expressed in the form of a
learning curve (also called an experience curve). A learning curve expresses
that the costs decrease by a constant fraction with each doubling of the total
number of units produced. For instance, for photovoltaic systems it has been
found that after each doubling of the cumulative production (generally
expressed in MW of capacity), the costs per unit of electricity production
capacity ($/Wp) decrease by approximately 20 per cent. The learning effect can
be described by the following expression:



Cp = cost per unit after the cumulative
production of P units

C1 = cost of the first unit

b = experience index

The price of the product is reduced by the same
percentage each time the cumulative production is doubled. This can easily be
shown as follows by comparing the price after P units with the price after 2·P
units. We take the ratio of these two price levels:


Hence, after each ‘doubling’, the price is multiplied by
the factor 2b. This factor is defined as the progress ratio. The
experience index b is generally negative, so 2b is smaller than one. The
reduction fraction after each doubling is 1 – 2b (called learning rate). In
Figure 11.1, the development of the price of a fictitious product with a
progress ratio of 0.8 is given. The price thus drops by 20 per cent for each
doubling. When both the price and the cumulative production are depicted on a
logarithmic scale, the learning curve becomes a straight line. In most cases
the cost of the first unit is unknown, or not well defined; in that case we can
write equation [11.10] as:


Note that in the learning curve equation there is no
relation between price and time. This means that the actual price development
will only depend on the rate of deployment of the new technology. Empirical
evidence indicates that for most technologies, the progress ratio is somewhere
between 0.7 and 0.95. In Figure 11.2 the price development for PV modules is
given. The learning rate (LR = 1 – 2b ) is 0.201, so the
progress ratio (2b ) is 0.799, and the experience index (b) equals
−0.324. Learning curves are a useful tool for making projections of the
cost price development of new technology, but the approach has its limitations.
It is not a ‘law’, just an empirical finding. Cost price reduction may
accelerate and slowdown and decrease again Technological breakthroughs and the
market situation will also influence costs and prices.


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