The One Equation that Describes the EV Revolution – that No-one Uses

Note – on the basis that each equation loses 50% of readers, I’ll assume only the hard-core readership remain.

EVs and other new energy technologies such as PV solar are disruptive.

But incumbent auto and oil companies, and industry analysts assume they will follow non-disruptive, simple growth curves. 

They won’t.

It’s time to model disruption, and confront reality.


The Mathematics of Disruption

Electric Vehicles (EVs*), like their gasoline predecessors, are a disruptive technology.

By 2020 or thereabouts, they are forecast to become cheaper than conventional internal combustion engines (ICE), and offer running and maintenance costs 90% lower. At this point, there could be a mass exodus from gasoline cars to electrified ones.

All disruptive technologies, in nature and economics, follow a prescribed pattern of growth. It’s commonly called the S curve due to the shape of expansion: slow incubation, sudden lift-off and break-neck growth, and a final mature plateau phase.

The examples are numerous:

S curves are based on two governing principles: rates of growth change quickly over time as demand or resources develop, but there is an upper limit to expansion as the resources get eaten up. So, disruptive populations or products start out small, explode in size, then level off as resources or demand are depleted.

The fearsome equation above – a Logistic Function – contains all these moving parts and so generates an S curve, with take-off point, dynamic rate change and upper limit; a disruptive piece of mathematics in its own right.

You would therefore expect forecasters of EV demand (or PV solar) to use S curve mathematics to try and pinpoint the time of lift-off, and the duration of the explosive, volatile phase of growth.

Not so.

Almost all mainstream analyses assume the exact opposite: they use an equation that has only one rigid moving part, a fixed growth rate, and no upper boundary whatsoever.

This is exponential growth, and is normally encountered in the less-natural worlds of finance, for example, to calculate compound interest on investments.

Proponents argue that although it’s a simple model, it mimics real-life quite well. But the exponential curve does not describe any real-life system; it is the antithesis of disruptive change, and creates massive errors in long-term forecasting.

A simple example below indicates the substantial difference between the two:

source: dollarsperbbl

Although both curves grow to a 100 units, after 10 time periods, any company or government unit looking to model disruption could seriously underestimate the speed and size of demand.

The S curve is sensitive to fast and changing growth rates, whilst the exponential curve smooths out demand, and avoids aggressive estimates, until later.

The Prevailing View: The Slow Rise of EVs and the Gentle Decline of Oil

At the risk of over-statement: the fate of two giant, global multi-trillion dollar industries, automotive and hydrocarbon, could lie in the difference between these two mathematical entities.

Global oil growth, already near-zero, could be tipped to swift decline if the growth of EVs is fast and disruptive.

How soon could that be?

Not that fast according to most analysts. Even those pre-disposed to alternative energies, but who still rely on a simple exponential growth curve.

Take Bloomberg New Energy for example. In the chart below they declare that EVs could reach sales “lift-off” by 2022. However, their analysis uses a static growth curve, showing no disruption.

No explosive phase occurs, and growth remains smooth and continuous to 2040 – at no point in this model do EVs even overtake incumbent gasoline vehicle sales.

Compare the Bloomberg chart with a model S curve chart below (derived in excel using the above equation). The S curve chart assumes the same lift-off around 2021-22, but that total EV sales reach 700-750m by about 2040. It fits actual sales data from 2010-16 reasonably well, and the forecasted peak of annual sales at 75m vehicles is feasible.

source; dollarsperbbl, MS excel

A disruptive chart has a totally different narrative to a smooth one. It allows for disruptive events such as the cross-over of EV and ICE gasoline sales – in this case 2029-30. And it is constrained by real-life upper levels such as market saturation, or policy.

A review of several forecasts indicates they almost all rely on simple exponential growth curves. As a result, if we take a key data point such as predicted total EV sales by 2030, most analyses predict at or below the average of 165million.

source: dollarsperbbl, BP, IEA, OPEC, CarbonTracker

Unsurprisingly the lower bound belongs to industry incumbents – OPEC, BP – advocacy masquerading as analysis perhaps.

The inclusion of some maths models also highlights a possible “herding” effect of analysts relying on simple growth rates. Only an S curve, or “strong EV” estimate predicts growth far above the mean value.

With a rigid growth rate, most analysts find it difficult to justify high values over many years. So they plump for reasonable ones, which then prevent any short-term explosive growth. And without upper limits, they have to constrain their models further to avoid unrealistically high numbers later on.

Result: the prevailing view of EV growth, a large disruptive technology, is that it will follow a non-disruptive path, with limited impact on the incumbent oil and auto industries, and a modest-sized market for new entrants.

Using The Maths of Disruption

The use of an S curve to predict growth in the EV and gasoline markets reveals very different outcomes – see charts below.

source: dollarsperbbl, and see post How to Underestimate EV Demand

EV Disruption – Higher Sales, Lower Investment

For any new entrant targeting 5% of EV market share, but using the simple growth curve, it means planning for a total of 1.3million EV sales by 2025, 5.5million in 2030, and 12.5million by 2035.

But the disruptive S curve predicts 3million in sales by 2025, 15m by 2030, and 30million by 2035 – a 2-3 times difference, and perhaps the destruction of an inaccurate business model.

In addition, relying on the simple growth curve, there is never a cross-over in demand with gasoline cars – so long-term parallel investment in both is necessary.

In the S curve case, however, EVs overtake gasoline cars around 2029-30; conventional production could be ramped down starting about now, for example.

Two different worlds: the smooth curve assumes lower demand and requires expensive duplicate investments, the S-curve predicts higher growth, and a sole focus on excellence in EVs.

Perhaps Tesla’s sky-high valuation has a point.

Gasoline Disruption – Zero Growth and Accelerating Decline

For oil industry incumbents, a focus on the simple growth curve is relaxing – a small amount of gasoline demand growth to 2030, with just 0.5mb/d decline by 2035,

If the S curve prevails, however, it’s almost zero growth to 2025, a 2mb/d decline by 2030 and an acceleration to 5.5mb/d lower by 2035.

As important is the disruptive storyline of peak demand.

With smooth EV growth, peak demand occurs around 2025-6; with an S-curve, it arrives as quickly as 2021-22.

Two different worlds: the exponential curve is a gentle transition, with perhaps two decades to adapt; the S-curve predicts near-term zero growth, and accelerating decline after 2022, a time horizon less than half a typical oil and gas project investment cycle.

Stress-Testing Realities

S curves are not a universal solution for predicting disruptive change – they can be highly inaccurate too.

But they ought to be used a lot more to stress-test scenarios, projections and forecasts.

Governments and corporations often try and define their own neat, orderly realities: but the disruptive maths of S curves helps us understand what reality actually is.

By abandoning the equation, and relying on synthetic exponential curves, analysts, policy-makers and incumbents are in opposition to the many examples of history, and the deeper forces of nature and mathematics.

Good luck with that.


To contact the author, email

* Throughout this post EV data refers to battery or plug-in EVs, not gasoline hybrids.

Note – if interested, in the S curve generated in the post, I used L the upper limit as 750, timeframe 2010-2050, k estimated at 0.4, and x0 equal to 1.

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2 Responses to The One Equation that Describes the EV Revolution – that No-one Uses

  1. Pingback: Energy’s Demand Shock, and The Acceleration of the Fossil Fuel Transition | dollarsperbbl

  2. Pingback: ¿Cómo será realmente la transición hacia un futuro automotriz eléctrico? - Autos RPM

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