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Is PV capacity growth logistic?

May 10, 2018Solar0 comments

The logistic growth function is a useful way to explore the growth rates of many natural phenomena. Many man-made developments can be described by a logistic function, including the diffusion of power technology, such as hydropower and nuclear. Logistic growth differs from exponential growth in that the growth rate declines to linear (constant) growth. There is currently interest in modelling the diffusion of renewable energy since some of these are expanding exponentially. However some observers have noted that some of the countries that adopted solar early have already moved into the linear phase of the logistic growth function, suggesting a boom-bust cycle (e.g. Hansen et al. and Evans). This implies that the long-term penetration may be much less than sometimes anticipated. In this post, I use data from the BP statistical review and a paper from last year to explore logistic growth in relation to solar PV.
In general, logistic growth is described by the equation (see more here) –

giving the following curve –

Logistic growth function.

The problem is the ‘L’ defines the curve’s maximum value, which is difficult to estimate ex-ante.
Firstly, I took the BP data and plotted the total capacity additions. I also included a stacked line graph for 7 of the leading PV nations, giving a perspective to the relative scale of each nation. I then normalised the national data to the year of the maximum capacity additions. This allows the plot to show the annual growth relative to the maximum year, given below –

Solar PV capacity additions in megawatts. Source: BP Statistical Review


Capacity additions for selected countries, normalised by year of maximum additions.

Sure enough, five of the countries have a declining growth rate, meaning that capacity is being installed, but at a declining rate. In contrast, the US and China have an accelerating growth rate. Not surprisingly, strong support brought PV installations forward in several countries, but with reducing support mechanisms, the decline in solar cost has not been sufficient to offset the decline. On the other hand, many countries that started deploying PV later show a curve more like the US and China. This list includes Netherlands, India, Mexico and Thailand for example.
Nextly, I took IEA-PVPS global installed solar data up to the end of 2017 and plotted on a log axis with an exponential trend line. The cumulative installed capacity can be described by an exponential. For the period 1990 to 2010, an exponential with doubling time 2.0 years provides an excellent fit, and post 2010, the doubling time has widened to 2.5 years. 

Global cumulative installed capacity, plotted on log axis.

Continuing an exponential with 2.5 year doubling will give 3.7 TW in 2025 and 15 TW in 2030. But a problem with interpreting global data is that China dominates, as it does with virtually all forms of power generation. My interpretation is that PV is likely to top out below the expectations of some national scenario roadmaps, but still has enormous global growth still to go.
Germany has already entered the mature declining growth phase. In 2017, solar PV contributed 7.2% of Germany’s electricity production, with an installed capacity of 42 GW. Other countries will follow. But with further global exponential expansion continuing for the foreseeable future, the current estimated total 450 GW will look small in retrospect. The more difficult question is evaluating the value that solar provides in the context of an energy transition.
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