Generally, a disruptive technology will replace the older technology, and the transition will follow an S-curve.

Figure 1: The S-curve shows how the market share of a disruptive technology evolves. The curve flattens when the market share reaches its saturation level. |

The curves showing the electricity production in the recent years by both wind and solar resemble the start of an S-curve. But there are topics for wind and solar that may cause the further evolution to differ from the S-curve. China, as the rest of the world, wants to phase out fossil fuels in its electricity production as fast as possible due to both global warming and local pollution. This may cause the transition to be even faster than the S-curve.

Wind and solar both have intermittency problems. We need electricity also when the wind does not blow and the sun does not shine. The intermittency problems become more and more serious as the shares of wind and solar increase. This may cause the transition to be slower than the S-curve.

There are some solutions to the intermittency problems, and new solutions will certainly be developed. Chris Goodall discusses these problems in his latest book The Switch. He writes that various solutions have to be applied, in part simultaneously. Many of the solutions are technical, but not all of them. A technical solution is to produce hydrogen and gas for later use when there is a surplus of solar and wind electricity. A non-technical solution is to control the demand for electricity using varying prices so that demand better matches production.

Solar, wind, hydro and the other renewables have different properties. Their saturation levels measured in percent of the total market will therefore vary around the world. The S-curve in Figure 1 may represent the electricity production by solar and wind, and the saturation level may be the production needed to totally phase out the electricity produced by fossil fuels.

The rest of the blog post deals with mathematical details about S-curves.

The S-curve is usually expressed mathematically as a Sigmoid function (1). The independent variable t in (1) may be scaled with a factor a and offset with T0, as shown in (2).

We often want to design the S-curve so that it is S1 at time t1 and S2 at time t2. Mathias has shown that (3) and (4) are the solution to this mathematical task.

The derivative of S(t) with respect to t tells the absolute increase in S(t) as a function of time. We apply the chain rule to calculate the derivative, as shown in (5).

Exponential growth means that the increase is a percentage of the current value. Figure 1 shows that there is a strong exponential growth in the first part of the S-curve. The exponential growth weakens as S increases. The exponential growth g is derived in (6) and (7), and the result shows that it is a function of time.

The left part of (6) is the derivative of S(t) with respect to time. It is calculated in (5).

(5) inserted into (6) gives (7).

Figure 2 shows the S-curve as calculated with (2), its derivative with respect to time as calculated with (5), and its exponential growth as calculated with (7). They are all functions of time. The a and T0 values of the S-curve are calculated with (3) and (4) so that the S-curve becomes 0.01 (1%) in 2016 and 0.99 (99%) in 2044. This gives a equal to 0.33 and T0 equal to 2030.

The S value in (2) is between zero and one. It indicates part of a max value, and it is therefore unitless. I set t1 equal to 2016 and t2 equal to 2030 when calculating a and T0. T0 therefore gets the unit year and a gets the unit year-1 (per year). The derivative of S with respect to time and the exponential growth of S get the same unit as a.

The exponential growth is strongest in the start of the S-curve. Then both the S(t) value and the derivative of S(t) with respect to time are small. The derivative of S(t) has its maximum value in the middle of the S-curve. Then the absolute increase per time unit has its maximum value, even though the exponential growth has decreased to half of its maximum value.

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