This will probably be a fairly long post mainly summarizing findings from my simple toy model….so proceed at your own peril.  For a while I have been interested in how the properties of the power source affect the end user. For the consumer different power sources deliver very different value, but the public discussion is typically centered (more or less honestly) on costs. I think one issue of great relevance is the capacity utilization and the aim of this post is to record my studies on the matter. In particular I wish to explore the variable power sources such as wind and solar in the context of capacity utilization. My thoughts are in the end closely related to “capacity factor rule” discussed by John Morgan, but I approach the issue from somewhat different angle.

What is a capacity utilization  and why it matters?

Capacity utilization compares realized production with what could be possible. The concept seems to be somewhat fuzzy since theoretically maximum output could be defined in different ways. However, for an advanced economy capacity utilization is high, for example, in EU it is typically higher than 80% with a scary dip during last financial crisis. In an undeveloped country capacity utilization is lower, for example around 55% in Bangladesh. This makes sense, since things like poor infrastructure hamper production that might have otherwise happened. High capacity utilization is needed especially when lots of capital is spent since otherwise production could not cover capital costs. If high capacity utilization cannot be ensured, investments requiring large amounts of capital will not happen (unless one finds someone to pay for the losses).

In a developed economy capacity utlization is not really limited by the power supply. We get power from the plug whenever we need it. Capacity utilization is limited more by things like rising labour costs if one aims for maximum production or perhaps uncertainty on whether or not a buyer can be found for the product. However, our electricity production follows the demand and not all power sources can do that. Some view it desirable that consumers should in fact adjust their consumption according to weather. This raises the question: “How will this limit the capacity utlizations?”

This is a hard question and I can only scratch the surface here. I assume a “machine” or factory that can use certain amount of power and what is produces is proportional to its electricity consumption. I will then either use wind power or solar power as a power source and also add a storage to help even out the power variations. If there is excess power and storage is not full, we fill it. If power supply is lacking, we drain the storage. (I assume 80% round trip efficiency.) How much power machine can use, is a variable. It probably makes no sense for this to be higher than the wind or solar capacity, but if it is reduced utilization rate for the machine can probably be increased. It should be noted that the estimates below do not (of course) use the economists definitions for capacity utilization. This is more likely to give an estimate on the additional limitations on capacity utilization on top of all those other factors that are operating in any case.

So let me quickly summarize what I find…

Figure 2: Wind power source limited capacity utilization as a function of “machine capacity” (i.e. what fraction of power source capacity it can use) and storage (days at average wind production). Wind power data from UK 2013.

Figure 3: same as Figure 1, but using solar power as a source. (Production data from Germany 2015.)

Figures 2 and 3 show my rough estimates for the “capacity utilization” as a function of machine capacity and amount of storage (hours of average power production). If machine capacity is equal to the capacity of the power source, capacity utlization is limited by the capacity factor of the power source. As machine capacity is reduced and/or storage is added capacity utilization can increase. However it is very hard to get to a situation where power source would not be a factor substantially limiting the overall capacity utilization.

In terms of capacity utilization wind power tends to beat solar power which has strong seasonal production profile. Removing that is hard since it would require massive amounts of seasonal storage which would (by definition) be used only by about once a year.

As machine capacity is reduced, the “factory”can run at a higher capacity utilization, but then certain fraction of the produced power will be wasted although waste can be reduced somewhat by storage. If we aim for high capacity utilization, wasted fraction can unfortunately be substantial. The unit cost of useful energy will rise with increasing waste.

Figure 4: Fraction of wasted wind output.

Figure 5: Fraction of wasted solar output. (Once daily variation is covered it is very hard to change things by adding even more storage.)

Waste can be reduced with storage, but then the question arises that how efficiently this storage is being utilized? Figures 6 and 7 illustrates this. If we add so much storage that capacity utilization is high and amount of wasted power is low, we tend to have a large amount of under utilized storage capacity lying around. Storage that is combined with solar power tends to be more efficiently used because of regular daily variation.

Figure 6: How efficiently storage is being utilized with wind power. (Here the scale is more arbitrary. I assumed full utilization amounts to one full cycle a day.)

Figure 7: Same as figure 5, but with solar power.

I suspect that these estimates are in fact too optimistic. If I choose a point from figure 2 with relatively high “capacity utilization”, the power supply for the machine is still quite erratic as seen in Figure 8.

Figure 8: Example power input to the machines when machines powered by wind had a capacity of 0.26 of wind capacity and system had 36 hours of storage at average wind output. Still a mess.

Maybe there are processes that do not mind this, but there are  also plenty of industrial processes where steady power supply is needed and where abrupt power cuts will undermine the economics of the plant. (It would be interesting to have real world examples of production economics as one changes between power sources. Do you know any? I suspect that current way of delivering power to industries in developed economies is close to optimal for their needs.)

I think I will stop here and discuss later what I think this will imply. Main point here is that nature of the power source will affect the capacity utilization and have economic consequences that are not accounted for when myopically computing the “cost” of electricity for the power sources.

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