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Obscure mathematics applied to battery modelling

If you are not aware that cheap rechargeable batteries are being touted as the solution to all things electrical energy related – you have not been paying attention.

Bruce Miller Service Line Lead

by Bruce Miller

Power Specialist Services Service Line Lead, Melbourne

07 September 2016
Math on blackboard

Whether it is the ‘Powerwall’ promoted by Tesla, the power sources for self-driving electric vehicles, or the introduction of large-scale energy storage devices for use on utility systems; batteries, particularly - but not exclusively of the Li-Ion variety, appear to be the ‘flavour of the month’.

Gartner’s Hype Cycle for Emerging Technologies is published each year, giving an idiosyncratic viewpoint on various, weird and wonderful ideas that appear in the science and technology media. Interestingly in the 2015 Cycle, rechargeable batteries do not seem to be listed, presumably because they are not considered to be that novel.

However, based on the fact that autonomous vehicles are near the ‘peak of inflated expectations’, we can assume that the necessary enabling technology of cheap rechargeable batteries is either falling into the ‘trough of disillusionment’ or more optimistically - and more likely in my opinion - climbing up the ‘slope of enlightenment’ to the ‘plateau of productivity’.

To me, the latter scenario appears most likely, if only because rechargeable batteries have been around for a quite a while and the only barrier to using them in high-energy applications is their cost. Fortunately, various sources are indicating that the cost is coming down rapidly.

For a thorough investigation of the technical and economic issues of battery storage as it applies to large-scale utility applications, refer to “Energy Storage for Renewable Integration South Australia (ESCRI) – General Project Report Phase 1” on the ARENA website knowledge bank.

Recent problems

As it seems likely low-cost energy storage, using electrochemical processes (i.e. batteries) will be with us in the near future. Perhaps we should try to understand how they interact with the various systems we are familiar with – in particular, electricity networks?
Recent problems with small scale battery storage devices sound a salutary warning that the engineering needs to be absolutely right before we scale up to utility sizes.

Phone failures may be personally expensive, even dangerous if they cause fires. But scale this up to utility size and the risks are obviously much greater.

Obscure Mathematics

Several years ago when I first looked at these issues I found an application for an obscure piece of mathematics that most people with a standard STEM education will not have heard of - or if they have heard of it, they may have promptly forgotten it as at first sight, it seems to be of little practical use in the real world.

The electrochemical kinetics of batteries is typically modelled using what is known in the field as the ‘Randles Circuit’. See below:

Randles Circuit

For any electrical engineer or technician there appears to be nothing difficult about this. We use and calculate with similar circuit diagrams all the time. If anything, the circuit above looks to be ludicrously simple to accurately reflect what physically happens inside a battery.

The key element is the Rw component known as the ‘Warburg Element’. It is unusual because its impedance depends inversely on the square root of frequency. As a result of this, it cannot be expressed as a combination of simple R (resistor), C (capacitance) or L (inductance) elements. It can only be expressed by an infinite series of these basic circuit elements.

This is because the impedance of capacitors depend inversely on frequency, whereas inductor impedances vary proportionally with frequency and resistors are ideally not dependent on frequency at all. Mathematically it is simply impossible to combine any of these components together to obtain a square root relationship. As you may be able to imagine, this makes the Warburg Element non-trivial to model.

The square root frequency relationship has a more interesting interpretation. In effect it is a fractional integration element.

If you can still remember your senior school or sophomore years (assuming you included maths in your education), you should be able to remember differential or integral calculus (e.g. things like d(x^2)/dx = 2 x etc.).

A standard STEM education would typically equip you with various formulas which would allow you to find areas under curves, tangents to curves, maximum and minimum points etc. which surprisingly turns out to be much more useful to science and engineering than appears reasonable to anyone working from a perspective of common sense.

A quick brush up for those whose memory of their late secondary or early tertiary education may be a bit foggy…

d(f[x])/dx gives the slope of the tangent to the curve f[x].

ʃ f[x] dx is used to find the area under the curve f[x]

So what does all this have to do with batteries?

It turns out that the half integral is what is required to model the Warburg Element in the circuit above.

What is a half integral?

It is not half the area under a curve, nor is it half a tangent slope. Unfortunately (as far as I am aware), there is no simple geometric interpretation for a half integral or half derivative. Even so, the basic mathematical rules of fractional calculus have now been known for several hundred years, almost as long as the discovery of calculus itself.

But - maybe because it has not yet found many applications - it usually doesn’t rate a mention in standard STEM education curriculums. Perhaps this will change now that the modelling of electrochemical processes, such as batteries, is poised to become much more important to our everyday lives.

The fact that an obscure piece of mathematics discovered several hundred years ago turns out to be ideal for modelling the electrical dynamics of batteries, demonstrated another example of the unreasonable effectiveness of mathematics.

The main thing that I am impressed by in all this is, despite the fact that mathematics is not designed to be useful (I concede that this is a somewhat controversial viewpoint - some people think it is not designed at all, it just is, others hold very strong views that it is definitely a subject that people create), it is essential for us to get our technology to work, and our science to explain things.

The fact that an obscure piece of mathematics discovered several hundred years ago turns out to be ideal for modelling the electrical dynamics of batteries, demonstrated another example of the unreasonable effectiveness of mathematics.

Who would have thought mathematics currently thought to be obscure and rarely if ever taught in our colleges and universities could be used to design electrical energy storage and help prevent battery fires from occurring?

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